論文 / 著書情報
Article / Book Information
題目(和文)

変分原理の良設定性及び三位一体重力理論の自由度勘定に関する研究

Title(English)

On well-posed variational principle and degrees of freedom counting ofgeometrical trinity of gravity

著者(和文)

友成恭介

Author(English)

Kyosuke Tomonari

出典(和文)

学位:博士(理学),学位授与機関:東京工業大学,
報告番号:甲第12639号,
授与年月日:2024年3月26日,
学位の種別:課程博士,
審査員:須山 輝明,伊藤 克司,今村 洋介,慈道 大介,関澤 一之,山口 昌英

Citation(English)

Degree:Doctor (Science),
Conferring organization: Tokyo Institute of Technology,
Report number:甲第12639号,
Conferred date:2024/3/26,
Degree Type:Course doctor,
Examiner:,,,,,

学位種別(和文)

博士論文

Type(English)

Doctoral Thesis

Powered by T2R2 (Tokyo Institute Research Repository)

DOCTORAL THESIS

On
well-posed variational principle
anddegrees of freedom counting ofgeometrical trinity of gravity
Kyosuke TOMONARI
Department of Physics, Tokyo Institute of Technology
Degree of Doctor of Science
February 28th, 2024

Supervisor of doctoral thesis

Co-Supervisor of doctoral thesis

Teruaki SUYAMA

Masahide YAMAGUCHI- Department of Physiscs, Tokyo Institute of Technology.

- Cosmology, Gravity and Astroparticle Physics Group, Center for Theoretical Physics of the Universe, Institute for Basic Science (IBS).
- Department of Physiscs, Tokyo Institute of Technology.

Committee of doctoral thesis (in alphabetical order)
Daisuke Jido (1)Katsushi Ito (1)Kazuyuki Sekizawa (1)Masahide Yamaguchi (1)(2)
Teruaki Suyama (1)Yosuke Imamura (1)Affiliations(1) Department of Physiscs, Tokyo Institute of Technology.
(2) Cosmology, Gravity and Astroparticle Physics Group, Center for Theoretical Physics of
the Universe, Institute for Basic Science (IBS).

Abstract
This thesis is devoted to searching for fundamental issues of constraint systems, aiming for
applications to theories of gravity. Theories of constraint systems play a crucial role inconstructing theories of gravity since these theories are constraint systems, and theories ofgravity provide the fundamentals of cosmology. Hence, it would be no exaggeration to saythat theories of constraint systems are a fundamental ground of cosmology.
A wide range of theories of modern physics is constructed upon the variational principle,
and so are the theories of gravity. The variational principle requires appropriate boundary
conditions to derive the equations of motion of a given theory. For non-degenerate systems,
the Dirichlet boundary conditions are imposed as usual. For degenerate systems, however,
the existence of constraints generically disturbs the consistency of the boundary conditions
with the possible dynamics. This means that an inappropriate choice of boundary conditions
is prone to restrict the possible solutions to a narrow region. Therefore, the well-posed
variational principle demands a methodology to compose valid boundary conditions without
such restriction to the solutions. In particular, since the theories of gravity have the propertyof higher-order derivative degenerate systems, the identification of such boundary conditions
is important for introducing counter-terms like the Gibbons-York-Hawking counter-terms.
The first topic of this thesis is the study of establishing the methodology.
Metric-affine gauge theories of gravity are one of the modified/extended theories of general relativity in a geometric and the most generic manner and describe gravity by using
curvature, torsion, and non-metricity, and it is expected to give perspectives into problems such as the late-time accelerated expansion of the universe (, or equivalently, darkenergy problem, cosmological constant problem), the origin of inflation from the viewpointof low-energy effective theories of quantum gravity based on the UV-competition, H0 - and/orii

σ8 -tension problems. These theories are classified into eight subclasses from viewpoints ofgauge structures in the internal space based on the Möbius representations. In particular,
the three subclasses of the so-called geometrical trinity of gravity, which is composed ofGeneral Relativity (GR), Teleparallel Equivalent to GR (TEGR), and Symmetric TEGR(STEGR), have gathered attention in recent days since the non-linear extensions of thesesubclasses have the potential for explaining problems given in above. However, the propagating Degrees of Freedom (pDoF) of those subclasses has not been unveiled yet, and thestudy of investigating these numbers is the second topic of this thesis.
One of the methods to identify the pDoF of a given theory is the use of the so-called DiracBergmann analysis. The heart of this analysis is very simple; the time evolution of a given
system is described by the so-called total Hamiltonian, which is composed of the Legendre
transformation of the Lagrangian of the given system such that it is compatible with theexistence of the primary constraints, and all constraints satisfy the so-called consistency
conditions, which demands all constraints to be static. For point particle systems, thisanalysis is mathematically well-formulated, meaning that the Poisson brackets are always
well-defined. For field theories, which include, of course, theories of gravity, however, thewell-definedness should be reconsidered since the existence of the Dirac delta function causes
problematic situations when determining the Lagrange multipliers such as contradictions in
the consistency conditions to the properties of the given system that are derived from its
action functional. In this thesis, the issues of the Dirac-Bergmann analysis in field theories
are scrutinized and clarified, and a prescription to make the pathological situation easier is
proposed. This prescription may restrict the solution space to a narrow region but makes
it possible to give a reliable result to each analysis. Then, based on the prescription, theanalysis is performed for each subclass in the geometrical trinity of gravity and its non-linear
extension.
Finally, the uncompleted works and future directions of this thesis are discussed.
iii

Impact statement
The new results of this thesis are based on two published papers and two peer-reviewing papers when this thesis was submitted but confirmed their validity by the committee of doctoral
thesis. This thesis is devoted to providing new perspectives for the following two problems:
the well-posedness of the variational principle in degenerate systems and the degrees of freedom counting in the use of the Dirac-Bergmann analysis of the modified/extended theories ofgravity, in particular, the so-called geometrical trinity of gravity and its non-linear extension.
There are three impacts of this thesis. The first impact is that a methodology for setting appropriate boundary conditions for the well-posed variational principle in degenerate
systems is established. Before the methodology appeared, boundary conditions of constraint
systems were naively determined as the apparent variables in the first-order variation of theaction functional, and there generically were incompatibilities with constraints of the given
system on the boundaries. This methodology would also provide the fundamentals of deriving the counter-terms in the action functionals of theories of gravity since a counter-term isderived only after appropriate boundary conditions are determined for the given theory.
The second impact is that the geometric quantities, i .e ., curvature, torsion, and nonmetricity, are formulated in a unified manner based on the Möbius representation. Before thisapproach appeared, these geometric quantities were separately introduced in each own andad hoc way. This unified perspective would provide a more comprehensive understanding ofgeometric quantities and gauge theories of gravity constructed by such geometric quantities,
and it might bring a discovery of new geometric quantities. If it is realized, the constructionof a new theory of gravity by using the new geometric quantity could be possible.
The third impact is that the issues of the Dirac-Bergmann analysis in field theories are
revealed and a prescription to make the analysis valid is established. Before the issues wereclarified and the prescription appeared, the result of the analysis of a given theory was basediv

on a set of PDEs with respect to the Lagrange multipliers which is not guaranteed whether
or not it is solvable, therefore, the degrees of freedom derived by the method was always
nothing more than a speculation. The prescription provides a reliable result of the analysis
and a set of concrete conditions to guarantee it.

v

Acknowledgement
First of all, I would like to thank Masahide Yamaguchi and Teruaki Suyama for their supervision of my Ph.D. work. In particular, I would like to thank Masahide Yamaguchi because
he kindly supported my Ph.D. work even after he moved to a foreign research institute.
I would like to thank all the collaborators, Keisuke Izumi, Keigo Shimada, and Sebastian
Bahamonde (in alphabetical order). Keisuke Izumi has been always helpful and given me
a lot of advice and ideas on physics study and research. Keigo Shimada has discussed
a lot with me and taught me fundamental attitudes to collaborative research. Sebastian
Bahamonde has given me a lot of ideas on research and supported me in writing scientificpapers. He has also taught me fundamental attitudes to collaborative research with foreign
researchers. His passion for physics has always encouraged me. I would like also to thankall my colleagues, in particular, Asuka Ito, Fumiya Sano, Jorge Gigante Valcarcel, Katsuki
Aoki, Korumilli Sravan Kumar, Luca Buoninfante, Mohammad Ali Gorji, Michiru Niibo,
Masaru Siino, Paolo Gondolo, Shin’ichi Hirano, Siyao Li, So Okano, So Tanaka, Taishi
Katsuragawa, Xin Ren, and Yuichi Miyashita (in alphabetical order) for fruitful discussions
and their thoughtfulness. In particular, Shin’ichi Hirano has given me a lot of advice on
improving my presentation skills in both oral ones and writing papers. Taishi Katsuragawa
has kindly heard my concerns and given me a lot of advice.
I would like to thank my friends, Kazuki Otsuka, Shintaro Nakamura, Tomoki Yatsuzuka,
and Yuuki Wakimoto (in alphabetical order). I would like to thank my family, Hiroko
Tomonari, Nobuyuki Tomonari, Kouki Tomonari, Ritsuko Watanabe, Tamotsu Watanabe,
Masaaki Kimura, and Koichi Harasaki for supporting my life. I would like also to thankmy family lawyers, Makoto Iwai and Han Teyong for supporting the negotiation of the job
retirement from the previous company. By virtue of their support, I could smoothly move to
my Ph.D. work. Also, when I got into a traffic accident, they supported me in the settlementvi

of it.
I would like to appreciate financial support from Tokyo Tech Fund Hidetoshi Kusama
Scholarship. I would like to thank the committee members of my doctoral thesis, in addition
to my supervisors, Daisuke Jido, Katsushi Ito, Kazuyuki Sekizawa, and Yosuke Imamura(in alphabetical order). In particular, Daisuke Jido has been caring about me since I was astudent in a master course.
Finally, I would like to thank all my loving things.

vii

To all my loving things...

viii

Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ii

Impact statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iv

Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

viChapter 1 : Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter 2 : Analytical mechanics on degenerate point particle systems . . .

12

2.1

Non-degenerate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

2.2

Degenerate systems and Dirac-Bergmann analysis . . . . . . . . . . . . . . .

19

2.3

Gauge symmetries and Dirac structures . . . . . . . . . . . . . . . . . . . . .

24

2.4

Propagating and gauge degrees of freedom . . . . . . . . . . . . . . . . . . .

31

2.5

Higher-order derivative systems and Ostrogradsky ghost degrees of freedom .

34

2.6

Ostrogradsky ghost-free theories . . . . . . . . . . . . . . . . . . . . . . . . .

41

2.7

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

Chapter 3 : The well-posed variational principle in degenerate point particle
systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

3.1

Cumbersome situations of the variational principle in degenerate systems . .

51

3.2

Dirichlet boundary conditions and second-order derivative equations of motions 54

3.3

Frobenius integrability of first-order time derivative systems . . . . . . . . .

58

3.4

Reformulation of the well-posed variational principle. . . . . . . . . . . . .

65

3.5

Embeddings of Symplectic manifolds . . . . . . . . . . . . . . . . . . . . . .

67

3.6

Well-posed variational principle 1: Cases of canonical embeddings . . . . . .

74

3.7

Well-posed variational principle 2: Cases of quasi-canonical embeddings . . .

78

ix

3.8

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

3.9

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

Chapter 4 : General theory of relativity . . . . . . . . . . . . . . . . . . . . . . 103
4.1

Spacetime and fundamental principles . . . . . . . . . . . . . . . . . . . . . . 103

4.2

Geodesic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.3

Einstein field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.4

ADM-foliation of spacetime and Einstein field equations

4.5

Reformulation based on the well-posed variational principle . . . . . . . . . . 123

4.6

Dirac-Bergmann analysis and degrees of freedom . . . . . . . . . . . . . . . . 130

4.7

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134. . . . . . . . . . . 115

Chapter 5 : Gauge approach to gravity . . . . . . . . . . . . . . . . . . . . . . . 135
5.1

Fiber bundles and G-bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.2

Principal G-bundles, Ehresmann connections, and curvatures . . . . . . . . . 137

5.3

Vector bundles, gauge fields, and curvatures . . . . . . . . . . . . . . . . . . 142

5.4

Internal space bundles, frame fields, and Weitzenböch connections . . . . . . 147

5.5

A derivation of Einstein field equations based on gauge approach to gravity . 153

5.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Chapter 6 : Metric-affine gauge theories of gravity . . . . . . . . . . . . . . . . 157
6.1

Curvatures, torsions, and non-metricities . . . . . . . . . . . . . . . . . . . . 157

6.2

Metric-affine gauge approach to gravity and its subclass . . . . . . . . . . . . 164

6.3

Geometrical trinity of gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.4

Modified theories of geometrical trinity of gravity . . . . . . . . . . . . . . . 184

6.5

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

xChapter 7 : Dirac-Bergmann analysis of geometrical trinity of gravity andits extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
7.1

Issues of Dirac-Bergmann analysis in field theories and a prescription . . . . 191

7.2

Degrees of freedom of TEGR . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

7.3

Degrees of freedom of Coincident GR . . . . . . . . . . . . . . . . . . . . . . 209

7.4

Degrees of freedom of non-linear extension of GR . . . . . . . . . . . . . . . 212

7.5

Degrees of freedom of non-linear extension of TEGR . . . . . . . . . . . . . . 217

7.6

Degrees of freedom of non-linear extension of CGR . . . . . . . . . . . . . . 228

7.7

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

Chapter 8 : Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
AppendixA : PB-algebras of coincident f (Q)-gravity in (n + 1)-dimensional spacetime 251
AppendixB : The explicit formulae of α and βji . . . . . . . . . . . . . . . . . . . . . 255
AppendixC : Inönü-Wigner contraction of metric-affine geometry and its extensions . 256
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

xiChapter 1
Introduction
The quest for the universe has fascinated mankind over the ages. In modern physics, thisquest is positioned as cosmology, and researchers have continued to devote all their efforts tounveiling the mysteries of the universe from both theoretical and phenomenological points ofview. The beginning of cosmology dates back to the era of A. Einstein, who established thegeneral theory of relativity. This theory is the most successful classical theory of gravity anddescribes how the universe evolves together with matter. Hence, combining the standard
model of particle physics, cosmology has challenged to unveil the history of the universe.
ΛCDM model is the standard model of cosmology based on FLRW spacetime with cosmological parameters that are determined by observations. (Stage 1 in Figure 1.1.) Thismodel became the standard model of cosmology around the beginning of twenty-first century based on the observations of the cosmic microwave background [Spergel et al. (2003)],the large-scale structure [Tegmark et al. (2004)], and the late-time accelerated expansion of
the universe [Riess et al. (1998); Perlmutter et al. (1999)]. The most recent observation at
the moment writing this thesis is provided in Planck 18 [Aghanim et al. (2020a,b)]. Theseobservations further indicated that the universe is anisotropic and inhomogeneous in the
distribution of photons and matter, respectively. In order to understand these observational
facts, it is mandatory to take the cosmological perturbation into account of the standard
model of cosmology together with the Big Bang and inflation hypothesis. A series of Boltzmann equations with respect to the perturbations of cosmic ingredients and Einstein field
equations describe how the ingredients develop with time under the influence of gravity.
(Stage 3 in Figure 1.1.) Based on this system, the power spectrum of each ingredient isderived and compared to the observations. (Stages 4 and 5 in Figure 1.1.) This is the basic
1

Figure 1.1: Modern framework for cosmology
formulation of the modern cosmology [Dodelson (2003); Mukhanov (2005); Weinberg (2008)].
The modern cosmology has also unveiled a series of new mysteries. The ingredients of
the universe should contain the so-called dark energy and dark matter in the percentileof about 95% but the true identity and the origin of these ingredients have not been unveiled yet. So is the mechanism of how inflation occurs and its origin. Furthermore, therecent observations point out the so-called tension problem in the cosmological parametersof σ8 and H0 [Aghanim et al. (2020b)], which is the discrepancy of the estimated values ofH0 and σ8 between the estimation based on the ΛCDM model and the model-independent
observation. H0 - and σ8 -tensions are confirmed in about 5σ and 3σ confidence level, respectively [Wong et al. (2020); Nunes and Vagnozzi (2021)]. In particular, Figure 1.2 indicatesthat the discrepancies of H0 between the value obtained from the CMB measurement (based2

Figure 1.2: The recent observations of Hubble parameter H0 and its discrepancy. The leftand right-panels are given in Ref [Di Valentino et al. (2021)] and Ref [Hu and Wang (2023)],
respectively.
on the ΛCDM model) and the Type Ia Supernovae (model-independent) have not gradually
been impossible to ignore. These problems may be related to the expansion of the universe,or equivalently, spacetime, and such phenomena could be ascribed to the nature of gravity. It would imply a possibility of the alternation of the basic theory of gravity: Stage 1
in Figure 1.1 should be scrutinized. Combined further with the recent development in theaccuracy of the observation of gravitational waves, the elaboration of modified/extended
theories of gravity is rapidly proceeding. This provides a natural motivation to investigate
the modified/extended theories of gravity.
The current thesis treats three topics. Metric-affine gauge theories of gravity are oneof the modified/extended theories of gravity, which describes gravity in terms of geometric3

quantities of curvature, torsion, and non-metricity [Blagojevic (2001)], and the Weitzenböch
connection introduces the covariant derivatives [Weitzenboch (1923)]. Since the metricaffine gauge theory of gravity is the most generic extension of the general theory of relativity
in a geometric manner, the extended theories have a possibility to explain gravitational
phenomena that are hard to treat within the general theory of relativity. The beginningof the gauge approach to gravity dates back to the era of the birth of Yang-Mills gauge
theory [Yang and Mills (1954)], which provides the foundation of modern particle physics.
Inspired by this theory, R. Utiyama extended it into describing gravity in a unificationmanner under the concept of gauge invariant principle [Utiyama (1956)]. Combined theseries of works by E. Cartan [Cartan (1924)] and H. Weyl [Weyl (1952); Hehl et al. (1986);
Blagojevic (2001)], the fundamentals of the theories are formulated as the formulation in thecurrent days. This is one of three topics of the present thesis.
A set of subclasses of the metric-affine gauge theories of gravity called the geometrical trinity of gravity is the most actively researched one on the purpose of the cosmological application for explaining the problems such as dark energy (the late-time accelerating expansion of the universe without the cosmological constant) and the origin of inflation [Beltrán Jiménez et al. (2019); Heisenberg (2023); Bahamonde et al. (2023a)]. This set
includes the general theory of relativity as a special case and these subclasses are equivalent
to one another up to boundary terms in action functional. The notable point is that thisequivalence gets changed when some extension or modification is performed and it gives◦

rise to the fertility for resolving the problems in cosmology. For instance, f (R)-gravity
is one of the non-linear extensions of the geometrical trinity of gravity, and it includes
the most successful inflation model so-called Starobinsky R2 -inflation model (Figure 1.3).
In addition, the non-linear extensions of the teleparallel and symmetric teleparallel gravity, which are also subclasses of the geometrical trinity of gravity, generically provide theso-called effective cosmological constant, and it can explain the late-time accelerated ex4

Figure 1.3: Tensor-to-scalar ratio [Aghanim et al. (2020a)].
pansion of the universe. Furthermore, those extensions in quadratic, non-linear, and nonlocal ways have also gathered attention as possible candidates for consistent low-energy
effective theories of quantum gravity from the viewpoint of UV-completion [Stelle (1978);Starobinsky (1980); Koshelev et al. (2023)], which are ways to grasp a clue for understanding
the mechanism of inflation [De Felice and Tsujikawa (2010); Capozziello and De Laurentis(2011); Donoghue and Menezes (2022)]. However, the cosmological application of the modified/extended theories of gravity demands unveiling such as the constraint structure of thosetheories such as gauge symmetry, propagating degrees of freedom, the existence of (Ostrigradski’s) ghost modes. The Hamiltonian/Dirac-Bergmann analysis makes it possible to unveil
such structures [Dirac (1950, 1958a,b); Anderson and Bergmann (1951); Bergmann et al.
(1950); Bergmann (1949); Bergmann and Brunings (1949)]. (Stage 2 in Figure 1.1.) For◦

instance, the analysis of f (R)-gravity unveils that this theory has diffeomorphism and local

5

Lorentz invariance, three possible propagating degrees of freedom, and no ghost degrees of
freedom. The first and third properties are completely the same as the general theory of
relativity. The different property is the second one, and it is important since one of three
propagating degrees of freedom can be used for generating inflation and the remaining two
degrees of freedom are precisely the same modes in the general theory of relativity. The
Dirac-Bergmann analyses of these theories are also one of three topics of the present thesis.
The last one of three topics of the present thesis is a consideration of the well-posednessof the variational principle. This topic is important in particular for the theories of gravity since these theories are degenerate systems and many of these theories include secondorder or higher-order derivative terms in those action functionals. The former property(degenerate system), on one hand, implies that the variational principle requires appropriate
boundary conditions that are consistent with the constraints of the theory. On the other
hand, the latter property (higher-order derivative system) indicates that the action functional demands appropriate counter-terms under the imposition of the well-posed boundary conditions of the variational principle known as the Gibbons-York-Hawking counterterms in the general theory of relativity [York (1972, 1986a); Gibbons and Hawking (1977);
Hawking and Horowitz (1996)]. It is well known that this sort of counter-terms plays a crucialrole in constructing theories of quantum gravity in the use of the path-integral formulation
and blackhole thermodynamics [Gibbons and Hawking (1977); York (1986b); Braden et al.
(1990); Brown and York (1993b,a)], which could contribute to understanding of the originof inflation. In this thesis, the well-posedness provides the action functionals of the theories
of gravity being suitable for the Dirac-Bergmann analysis, in which the absence of Ostrogradsky’s ghost modes is guaranteed by virtue of the work of the Gibbons-York-Hawking
counter-term.
Reassembling these three topics in theoretical order, the current thesis aims to give newperspectives for the following three problems:61. Well-posedness of the variational principle
2. Classification of the metric-affine gauge theory of gravity
3. Degrees of freedom counting of the geometrical trinity of gravity and its non-linear
extension with the Dirac-Bergmann analysis
Based on these problems, the current thesis is composed as follows:
The first two chapters are devoted to a review of degenerate systems and a method of making
the variational principle well-posed, respectively.
In chapter 2, a theory of analytical mechanics for theories of gravity is provided in a selfcontained manner. Theories of gravity have the properties of degenerate and higher-orderderivative systems. Starting with reviewing the basic concepts of non-degenerate point particle systems, a theory of degenerate point particle systems is introduced. In particular, theDirac-Bergmann analysis is reconstructed in detail. Based on these ingredients, the concepts of gauge symmetry/transformation, Dirac structure, propagating degrees of freedom,and gauge degrees of freedom are established. Finally, higher-order derivative point particle
systems are introduced. The problematic situations caused by the Ostrogradsky instability/degrees of freedom are pointed out, and a methodology for removing these problematic
degrees of freedom is provided.
In chapter 3, a methodology for making the variational principle in point particle systems well-posed is established. The extension to field theories has been developing, butsome specific cases will be provided in the succeeding chapters of the present thesis. Firstof all, the problems of degenerate systems in the variational principle are explained. It isshown that the valid boundary conditions in a higher-order time derivative system but which
has equations of motion up to second-order time derivative are restricted to position-fixing
boundary conditions. Based on the Pons formulation, which is introduced in Section 2.5,
7

Frobenius integrability conditions of first-order degenerate time derivative systems are investigated, and the concrete formulae of the conditions are derived. After these preparations,
the well-posedness of the variational principle is reformulated. Introducing two types of
embeddings of the symplectic manifold based on the Shanmugadhasan-Maskawa-Nakajima
theorem, i .e ., quasi-canonical embeddings and canonical embeddings, the well-posedness isinvestigated. As a result, appropriate boundary conditions are specified. Finally, examples
are given, and this chapter is summarized with the outlook for these works.
Chapter 2 is based on• K. Tomonari, “On the well-posed variational principle in degenerate point particle
systems using embeddings of the symplectic manifold”, PTEP 2023 6 063A05.
DOI: 10.1093/ptep/ptad073, arXiv:2304.00877 [math-ph].
Copyright (2023) by the Physical Society of Japan [Tomonari (2023b)].
• K. Izumi, K. Shimada, K. Tomonari, and M. Yamaguchi, “Boundary Conditions forConstraint Systems in Variational Principle”, PTEP 2023 10 103E03.
DOI: 10.1093/ptep/ptad122, arXiv:2303.03805 [hep-th].
Copyright (2023) by the Physical Society of Japan [Izumi et al. (2023)].
The next two chapters provide a review of the general theory of relativity and bundle
theory in a self-contained form. Many of the recent works of the metric-affine gauge theories
of gravity are discussed just by imposing some conditions on three fundamental geometricquantities: curvature, torsion, and non-metricity, but, in the author’s opinion, it is oversimplified. These geometric quantities and conditions are closely and precisely related to
the gauge structures of internal space, and, in order to unveil the relations and structures,
it is mandatory to discuss them based on bundle theories. Otherwise, the fertility of themetric-affine gauge theories of gravity would be lost.
In chapter 4, the general theory of relativity, which is the most successful classical theory8

of gravity in the modern physics, is established. First of all, the fundamental concepts ofthe theory such as spacetime, weak/Einstein/strong equivalence principles are introduced.
Based on these concepts, the geodesic equations and the Einstein field equations are derived
from both viewpoints of physical considerations (bottom-up approach) and the variational
principle (top-down approach). As mentioned in Section 3.9, we assume that the wellposedness of the variational principle demands consistency with the primary constraint(s):C(1) . In particular, before deriving the Einstein field equations by using the variational
principle, the ADM-foliation/decomposition of the spacetime is introduced. This technique ismandatory not only to apply the variational principle in a well-posed manner being consistent
with C(1) but also to perform the Dirac-Bergmann analysis. Finally, performing the DiracBergmann analysis, the propagating and gauge degrees of freedom of the theory are clarified.
In chapter 5, a mathematical framework for gauge approach to gravity is constructed.
First of all, three fundamental bundles, i .e ., fiber bundles, G-bundles, and principal Gbundles are defined. For principal G-bundles, the Ehresmann connection and its curvature
are introduced. After that, vector bundles together with gauge fields and those curvature(field strengths) are derived. Internal space bundles, which play a crucial role in constructing
the metric-affine gauge theories of gravity, are also derived, in particular, focusing on frame
fields and Weitzenböch connection. Finally, a gauge-approach derivation of the Einstein field
equations is described. This chapter gives mathematical basics for the succeeding chapters.
Based on the previous chapters, the final two chapters are devoted to formulating themetric-affine gauge theory of gravity, in particular, focusing on the geometrical trinity ofgravity and those non-linear extension, and performing the Dirac-Bergmann analysis of these
theories.
In chapter 6, the metric-affine gauge theories of gravity are constructed. First, based on
Sections 5.2 and 5.4, three fundamental geometric quantities, i .e ., curvature, torsion, andnon-metricity are introduced. Second, using the so-called Möbius representation of a gauge
9group, the metric-affine geometries are classified into eight subclasses: Riemann-Cartan,
teleparallel, Riemann, Minkowski, Weyl-Cartan, Weyl, metric-affine, and symmetric teleparallel geometries. In this formulation, each geometric quantity is given by the exterior covariant derivative of each corresponding potential. Third, The geometrical trinity of gravity,
which is composed of the three subclasses: Riemann (general relativity), symmetric teleparallel, and torsion teleparallel, is established. These three subclasses are equivalent up toboundary terms in the meaning of action functional. Finally, two extensions of the geometrical trinity of gravity, that is, quadratic extension and non-linear extension, are introduced.
The latter extension plays a main role in the next section.
In chapter 7, the Dirac-Bergmann analysis is performed for each subclass of the geometrical trinity of gravity and its non-linear extension. In order to apply the analysis, which
was introduced in Section 2.2 but restricted to point particle systems, to field theories, threefundamental issues are clarified. These issues are commonly caused by the spatial derivativesof the Dirac delta function. And then a prescription that circumvents the issues is proposed.
Based on the prescription, the analysis is performed for TEGR, CGR, a non-linear extensionof GR, TEGR, and CGR, and the propagating degrees of freedom of each theory is unveiled.
Throughout these analyses, the three issues in field theories of the Dirac-Bergmann analysis
are concretely clarified for each subclass, and the reason why the prescription is valid forremoving the issues is investigated. I emphasize that these analyses are not a complete manipulation of the issues; it is just a prescription to circumvent the issues and make it possible
to perform the Dirac-Bergmann analysis. Therefore, the results could be valid only for someclasses of solutions. Finally, this chapter is summarized with the outlook for these works.
Chapter 6 is based on• K. Tomonari , “A unified description of curvature, torsion, and non-metricity of themetric-affine geometry with the Möbius representation”, arXiv:2312.11558 [gr-qc]. [Tomonari(2023a)].
10Chapter 7 is based on• K. Tomonari and S. Bahamonde, “Dirac-Bergmann analysis and Degrees of Freedomof Coincident f (Q)-gravity”, arXiv:2308.06469 [gr-qc].

[Tomonari and Bahamonde(2023)].
Finally, in Chapter 8, I summarize the conclusions of this thesis.

11

Chapter 2
Analytical mechanics on degenerate point
particle systems
Chapter abstract
In this chapter, a theory of analytical mechanics for theories of gravity is provided in a selfcontained manner. Theories of gravity have the properties of degenerate and higher-orderderivative systems. Starting with reviewing the basic concepts of non-degenerate point particle systems, a theory of degenerate point particle systems is introduced. In particular, theDirac-Bergmann analysis is reconstructed in detail. Based on these ingredients, the concepts of gauge symmetry/transformation, Dirac structure, propagating degrees of freedom,and gauge degrees of freedom are established. Finally, higher-order derivative point particle
systems are introduced. The problematic situations caused by the Ostrogradsky instability/degrees of freedom are pointed out, and a methodology for removing these problematic
degrees of freedom is provided.

2.1. Non-degenerate systems
In this section, the basics of the Lagrangian and Hamiltonian formalism of non-degenerate
point particle systems are briefly reviewed. First of all, let us introduce the Lagrangian
formalism.
The principle of least action/variational principle plays the most fundamental role in thisformalism. This principle derives the equations of motion of a given system with respect tothe generic variations: not only to the variations of the configuration variables and thosevelocity variables but also to the time variable. Assume that it exists that a system which12

is described by a Lagrangian L = L(q̇ i , q i , t) (i ∈ {1 , 2 , · · · , N }) defined on the velocity
phase space T Q × R of a configuration space Q. Here, q i = q i (t) are a coordinate system forthe configuration space. q̇ i = q̇ i (t) are the velocity variables of q i (t), respectively, and then,q̇ i and q i give a coordinate system for the velocity phase space. A non-degenerate system
satisfies a condition that the rank of the kinetic (or Hessian) matrix, i .e .,

Kij =

∂ 2L
∂ q̇ i ∂ q̇ j

(2.1)

is full rank. Then the action functional is introduced as follows:
Z t2i

I[q , t] =L(q̇ i , q i , t) dt ,

(2.2)

t1

where “ [q i , t] ” denotes that the right-hand side of the equation is a functional defined on
the configuration space T Q × R. The principle of least action/variational principle, statingthat the path in nature locally minimizes the value of the action functional, leads to a setof equations of motion. Consider the following variations:q i (t) → q ′i (t′ ) = q i (t) + δq i (t) ,

(2.3)

′

t → t = t + δt(t).
Since the time variable is also varied, the commutativity of the variational operation δ andthe time derivative operation d/dt is lost; it indicates the necessity to reconsider the variationof the time variables. Introducing the equal-time variation, i .e .,δ̄q i = q ′i (t) − q i (t) ,

13

(2.4)

the following variation formula is derived:(2.5)

δ q̇ i (t) = δ̄ q̇ i (t) + q̈ i (t)δt ,where second-order variational terms are neglected. In the same manner, the variation of
the configurations are also calculated as follows:(2.6)

δq i (t) = δ̄q i (t) + q̇ i (t)δt.

Then the first variation of the action functional (2.2) is calculated as follows:i

Z t2 

δI[q , t] =
t1

∂L
d
−
i
∂q
dt



∂L
∂ q̇ i



d
δ̄q +
dt
i



∂L
∂ q̇ iiδ̄q + Lδt

dt .

(2.7)

Under the imposition of the Dirichlet boundary conditions, i .e .,δ̄q i (t2 ) = δ̄q i (t1 ) = 0 ,δt(t2 ) = δt(t1 ) = 0 ,(2.8)

then the variational principle, δI[q i , t] = 0, leads to the equations of motion, i ,e ., EulerLagrange equations, as follows:
∂L
d
−
i
∂q
dt



∂L
∂ q̇ i


= 0.

(2.9)

The above derivation is based on the variational principle with respect to the generic variations Eq (2.5). Without the time variation, the boundary conditions δt(t2 ) = δt(t1 ) = 0
are satisfied by definition, and the derivation results back to the ordinary one in which thevariations are taken only to the configuration and those velocity variables.
Second, assuming that the Euler-Lagrange equations hold, Noether’s first theorem is14derived. Then, using Eq (2.6), Eq (2.7) becomes as follows:
Z t2dΘ(δq i (t) , δt(t) , t) ,
dt
t1





∂L
∂L
i
i
i
Θ(δq , δt , t) =δq −q̇ − L δt .
∂ q̇ i∂ q̇ ii

δI[q , t] =(2.10)

Since boundary terms do not affect to the Euler-Lagrange equations, the Lagrangian has anarbitrariness of the boundary terms. That is,
L → L′ = L +d
W (q i , t) ,dt

(2.11)and this symmetry is called quasi-invariance of the system. Therefore, the following formula
holds:d 
Θ(δq i (t) , δt(t) , t) − W (q i (t) , t) = 0 .
dt(2.12)

This formula indicates that the quantity Θ − W is invariant with respect to the variation
Eq (2.3) in the time evolution of the system; Θ − W is a conserved quantity of the system.
This is a result of the quasi-invariance given in Eq (2.11).
Third, to exist a Lagrangian for given equations of motion, a set of relations have to besatisfied as necessary and sufficient conditions. This theorem is crucial to validate Noether’s
first theorem since it is meaningless without the existence of a Lagrangian for the equations of
motion even if symmetries exist. All physics of a given system is engraved into a Lagrangian
excepting initial conditions. Assume that a set of equations of motion is given such thatthese equations include up to second-order time derivative terms of the configurations asfollows:[EoM]i (q̈ k , q̇ k , q k , t) = Kij (q̇ k , q k , t)q̈ j + Ai (q̇ k , q k , t) = 0 ,(2.13)

where Kij is the kinetic matrix defined in Eq (2.1) and Ai are arbitrary functions. The

15

well-posedness of the variational principle under the imposition of the Dirichlet boundary
conditions guarantees the validity of the generic expression of these equations of motion
above. This point will be discussed in Section 3.2. The necessary conditions are nothing but
the satisfaction of the following formula:d
Kij (q̇ , q , t)q̈ + Ai (q̇ , q , t) =
dt
k

k

j

k

k



∂L
∂ q̇ i


−

∂L
.
∂q i

(2.14)

Comparing the coefficients of q̈ i , q̇ i , q i , and other terms, it leads to the following equivalent
formulae:
Kij = Kji ,
∂Kki
∂Kjk
=
,
∂ q̇ i∂ q̇ j∂∂Ai ∂Ajk ∂+= 2 q̇
+Kij ,∂ q̇ j∂ q̇ i∂q k ∂t

 


∂
∂Ai ∂Aj∂Ai ∂Aj
k ∂
−
+
−
2
= q̇
.
∂q j
∂q i
∂q k ∂t∂ q̇ j∂ q̇ i(2.15)

Therefore, for a set of the equations of motion (2.13), if a Lagrangian exists then the above
relations have to be satisfied.
Conversely, if the above relations hold then the Lagrangian can be derived as follows:
L = K(q̇ j , q j , t) + Di (q j , t)q̇ i + C(q j , t) ,(2.16)

where K, Di , and C are given as follows:
i

i

Z 1

K(q̇ , q , t) =q̇ i Hi (q̇ j τ , q j , t) dτ + W1 (q j , t) ,Z 01

Di (q k , t) = q jτ Z[ij] (q k τ, t) dτ + W2 (q̇ k , t) ,Z 01
C(q i , t) = q i
Yi (q j τ , t) dτ + W3 (q̇ j , t) ,016

(2.17)

where W1 , W2 , and W3 are arbitrary functions and Hi , Z[ij] , and Yi are defined as follows:
Z 1q̇ j Kij (q̇ k τ, q k , t) dτ ,01 ∂Ai ∂Aj
∂ 2K
∂ 2K
Z[ij] =+−−,
2 ∂ q̇ j∂ q̇ i∂q i ∂ q̇ j
∂q j ∂ q̇ i


 2
∂K
∂Di
∂ K
1 ∂Ai ∂Aj
∂ 2K
jq̇ − i − Ai + i +
.
Yi =
+
−
i
j
j
i
∂q q̇
2 ∂ q̇∂ q̇
∂q∂ q̇ ∂t∂t

Hi =(2.18)

These arbitrary functions can be neglected as total divergent terms. This statement isshown as follows: Without any loss of generality, these functions are encapsulated as K ′ =K +Ui (q j , t)q̇ i +V (q j , t) for appropriate functions Ui and V . Using K ′ , Eq (2.16) is rewritten
as L′ = K ′ + Di′ q̇ i + C ′ = L + D̄i q̇ i + C̄, where D̄i = Di′ + Ui − Di and C̄ = C ′ + V − C. Since
Eq (2.14) leads to ∂ D̄i /∂q j − ∂ D̄j /∂q i = 0 and ∂ C̄/∂q i − ∂ D̄i /∂t = 0, it exists a function
W = W (q i , t) such that D̄i = ∂W/∂q i and C̄ = ∂W/∂t, and these conditions indicate that
L′ = L + dW/dt.
There is a remark; Eq (2.13) has an arbitrariness of the multiplication of a non-singularmatrix Xi j : [EoM]′i = Xi j [EoM]j = 0. It indicates that the correspondence between a
Lagrangian and a set of the equations of motion, Eq (2.13), is not generically determineduniquely and also implies that there is a possibility for the existence of a Lagrangian eventhough once the initial set of Eq (2.13) does not satisfy the necessary and sufficient conditions
given in Eq (2.15). That is, it is enough to satisfy the conditions (2.15) up to this arbitrarinessto exist a Lagrangian for Eq (2.13).
Finally, Hamiltonian formalism is introduced. The second equation in Eq (2.10) introduces canonical momentum variables as follows:

pi =

∂L
.
∂ q̇ i(2.19)

Then q i and pi span a coordinate system for the phase space T ∗ Q × R by virtue of the17

implicit function theorem since the system is non-degenerate. For the coordinate system,
the Hamiltonian is introduced as follows:H(q j , pj , t) = pi q̇ i − L(q̇ j , q j , t) q̇j =F j (qk ,p ,t) ,k(2.20)

where F i (q j , pj , t) are a set of functions that are determined by the implicit function theorem.
This is nothing but the Legendre transformation of the Lagrangian L = L(q̇ i , q i , t). The variation of the Hamiltonian under the Euler-Lagrange equations leads to δH = F i (q j , pj , t)δpi −pi δq i , and it guarantees that the Hamiltonian formalism is described in the phase space
T ∗ Q × R in a well-defined manner. The action functional in the phase space is given as
follows:
∗ I[q

i

Z t2
, pi , t] =pi q̇ i − H(q j , pj , t) dt ,(2.21)

t1

where the left under “ ∗ ” of ∗ I[q i , pi , t] denotes a quantity in the phase space. The variational principle under the Dirichlet boundary conditions, δq i (t2 ) = δq i (t1 ) = 0, leads to the
Hamilton equations as follows:q̇ i =∂H,∂piṗi = −

∂H.
∂q i

(2.22)

Under the satisfaction of the Legendre transformation Eq (2.20) and its inverse transformation, the first equations reproduce the relations q̇ j = F j (q k , pk , t) and the second equations
are equivalent to the Euler-Lagrange equations given in Eq (2.9). This derivation is valid
only in the case that δq i and δpi are independent. If a set of constraints to the phase space
exist then the derivation has to be reconsidered. We need a theory on constraint/degenerate
systems. The derivation of the Hamilton equations under the existence of constraints will
be discussed in Sections 3.6 and 3.7.

18

2.2. Degenerate systems and Dirac-Bergmann analysis
In this section, degenerate systems are defined, and a methodology to determine the time evolution of a given system, i .e ., the Dirac-Bergmann analysis [Dirac (1950, 1958a,b); Bergmann(1949); Bergmann and Brunings (1949); Bergmann et al. (1950); Anderson and Bergmann(1951)], is introduced. First of all, let us introduce the Hamiltonian formulation of degenerate systems.
The rank of the kinetic matrix of the system, i .e ., Eq (2.1), discriminates the degeneracy/singularity of a given system from the non-degenerate systems. Assume that the rankof the kinematic matrix is N − R: rank K = N − R (R ≥ 1). Then the number of R primaryconstraints exist, and such system is defined as a degenerate/singular system. Let us denote(1)

these primary constraints as ∗ ϕα (q i , pi ) := 0 (α ∈ {1 , 2 , · · · , R}) and the phase subspace(1)

which is restricted by the constraints as C(1) (t) = {P ∈ T ∗ Q × {t}|∗ ϕα (q i (t) , pi (t)) := 0}(t ∈ R), where P is a point in T ∗ Q × R at a time t; P is coordinated by q i and pi un(1)der the satisfaction of ∗ ϕα (q i , pi ) := 0, and “ := ” denotes the imposition as a condition. The degeneracy also implies the existence of the number of r zero-eigenvalue vectors of the kinetic matrix. Based on this, taking into account that the number of N − R
velocity variables q̇ a (a ∈ {1 , 2 , · · · , N − R}) can be expressed as q̇ a = F a (q i , pb ; q̇ α )(b ∈ {1 , 2 , · · · , N − R} α ∈ {N − R + 1, N − R + 2, · · · , N }) by virtue of the implicit
function theorem, it is shown that the Lagrangian depends linearly on the velocity variablesq̇ α (α ∈ {N − R + 1 , N − R + 2 , · · · , N }) [Saito et al. (1989, 1993)]
∂ ∂L
= 0.
∂ q̇ α ∂ q̇ β q̇a =F a (qi ,pb ;q̇α )(2.23)

Therefore, the Legendre transformation of the Lagrangian becomes as follows:
H = H0 + q̇ α ∗ ϕ(1)α ,H0 = pa q̇ a + pα q̇ α − L(q i , F a (q i , pa ; q̇ α ); q̇ α ) q̇α :=019(2.24)

(1)under the imposition of q̇ α := 0, where ∗ ϕα = pα − fα (q i , pi ) := 0 for some functions
fα (q i , pi ) which are determined by the given Lagrangian. If the Legendre transformationis restricted to the subspace C(1) (t) then H ≈ H0 , where “ ≈ ” denotes the weak equality:
equality under the imposition of all the constraints. It indicates that q̇ α can be expressed by
the phase space variables of a neighbor region of C(1) (t) in T ∗ Q × R, denoted by O(C(1) (t)),
and therefore q̇ α in Eq (2.24) can be replaced as a set of Lagrange multipliers, denoted by
λα = λα (q i , pi ), as follows:
HT = H0 + λα ∗ ϕ(1)α .

(2.25)

This quantity is noting but the so-called total Hamiltonian of the system, and this quantity
governs the time evolution of the system. H0 is called as the kernel Hamiltonian of thesystem. Remark that the total Hamiltonian is defined in O(C(1) (t)), which is a region ofT ∗ Q × R, not in the subspace C(1) (t) of T ∗ Q × R.
In order to reveal the dynamics of the system, it is necessary to determine the multipliers.
For this purpose, let us introduce the Dirac-Bergmann analysis. A natural way to resolve
this problem is that, since the constraints restrict the phase space to a possible subspacein which the dynamics live, the constraints should not develop according to passing time.
C(1)Expressing in a formula, denoting “ ≈ ” as the weak equality under the imposition of the
primary constraints, it becomes as follows:
C(1)
(1)
(1)
(1)
β
∗ ϕ̇α = {∗ ϕα , HT }P.b. ≈ {∗ ϕα , H0 }P.b. + λ Dαβ := 0 ,(2.26)

these equations are called consistency conditions, where “ { · , · }P.b. ” denotes the Poissonbracket and Dαβ is called as Dirac matrix which is defined as follows:(1)

Dαβ = {∗ ϕ(1)α , ∗ ϕβ }P.b. .

20(2.27)

Therefore, the consistency conditions determine the multipliers depending on the number ofthe rank of the Dirac matrix. Assume that the rank of the Dirac matrix is R − R1 (R1 ≥ 1).
Then the number of R−R1 multipliers are determined but that of R1 multipliers remain arbitrary. It indicates that the same number of new constraints, higher-order constraints, appear.
The fundamental matrix transformations can split out the higher-order constraints from theconsistency conditions. This manipulation is allowed since the constraints can be reorganized
up to linear combinations. Applying this manipulation, the Dirac matrix is transformed as′(1)

follows: Dαβ = P (1) D(1) Q(1) for appropriate non-degenerate matrices P (1) and Q(1) . The′(1)shape of the transformed matrix Dαβ is free to choose for the purpose of the analysis. In
′(1)

(1)

(1)

(1)

this section, let us choose the shape as Dαβ = diag(τ1 , τ2 , · · · , τR−R1 , 0 , 0 , · · · , 0). Using
these ingredients, the consistency conditions are transformed as follows:
′(1)

(1)

β

P (1) α β {∗ ϕβ , H0 }P.b. + Dαβ λ′(1) := 0 ,αwhere λ′(1) = Q(1)

−1 α

β
βλ .

(2.28)

Eq (2.26) is equivalent to Eq (2.28) by virtue of the regularityof the matrices P (1) and Q(1) . For the indices a(1) ∈ {1 , 2 , · · · , R − R1 }, on one hand, theβ(1)multipliers are determined as λ′ a(1) = −τ (1) a(1) P (1) a(1) {∗ ϕβ , H0 }. Remark that we do not
sum over with respect to a(1) . On the other hand, for the indices α(1) ∈ {R − R1 + 1 , R −R1 + 2 , · · · , R}, new constraints, i. e ., secondary constraints, have to be imposed as follows:
(2)
(1)
(1) β
∗ ϕα(1) = P
α(1) {∗ ϕβ , H0 }P.b. := 0 .

(2.29)In addition, there is a case that, in Eq (2.28), removing some multipliers from an equation
using other equations, a new constraint appears. In such cases, these constrains include in(2)(1)∗ ϕα(1) extending the range of the index α . Then the total Hamiltonian can be reassembled

as follows:
(1)

(1)

HT = H0 + λα
21

(1)

(1)
∗ Φα(1) ,(2.30)

(1)(1)

where H0 and ∗ Φα(1) are defined as follows:
(1)

(1)

H0 = H0 + λa

(1)(1) (1) β∗ Φα(1) = ∗ ϕβ Qα(1) .

(1)∗ ϕa(1) ,(2.31)(1)

That is, the determined terms are encapsulated as the first term H0 in the reassembled-total
(1)

(1)

Hamiltonian HT . The second term in HT still has arbitrariness.
Repeating the same procedure for the secondary constraints under the reassembled-total
Hamiltonian Eq (2.30), and if it gives rise to new constraints then repeat over the sameprocess until all the multipliers are determined or the new constraint does not appear up
to linear combinations. Let us assume the process stops by S-steps. Then we obtain the
(1)

(S)

(2)

final results; the constraints ∗ ϕα := 0, ∗ ϕα(1) := 0, · · · , ∗ ϕα(S−1) := 0 appear, the multipliers
(1)

(2)

(S)

λa , λa , · · · , λa

are determined, the reassembled-total Hamiltonian
(S)

HT

(S)

= H0

+ λα

(S)

(S)
∗ Φα(S) ,(2.32)

where
(S)

H0

(S−1)

= H0

+ λα

(S)

(1)
∗ ϕα(S) ,(S)(S) (S) β∗ Φα(S) = ∗ ϕβ Qα(S)(2.33)

is derived, and the dynamics of the system is restricted to the phase subspace C(S) (t) = {P ∈
(1)

(2)

(S)

T ∗ Q × {t}|∗ ϕα := 0, ∗ ϕα(1) := 0, · · · , ∗ ϕα(S−1) := 0}. The multipliers λα(S)remain arbitrary

and the existence of such multipliers then implies that this system has gauge symmetry. This
point will be discussed in detail in the next Section 2.3. There is a remark; as long as all
the consistency conditions for up to sth-order constraints hold, where s ∈ {1, 2, · · · , S}, theargument “ (t) ” can be removed from the subspaces C(1) (t), C(2) (t), · · · , C(s) (t) since all theconstraints up to sth-order are static.
The Dirac-Bergmann analysis reveals all the constraints of the system. These constraints
are classified into two classes: first-class and second-class. The former is defined as a set of22

constraints that are commutative with all other constraints in the phase subspace C(S) withrespect to the Poisson bracket. The number of the set of first-class constraints has to be taken
into the maximum number. Otherwise, the constraints are classified into second-class and thetotal number of second-class constraints is always even as long as the first-class constraints
are taken in the maximum number [Castellani (1982)]. The constraints satisfy the following
propositions; (i) A Poisson bracket between two first-class constraints is also classified into
first-class constraint; (ii) The product of two constraints is classified into first-class constraint.
For a function f in T ∗ Q × R such that f = 0 in C(S) , (iii) f is expressed in a neighbor
O(C(S) ) as a linear combination of the constraints with appropriate coefficient functions:(s−1)α
f = g(s)(s)∗ ϕα(s−1) , where α0 = α.

C(S)In more detail, for first-class constraints ∗ χv ≈ 0 (v ∈C(S){1 , 2 , · · · , V }) and second-class constraints ∗ θw ≈ 0 (w ∈ {1 , 2 , · · · , 2W }), (iv) A firstclass function f is expressed in a neighbor O(C(S) ) as a linear combination of the first-classconstraints with appropriate coefficient functions: f = g v ∗ χv ; (v) A second-class function f
is expressed in a neighbor O(C(S) ) as a linear combination of both the first-class and secondclass constraints with appropriate coefficient functions: f = g v ∗ χv + hw ∗ θw . In particular,
(S+1)

(S)

Proposition (ii) indicates that the first-class constraints ∗ ϕα(S) = {∗ ϕα(S−1) , HT }P.b. can be(s)expressed as a linear combination of the first-class constraints ∗ ϕα(s−1) (s ∈ {1 , 2 , · · · , S},(S+1)α(0) = α) with appropriate coefficient functions: ∗ ϕα(S) = C α

(s)

(s−1)

α(S) (s)C(S)∗ ϕα(s−1) ≈ 0.

Based on the above manipulations for the system, the time evolution of a quantity,(S)denoted by f , of the system is described by the reassembled-total Hamiltonian HT

as

(S)
(S)
(S)
follows: f˙ = {f , HT }P.b. = XT f = LX (S) f , where XT and LX (S) denote the Hamiltonian
T

T

(S)
vector field with respect to HT , which is defined by
(S)
∂HT
(S)
XT =
i

∂q

(S)

∂
∂HT ∂
∂
−
+
,
i
∂pi
∂pi ∂q∂t(S)(2.34)and the Lie derivative with respect to XT , respectively. The remarkable point here is that,23

in the cases that the Dirac-Bergmann analysis does not determine all the multipliers, thetime evolution of quantities of the system cannot be uniquely determined, and it is causedby the existence of first-class constraints. Therefore, it needs to construct a theory to resolve
this situation and to identify the unique time evolution.

2.3. Gauge symmetries and Dirac structures
In this section, based on the properties of constraints of a given degenerate system, gauge
symmetry is formulated [Sugano and Kimura (1983); Sugano et al. (1986); Sugano and Kimura(1990a,b); Sugano and Kagraoka (1991a,b)]. Then, the Dirac structure [Dirac (1950, 1958b)],which identifies the time evolution of dynamical/physical quantities of the system, is introduced. First of all, let us formulate the gauge symmetry.
In order to formulate the change of quantities of a given system, let us introduce theconcept of infinitesimal canonical transformation. For a canonical coordinate system q i andpi , denoted by {q i , pi }, consider a canonical transformation from {q i , pi } to {Qi , Pi } whichis generated by a generator: W = −pi Qi + ϵG(Qj , Pj , t), where ϵ and G are an infinitesimal
parameter and an arbitrary function of Qi , Pi , and t, respectively. Then, taking into accountup to first-order terms of ϵ, q i and pi are transformed as follows:
q i → Qi = q i + ϵ∂G(q j , pj , t),∂pipi → Pi = pi − ϵ∂G(q j , pj , t).
∂q i(2.35)

Remark that, in Eq (2.35), G becomes a function of q i , pi , and t in the first-order approximation with respect to ϵ due to the existence of ϵ in the front of the partial derivatives. For thisinfinitesimal canonical transformation, a quantity of the system, denoted by f = f (q i , pi , t),
is transformed into as follows:δG f = {f , G}P.b. = XG f = LXG f .

24(2.36)

In the above manipulation, G governs the transformation; G can be regarded as the generatorof the transformation rather than W . Replacing the generator G by the total Hamiltonian
HT , the Hamilton equations are derived; the time evolution of the system is nothing but
the sequences of the infinitesimal canonical transformations that are generated by the total
Hamiltonian.
The perspective described above for the change of quantities of the system reveals therelation between first-class constraints and gauge transformations. For simplicity, assume
(s)

C(s)

that a degenerate system has only first-class constraints: ∗ Ψα(s−1) ≈ 0 (s ∈ {1 , 2 , · · · , S},
α(0) = α ∈ {1 , 2 , · · · , N − rank K}). Then the total Hamiltonian is given by HT = H0 +(1)λα ∗ Ψα . Since the multipliers λα can be arbitrarily determined, for the transformation of agenerator G and a quantity f , it is possible to impose the following relation:

LXG (XT f ) := Xλα ∗ Ψ(1)f,α(2.37)

or equivalently, it is calculated as follows:{G , HT }P.b. +C(1)∂G:= −λα ∗ Ψ(1)
≈ 0.
α
∂t

(2.38)

That is, the generator G is static under the imposition of the primary constraints; it indicates that δG f = LXG f is nothing but a gauge transformation of the quantity f and G
is the generator of the gauge transformation, and the system is, of course, invariant under the satisfaction of the relation Eq (2.38). That is, using the Hamilton equations andthe relations from the Legendre transformation, the variation of the Lagrangian L with respect to the gauge transformation in the phase subspace C(1) , i .e ., ∗ δG L, is calculated as
(1)
(1) C
i
α
≈ d(pi δG q i − G)/dt, and it varies up to a total divergent∗ δG L = d(pi δG q − G)/dt − λ ∗ Ψα25term: ∗ δG L = dF/dt. Therefore, the following relation is derived:C(1)F ≈ pi

∂G
− G.
∂pi

(2.39)

This relation implies a possible explicit form of the generator G.
In order to derive a concrete form of the generator G, let us consider an infinitesimaltransformation for the Lagrangian formulation which includes a set of arbitrary functions:δϵ q i = (Dlα ϵα (t))g i α,lα (q̇ j , q j ) ,(2.40)

where Dlα = dlα /dtlα , ϵα (t) are arbitrary functions of time, g i α,lα (q̇ j , q j ) are determinedby the Lagrangian, and lα ∈ {0 , 1 , · · · , Mα − 1}. Dlα ϵα (t) are independent functions withrespect to lα and α. The invariance of the system to the above transformation indicates that
the Lagrangian varies up to a total divergent term: δϵ L = DF , where F is defined by
F = Dlα ϵα (t)fα,lα (q̇ j , q j ) .

(2.41)

The functions fα,lα (q̇ j , q j ) are also determined by the Lagrangian. Since δϵ L = [EoM]i δϵ q i +D(pi δϵ q i ) = DF , the following identity is derived:[EoM]i (Dlα ϵα )g i α,lα + D{(Dlα ϵα )∆α,lα } = 0 ,(2.42)

i∆α,lα = pi g α,lα − fα,lα ,where [EoM]i denote the equations of motion in Lagrange formulation. Integrating by parts in
the range of (t1 , t2 ) under the imposition of the boundary conditions Dlα ϵα (t2 ) = Dlα ϵα (t1 ) =26

0, the following identities are derived:
M
α −1
X

(2.43)(−1)lα Dlα ([EoM]i g i α,lα ) = 0 ,

lα =0

and
∆α,Mα −1 = 0 ,∆α,lα + D∆α,lα +1 + [EoM]i g i α,lα +1 = 0 ,D∆α0 −(lα ∈ {1 , 2 , · · · , Mα − 2})(2.44)M
α −1X(−1)lα Dlα ([EoM]i g i α,lα ]) = 0 .

lα =1

Eq (2.43) and the third identity of Eq (2.44) lead to the following identity:(2.45)

D∆α0 + [EoM]i g i α,lα −1 = 0 .

The first identity of Eq (2.44) is nothing but the primary constraints in the Lagrangian
formulation:

(1)
∗ Ψα

= ∆α,Mα −1 = 0. The second identity of Eq (2.44) leads to higher-

order constraints in the Lagrangian formulation under the imposition of the Euler-Lagrange(M −lα )equations: ∗ Ψα α

(M −lα −1)= ∆α,lα = −D∆α,lα +1 −[EoM]i g i α,lα +1 = −D∗ Ψα α

= 0, applying

sequentially in the order of lα = Mα − 2, Mα − 3, · · · , 0. This sequence implies that(lα )= ∆α,Mα −lα = 0 ,∗ Ψα(lα ∈ {1, 2, · · · , Mα }) .

(2.46)In the Hamiltonian formulation, using the Hamilton equations, these constraints are converted intoC(lα )(lα +1)= ∆α,Mα −lα −1 = {∗ Ψ(lα α ) , HT }P.b. ≈ 0 .
∗ Ψα(2.47)

This indicates Mα = Sα . Based on these ingredients, Eq (2.40), Eq (2.41), the definition of27

∆α,lα given in Eq (2.42), and Eq (2.47) derive the following expression of the generator G:

G=

M
α −1
X

C(1)

lα α

(D ϵ )∆α,lα ≈

lα =0

M
α −1
X

α −lα )
(DIα ϵα )∗ Ψ(M= pi δϵ q i − F
α(2.48)

lα =0

in a consistent manner with Eq (2.39). Therefore, since Dlα ϵα can be regarded as independentfunctions ϵαlα , a possible expression of the generator G is nothing but a linear combination(s)of the first-class constraints ∗ Ψα with coefficient functions ϵαlα (t)(2.49)

G = ϵαlα (t)∗ Ψ(lα α ) .

That is, the first-class constraints dominate gauge transformations under the satisfaction of
the condition of Eq (2.38).
The coefficient functions ϵαlα (t) are restricted from the conditions of Eq (2.38) as follows:{G , H0 }P.b. +C(1)∂G= −λα ∗ Ψ(1)
≈ 0,
α
∂t

(2.50)C(1)≈ 0.
{G , ∗ Ψ(1)
}
P.b.
α

Taking into account Eq (2.49), the first relations above lead to
Sα
X
lα =1

ϵ̇αlα ∗ Ψ(lα α ) +

SX
α −1

ϵαlα ∗ Ψ(lα α +1) +

lα =1

Sα
X

(1)
(l ) CϵαSα C β α,lα ∗ Ψβ α

≈ 0,(2.51)

lα =1

where the Greek letters run from 1 to the total number of the primary first-class constraints.
Therefore, a set of recurrence relations hold as follow:
ϵ̇αlα + ϵαlα −1 + ϵβSα C α β,lα = 0 ,(2.52)

where lα ∈ {2 , 3 , · · · , Sα }. The recurrence relations indicate that, providing the number of28the primary first-class constraints functions ϵβSα under the satisfaction of the conditions in
Eq (2.50), all the remaining functions ϵβlα are sequentially determined. That is, the generator
of a gauge transformation in Eq (2.49) is determined from the recurrence relations Eq (2.52)
under the satisfaction of the second conditions in Eq (2.50), and for this transformation, thesystem is invariant; gauge symmetries are now formulated. The Poisson bracket algebra of
the first-class constraints, {∗ Ψα , ∗ Ψβ }P.b. = C γ αβ ∗ Ψγ , characterise the symmetries. In theexistence of second-class constraints, the above construction is valid for the reassembled-total(S)

Hamiltonian HT

(S)

= H0

+ λα

(S)

(S)
(S)
∗ Ψα(S) , recomposing the first-class constraints ∗ Ψα(S)

into

(l )

the form ∗ Ψα α up to linear combinations.
There is a remark on Eq (2.44); if Sα = 2, the identities becomes as follows:∆α,1 = 0 ,(2.53)ii∆α,0 + D∆α,1 + [EoM]i g α,1 = ∆α,0 + [EoM]i g α,1 = 0 .
These identities are noting but Noether’s second theorem with no time variation. Considering
time variation, δϵ t = ϵα (t)τα (q̇ i , q i , t), the second identity becomes as follows:∆α,0 − Eτα + [EoM]i g i α,1 = 0 ,ii(2.54)i

E = pi q̇ − L(q̇ , q , t) .
The first identity in Eq (2.53) and Eq (2.54) are nothing but Noether’s second theorem inthe most general form.
So far all the quantities of the system are arbitrary functions that are defined in thephase space T ∗ Q × R. A dynamical/physical quantity, denoted by f , is restricted to a
1st

1st

gauge invariant quantity: δG f = LXG f = {f , G}P.b. ≈ 0, or equivalently, {f , ∗ Ψα }P.b. ≈ 0
1st

1st

for all the first-class constraints ∗ Ψα ≈ 0 (α ∈ {1 , 2 , · · · , U }), where “ ≈ ” denotes theequality under the satisfaction of all the first-class constraints. In addition to the first-class29

constraints, hereinafter in the rest of the section, assume that second-class constraints exist:
∗ Θa

2nd

2nd

≈ 0 (a ∈ {1 , 2 , · · · , 2V }), where “ ≈ ” denotes the equality under the imposition of all
1st

the second-class constraints. The Poisson brackets among f and ∗ Ψα ≈ 0 are already weak
2nd

equal to zero by its definition, but that among f and ∗ Θa ≈ 0 do generically not vanish in
the weak equality even under the imposition of all the constraints. The Dirac bracket realizes
the weak equal vanishing not only for the first-class constraints but also for the second-classconstraints. Let us consider the following quantity:
∗

f =f−

2V
X

−1

{f , ∗ Θa }P.b. (D )ab ∗ Θb +UXAα ∗ Ψα ,(2.55)

α=1

a=1,b=1

where Aα are arbitrary functions. This quantity satisfies not only {f ∗ , ∗ Ψα }P.b. ≈ {f , ∗ Ψα }P.b. ≈1st

0 but also {f ∗ , ∗ Θa }P.b. ≈ 0. Since f ∗∗ ≈ f , the map f 7→ f ∗ is interpreted as a projection
map under the imposition of all the first-class constraints. Considering another physicalquantity g, the Dirac bracket, denoted by “ { · , · }D.b. ”, between f and g is introduced asfollows:∗∗{f , g}D.b. = {f , g }P.b. ≈ {f , g}P.b. −2v
X{f , ∗ Θa }P.b. (D−1 )ab {∗ Θb , g}P.b .

(2.56)

a=1,b=1

The Dirac bracket inherits all algebraic properties of the Poisson bracket and satisfies thefollowing properties: {f , g}D.b. ≈ {f ∗ , g}P.b. ≈ {f , g ∗ }P.b. . Then the Dirac structure is
introduced as follows: For the linear transformation of the constraints′α∗ Ψα → ∗ Ψα = X ∗ Ψα ,′a
α∗ Θa → ∗ Θa = Y ∗ Θa + Z ∗ Ψα ,(2.57)

where X α , Y a , and Z α are arbitrary functions, the Dirac bracket is invariant{f , g}′D.b. ≈ {f , g}D.b. .
30(2.58)

Using the Dirac bracket, the time evolution of the physical quantity f is described as follows(S)
f˙ = {f , HT }D.b ≈ {f , H0 }D.b ,(S)

where H0(S)

H0 + λα(S)(2.59)(S)

is the determined part of the reassembled-total Hamiltonian HT = HT=(S)
∗ Φα(S) . That is, for a gauge invariant quantity, i .e ., a physical quantity, the time

evolution is given by Eq (2.59). The Dirac structure introduces the dynamical/propagating
degrees of freedom and the gauge degrees of freedom.

2.4. Propagating and gauge degrees of freedom
In this section, the concepts of dynamical/propagating and gauge degrees of freedom are introduced. The gauge Degrees of Freedom (gDoF) of a given system, on one hand, is already
implied in Section 2.3; the number of the undetermined multipliers λα(S)(c .f . Eq (2.32) andEq (2.33) in Section 2.2), or equivalently, that of the functions ϵα = ϵαSα (c .f . Eqs (2.52) in
Section 2.3) is nothing but the gDoF. On the other hand, the propagating Degrees of Freedom(pDoF) can be introduced in two different ways: based either on the Dirac structure or a theorem known as the Shanmugadhasan-Maskawa-Nakajima theorem [Shanmugadhasan (1973);
Dominici and Gomis (1980, 1982); Maskawa and Nakajima (1976); Tomonari (2023b)]. Firstof all, let us consider the first case.
Assume that only second-class constraints, ∗ Θa ≈ 0, exist. Then the Dirac structureholds for any quantities of the system, and then the time evolutions of these quantities
are uniquely determined by Eq (2.59). Since the second-class constraints are static for theunique time evolution, the number of possible canonical coordinate variables to describe thesystem, i .e ., propagating Degrees of Freedom (pDoF), is counted out as follows: pDoF =(2N − 2V )/2 = N − V . In the case of existing first-class constraints, ∗ Ψα ≈ 0, the situation
is different; the time evolution is no longer uniquely determined until the gauge degrees of
freedom are removed. One of the methods to realize this removal is the gauge fixing method.
31

This method turns the first-class constraints into second-class constraints by imposing a setof additional constraints.
(1)

(2)

(1)

(S)

Assume that the gDoF is T and ∗ Ψα ≈ 0 , ∗ Ψα = {∗ Ψα , HT }P.b. ≈ 0 , · · · , ∗ Ψα
(S−1)

{∗ Ψα

=

, HT }P.b. ≈ 0 (α = 1 , 2 , · · · , T ) are first-class constraints. Since the total Hamilto-

nian contains the number of T undetermined multipliers, only the number of T additional(1)

constraints can be imposed. Let us denote these T additional constraints as ∗ Ξα := 0. Thena gauge fixing condition is given as follows [Sugano et al. (1992)]:
1st;1

(1)

{∗ Ψ(k)α , ∗ Ξβ }P.b. ≈ 0 ,(1)

det ({∗ Ψ(K)α , ∗ Ξβ }P.b. )(1){∗ Ξβ , H0 }P.b.
1st;1

where “ ≈ ” and “

NOT:1st;1≈

NOT:1st;1≈

NOT:1st;1≈(2.60)

0,0,” denote the weak equality and the weak not equality, respectively,(1)

under the imposition of all the first-class constraints and ∗ Ξα := 0. The remaining T ×(s+1)(S − 1) additional constraints are automatically derived as ∗ Ξα
(s)

(s)

= {∗ Ξα , HT }P.b.

1st;1 ,2 ,··· ,s≈(s){∗ Ξα , H0 }P.b. . Then, impose: ∗ Ξα := 0 (s ∈ {1 , 2 , · · · , S}). Using the Jacobi identity, it is
(s)

(s)

shown that ∗ Ψα ≈ 0 and ∗ Ξα ≈ 0 satisfy the following relation:(S−s+1)det ({∗ Ψ(s)}P.b. )α , ∗ Ξα(s)

NOT:1st;1 ,2 ,··· ,S≈(2.61)

0,(s)otherwise the determinant among ∗ Ψα and ∗ Ξα are weak equal to zero under the impositionof all the first-class constraints and all the additional constraints, as desired; This relation
(s)

(s)

indicates that ∗ Ψα ≈ 0 and ∗ Ξα ≈ 0 are now classified into second-class constraint. The
third conditions in Eq (2.60) determine the multipliers λα .
(1)

(2)

The pDoF in the case of existing the first-class constraints, i .e ., ∗ Ψα ≈ 0 , ∗ Ψα =
(1)

(S)

(S−1)

{∗ Ψα , HT }P.b. ≈ 0 , · · · , ∗ Ψα = {∗ Ψα, HT }P.b. ≈ 0 (α = 1 , 2 , · · · , T ), or equivalently,

32gathering these into ∗ Ψα ≈ 0 (α ∈ {1, 2, · · · , V = T × S}), is now obvious by applying the
pDoF in the case of existing only second-class constraints: pDoF = (2N − (V + V ))/2 =N − V . In addition, in the case of existing the second-class constraints ∗ Θa ≈ 0, the pDoF
is (2N − 2U − 2V )/2 = N − U − V .
Finally, let us introduce the pDoF in a direct way based on the ShanmugadhasanMaskawa-Nakajima theorem. The statement of the theorem is expressed as follows:
Theorem 1.
For a symplectic 2-form of the given system: Ω = dq i ∧ dpi (i ∈ {1 , 2 , · · · , N }), it exists a canonical coordinate system such that Ω = dQI ∧ dPI + d∗ Θa ∧ d∗ Θa + d∗ Ξα ∧ d∗ Ψα(I ∈ {1 , 2 , · · · , N − V − U }; a ∈ {1 , 2 , · · · , V }; α ∈ {1 , 2 , · · · , U }), where ∗ Θa ≈ 0 and∗ Θa ≈ 0 are composed only of all the 2V

second-class constraints, ∗ Ψa ≈ 0 are composed

only of all the U first-class constraints.
■
Applying this theorem, the pDoF and gDoF of a given system are counted out. A point
which should be careful is that the conjugate pairs of the second-class constraints, i .e .,a∗Θ≈ 0 and ∗ Θa ≈ 0, are static in the time evolution, but so are not the configuration

variables for the first-class constraints, i .e ., ∗ Ξα , unless appropriate gauge fixing conditions
are imposed. Therefore, on one hand, if the system has only second-class constraints thenthe counting is performed in the same manner as the previous paragraph. On the other hand,
if the system has first-class constraints then the counting for pDoF in the phase space turns
out to be ∗ pDoF = 2N − 2V − U . In order to convert ∗ pDoF into that of the velocity-phasespace, i .e ., pDoF, speculating the possibility of gauge fixing, pDoF should be counted out asfollows: pDoF = (∗ pDoF − U )/2 = N − V − U . This means that, when counting out pDoF
based on the above theorem, the resulting number of pDoF is generically different from thenumber of actual propagating modes since the set of degrees of freedom which evolves intime, i .e,., ∗ Ξα , is removed out. That is, there may occur a situation such that the pDoF
33

counted out based on the above theorem is less than the number of actual propagating
modes. If encountering such a situation, we should perform gauge fixing and check whether
this discrepancy is reconciled or not.
There is a remark; Taking into account the rank of the kinetic matrix Eq (2.1), it gives
the upper limit of the pDoF of the given system. That is, let us assume rank K = N − R. If
R is an even number then the theorem indicates that the upper limit becomes (2N − R)/2:
pDoF ≤ N − R/2. If R is an odd number then the theorem implies that the limit becomes(2N − R − 1) = (2N − 2R′ − 2)/2: pDoF ≤ N − R′ − 1, where R = 2R′ + 1. The pDoF
counted out by the Dirac-Bergmann analysis has to satisfy this upper limit as consistency
with existing constraints.

2.5. Higher-order derivative systems and Ostrogradsky ghost degrees
of freedom
In this section, higher-order derivative systems, of which Lagrangian contains higher-ordertime derivatives, are introduced. Such systems generically have the so-called Ostrogradsky
instability. There are two ways to formulate such systems: Ostrogradsky’s original formulation [Ostrogradsky (1850); Woodard (2015); Saito et al. (1989, 1993)] and Pons’ modified
formulation [Pons (1989)]. First of all, let us introduce Ostrogradsky’s formulation.
Assume that a Lagrangian L = L(q i , Dq i , · · · , DM q i ) (i ∈ {1 , 2 , · · · , N } , D = d/dt)
exists. Then the kinetic matrix of the system is given as follows:
(M )

Kij

=

∂ 2L
,
∂(DQi(M ) )∂(DQj(M ) )(2.62)and the action functional is introduced as follows:
I[Qi(1) , Qi(2) , · · · , Qi(M ) ] =Z t2

L(Qi(1) , Qi(2) , · · · , Qi(M ) , DQi(M ) ) dt ,t1

34

(2.63)

whereQi(s) = Ds−1 q i ,DQi(M ) = DM q i(s ∈ {1 , 2 , · · · , M }) .

(2.64)First, let us consider the non-degenerate case: det K (M ) ̸= 0.
The variational principle is applied under the imposition of the Dirichlet boundary conditions, Qi(s) (t2 ) = Qi(s) (t1 ) = 0; the variation of the action functional with respect to theconfigurations Qi(s) = Ds−1 q i is computed as follows:
Z t2 
δI =
t1

∂L
(1)
− Ṗi
i
∂Q



h
it2
(s)
δ Qi dt + Pi δ Qi(s) ,t1(2.65)(s)where Pi , canonical momentum variables with respect to Qi(s) , are defined as follows:

(s)

Pi

=

M
X

(−1)r−s Dr−sr=s∂L
∂DQi(r)!
.

(2.66)

These definitions can be reassembled as a reccurence relation, the so-called Ostogradskytransformation, as follows:
(M )

Pi

=

∂L
,
∂(DQi(M ) )

(s−1)

Pi

=

∂L
(s)
− Ṗi ,i
∂DQ(s−1)(2.67)

together with Eq (2.64), where s ∈ {2 , 3 , · · · , M }. Therefore, the variational principle,
δI := 0, the Euler-Lagrange equations are derived as follows:∂L(1)− Ṗi = 0 .
i∂Q(2.68)

Remark that the second equations in Eq (2.67) do not describe the time evolution of thesystem, although each equation contains the time derivative of the canonical momentum
variable. From the viewpoint of the variational principle, the equations of motion are given
only as Eq (2.68).
35

In order to introduce the Hamiltonian formalism, let us consider the canonical variables(s){Qi(s) , Pi } (s ∈ {1 , 2 , · · · , M }) and the Legendre transformation, the so-called Ostrogradsky Hamiltonian, given as follows:(s)
H(Qi(s) , Pi ) =

M
−1
X

(M )

(s)

Pi DQi(s) + Pi

F i − L(Qi(1) , Qi(2) , · · · , Qi(M ) ; F i ) ,(2.69)

s=1

where
(M )

F i = DQi(M ) = F i (Qi(1) , Qi(2) , · · · , Qi(M ) ; Pi

(2.70))

is determined by the implicit function theorem for Eq (2.62). Notice that all the velocity
(s)

(s)

variables are removed from the theory. The first term depends linearly on Pi . Since Pi(s ∈ {1 , 2 , · · · , M − 1}) can take the range of arbitrary value, on one hand, H would not bebounded from below; the system may have an instability. If this is the case, it is nothing but
the Ostrogradsky instability and the corresponding degrees of freedom is called Ostrogradsky
ghosts, the total number of them is N × (M − 1) in this case. On the other hand, since(M )the canonical momentum variables Pi

can be bounded from below via the relation (2.70),

therefore, if the first terms are absent, H is bounded from below. The variational principleof the Ostrogradsky Hamiltonian derives the equations of motion as follows:Q̇i(s) =

∂H
(s)
∂Pi

,

(s)

Ṗi

=−

∂H
.
∂Qi(s)(2.71)

The first equations for s ∈ {1 , 2 , · · · , M −1} and the second equations for s ∈ {2 , 3 , · · · , M }reproduce the Ostrogradski transformation. The first equations for s = M lead to therelations in Eq (2.70). The second equations for s = 1 derive the Euler-Lagrange equations (2.68).
The next stage is the degenerate cases but it is easy to describe from the knowledge in
Section 2.2. Let us assume that the rank of the kinetic matrix is N − R: rank K (M ) = N − R.
36

In the same manner as Section 2.2, the total Hamiltonian can be introduced as follows:(2.72)HT = H0 + λα ∗ ϕ(1)
α ,
(1)

(1)

(M )

where ∗ ϕα = ∗ ϕα (Qi(1) , Qi(2) , · · · , Qi(M ) ; Pi

) (α ∈ {1 , 2 , · · · , R}). The Dirac-Bergmannanalysis is also applied in the same manner as Section 2.2. The time evolution of the canonicalvariables is calculated as follows:
DQi(s) = {Qi(s) , HT }P.b. ≈ Qi(s+1)

s ∈ {1 , 2 , · · · , M − 1} ,(M )DQa(M ) ≈ F a (Qi(1) , Qi(2) , · · · , Qi(M ) ; Pb

; λα ) ,DQα(M ) ≈ λα ,∂L,
∂Qi(1)

(1)

≈

(s)

≈ −Pi

DPi
DPi

(2.73)

(s−1)

+

∂L
∂DQi(s−1)s ∈ {2 , 3 , · · · , M } .

The fourth equations describe the time evolution of the system. The others reproduce theOstrogradsky transformation.
Finally, let us introduce Pons’ formulation; a method to decompose a higher-order derivative system into a first-order system with the corresponding second-class constraints using
a set of auxiliary variables. This formulation is frequently used in the studies of modified/extended theories of gravity. Assume that a Lagrangian L = L(q i , Dq i , · · · , DM q i )(i ∈ {1 , 2 , · · · , N } , D = d/dt) exists. Then let us require the first relations of the Ostrogradsky transformation Eq (2.64) as a set of constraints:
DQi(s) − Qi(s+1) := 0 ,(s ∈ {1 , 2 , · · · , M − 1}) .

37(2.74)

These constraints decompose the Lagrangian into
N M
−1
X
X

λi(s) (DQi(s) − Qi(s+1) ) ,(2.75)

L → L′ = L′ (Qi(1) , Qi(2) , · · · , Qi(M ) ; DQi(1) , DQi(2) , · · · , DQi(M ) ; Dλi(s) , λi(s) ) .

(2.76)

L→L′′= L (Qi(1) , Qi(2) , · · · , Qi(M ) , DQi(M ) ) +i=1 s=1

or schematically,

That is, the original system is decomposed into the first-order derivative system with a setof auxiliary variables. Remark that, in the Ostrogradsky formulation, Eq (2.74) is not a setof constraints but just that of replacement Eq (2.64); the given system remains M th-orderderivative system. First, let us consider the case: det K (M ) ̸= 0 for the kernel kinetic matrix
K (M ) =∂ 2 L′,∂(DQi(M ) )∂(DQj(M ) )(2.77)

but det K̃ (M ) = 0 for the entire kinetic matrix
K̃ (M ) =∂ 2 L′,∂(DX i′ )∂(DX j ′ )(2.78)′

where X i = {Qi(M ) , λi(s) }; the sector for the configurations Qi(M ) is non-degenerate but the′whole sector for X i is degenerate due to the existence of the configurations λi(s) . It indicatesthat there is a set of primary constraints as follows:

∗θ

(s)
(1) s
i = πi := 0 ,

∗θ

(s)
(s)
(1) M −1+s
= Pi − λi := 0 ,i38(2.79)

(s)(s)

where πiand Piare canonical momentum variables
(s)

πi =

∂L′
= 0,
∂Dλi(s)

(s)

Pi

=

∂L′
(s)
= λi .
∂DQi(s)(2.80)

Since the sector for the configurations Qi(M ) is non-degenerate, the system does not give rise
M −1+ssto another primary constraint. The Poisson brackets among ∗ θ(1) i and ∗ θ(1) iM −1+vu{∗ θ(1) i , ∗ θ(1) jare}P.b. = δij δ uv ; these constraints are classified into second class constraint.

The total Hamiltonian is derived as follows:
(s)

(M )

HT′ =Pi DQi(s) + Pi
−

N M
−1
X
X

F i − L′ (Qi(1) , Qi(2) , · · · , Qi(M ) , F i )s
M −1+sλi(s) (DQi(s) − Qi(s+1) ) + σsi ∗ θ(1) i + τsi ∗ θ(1) i,(2.81)

i=1 s=1

where σsi and τsi are multipliers, F i are implied from the implicit functions theorem for K̃ (M )as follows:(M )F i = DQi(M ) = F i (Qi(1) , Qi(2) , · · · , Qi(M ) , Pi
(M )and Pi(2.82)),are canonical momentum variables as follows:
(M )

Pi

(s)

The linear dependence on Pi

=

∂L′
.
∂DQi(M )(2.83)

that is not bounded from below gives rise to the Ostrogradsky
(s)

(s)

instability. Since the first three terms in Eq (2.81) do not contain πi and λi , the consistency
conditions determine all multipliers as
C(1)

(s−1)

σsi ≈ −Pi+

∂L′,
∂DQi(s−1)C(1)τsi ≈ DQi(s) − Qi(s+1) .

(2.84)

Therefore, the Pons formulation is completely equivalent to the Ostrogradsky formulation39

in terms of Dirac brackets. In particular, applying the equations of motion with respect to
C(1)

(s)

λi , i .e ., DQi(s) − Qi(s+1) ≈ 0, then τsi ≈ 0.
The next consideration is a degenerate case for the configurations Qi(M ) ; assume thatrank K̃ (M ) = N − R. Then the number of R primary constraints
(M )
(1)
(1)
i
i
i
) := 0∗ ϕα = ∗ ϕα (Q(1) , Q(2) , · · · , Q(M ) , Pi

α ∈ {1 , 2 , · · · , R}

(2.85)

are added to the above consideration. Then the total Hamiltonian (2.81) becomes as follows:(s)HT′ = H0 + Pi DQi(s) −

N M
−1
X
X

s

M −1+sλi(s) (DQi(s) − Qi(s+1) ) + σsi ∗ θ(1) i + τsi ∗ θ(1) i+ κα ∗ ϕ(1)α ,i=1 s=1

H0′ = Pa(M ) F a − L′ (Qi(1) , Qi(2) , · · · , Qi(M ) , F b ) ,(2.86)(s)where a ∈ {N − R + 1 , N − R + 2 , · · · , N }. The linear dependence on Pi

that is notbounded from below gives rise to the Ostrogradsky instability also in the degenerate case.
sC(1)M −1+s

The consistency conditions for ∗ θ(1) i ≈ 0 and ∗ θ(1) iC(1)≈ 0 determine the multipliers

τsi and σsi as follows:
C(1)
(s−1)
+
σsi ≈ −Pi

(1)

∂ϕα
∂L′
− κα i ,i∂DQ(s−1)
∂Q(s)

C(1)

τsi ≈ DQi(s) − Qi(s+1) .

(2.87)

In particular, the second equations turn into being weak equal zero under the equationsof motion with respect to λi(s) . This situation is different from the non-degenerate case;the multipliers σsi are not determined until the multipliers κα are fixed. It indicates that(1)

the constraint structure, which is given by the primary constraints ∗ ϕα ≈ 0 and thoseconsistency conditions, dominates all physics of the system. Notice that if it exists firstclass constraints then the equivalence in terms of Dirac brackets between the Pons and40

Ostrogradsky formulation only holds for physical quantities (gauge invariant quantities).
Only the cases of existing second-class constraints hold the complete equivalence.

2.6. Ostrogradsky ghost-free theories
In this section, a set of conditions to remove the Ostrogradsky instability is introduced. The
most general theorem is very complicated, therefore, let us first discuss two simple cases:
L = L(ϕ̈ , ϕ̇ , ϕ; q̇, q) and L = L(ϕ̈a , ϕ̇a , ϕa ; q̇ i , q i ) (a ∈ {1 , 2 , · · · , A}; i ∈ {1 , 2 , · · · , N }) to
understand the essential points. After that, the general statement is just given without proof.
The detailed considerations including the explicit proofs are given in Refs [Motohashi et al.
(2016, 2018a,b)].
The first case is L = L(ϕ̈ , ϕ̇ , ϕ; q̇ , q); the Lagrangian contains at most second-order time
derivative and first-order time derivative of ϕ = ϕ(t) and q = q(t), respectively. The Ostrogradsky instability is generated by the variable ϕ. In Pons’ formulation given in Section 2.5,
the Lagrangian is decomposed by using an auxiliary variable as follows:
L → L′ = L′ (Q̇ , Q , ϕ; q̇ , q) + λ(ϕ̇ − Q) ,(2.88)

where Q is an auxiliary variable, λ is a multiplier. The total Hamiltonian is derived asfollows:
HT′ = H0′ + λQ + µ∗ Φ + ν ∗ Ψ ,H0′ = P F + pf − L′ (F , Q , ϕ; f , q) ,(2.89)

where P , p, π, and ρ are canonical momentum variables with respect to Q, q, ϕ, and λ,respectively. µ and ν are multipliers. ∗ Φ = π − λ ≈ 0 and ∗ Ψ = ρ ≈ 0 are primary secondclass constraints. Remark that the multiplier λ is regarded as a configuration variable.
F = F (Q , P ; q , p)|ϕ,π,λ,ρ and f = f (q , p; Q , P )|ϕ,π,λ,ρ are functions that are determined fromthe implicit function theorem under arbitrarily fixing values of ϕ, π, λ, ρ for the kernel kinetic41

matrix given byLQ̇Q̇ Lq̇Q̇ 
K (2) = ,LQ̇q̇ Lq̇q̇

LXY =

∂ 2L
.
∂X∂Y

(2.90)

Here, the kernel kinetic matrix is assumed as non-degenerate. The entire kinetic matrix
K̃ (2) = LXY , where X, Y ∈ {Q̇ , q̇ , ϕ̇ , λ̇}, has its rank of 4 − 2 = 2; this is consistent with theexistence of the two primary constraints. Then the consistency conditions for ∗ Φ ≈ 0 and∗ Ψ ≈ 0 determine all the multipliers µ and ν as follows:

C(1)

C(1)

µ ≈ 0 and ν ≈ {∗ Φ , H0′ }P.b. .

That is, there is no secondary constraint. Therefore, the degrees of freedom of the system is(4 × 2 − 2)/2 = 2 + 1. The extra degrees of freedom is caused by the term λQ in Eq (2.89)
C(1)since λ ≈ π is unbounded from below; this is nothing but an Ostrogradsky ghost degrees
of freedom.
In order to remove this difficulty, an extra condition has to be imposed as follows:

R(P , p ; Q , ϕ, q) = 0 .

(2.91)In a local region, this additional condition leads to the following conditions by virtue of theimplicit function theorem: either P = G1 (p ; Q , ϕ , q) or p = G2 (P ; Q , ϕ , q). Therefore, thefollowing additional primary constraint can be imposed; either∗ Ξ = P − G1 (p ; Q , ϕ , q) := 0(2.92)∗ Π = p − G2 (P ; Q , ϕ , q) := 0 .

(2.93)orLet us consider the first condition (2.92). The total Hamiltonian of the system is modified
as follows:
HT′′ = H0′ + λQ + µ ∗ Φ + ν ∗ Ψ + ξ ∗ Ξ ,42(2.94)

where H0′ is given in Eq (2.89). The consistency conditions for these primary constraints,∗ Φ := 0, ∗ Ψ := 0, and ∗ Ξ := 0, determine the two multipliers µ and νC(1)as µ ≈ 0 andC(1)ν ≈ {∗ Φ , H0′ }P.b. + ξ(∂G1 /∂ϕ), respectively, and give rise to a secondary constraint∂G1′′′:= 0 .
∗ Θ = {∗ Ξ , HT }P.b. = −π + {∗ Ξ , H0 }P.b. − Q∂ϕ(2.95)

C(1)

Notice that the momentum variable π ≈ λ is now related to other phase space variables
by virtue of Eq (2.95). Furthermore, since all constraints that appeared here are classified
into second-class constraints, therefore, the degrees of freedom is at least (4 × 2 − 4)/2 = 2.
C(1)Taking into account the fact that π ≈ λ is bounded with other phase space variables,
the Ostrogradsky instability is now removed from the system. Remark that the second
condition (2.93) concludes the same result.
There is an important theorem that relates either the additional condition (2.92) or (2.93)
to the degeneracy of the system and vice versa, stated as Condition (2.92) is equivalent to
det K (2) = LQ̇Q̇ Lq̇q̇ − (LQ̇q̇ )2 = 0(2.96)

under the imposition of Lq̇q̇ ̸= 0. The same statement holds for Condition (2.93) under theimposition of LQ̇Q̇ ̸= 0. If Lq̇q̇ = Lq̇q̇ = 0 then the dependence on p and P is eliminated fromboth the conditions and these coincide one another. This theorem can be shown by using
the implicit function theorem. Using this theorem and the fact that the degeneracy of the
kinetic matrix K (2) leads to the existence of its null-eigenvalue vector, it is shown that theEuler-Lagrange equations of the system contain at most second-order time derivative not
only for q but also for ϕ. This is another signal for removing the Ostrogradsky instability
from the system.
The above arguments can be generalized into the case of existing multiple variables: L =43

L(ϕ̈a , ϕ̇a , ϕa ; q̇ i , q i ) (a ∈ {1 , 2 , · · · , A ≥ 2}; i ∈ {1 , 2 , · · · , N }). The original Lagrangian isdecomposed as follows:
L → L′ = L′ (Q̇a , Qa ; ϕ̇a , ϕa ; q̇ i , q i , λa ) + λa (ϕ̇a − Qa ) ,(2.97)

where Qa are auxiliary variables and λa are multipliers. The total Hamiltonian is derived asfollows:
HT′′ = H0′ +λa Qa +µa ∗ Φa +ν a ∗ Ψa +ξ a ∗ Ξa ,H0′ = Pa F a +pi f i −L′ (F a , Qa , ϕa ; f i , q i ) , (2.98)

where Pa , pi , π a , and ρa are canonical momentum variables with respect to Qa , q i , ϕa ,and λa , respectively. F a = F (Qa , Pa ; q i , pi )|ϕa ,πa ,λa ,ρa and f a = f (q i , pi ; Qa , Pa )|ϕa ,πa ,λa ,ρa arefunctions that are determined from the implicit function theorem. ∗ Φa = πa − λa := 0 and∗ Ψa = ρa := 0 are primary constraints, and∗ Ξa = Pa − Ga (pi , qi, Qb , ϕb ) := 0

(2.99)

are additional primary constraints that need to remove the Ostrogradsky instability, andthat are equivalent to the following condition:
det K (2) = Lab − Lai Lij Ljb = 0 ,(2.100)where K (2) is defined as follows:Lab Laj 
K (2) = ,
Lib Lij(2.101)

where Lab = ∂ 2 L/∂ Q̇a ∂ Q̇b , Laj = ∂ 2 L/∂ Q̇a ∂ q̇ j , and Lij = ∂ 2 L/∂ q̇ i ∂ q̇ j . Then the consistency
44

conditions for ∗ Φa := 0, ∗ Ψa := 0, and ∗ Ξa := 0 determine the multipliers µa and ν a as
C(1)

C(1)

µa ≈ 0 and ν a ≈ {∗ Φa , H0′ }P.b. + ξ a (∂G/∂ϕa ), respectively, and give rise to secondary
constraints as follows:′b ∂Ga+ ξ b {∗ Ξa , ∗ Ξb }P.b. := 0 .
∗ Ξ̇a = −πa + {∗ Ξa , H0 }P.b. − Qb∂ϕ(2.102)

Here, a departure from the single variable case appears; the rank of the matrix ∆ab ={∗ Ξa , ∗ Ξb }P.b. in the last terms dominates whether or not the above conditions derive secondary constraints. To remove the Ostrigradsky instabilities, it is necessary and sufficientconditions to impose the rank of ∆ab is zero, or equivalently,∆ab = {∗ Ξa , ∗ Ξb }P.b. := 0 .

(2.103)

These conditions are new additional conditions when existing multiple second-order time
derivative variables. Then the number of A secondary constraints appear, that is′b ∂Ga:= 0 ,∗ Φa = −πa + {∗ Ξa , H0 }P.b. − Qb∂ϕ(2.104)and the degrees of freedom are now (6A+2N −2A−2A)/2 = A+N . Taking into account thatEq (2.102) under Eq (2.103) relate πa to the other phase space variables, the Ostrogradsky
instability is now removed from the system. Then the additional constraints ∗ Ξa := 0 andthe degeneracy of K (2) lead to the fact that the Euler-Lagrange equations contain at mostsecond-order time derivative terms.
Finally, let us state a most general case: L = L(ϕi0 , Dϕi0 ; ϕi1 , Dϕi1 , D2 ϕi1 ; · · · ;ϕid , Dϕid , D2 ϕid , · · · , Dd+1 ϕid ). That is, a system which includes at most (d + 1)-order time
derivative terms. Here, the indices ik (k ∈ {0 , 1 , 2 , · · · , d}) run the range of as follows:
P
Pd−1
Pd−1
i0 = 1 , 2 , · · · , n0 and ik = d−1
(n
+
1)
,
(n
+
2)
,
·
·
·
,
(k ≥ 1). In
l
l
l=0
l=0
l=0 (nl + nk )45

order to apply Pons’ formulation in Section 2.5, the following configurations are introduced:
Qi000 = ϕi0 ,Qi101 = ϕi1 ,(2.105)

Qid0d = ϕid .

···Then the original Lagrangian is decomposed as follows:
L′ = L(Qi000 , Q̇i000 ; Qi101 , Qi111 , Q̇i111 ; Qi202 , Qi212 , Qi222 , Q̇i222 ; · · · ; Qid0d , Qid1d , · · · , Qiddd , Q̇iddd )+ λi101 (Q̇i101 − Qi111 )+ λi202 (Q̇i202 − Qi212 ) + λi212 (Q̇i212 − Qi222 )
+ ···
d
d
− Qiddd ) .
(Q̇id,d−1+ λid0d (Q̇id0d − Qid1d ) + λid1d (Q̇id1d − Qid2d ) + · · · + λid,d−1

The configurations of the system, i .e ., Qiklk , λiklk lead to the corresponding canonical moi0
mentum variables; let us denote them as Pklik , ρiklk . In particular, q i ∈ {Qi000 }, pi ∈ {P00
}=
(0)

(0)

id
i1
i2
{Li }, Q̃I1 ∈ {Qi111 , Qi222 , · · · , Qiddd }, and P̃I1 ∈ {P11, P22
, · · · , Pdd} = {LI1 }, where I1 ∈(0)

{i1 , i2 , · · · , id }. Then the following notations are introduced: Q ∈ {Qiklk } \ {Q̃I1 }, λ ∈(0){λiklk } \ {λikl0 }, P = {Pklik } \ {P̃I1 } = λ, and ρ ∈ {ρiklk } \ {ρikl0 } = 0. Therefore, the dimensionPPof the entire phase space is Ntotal = 2 dk=0 nk +4 dk=1 knk . Assembling further notations as
(2)

(1)

(d−1)

i

d−1
d
d
, Qid1d },}, Q̃I2 ∈ {Qi202 , Qi313 , · · · , Qid,d−2}, Q̃Id−1 ∈ {Qd−1,0Q̃I1 ∈ {(Qi101 , Qi212 , Qi323 , · · · , Qid,d−1

(d)

(k)

(k)

and Q̃Id ∈ {Qid0d }, where Ik ∈ {ik , · · · , id }, and then Q̃Ik ∈ {Qik0k , Q̃Ik+1 },
i

(k)

i

i

(k)

ik
id
k+1
k+2
k+1
d
Q̃Ik+1 ∈ {Qk+1,1, Qk+2,2, · · · , Qid,d−k}, and P̃Ik ∈ {Pk0
, Pk+1,1, · · · , Pd,d−k}, the above La-

grangian is arranged as follows:′′

L = L(q

i

(0)
(0)
(1)
(2)
(d)
, q̇ ; Q̃I1 , Q̃˙ I1 ; Q̃I1 , Q̃I2 , · · · , Q̃Id ) +
i

d
X
k=1

46

(k)
(k)
(k−1)
λ̃Ik (Q̃˙ Ik − Q̃Ik ) .

(2.106)

Therefore, the total Hamiltonian is derived as follows:
(0)

(0)

HT′′ = H ′′ + µα ∗ Φα + µ̄α ∗ Φ̄α + ν̃I1 ∗ Ψ̃I1 ,
(0)

(1)

(1)

(2)

(d−1)

H ′′ = H0′′ + Q̃I1 P̃I1 + Q̃I2 P̃I2 + · · · + Q̃Id

(d)

P̃Id ,(2.107)(0) (0)H0′′ = Q̃˙ I1 P̃I1 + q̇ i pi − L′′ ,

C(1)

C(1)

where ∗ Φα ∈ {P − λ} ≈ 0 and ∗ Φ̄α ∈ {ρ} ≈ 0. µα and µ̄α are Lagrange multipliers.
(k−1)

Notice that the linear terms Q̃Ik

(k)

P̃Ik cause the Ostrogradsky instability.

The degeneracy conditions that have to be imposed to eliminate the Ostrogradsky instabilities are derived by performing the Dirac-Bergmann analysis. The same considerations of
the case as L = L(ϕ̈a , ϕ̇a , ϕa ; q̇ i , q i ) (a ∈ {1 , 2 , · · · , A ≥ 2}; i ∈ {1 , 2 , · · · , N }) leads to thefollowing degeneracy conditions:
LI1 J1 − LI1 i Lij LjJ1 := 0 ,(0)(0){∗ Ψ̃I1 , ∗ Ψ̃J1 }P.b. := 0 ,(2.108)(0)

where ∗ Ψ̃I1 ∈ {∗ Ψi111 , ∗ Ψi222 , · · · , ∗ Ψiddd } are defined as follows:
C(1)
(0)
(0)
(0)
(0)
i
∗ Ψ̃I1 = P̃I1 − F̃I1 (pi , q , Q̃I1 , Q) ≈ 0 .

(2.109)The higher-order structure which is more than third-order requires further degeneracy conditions to remove the Ostrogradsky instability. The consistency conditions for the secondary(1)dconstraints, ∗ Υ̃I1 ∈ {∗ Υi101 , ∗ Υi212 , · · · , ∗ Υid,d−1}, that is,

C(2)
(1)
(0)
(1)
(1) i
(0)
′′
i
∗ Υ̃I1 = −{∗ Ψ̃I1 , H }P.b. = P̃I1 − GI1 (p , q , Q̃I1 , Q) ≈ 0

47(2.110)demand the following degeneracy conditions:(0){∗ Υ , ∗ Ψ̃J1 }P.b. := 0 except

i1
∗ Υ10

(2.111)

where ∗ Υ is defined as follows:
C(k)
(0)
(k)
(k)
(k−1)
(k)
, H ′′ }P.b. = P̃Ik − G̃Ik (pi , q i , Q̃I1 , Q) ≈ 0∗ Υ̃Ik ≡ −{∗ Υ̃Ik

C(k)

(0)

for k ∈ {1 , 2 , · · · , d}. {∗ Υik0k , ∗ Ψ̃I1 }P.b. ≈ 0(2.112)(k ∈ {2 , 3 , · · · , d}) are included since theseC(k)conditions will be required below. Notice that ∗ Υ ≈ 0 fix the linear dependence of thecanonical momentum in Eq (2.107). The counterpart of the consistency conditions are given
as follows:
(0)

C(k)

(0)

{∗ Υik0k , H ′′ }P.b. + ν̃I1 {∗ Υik0k , ∗ Ψ̃I1 }P.b. ≈ 0 (k ∈ {1 , 2 , · · · , d})
(0)

C(k)

Imposing {∗ Υik0k , ∗ Ψ̃I1 }P.b. ≈ 0(2.113)(k ∈ {2 , 3 , · · · , d}), these conditions yield the following

additional constraints:
C(k)
ik
ik
′′
∗ Ωk0 = −{∗ Υk0 , H }P.b. ≈ 0(k ∈ {2 , 3 , · · · , d}) ,(2.114)C(k)and the consistency conditions for ∗ Ωik0k ≈ 0 demand the following degeneracy conditions(0){∗ Ω , ∗ Ψ̃J1 }P.b. := 0 except(2)∗ Ω̃I2 ,(2.115)C(k)kwhere ∗ Ω ∈ {−{∗ Ωik,l−1, H ′′ }P.b. ≈ 0} (k ∈ {l + 1 , l + 2 , · · · , d}; l ∈ {2 , 3 , · · · , d − 3}).

Then, since all the constraints are classified into second-class constraints, therefore, theP′number of degrees of freedom is Ntotal= dk=0 nk ; the number of the Ostrogradsky ghost48

degrees of freedom is now removed out.
These conditions correspond to Eq (2.96) in the single second-order time derivative systems, or Eq (2.100) in the multiple second-order time derivative systems, and Eq (2.103),
respectively. The existence of the higher-order time derivative terms that are more than
third-order requires the additional degeneracy conditions, that is, the third and the fourth
conditions, Eq (2.111) and Eq (2.115), to remove the Ostrogradsky instability. Furthermore,
under these conditions, it is shown that the Euler-Lagrange equations of the system include
the time derivative terms up to second-order by applying the implicit function theorem.

2.7. Summary
In this chapter, a theory of analytical mechanics for theories of gravity, focusing on degenerate and higher-order derivative point particle systems, was established. The Dirac-Bergmannanalysis reveals the constraint structure of a given system. It also provides a set of fundamental concepts that play the main role of the present thesis: the propagating degrees of
freedom, the gauge degrees of freedom, and the methodology of removing the Ostrogradsky
degrees of freedom. Since theories of gravity have the properties of degenerate and higherorder derivative systems, therefore, a theory established in this chapter provides fundamental
tools for treating theories of gravity.

49

Chapter 3
The well-posed variational principle indegenerate point particle systems
Chapter abstract
In this chapter, a methodology for making the variational principle in point particle systems well-posed is established. The extension to field theories has been developing, butsome specific cases will be provided in the succeeding chapters of the present thesis. Firstof all, the problems of degenerate systems in the variational principle are explained. It isshown that the valid boundary conditions in a higher-order time derivative system but which
has equations of motion up to second-order time derivative are restricted to position-fixing
boundary conditions. Based on the Pons formulation, which is introduced in Section 2.5,
Frobenius integrability conditions of first-order degenerate time derivative systems are investigated, and the concrete formulae of the conditions are derived. After these preparations,
the well-posedness of the variational principle is reformulated. Introducing two types of
embeddings of the symplectic manifold based on the Shanmugadhasan-Maskawa-Nakajima
theorem, i .e ., quasi-canonical embeddings and canonical embeddings, the well-posedness isinvestigated. As a result, appropriate boundary conditions are specified. Finally, examples
are given, and this chapter is summarized with the outlook for these works.

50

3.1. Cumbersome situations of the variational principle in degenerate
systems
In this section, the difficulties of applying the variational principle to degenerate point particle systems are clarified. Let us start with considering four examples
1
1
L1 = ẋ2 + ẏ 2 + λ(x + y) ,2
2
1
L2 = q̇ 1 q̇ 3 + q 2 (q 3 )2 ,
2
1 2

2 1

1 2

(3.1)
2 2

L3 = q q̇ − q q̇ − (q ) − (q ) ,1
1
L4 = − q q̈ − q 2 .
22
The first system is a textbook example of existing an exterior constraint that is taken
into account by a Lagrange multiplier. The second and third ones are systems only with
first-class constraints and only with second-class constraints, respectively [Cawley (1979);
Dirac (1928)]. The last one is an example of existing with a second-order time derivative
term [Dyer and Hinterbichler (2009); Izumi et al. (2023)].
The first-order variation of these systems are computed as follows:d  1 1
δL1 = λ − q̈ 1 δq 1 + λ − q̈ 2 δq 2 + (q 1 + q 2 )δλ +q̇ δq + q̇ 2 δq 2 ,
dt


 3


d  3 11
(q 3 )2 δq 2 + −q̈ 1 + q 2 q 3 δq 3 +q̇ δq + q̇ 1 δq 3 ,δL2 = −q̈ δq 1 +
2
dt



d 
(−q 2 )δq 1 + q 1 δq 2 ,δL3 = 2q̇ 2 − 2q 1 δq 1 + −2q̇ 1 − 2q 2 δq 2 +
dt


d 1
1
δL4 = {−q̈ − q} δq +q̇δq − qδ q̇ ,
dt 2
2

(3.2)

where in the first system the multiplier is regarded as a configuration. Integrating theseover an open interval (t1 , t2 ), and imposing appropriate boundary conditions at each endpoint t1 and t2 , the variational principle leads to the Euler-Lagrange equations for each51

system are obtained. It is easy to confirm that the naive imposition of Dirichlet boundary
conditions for all the configuration variables in the boundary term does not work well; For
the first system, the boundary conditions δq 1 (t2 ) = δq 1 (t1 ) := 0 and δq 2 (t2 ) = δq 2 (t1 ) := 0
are equivalent to q 1 (tτ ) := a1τ and q 2 (tτ ) := a2τ (τ = 1 , 2), where a1τ and a2τ are arbitrary
constants. Then the variational principle leads to the Euler-Lagrange equations of motion
but one of the equations, i .e ., q 1 + q 2 = 0, it is nothing but an exterior constraint, isgenerically inconsistent with the boundary conditions at each end-point t1 and t2 ; For thesecond system, in the same manner as the first case, it is shown that the naive boundary
conditions δq 1 (t2 ) = δq 1 (t1 ) := 0 and δq 3 (t2 ) = δq 3 (t1 ) := 0 are inconsistent at least withthe constraint q 3 = 0; For the third system, the system has only one propagating degrees
of freedom since the equations of motion are either q̈ 1 + q 1 = 0 or q̈ 2 + q 2 = 0. Therefore,
the naive boundary conditions δq 1 (t2 ) = δq 1 (t1 ) = 0 and δq 2 (t2 ) = δq 2 (t1 ) = 0 become
over-imposing, and these boundary conditions cannot uniquely determine the two initialconditions in the solution; the dynamics of the system cannot be determined uniquely; For
the last system, the naive boundary conditions δq(t2 ) = δq(t1 ) = 0 and δ q̇(t2 ) = δ q̇(t1 ) = 0
are themselves inconsistent in general since there is one equation of motion and therefore
the velocities at the boundaries are of course determined by the derivative of the solution
q = q(t); it is impossible to fix the configuration and its velocity variable at the boundaries
in an independent and arbitrary manner. These cumbersome situations are caused by theconstraint structure of each system. That is, in order to make the variational principle wellposed in degenerate systems, a methodology for taking appropriate boundary conditions is
mandatory.
A clue to formulating the well-posed variational principle is the coincidence between
the number of boundary conditions and that of initial conditions, and the half numbersof them are also equivalent to the propagating degrees of freedom. Therefore, in order toapproach the problem, the following guiding principle is assumed, stated as “as many inte52

gral constants (/initial conditions) in the solutions of a given system as possible should be
uniquely determined through the boundary conditions”. While, as mentioned in Section 2.4,
the Shanmugadhasan-Maskawa-Nakajima theorem decomposes the entire phase space into aphysical subspace and a constraint subspace, and it indicates that the existence of a canonicalcoordinate system that spans each subspace and that the original canonical coordinate system
can be transformed into the new canonical coordinate system by performing the corresponding canonical transformation. Therefore, it would be expected that fixing the configurations
for the physical space is a possible candidate for appropriate boundary conditions that are
to make the variational principle well-posed from the viewpoints of the coincidence of thenumbers. In fact, this insight gives a correct way, although the cases of existing first-classconstraints need more careful considerations. In order to proceed with this strategy, theDirac-Bergmann analysis has to be valid without depending on the well-posedness of thevariational principle. Finally, let us confirm this statement.
This statement can be proved by noticing that the well-posedness relates only to the symplectic potential ω = pi dq i ; this quantity is nothing but the boundary term of the first variation of a Lagrangian with pi = ∂L/∂ q̇ i . That is, after performing the appropriate canonicaltransformation, when varying, the well-posed variational principle should give rise to a symplectic potential schematically such that ω ′ = PI dQI + Constraint Structure in 1-form + δW
as its boundary term up to some total divergent term W . While this transformation does
C.T.

not affect the symplectic structure since Ω = δω = δq i ∧ δpi = δω ′ = δQI ∧ δPI +C.T.

Constraint Structure in 2-form = Ω′ , where “ = ” denotes the equality under a canonicaltransformation; canonical transformations never change the symptectic structure, or equivalently, the symplectic 2-form. Therefore, taking into account the following two facts; (i)The definition of the Poisson bracket: {f , g}P.b. = Ω(Xf , Xg ), where Xf and Xg are the
Hamiltonian vector fields with respect to some functions f and g, respectively; (ii) The timedevelopment of a quantity f of the system is given as f˙ = {f , HT }P.b. , where HT is the total
53

Hamiltonian of the given system, it shows that the statement holds.

3.2. Dirichlet boundary conditions and second-order derivative equations of motions
In this section, the meaning of Dirichlet/position-fixing boundary conditions is clarified in thevariational principle [Tomonari (2023b)]. In higher-order derivative systems, the first-ordervariation of the Lagrangian of a given system gives generically rise not only to the first-ordervariation of position variables but also to that of the derivatives of these variables at theboundaries. In such theories, the Euler-Lagrange equations generically include higher-orderderivative terms that are more than second-order. However, in order to make sense of theequations as physical meanings, these equations have to be at most second-order differential
equations. This ansatz restricts the boundary conditions in the variational principle.
Let us first consider a second-order derivative system given as follows:

I

(2)

i

Z t2

i

[q , q̇ , t] =(3.3)

L(q̈ i , q̇ i , q i , t) dt ,t1

where i ∈ {1 , 2 , · · · , N }. This is a special case of Eq (2.63) with M = 2. The first-ordervariation is given as follows:

δI

(2)

Z t2 XN d2 ∂L(2)+ 2
δq i dt=i
dt∂ q̈
t1 i=1
" N 


 #t2
N 
(2)
X
X ∂L(2)d ∂L(2)∂Lδq i +
δ q̇ i
,
+
−
i
i
i
∂
q̇
dt
∂
q̈
∂
q̈
i=1
i=1
d
∂L(2)
−
i
∂q
dt



∂L(2)
∂ q̇ i



(3.4)

t1

check also Eq (2.65). The variational principle leads to the following Euler-Lagrange equations (check also Eq (2.66) and Eq (2.68)):
∂L(2)
d
−
i
∂q
dt



∂L(2)
∂ q̇ i



d2
+ 2
dt

54



∂L(2)
∂ q̈ i


=0

(3.5)

under fixing both the position and velocity coordinates as follows:δq i (t1 ) =δq i (t2 ) = 0,
i

(3.6)

i

δ q̇ (t1 ) =δ q̇ (t2 ) = 0
if the system is non-degenerate, i .e ., K (2) = ∂ 2 L(2) /∂ q̈ i ∂ q̈ j is full rank, but this condition is
not satisfied due to the ansatz; the order of the Euler-Lagrange equations has to be at mostsecond-order in time derivative. That is, the Euler-Lagrange equations (3.5) are rewritten
as follows:(2)....
(2) ...
Kij q j + Eij q j + (up to 2nd-order terms) = 0(3.7)

where we defined
(2)

Kij =

∂ 2 L(2),∂ q̈ i ∂ q̈ j

(2)

Eij =

∂ 2 L(2)∂ 2 L(2)−.
∂ q̈ i ∂ q̇ j
∂ q̈ j ∂ q̇ i

(3.8)

To turn the equations into at most second-order time differential equations, the matrices
K (2) and E (2) have to be zero-matrices:
K (2) = 0 ,E (2) = 0.

(3.9)

Then the Euler-Lagrange equations have at most second-order time derivative. The first
condition in (3.9) indicates that this system has to be degenerate as the second-order derivative theory. Notice also that Condition (3.9) are noting but the Ostrogradsky ghost-free
conditions given in Section 2.6.
In addition, on one hand, the first condition of Eq (3.9) leads to a specific form of
Lagrangian [Motohashi and Suyama (2015); Motohashi et al. (2016)]:

(2)

L

=

N
X

fi (q̇ j , q j )q̈ i + g(q̇ j , q j ).

i=1

55

(3.10)

On the other hand, this second-order time derivative system should be equivalent to a firstorder time derivative system discussed in Section 2.1. It indicates that the Lagrangian of
the system should be equivalent up to surface terms. That is,L(2) → L′(1) = L(2) +

dW
,
dt

(3.11)

where W = W (q̇ i , q i ). If W satisfies the following conditions:

fi +

∂W
=0
∂ q̇ i(3.12)

or equivalently,
W =−

N Z
X

(3.13)

fi dq̇ i + C(q j ),i=1

where C is an arbitrary function of position coordinates, the first terms in the Lagrangian (3.10)
vanish, and the system turns into a first-order time derivative system. Note, here, thatEq (3.13) is nothing but a counter-term in the second-order time derivative system. Furthermore, the first-order variation of the action functional of L′(1) becomes as follows:δI ′(1) = the same terms to δI (2) +" N X ∂L(2)∂ q̇ i

i=1

d
−
dt



∂L(2)
∂ q̈ i



∂W
+
∂q i



#t2
(3.14)

δq i
t1

Therefore, the boundary condition (3.6) for the variational principle now turns into positionfixing boundary conditions:δq i (t1 ) = δq i (t2 ) = 0(3.15)(1)

if the kinetic matrix Kij = ∂ 2 L′(1) /∂ q̇ i ∂ q̇ j is non-degenerate. This coincides with the boundary conditions for the first-order time derivative systems: Eq (2.8) with no time variation. It
makes sense since boundary conditions should uniquely determine the dynamics. From these
considerations, the ansatz reveals that appropriate boundary conditions for the variational56

principle in second-order time derivative theories are position-fixing boundary conditions.
In general, the action functional for the systems is given as follows:

I

(M )

Z t2
=


L(M ) DM q i , · · · , Dq i , q i , t dt ,(3.16)

t1

this is nothing but Eq (2.63). The first-order variation of the action functional is given as
follows (check Eq (2.65)):

δI (M ) =

#
" N M
#t2
(M )
(M )
X X
∂L
∂L
(−1)r Drδq i dt+(−1)r−s Dr−s
δ(Ds−1 q i )
.
r qi)
r qi)
∂(D
∂(D
r=0
i=1 r=s≥1"M
Z t2 X
N
X
t1

i=1

t1

(3.17)
If the following differential equations for W = W (q i , Dq i , · · · , DM −1 q i ):
M 
X

r−s

(−1)

D

r=s≥2

r−s


∂L(M )∂W+= 0,∂(Dr q i ) ∂(Ds−1 q i )(3.18)

are solvable, the Euler-Lagrange equations are derived as follows (check also Eq (2.66) andEq (2.68)):
M
X
∂L(1)
d ∂L(1)
∂L(1)
=
−
= 0,
(−1)r Dr
r qi)
i
i
∂(D
∂q
dt
∂
q̇
r=0

(3.19)

where L(1) = L(d) + dW/dt, under position-fixing boundary conditions:δq i (t1 ) = δq i (t2 ) = 0 ,(3.20)(1)

if the kinetic matrix Kij = ∂ 2 L′(1) /∂ q̇ i ∂ q̇ j is non-degenerate. Therefore, the ansatz indicatesthat appropriate boundary conditions for the variational principle in M th-order derivative
theories are nothing but position-fixing boundary conditions. Remark, finally, that theOstrogradsky ghost-free conditions given in Section 2.6 are also verified by a straightforward
computation although it is so tedious.
57Based on the above considerations, as long as the ansatz, stated as the order of theEuler-Lagrange equations has to be at most second-order in time derivative, is assumed,
the higher-order time derivative systems need the position-fixing boundary conditions when
applying the variational principle. Therefore, the problem is a methodology to identify
a set of position variables that should be fixed on the boundaries when the variational
principle applies in the first-order time derivative degenerate systems under the addition ofan appropriate counter-term if needed.

3.3. Frobenius integrability of first-order time derivative systems
In this section, based on the Frobenius theorem, the existence of solutions in both nondegenerate and degenerate systems is considered [Sugano and Kamo (1982)]. This existencealso means that a set of integral constants is derived, and these play a crucial role in formulating the well-posed variational principle [Tomonari (2023b)]; it relates to the realization of theguiding principle, stated as “as many integral constants (/initial conditions) in the solutionsof a given system as possible should be uniquely determined through the boundary conditions”.
From the previous section 3.2, introducing appropriate counter-terms, it revealed that theLagrangian describing a system of which equations of motion contain at most second-ordertime derivative terms becomes first-order. This indicates that it is enough to investigate thecases that Lagrangians contain at most first-order time derivative terms. Let us start withintroducing the Pfaffian system of the Lagrangian formulation in a non-degenerate system
and investigating its Frobenius integrability.
The Euler-Lagrange equations, Eq (2.9), are given as follows:
d
∂L
−
i
∂q
dt



∂L
∂ q̇ i58= 0.

(3.21)

Pfaffian system is introduced by splitting the equations into 1-forms as follows:ρi = dq i − v i dt = 0 ,(3.22)iθi = Kij dv − Si dt = 0 ,where Kij and Si are defined as follows:

Kij =

∂ 2L
,
∂ q̇ i ∂ q̇ j

Si = −v i

∂ 2L
∂ 2L
∂L
−
+ i.
i
i
i
∂q ∂v
∂t∂v
∂q

(3.23)

Remark that the 1-forms in Eq (3.22) can be derived by calculating the exterior derivativeof the second equation in Eq (2.10). It implies another viewpoint from the Lagrangian
formulation for the validity of the Dirac-Bergmann analysis without depending on the wellposedness of the variational principle as mentioned in Section 3.1. The Frobenius theorem
states that the Pfaffian system is Frobenius integrable, or equivalently, the system describedby Eq (2.9) has solutions, if and only if a system of generators for the Pfaffian system,that is, a closed algebra in the Lie bracket composed of vector fields that are orthogonal to
the Pfaffian system, exists. Since the Pfaffian system spans 2N -dimensional subspace ofT ∗ (T Q × R), the system of generators is 1-dimensional subspace of T (T Q × R) as follows:
X t = ai∂∂∂
+ bi i +
,
i
∂v
∂q
∂t

ai = (K −1 )ij Sj ,(3.24)

bi = v i .

Then the Frobenius theorem indicates the existence of the number of 2n functions f I and
integral constants cI such thatf I (q i , v i , t) = cI ,(3.25)

where I ∈ {1 , 2 , · · · , 2N }. Therefore, the implicit function theorem leads to the solutions59

as follows:
q i = q i (t , cI ) ,v i = v i (t , cI ).

(3.26)

Under the solutions of the first equation in Eq (3.22), the second solutions above can be
rewritten as q̇ i = q̇ i (t , cI ).
In the Hamiltonian formulation, the Hamilton equations, Eq (2.22), are given as follows:q̇ i =∂H,∂piṗi = −

∂H∂q i

(3.27)and expressed as the following Pfaffian system:∂Hiidt = 0 ,∗ ρ = dq −∂pi∂H
dt = 0 .
∗ θi = dpi +∂q i(3.28)The same considerations as the Lagrangian formulations but in the phase space T ∗ Q × R
lead to the following generator:
∂
∂
∂
+
b
+
,
∗
i
∂q i
∂pi ∂t∂H∂Hi, ∗ bi = − i .
∗a =
∂pi
∂q
∗ Xt = ∗ ai(3.29)

Then the Frobenius theorem leads to the following relations:∗fI(q i , pi , t) = ∗ cI ,(3.30)

where I runs from 1 to 2N , ∗ f I and ∗ cI are functions and integral constants, respectively,
which are implied from this theorem. Therefore, the implicit function theorem gives thesolutions:q i = ∗ q i (t , ∗ cI ) ,60

pi = pi (t , ∗ cI ) .

(3.31)

Since the configurations are given by the first formulae in Eq (3.26) and both the formulations have a common configuration space, ∗ q i (t , ∗ cI ) should exactly be the same as q i (t , cI ).
Therefore, ∗ cI = cI are concluded, and the solutions become
q i = q i (t , cI ) ,pi = pi (t , cI ) .

(3.32)The non-degeneracy, det K ̸= 0, implies the equivalence between the solution (3.26) and(3.32) by virtue of the implicit function theorem. This is also a conclusion from the following
one-to-one correspondences:
∂
∂pj ∂
∂
↔ i+ i,i∂q∂q∂q ∂pj
∂pj ∂
∂
∂
↔ i
= Kij
,
i
∂v
∂v ∂pj
∂pj
∂pi ∂
∂
∂
↔
,
+
∂t
∂t ∂pi ∂t(3.33)

where the left-hand side and the right-hand side are composed of the coordinate basis of
T (T Q × R) and T (T ∗ Q × R), respectively.
Next, let us consider the Frobenius integrability of the given degenerate system in the Lagrangian formulation, i .e ., in the velocity phase space T Q × R. Assume that the degeneracy
condition rank K = R(< N ), or equivalently, the following condition:(3.34)

Kij ταj = 0 ,is taken into account, where α ∈ {1 , 2 , · · · , N − R} and ταi are zero-eigenvalue vectors. The(1)given system has the number of R primary constraints: ϕα = 0, and these are satisfied as
identity. This property and the second equation in Eq (3.22) lead to the following secondary
constraints:iϕ(2)α = τα Si := 0 .

61(3.35)

The completeness of the zero-eigenvalue vectors implies the existence of a set of vectors, η i ,such thatC(2)(3.36)

Si ≈ Kij η j .
The generator of the Pfaffian system (3.22) are derived as follows:(3.37)

XT = Xt + ζ α Yα + ξ α Zα ,where ζ α and ξ α are multipliers, Xt , Zα , and Yα are defined as follows:
Xt = η i∂∂∂+ vi i + ,
i
∂v
∂q
∂t

Zα = −ταi

∂
,
∂v i

Yα = [Xt , Zα ] = ταi

 ∂
∂
i
i
+
−Z
η
−
X
τ
.
α
t
α
∂q i
∂v i
(3.38)(s)

In the presence of the constraints, ϕα ≈ 0 (s ∈ {1 , 2}), the space in where the dynamics
processes is restricted into a subspace of the entire phase space, T ∗ Q × R. It implies thatthe generators of the Pfaffian system have to be formed with respect not to the entire phasespace but to the constraint subspace that is restricted by all the constraints. In order to findall other constraints, the following condition has to be imposed:(3.39)dϕ(2)α (XT ) := 0 .
(2)

(1)

If dϕα (Yβ ) ̸= 0 hold for a β then this condition determines a multiplier ζ α . Otherwise(3)

it gives rise to a new constraint: ϕα

(2)

(2)

= dϕα (XT ) := 0. Remark that dϕα (Zβ ) = 0

are automatically satisfied therefore the multipliers ξ α remain arbitrary. Repeating thisprocedure, all constraints are derived as follows:ϕ(s+1)= dϕ(s)αα (XT ) := 0(s = 2, 3, · · · , Sα − 1),

(Sα )
(Sα )
β (s) C
α +1)
ϕ(S
=
dϕ
(X
)
=
C
ϕ
≈
0,
T
α
α
αs β

62

(3.40)

β
where Cαs
are some coefficients functions. The above conditions are called consistency con-ditions in the Lagrangian formulation. Under these constraints, if the following conditions
are satisfied then the system is Frobenius integrable:
C(Sα )(3.41)

θi (XT ) = −ζ α Kij (Zα η j + Xt ταj ) ≈ 0 .

ρi (XT )C(Sα )≈

0 are automatically satisfied by virtue of ρi := Kij ρj . Then the number of(2N − all constraints) functions and integral constraints exist, and the implicit function
theorem derives the solution of the Euler-Lagrange equations (3.22) of the system, in thesame manner as the non-degenerate case, under the imposition of the Frobenius integrablity
condition (3.41).
Finally, let us consider the Frobenius integrability of the given system in the Hamiltonian formulation. Under the degeneracy condition rank K = R(< N ), the Dirac-Bergmann(1)analysis in Section 2.2 leads to all constraints from the primary constraints ∗ ϕα := 0 as
follows:′)∂ ∗ ϕ(s(s′ +1)(s′ )α= {∗ ϕα , HT }P.b. +:= 0 ,
∗ ϕα

∂t

′

(S )
′ +1)
′
(s′ ) C α(Sαα)
= ∗ X HT ∗ ϕ(S= ∗ C βαs′ ∗ Φβ
≈ 0,
∗ ϕα
α

(3.42)(1)

where s′ ∈ {1 , 2 , · · · , Sα′ − 1} and the total Hamiltonian HT is given by HT = H0 + ζ α ∗ ϕα .
Remark that all indices are primed since the equivalence of constraints between the Lagrangian and Hamiltonian formulations is not proved yet (it will be proved later). Under
these constraints, for the Pfaffian system of Eq (3.28), if the following conditions are satisfied
then the system is Frobenius integrable:(1)α ∂ ∗ ϕα∗ θ i (∗ X T ) = −ζ
∂q i

63

′

C(Sα )≈ 0,(3.43)

′∗ ρ i (∗ X T )C(Sα )≈

0 are automatically satisfied, where ∗ ρi = Kij ∗ ρj . Then the number of(2N − all constraints) functions and integral constraints exist, and the implicit function
theorem derives the solution of the Hamilton equations (3.28) of the system, in the samemanner as the non-degenerate case, under the imposition of the Frobenius integrability
conditions (3.43).
In a non-degenearate case, a one-to-one correspondence of vector fields between T (T Q ×R) and T (T ∗ Q × R) given as Eq (3.33) exists, but in a degenerate case such correspondenceis violated due to the degeneracy of the kinetic matrix Kij :
∂pj ∂
∂
∂
→ i
= Kij
.
i
∂v
∂v ∂pj∂pj(3.44)

Therefore, the Hamiltonian formulation cannot be transformed into the Lagrangian formulation but the opposite is possible as follows:
Xt → ∗ X t =

∂H0 ∂
∂
∂
∂H0 ∂
−
+
= XH 0 + ,ii∂pi ∂q∂q ∂pi ∂t∂t(3.45)Zα → ∗ Z α = 0,
Yα → ∗ Y α =∂ ∗ ϕ(1)∂ ∗ ϕ(1)∂∂αα−= X∗ ϕ(1).
iiα∂pi ∂q∂q ∂pi

Therefore, XHT → ∗ X HT becomes
XT → ∗ X HT = ∗ X H0 + ζ α ∗ X ∗ ϕ(1)+α∂∂= ∗ X HT + ,∂t∂t(3.46)

where HT = H0 + ζ α ∗ ϕ(1)α is nothing but the total Hamiltonian of the system. Therefore,
(s)

(s′ )

the indices s and s′ are identified and so are Sα and Sα′ ; ϕα and ∗ ϕαcorrespondence, respectively.

64

have a one-to-one

3.4. Reformulation of the well-posed variational principle
In this section, introducing three fundamental maps that are correspondences between integral constants from the Frobenius integrability and position-fixing boundary conditions in thevariational principle, velocity variables and momentum variables, and the Hamiltonian vector fields in the Lagrangian and Hamiltonian formulations, respectively, the well-posednessof the variational principle is reformulated [Tomonari (2023b)]. First of all, let us start byintroducing the three fundamental maps scrutinizing a non-degenerate system.
Based on the considerations in the previous section, a non-degenerate system is Frobenius
integrable and has the solutions Eq (3.26) in the Lagrangian formulation, or equivalently
Eq (3.32) in the Hamiltonian formulation, and vice versa. The solutions of configurations
are common in both the formulations as follows: q A = q A (t , cB ) (A, B ∈ {1 , 2 , · · · , 2N }).
In a local open region U of the configuration space Q, regarding the integral constants, cI ,as continuous parameters and assuming ∂q A /∂cB ̸= 0 on U , the implicit function theorem
becomes applicable. This leads to a map ι that maps the configurations at times t1 and t2
to a 2N -dimensional parameter space C which is spanned by all the integral constants:
ι : Q[t1 ]×Q[t2 ] → C; (q 1 (t1 ) , q 2 (t1 ) , · · · , q N (t1 ) , q 1 (t2 ) , q 2 (t2 ) , · · · , q N (t2 )) 7→ (c1 , c2 , · · · , c2N ) ,(3.47)
where Q[t] is the space spanned by the configurations at a time t and q i (t1 ) are the values ofconfiguration coordinates at the time t1 . q i (t2 ) are defined in the same manner. The space
Q[t] is usually a local region since the variational principle leads to a local minimum of theaction functional. The important point is that map ι is invertible by virtue of the implicit
function theorem. In other words, the position-fixing boundary conditions can be replacedby the integral constants in the solutions and determine the dynamics uniquely when the
propagating degrees of freedom matches the number of the independent integral constants.

65

This is the first correspondence and nothing but the realization of the guiding principle,
that is, “as many integral constants (/initial conditions) in the solutions of a given system
as possible should be uniquely determined through the boundary conditions”.
The second correspondence is introduced by a map as follows:κ : T Q × R → T ∗ Q × R; q̇ i (t) 7→ pi (t) =∂L.
∂ q̇ i(3.48)In the non-degenerate system, that is, det K ̸= 0 implies that map κ is invertible by virtueof the implicity function theorem:
κ−1 : T ∗ Q × R → T Q × R; pi (t) 7→ q̇ i (t) = v i (q j , pj , t) .

(3.49)

This property is of course satisfied in a local region.
The final correspondence is expressed in a map as follows:
O : G[T (T Q × R)] → G[T (T ∗ Q × R)]; XT 7→ ∗ X HT ,(3.50)

where G[T (T Q × R)] and G[T ∗ (T Q × R)] denote spaces of the system of generator for thePfaffian systems (3.22) and (3.28), respectively. The Hamiltonian vector fields Xt and ∗ Xt
are defined in Eq (3.37) and Eq (3.46), respectively. Note that Eq (3.46) is directly indicatesthe existence of the above map O and its well-definedness. The existence of its inverse map
is guranteed from Eq (3.33).
Finally, let us define the well-posedness of the variational principle as follows:
Definition 1.
The variational principle is well-posed if and only if the three fundamental maps ι, κ, andO are well-defined and invertible on a phase (sub)space in which the physical phase space
exists.
66

■In a degenerate system, the degeneracy makes it generically impossible to introduce
map ι in a well-defined manner. Furthermore, maps κ and O are always introduced ina well-defined manner, but their inverses generically do not exist. Therefore, in order toestablish a well-posed variational principle in the degenerate system, we have to remove theseincompatibilities from the theory. The strategy to tackle this problem is that we restrict theentire phase space to a subspace such that each invertible map for κ and O exists, andthen we construct a well-defined map ι in the subspace of which the inverse map exists.
The construction of subspaces of phase space, however, differs from the cases of ordinary
differential manifolds. That is, arbitrary restrictions, or strictly speaking, embeddings, are
not allowed, unlike ordinary differential manifolds.

3.5. Embeddings of Symplectic manifolds
In this section, based on the Shanmugadhasan-Maskawa-Nakajima theorem, which is alreadystated in Section 2.4, the embeddings of a symplectic manifold are introduced [Tomonari(2023b)]. The embeddings are classified into two types, “canonical” or “quasi-canonical”
embeddings, according to the existence of first-class constraints. In order to formulate theembeddings, a method of fixing coordinate functions is applied. This method introduces
an embedding and a submanifold simultaneously. Let us start by reviewing this method in
ordinary differential manifolds by using examples.
Assume that an abstract embedding “ψ” : “S” → R3 for some differential submanifold S of R3 exists. The submanifold “S” is also abstract one; it symbolically denotes all
possible submanifolds of R3 . That is, any concrete embedding and submanifold are determined yet; there is only a structure (“ψ” , “S”). Now, let us equip R3 with a coordinate
system. For instance, assume that R3 is equipped with polar-coordinates (r , θ , φ). Thenthe abstract embedding becomes ψ : “S” → R3 ; (ψ ∗ r , ψ ∗ θ , ψ ∗ φ) 7→ (r , θ , φ), where ψ ∗ is67the pullback operator of ψ; the structure can be identified by fixing a set of pullback of
the coordinate functions by ψ. If ψ ∗ r is fixed as 1 then the abstract embedding turns into
ψ : S → R3 ; (ψ ∗ r := 1 , ψ ∗ θ , ψ ∗ φ) 7→ (r , θ , φ); this is nothing but the embedding of the unit
sphere. Similarly, fixing either ψ ∗ θ or ψ ∗ φ leads to an embedding of a 2-dimensional plane,
respectively, fixing both ψ ∗ θ and ψ ∗ φ derives an embedding of a line. In general, for a differential manifold, M and a possible submanifold “S”, an abstract embedding “ψ” : “S” → M
can be introduced, and it forms a structure (“ψ” , “S”). Then, equipping a coordinate system with M and fixing a set of the pullbacks of the coordinate functions, the structure is
identified: (ψ , S).
In the case of phase space, however, embeddings by fixing coordinate functions are generically restricted. For instance, any odd-number dimensional subspaces cannot be embedded
into the entire phase space. Therefore, this restriction has to be realized as an embedding
into the entire phase space such that the canonical structure holds at least for the subspacein which the dynamics lives. Let us call such embedding “canonical” if it exists. To consider
canonical embeddings, we need to introduce the concept of symplectic manifolds.
At each time t ∈ I, where I ⊂ R is an open interval of the time variable, a space T ∗ Q × R
is equivalent to T ∗ Q. Then a symplectic manifold of T ∗ Q is defined as the structure (T ∗ Q , Ω)equipped with a 2-form Ω on T ∗ Q represented by1Ω := ΩAB dz A ∧ dz B
2

(3.51)

satisfying the following three conditions:(i). ΩAB = −ΩBA(ii). det ΩAB ̸= 0(iii). dΩ = 0 ,68(3.52)

where A , B ∈ {1 , 2 , · · · , 2N }. The Darboux theorem under the conditions in Eq (3.52)
leads to the canonical structure:1Ω := JAB dz A ∧ dz B = dq i ∧ dpi
2(3.53)

at least in a local region on the symplectic manifold (T ∗ Q , Ω), where (z i , z N +i ) = (q i , pi )
and





IN ×N  0
J =.
−IN ×N0(3.54)

IN ×N is an N × N unit matrix. Then the Poisson bracket is defined as follows:{f , g}P.b. := Ω(Xf , Xg ) ,(3.55)

where Xf and Xg are the Hamiltonian vector fields of f and g, respectively. Using theseconcepts, we can define a canonical transformation as a coordinate transformation ϕ : T ∗ Q →T ∗ Q; (q i , pi ) 7→ (Qi , Pi ) such that the symplectic 2-form is invariant under the pullback
operation ϕ∗ :ϕ∗ ΩQ ,P = Ωq ,p(3.56)

1Ωq ,p := dq i ∧ dpi = JAB dz A ∧ dz B ,21ΩQ ,P := dQi ∧ dPi = JAB dZ A ∧ dZ B ,2(3.57)

whereand (Z i , Z N +i ) = (Qi , Pi ). The condition (3.56) is equivalent to
S T JS = J,

69

(3.58)

where S T is the transpose of S, and S is given as follows:ϕ∗ dZ A = S A B dz B ,(3.59)or more concretely,
S A B :=

∂Z A
∂z B

(3.60)

for Z A = (ϕ−1 )∗ z A . The definition of Poisson bracket given in Eq (3.55) implies that acanonical transformation does not change the Poisson bracket. That is, the equations of
motion are invariant; the dynamics remains unchanged before and after the transformation.
Conversely, since the time evolution is decomposed into a series of infinitesimal canonicaltransformations, the symplectic 2-form in Eq (3.52), or more general form Eq (3.51), is invariant under the time evolution. Note that when we consider a subspace as an embedding into
the symplectic manifold (T ∗ Q , Ω) by fixing some coordinates, a set of conjugate variables,i .e . pairs: q i and pi , have to be fixed simultaneously. Fixing either q i or pi would destroy
the symplectic structure of the phase space. That is, the first and/or second conditions in
the definition (3.52) would be violated. This is the peculiar property of symplectic manifolds
differing from ordinary differential manifolds.
Now, let us introduce embeddings of a given symplectic manifold. The method needs tointroduce an abstract embedding “σ” : “S” × R → T ∗ Q × R and a coordinate system forthe entire phase space T ∗ Q × R. The required coordinate system is of course provided by
the Shanmugadhasan-Maskawa-Nakajima theorem. Therefore, the abstract embedding turns
into σ(t) : “S” × R → T ∗ Q × R; (σ ∗ (t)QI , σ ∗ (t)PI , σ ∗ (t) ∗ Θa , σ ∗ (t) ∗ Θa , σ ∗ (t) ∗ Ξα , σ ∗ (t) ∗ Ψα ,σ ∗ (t)u = t) 7→ (QI , PI , ∗ Θa , ∗ Θa , ∗ Ξα , ∗ Ψα , u), where I ∈ {1 , 2 , · · · , N − V − U }; a ∈{1 , 2 , · · · , V }; α ∈ {1, 2, · · · , U }. Hereinafter, we abbreviate the notation of “ ∗ ” for the phasespace variables if it is apparent. The possible set of pullback of the coordinate functions isonly constraint coordinate functions, i .e ., Θa , Θa , and Ψα , since other coordinate functions
70

evolve in time passing. In other words, the consistency conditions for Θa , Θa , and Ψα make itpossible to fix for incarnating the abstract embedding. In addition, fixing Ξα is also possible
since this manipulation corresponds to gauge fixing. Therefore, there are eight possibilities
to incarnate the abstract embedding; (i) Fix Θa and Θa when Ψα do not exist; (ii) Impose
gauge fixing conditions to Ξα , and fix Ξα and Ψα when Θa and Θa do not exist; (iii) Impose
gauge fixing conditions to Ξα , and fix Θa , Θa , Ξα , and Ψα ; (iv) Fix Ψα when Θa and Θa do
not exist; (v) Fix Θa , Θa , and Ψα ; (i)’ Fix only Θa and Θa when Ψα do exist; (ii)’ Impose
gauge fixing conditions to Ξα , and fix only Ξα and Ψα when Θa and Θa do exist; (iv)’ Fix
only Ψα when Θa and Θa do exist; (vi) Impose gauge fixing conditions to Ξα , and fix only Θaand Θa . Here, the prime “ ’ ” of (i)’ denotes the existence of Ψα . If this is not the case thenwe remove the prime “ ’ ” from it. So are for the prime “ ’ ” of (ii)’ and (iv)’ with respect toΘa and Θa . Remark that imposing gauge fixing conditions to Ξα means that these variables
turn into a set of constraints and satisfy those consistency conditions: Ξ̇α“

all constraints≈

all constraints≈

0, where” denotes the weak equality under the imposition of all the constraints including

the gauge fixing conditions.
First, let us consider Cases (i), (ii), and (iii). Case (i) leads to the following embedding:
σ2 (t) : T ∗ Q|Q ,P × R → T ∗ Q × R; (σ2∗ (t)QI , σ2∗ (t)PI , σ2∗ (t)Θa := ϵa , σ2∗ (t)Θa := ϵa , σ2∗ (t)u = t) 7→ (QI , PI , Θa , Θa , u) ,(3.61)
where ϵa and ϵa are constant parameters, and T ∗ Q|Q ,P denotes that the phase subspacecoordinated by QI and PI . This embedding pulls back the entire symplectic manifold,(T ∗ Q × R , Ω = dq i (u) ∧ dpi (u)), to the submanifold, (T ∗ Q|Q ,P × R , ΩQ ,P = dσ2∗ (t)QI ∧dσ2∗ (t)PI ): σ2∗ (t)(T ∗ Q×R , Ω = dq i (u)∧dpi (u)) = (T ∗ Q|Q ,P ×R , ΩQ,P = dσ2∗ (t)QI ∧dσ2∗ (t)PI ).
ΩΘ (t) = dσ2∗ (t)Θa ∧ dσ2∗ (t)Θa = 0 on T ∗ Q|Θ × R throughout all time implies that it does71

not relate to the dynamics, but ΩQ,P (t) ̸= 0 on T ∗ M |Q ,P × R with the symplectic structure
describes the dynamics, and this subspace is nothing but the physical one. Notice that taking
limits of ϵa → 0 and ϵa → 0, σ2 (t) restores the constraint space. Therefore, Case (i) leads
to a canonical embedding: σ2 (t). σ2 (t) occupies 2V integral constants that are indicated by
the Frobenius integrability in Section 3.3.
In the same manner, Cases (ii) and (iii) lead to the following embeddings:
σ1 (t) : T ∗ Q|Q ,P × R → T ∗ Q × R; (σ1∗ (t)QI , σ1∗ (t)PI , σ1∗ (t)Ξα := ϵα , σ1∗ (t)Ψα := ϵα , σ1∗ (t)u = t) 7→ (QI , PI , Ξα , Ψα , u) ,(3.62)andσ3 (t) : T ∗ Q|Q ,P × R → T ∗ Q × R; (σ3∗ (t)QI , σ3∗ (t)PI , σ3∗ (t)Ξα := ϵα , σ3∗ (t)Ψα := ϵα , σ3∗ (t)Θa := ϵa , σ3∗ (t)Θa := ϵa , σ3∗ (t)u = t)
7→ (QI , PI , Ξα , Ψα , Θa , Θa , u) .
(3.63)In the both cases, σ1 (t) and σ3 (t), the original symplectic structure, (T ∗ Q × R , Ω = dq i (u) ∧dpi (u)), is pulled back to (T ∗ Q|Q ,P × R , ΩQ ,P = dσ2∗ (t)QI ∧ dσ2∗ (t)PI ); it is the same conclusion as σ2 (t). Therefore, Cases (ii) and (iii) also lead to the canonical embeddings: στ (t)(τ ∈ {1 , 2 , 3}), and occupy 2V integral constants that are indicated by the Frobenius integrability in Section 3.3.

72

Second, let us consider Cases (iv) and (v). Case (iv) leads to the following embedding:σ̃1 (t) : T ∗ Q × R → T ∗ Q × R; (σ1∗ (t)QI , σ1∗ (t)PI , σ1∗ (t)Ξα , σ1∗ (t)Ψα := ϵα , σ1∗ (t)u = t) 7→ (QI , PI , Ξα , Ψα , u) .
(3.64)
Remark that σ1∗ (t)Ξα cannot be fixed since the corresponding variables Ξα evolve in time
unless gauge fixing conditions, or equivalently, a set of consistency conditions, are imposed.
In the case of the above embedding, the symplectic 2-form is pulled back as follows:σ̃1∗ (t)Ω = dσ̃1∗ (t)QI ∧ dσ̃1∗ (t)PI + dσ̃1∗ (t)Ξα ∧ 0 .

(3.65)

This result indicates that the case without the consistency conditions for Ξα does not reduce
the phase space into the physical space. In addition, since the first and/or the second condition of the definition for symplectic 2-form in Eq (3.52) would be violated, this embedding is
not canonical as it is. However, we will confirm that this embedding can be canonical as we
impose gauge fixing conditions. This is a peculiar feature of systems existing of first-classconstraints. Let us call this sort of embeddings being “quasi-canonical”. Notice that if wetake the limit of ϵα → 0, then σ̃1 (t) restores the constraint space wherein all the constraints
are satisfied; this is another reason why σ1∗ (t)Ξα cannot be fixed in σ1∗ (t). Ξα do not restrict
the constraint space at all. Notice also that σ̃1 (t) occupies U integral constants that are
indicated by the Frobenius integrability given in Section 3.3.

73

In the same manner, Case (v) leads to the following embedding:σ̃3 (t) : T ∗ Q|Ξ ,Ψ × T ∗ Q|Q ,P × R → T ∗ Q × R; (σ̃3∗ (t)QI , σ̃3∗ (t)PI , σ̃3∗ (t)Ξα , σ̃3∗ (t)Ψα := ϵα , σ̃3∗ (t)Θa := ϵa , σ̃3∗ (t)Θa := ϵa , σ̃3∗ (t)u = t)
7→ (QI , PI , Ξα , Ψα , Θa , Θa , u) .
(3.66)
This embedding is also quasi-canonical and occupies 2V + U integral constants that are
indicated by the Frobenius integrability given in Section 3.3.
Finally, let us consider Cases (i)’, (ii)’, (iv)’, and (vi). Considering in the same manner
as above, it is shown that (i)’, (ii)’, and (vi) become canonical embeddings, and (iv)’ becomes a quasi-canonical embedding. Cases (i)’, (ii)’, (iv)’, and (vi) occupy 2U , 2V , V , 2U
integral constants that are indicated by the Frobenius integrability in Section 3.3. However,
it is also shown that these embeddings cannot introduce the well-posed variational principle [Tomonari (2023b)], although these embeddings are also well-defined ones as embeddingsof the symplectic manifold. A way to grasp this point is that if all the parameters are taken
as the limit vanishing, the embeddings have to restore the original constraint subspace; However, the embeddings above cannot realize this specific case. Therefore, as the results, only
the embeddings στ (t) (τ ∈ {1 , 2 , 3}), σ̃1 (t), and σ̃3 (t) are valid to formulate the well-posed
variational principle.

3.6. Well-posed variational principle 1: Cases of canonical embeddings
In this section, based on Definition 1 and the canonical embeddings, στ (t) (τ ∈ {1, 2, 3}),
which are introduced in the previous section 3.5, the well-posed variational principle for asystem that includes second-class constraints or first-class constraints under the imposition74

of gauge fixing conditions is formulated [Tomonari (2023b)]. Hereinafter, we abbreviate thenotation of “ ∗ ” for the phase space variables if it is apparent. First of all, let us show that,in the pullback of the entire symplectic manifold (T ∗ Q , Ω) by στ (t), στ∗ (t)(T ∗ Q × R , Ω) =(T Q|Q ,P × R , ΩQ,P (t) = dQI (t) ∧ dPI (t)), it restores the existence of the inverse of κ and O,
respectively.
The dual bundle of T ∗ Q|Q ,P × R = στ∗ (t)(T ∗ Q × R) is determined as T Q|Q ,R × R up
to isomorphism, where RI (t) are a set of contra-variant vector components on Q|Q , where|Q denotes the configuration space Q equipped with the coordinates QI . Then assume that
there is a function L = L(QI (t) , RI (t) , t) defined on T Q|Q ,R × R such that the following
conditions are satisfied:∂L,
PI (t) =∂RI (t)det∂PI (t)∂RJ (t)
̸= 0 .

(3.67)

Using the implicit function theorem, we get a set of functions: RI = RI (QI (t) , P I (t) , t).
Since Q̇I (t) are contra-variant vector components on Q|Q as well, without any loss of generality, RI can be identified as Q̇I (t). Therefore, we acquire a coordinate system for the velocity
phase space T Q: QI (t) and Q̇I (t). This construction leads to a one-to-one correspondence
between PI (t) and Q̇I (t). That is, the following map is a well-defined and invertible map:
κ|στ (t) : T Q|Q ,Q̇ × R → T ∗ Q|Q ,P × R; Q̇I (t) 7→ PI (t) ,(3.68)

which restricts the domain T Q × R and the range T ∗ Q × R of κ to στ∗ (t)(T Q × R) =T Q|Q ,Q̇ × R and στ∗ (t)(T ∗ Q × R) = T ∗ Q|Q,P × R, respectively. Of course, there is a relation:
T Q|Q ,Q̇ × R ≃ T ∗ Q|Q ,P × R, and it implies that T Q|Q ,Q̇ × T ∗ Q|Q ,P × R is equivalent to
T Q|Q ,Q̇ × R and T ∗ Q|Q ,P × R.
Second, we define a Lagrangian in the space T Q|Q ,Q̇ × T ∗ Q|Q ,P × R, denoted by LT ,75

which corresponds to the total Hamiltonian HT , as follows:hi
LT := στ∗ (t) PI Q̇I + Θa Θ̇a − HT (Θa , Θa , QI , PI )Iaa

(3.69)I= PI (t)Q̇ (t) − HT (Θ (t) = ϵ , Θa (t) = ϵa , Q (t) , PI (t)) .
That is, L = LT and PI (t) are introduced by Eq (3.67). Here, στ∗ (t)(dX/ds) = d(στ∗ (t)X)/d(στ∗ (t)s) =dX(t)/dt is applied, replacing X by QI and Θa . This leads to στ∗ (t)Θ̇a = στ∗ (t){Θa , HT }P.b. =στ∗ (t)F a (Θ) = F a (στ∗ (t)Θ) = constant, where F a (Θ) denote a function depending only on
the constraint coordinates: Θa and Θa . It is used also that, for τ ∈ {1 , 3}, all of Ξα and Ψα
in the Shanmugadhasan-Maskawa-Nakajima theorem turn into second-class by virtue of thegauge fixing conditions. Based on this, these variables are gathered together into Θa andΘa . The Lagrangian LT is uniquely determined by its construction up to surface terms. Let
us define the pullback of “total Lagrangian” by στ∗ (t) as in the above. Taking into accountthe κ|στ (t) , it suggests that the following map is a well-defined and invertible map:
O|στ (t) : G[T (T Q|Q ,Q̇ × R)] → G[T (T ∗ Q|Q ,P × R)]; Xt |Q ,Q̇ 7→ ∗ X t |Q ,P(3.70)

where Xt |Q ,Q̇ and ∗ X t |Q ,P are Hamiltonian vector fields restricted to T (T Q|Q ,Q̇ × R) andT (T ∗ Q|Q ,P × R), respectively, and these corresponds in a one-to-one manner.
Finally, let us consider the map ι. When varied, Eq (3.69) becomes as follows;






∂HT
∂HT
d 
I
I
δLT = Q̇ −
δPI + −ṖI −
δQ
+
P
δQ
I
∂PI
∂QI
dt
I

(3.71)

where the argument t in QI (t) and PI (t) are abbreviated. In the form of action functional,
it can be written as follows:δ (στ∗ (t)I) =Z t2 
t1





t2
∂HT
∂HT
Q̇ −
δPI dt + −ṖI −δQI dt + PI δQI t1 .
I
∂PI
∂Q
I

76

(3.72)

In the subspace T Q|Q ,Q̇ × T ∗ Q|Q ,P × R, the formulas in Eq (3.67) or the invertible map
κ|στ (t) can be used. Therefore, the above formula becomes as follows:δ (στ∗ (t)I) =Z t2 
t1







t
d ∂LT
∂LT I 2∂HT∂HTI
δPI dt+ −δQ dt+Q̇ −−δQ. (3.73)∂PI
dt ∂ Q̇I
∂QI
∂ Q̇I
t1
I

The one-to-one correspondence between στ∗ (t)HT and LT indicates that
(3.74)στ∗ (t)HT := PI Q̇I − LT (QI , Q̇I )in the subspace T Q|Q ,Q̇ × T ∗ Q|Q ,P × R under Eq (3.67). Therefore, Eq (3.73) becomes asfollows:δ (στ∗ (t)I) =Z t2 
t1

d
−
dt



∂LT
∂ Q̇I





t
∂LT
∂LT I 2I+
δQ dt +
δQ
,
∂QI
∂ Q̇I
t1

(3.75)

which is now defined in the subspace T Q|Q ,Q̇ × R. The second condition of Eq (3.67) implies
that LT is non-degenerate. Therefore, the boundary conditions are given as follows:(3.76)δQI (t1 ) = δQI (t2 ) = 0 .

Here, notice that in the subspace T Q|Q ,Q̇ × R this system always holds the Frobenius integrability based on the consideration given in Section 3.3. In this formulation, map ι can be
introduced in a well-defined manner as an invertible map automatically:′ι|στ : Q|Q [t1 ] × Q|Q [t2 ] → C|στ ; (QI (t1 ) , QI (t2 )) 7→ cA ,(3.77)

where Q|Q [t] is the configuration subspace restricted by the canonical embedding στ andC|στ is a parameter space spanned by the independent integral constants restricted by all
the constraints. A′ runs from 1 to the twice number of QI . Therefore, based on Definition
1, δ (στ∗ (t)I) := 0 is the well-posed variational principle.
77Summarising, the well-posed variational principle isδ (στ∗ (t)I) := 0

(3.78)

under the boundary condition (3.76).

3.7. Well-posed variational principle 2: Cases of quasi-canonical embeddings
In this section, based on Definition 1 and the quasi-canonical embeddings, σ̃1 (t) and σ̃3 (t),which are introduced in the previous section 3.5, the well-posed variational principle for asystem that includes first-class constraints without any gauge fixing conditions is formulated [Tomonari (2023b)]. Hereinafter, we abbreviate the notation of “ ∗ ” for the phasespace variables if it is apparent. First of all, let us consider the case of σ̃3 (t). The sameconsiderations are applicable to σ̃1 (t).
The quasi-canonical embedding σ̃3 (t) leads to σ̃3∗ (t)(T ∗ Q×R , Ω) = (T ∗ Q|Ξ ,Ψ ×T ∗ Q|Q ,P ×
R , ΩQ ,P (t) + ΩΨ ,Ξ (t) = dQI (t) ∧ dPI (t) + dΞα (t) ∧ dΨα (t)). For T ∗ Q|Ξ ,Ψ × T ∗ Q|Q ,P × R, afunction L = L(Ξα (t) , Rα (t) , QI (t) , Q̇I (t) , t) defined in T Q|Ξ ,R × T Q|Q ,Q̇ × R such that∂L
= Ψα (t) = constant∂Rα (t)(3.79)

can be introduced, but, of course,∂Ψα (t)= 0.
∂Rβ (t)(3.80)

This indicates that invertible map κ does not exist even if the entire phase space T ∗ Q × R is
restricted to the subspace σ̃3∗ (t)(T ∗ Q × R). However, if T ∗ Q × R is restricted to T ∗ Q|Q,P × R,
then PI and Q̇I correspond to one another in a one-to-one manner. Then an invertible map78

κ|σ̃3 (t) between T Q|Q ,Q̇ × R and T ∗ Q|Q ,P × R is introduced as follows:κ|σ̃3 (t) : T Q|Q ,Q̇ × R → T ∗ Q|Q ,P × R; Q̇I (t) 7→ PI (t)(3.81)

in a well-defined manner. In addition, under the same restriction of the phase space T ∗ Q×R,
an invertible map O|σ̃3 (t) : G[T (T Q|Q ,Q̇ × R)] → G[T (T ∗ Q|Q ,P × R)]; Xt |Q ,Q̇ 7→ ∗ X t |Q ,P isintroduced in a well-defined manner.
Now, let us consider the map ι. The pullback of the total Lagrangian by σ̃3 (t) corresponding to the total Hamiltonian HT is defined as follows:
LT =PI (t)Q̇I (t) + Ψα (t)Ξ̇α (t) + Θa (t)Θ̇a (t)− HT (Ξα (t) , Ψα (t) = ϵα , Θa (t) = ϵa , Θa (t) = ϵa , ζ α1 , QI (t) , PI (t))=PI (t)Q̇I (t) − HT (Ξα (t) , Ψα (t) = ϵα , Θa (t) = ϵa , Θa (t) = ϵa , ζ α1 , QI (t) , PI (t)) +d[Ψα (t)Ξα (t)]dt
(3.82)

in the space T Q|Q ,Q̇ × T ∗ Q|Q ,P × T Q|Ξ ,Ξ̇ × T ∗ Q|Ξ ,Ψ × R, where α1 denotes the indices that
correspond to the primary first-class constraints. Remark that Condition (3.67) on QI andPI holds in the subspace T Q|Q ,Q̇ × T ∗ Q|Q ,P × R as well. ζ α are Lagrange multipliers and α
runs from 1 to the number of the primary first-class constraints. Hereinafter, the argument
t in QI (t), PI (t), Ξα (t) and Ψα (t) is abbreviated. Since PI and Q̇I have the one-to-one
correspondence by virtue of det(∂PI /∂ Q̇J ) ̸= 0 in Eq (3.67), when varying Eq (3.82) in theaction functional form, it becomes as follows:δ (σ̃3∗ (t)I) =Z t2 




∂HT
∂HT
Q̇ −
δQI dtδPI dt + −ṖI −
I
∂PI
∂Q
t1


∂HT α
∂HT α1
I
α t2
−
δΞ
dt
−
δζ
dt
+
P
δQ
+
Ψ
δΞ
.
I
α
t1
∂Ξα
∂ζ α1
I

(3.83)

Note, here, that ∂HT /∂ζ α1 correspond to the pullback of the primary first-class constraints
79

by σ̃3 (t). Therefore, we getδ (σ̃3∗ (t)I) =Z t2 
t1

d
−
dt

Z t2
+
t1



∂LT
∂ Q̇I





t2
∂LT α′
∂LT
∂LT I
I
α′
δQ dt ++
δΞ dt +
δQ + Ψα′ δΞ
∂QI
∂Ξα′
∂ Q̇I
t1

∂LT
∂LT α1α1δΞdt+
δζ dt + [Ψα1 δΞα1 ]tt21
α
∂Ξ 1
∂ζ α1
(3.84)

in the subspace T Q|Q ,Q̇ ×T Q|Ξ ,Ξ̇ ×T ∗ Q|Ξ ,Ψ ×R, where we used the one-to-one correspondence
between Q̇I and PI which is described by κ|σ̃3 (t) . α′ denotes the indices that eliminate therange of α1 form {1 , 2 , · · · , U }. Notice that in this subspace the symplectic structure breaks
down: ΩQ ,P (t) satisfies the definition (3.52), but so is not ΩΞ ,Ψ (t) as mention in Section 3.5.
Here, let us define the “effective first-order variation” of the action functional as follows:δeffective (σ̃3∗ (t)I) :=Z t2 
t1

d
−
dt



∂LT
∂ Q̇I





t2
∂LT
∂LT α′∂LT I
Iα′+
δQ dt+ α′ δΞ dt+
δQ + Ψα′ δΞ
.
∂QI
∂Ξ
∂ Q̇I
t1
(3.85)′

The coordinate variables QI and Ξα are now independent. The reason why the terms
concerning primary first-class constraints are split out will be revealed soon. Therefore, tovanish the effective first-order variation, all the position variables QI have to be fixed atboth t = t1 and t = t2 as boundary conditions:(3.86)δQI (t2 ) = δQI (t1 ) := 0and, since Ψα′ are constant, we have to impose′′(3.87)

δΞα (t2 ) = δΞα (t1 ) .
′′Taking into account the equations of motion Ξ̇α = {Ξα , HT }P.b. , appropriate boundary80′

conditions for these variables Ξα have to be either′(3.88)

δΞα (t1 ) := 0 ,′then δΞα (t2 ) = 0 are automatically satisfied, or′(3.89)

δΞα (t2 ) := 0 ,′′′then δΞα (t1 ) = 0 are automatically satisfied, since each solution of Ξ̇α = {Ξα , HT }P.b. has
one integral constant, respectively. That is, each solution with a given integral constant on
a boundary determines the value on the other boundary. Therefore, Eq (3.87) is satisfied
and becomes zero if either Eq (3.88) or Eq (3.89) is imposed. In this work, we adopt thefirst choice; this choice is none other than setting initial conditions. Here, notice that for LT
on T Q|Q ,Q̇ × T Q|Ξ ,Ξ̇ × R the Frobenius integrability (3.43) has to be imposed. That is,θI (XT ) = δI i ζ α1 ταα1 Kij∂η j:≈ 0 ,∂ Ξ̇α(3.90)

where I runs the range of indices eliminating the ones for the second-class constraint coorij

dinates, τβα are zero-eigenvalue vectors of the kinetic matrix, and η i ≈ (K −1 ) Sj . Remarkthat the (I , J)-block of the kinetic matrix is non-degenerate. In contrast, for σ̃3∗ (t)HT on
T ∗ Q|Q ,P ×T ∗ Q|Ξ ,Ψ ×R, the Frobenius integrability (3.43) is automatically satisfied as follows:∗ θ I (∗ X T ) = −δIα ζ∂Ψα1α1 ∂Ψα1− δIJ ζ α1
≈ 0,
α
∂Ξ
∂QJ

(3.91)

since Ξα and Ψα form a part of the entire canonical coordinate system and all Ψα arecanonical momentum variables with respect to Ξα , respectively.
In order to introduce the map ι in a well-defined manner, the following two conditions

81

have to be taken into account; The first is that if gauge fixing conditions are imposed thenσ̃3 (t) turns into σ3 (t); The second is that the dimension of a parameter space that is spannedby all independent integral constants is up to 2N −2V −U . Then the map ι can be introduced
as follows:′′ι|σ̃3 : Q|Q ,Ξ [t1 ] × Q|Q [t2 ] → C|σ̃3 ; (QI (t1 ) , Ξα (t1 ) , QI (t2 )) 7→ cA ,(3.92)′

δΞα (t2 ) = 0 are automatically satisfied, where Q|Q and Q|Q ,Ξ are the configuration subspaceof T ∗ Q|Q ,P and T ∗ Q|Q ,P × T ∗ Q|Ξ ,Ψ , respectively, and C|σ̃3 is the parameter space spannedby the independent integral constants restricted by all the constraints. A′ runs from 1 to the′sum of twice number of QI and Ξα . Then map ι is invertible. Therefore, based on Definition
1, δeffective (σ̃3∗ (t)I) := 0 is the well-posed variational principle.
Under the well-posed variational principle δeffective (σ̃3∗ (t)I) := 0 with the boundary conditions (3.86) and (3.88), the original first-order variation of the action functional (3.84)
becomes as follows:δ (σ̃3∗ (t)I) =

Z t2
t1

∂LT
∂LT
δΞα1 dt + α1 δζ α1 dt + [Ψα1 δΞα1 ]tt21 .
α
1
∂Ξ
∂ζ

(3.93)

Applying the variational principle: δ (σ̃3∗ (t)I) := 0, we derive∂HT∂LT= − α1 = 0
α
1
∂Ξ
∂Ξ

(3.94)

if the following equations are identically satisfied:∂LT∂HT= − α1 = Ψα1 (t) = 0 .
α
1
∂ζ
∂ζ

(3.95)

In fact, the boundary conditions δΞα1 (t2 ) = δΞα1 (t1 ) are not satisfied since Ξ̇α1 = {Ξα1 , HT }P.b. ≈82

ζ α1 + f (QI , PI ) leads to
α1

α1

Z t2

δΞ (t2 ) − δΞ (t1 ) = δ α1
ζ + f (QI , PI ) dt = δZ t2

ζ α1 dt ,(3.96)

t1

t1

where f is the definite function determined by HT and we used the fact that the integral on
the interval t1 ≤ t ≤ t2 of f has a definite value since QI and PI are propagating degrees
of freedom and the equations of motion for these variables are already derived by virtueof the well-posed variational principle δeffective (σ̃3∗ (t)I) := 0. This result indicates that theconfigurations corresponding to the primary first-class constraints cannot be fixed on theboundaries. Therefore, we have to impose Ψα1 (t) = constant = 0 in advance and this meansthat the existence of first-class constraints restricts the possible embedding σ̃3 (t). Therefore,
Eq (3.93) becomes as follows:δ (σ̃3∗ (t)I) =

Z t2
t1

∂LT
δΞα1 dt .
∂Ξα1(3.97)The variational principle, δ (σ̃3∗ (t)I) := 0, derives the equations of motion, ∂LT /∂Ξα1 = 0,
without any boundary condition. Under this assumption, if the effective first-order variation
vanishes: δeffective (σ̃3∗ (t)I) := 0, the variational principle is applied to the entire phase space,and vice versa. As another aspect, it would be convenient to introduce the “effective-total
Hamiltonian” as follows:
Heffective := HT |Ψα1 :=0 .

(3.98)For this Heffective , repeating the same consideration going back to Eq (3.82), Eq (3.85) is
directly derived. Of course, the appropriate boundary conditions are given by Eq (3.86) andEq (3.88).

83

There is a remark. Eq (3.82) can be rewritten as follows:
L′T (QI , PI , Ξα , Ψα ) := PI Q̇I − σ̃3∗ (t)HT(3.99)

where
L′T = LT −d α
[Ψ Ξα ] ,dt

(3.100)and we abbreviatied Θa and Θa . L′T in Eq (3.100) has the arbitrariness of continuous infinite
since Ψα can be regarded as continuous parameters. In other words, L′T is parametrized byΨα . The arbitrariness is not the one deriving from a canonical transformation on T ∗ Q|Q ,P ×
R. That is, for a total Hamiltonian HT , the corresponding Lagrangian is not uniquelydetermined, unlike the case of canonical embeddings. It is another aspect of the absence of
the inverse map O−1 in the entire space T ∗ Q × R.
Summarising, the well-posed variational principle isδeffective (σ̃3∗ (t)I) := 0

(3.101)

under the boundary conditions (3.86) and (3.88). The important result is that the configurations corresponding to the primary first-class constraint coordinates never be fixed on theboundaries until some gauge fixing conditions are imposed.

3.8. Examples
In this section, the difficulties in the examples given in Section 3.1 are resolved by applying the theory that is established throughout this chapter [Izumi et al. (2023); Tomonari(2023b)]. Hereinafter, we abbreviate the notation of “ ∗ ” for the phase space variables if it
is apparent.

84

The first examples is the Lagrangian L1 in Eq (3.1):
1
1
L1 = ẋ2 + ẏ 2 + λ(x + y) .
2
2
The kinetic matrix is1 0 0
K=0 1 0 ,


0 0 0

(3.102)(3.103)where the following canonical momentum variables are used: p1 = ẋ, p2 = ẏ, pλ = 0. Remarkthat the Lagrange multiplier λ is regarded as a configuration variable. Since the rank of the
kinetic matrix is 2, there is a primary constraint as follows:ϕ(1) = pλ := 0 .

(3.104)

The total Hamiltonian is given as follows:
HT = H0 + ζϕ(1) ,1
1
H0 = p2x + p2y − λ(x + y) .
22

(3.105)

The Dirac-Bergmann procedure derives a secondary, a tertiary, and a quaternary constraint
as follows:ϕ(2) = x + y := 0 ,1
ϕ(3) = (px + py ) := 0 ,2(3.106)

ϕ(4) = λ := 0 .
The consistency for ϕ(4) is automatically satisfied. Therefore, the procedure stops here. The
Poisson brackets are {ϕ(1) , ϕ(4) }P.b. = 1, {ϕ(2) , ϕ(3) }P.b. = 1, otherwise vanish; all constraints
are classified into second-class constraint and the propagating degrees of freedom of the85system is (3 × 2 − 4)/2 = 1. The Shanmugadhasan-Maskawa-Nakajima theorem indicatesthat the system has a canonical transformation from the original coordinates to the ones suchthat a part of the entire canonical coordinates are represented by the linear combination of
the second-class constraints. In fact, the symplectic 2-form is computed as follows:Ω = dq i ∧ dpi = dQ ∧ dP + Θa ∧ Θa ,(3.107)

where Q, P , Θa , and Θa (a ∈ {1 , 2}) are defined as follows:1
Q = (x − y) ,2

P = px − py

Θ1 = ϕ(4) ,Θ1 = ϕ(1)Θ2 = ϕ(2) ,Θ2 = ϕ(3) .

(3.108)

Notice that the second equality of Ω is the strong equality, not weak equality. In terms ofthese canonical variables, the total Hamiltonian is transformed as follows:1
HT = P 2 + (Θ2 )2 − Θ1 Θ2 .
4

(3.109)The pullback of total Lagrangian by the canonical embedding σ2 (t) is derived as follows:ihLT = σ2∗ (t) P Q̇ + Θa Θ̇a − HT1
= P Q̇ − P 2 − (Θ2 )2 − Θ1 Θ2 ,4
1
∴ LT = P Q̇ − P 2 + constant ,4(3.110)which is defined in the subspace T Q|Q ,Q̇ × T ∗ Q|Q ,P × R. Here, we abbreviated the pullback
operator σ2∗ (t), which is defined in Section 3.5, in the second and the last line. Remarkthat the pullback of HT by σ2 (t) above is defined in the symplectic submanifold (T ∗ Q|Q ,P ×

86

R , ΩQ ,P = dQ ∧ dP ). This system is Frobenius integrable as mentioned in Section 3.6. In
fact, it can be shown that this system has a unique solution.
The first-order variation of the action functional is calculated as follows:δ(σ2∗ (t)I) =Z t2 
t1

hi
1− P + Q̇ δP dt + −Ṗ δQ dt + [P δQ]tt21 .
2(3.111)

The appropriate boudnary conditions are set as follows:δQ(t2 ) = δQ(t1 ) := 0

(3.112)

Then the variational principle leads to the following equations:1− P + Q̇ = 0 ,2−Ṗ = 0 .

(3.113)

The first equation gives the explicit form the canonical momentum variable P . Therefore,
LT is rewritten as follows:
LT = 2Q̇2 + constant ,(3.114)

which is now defined in the subspace T Q|Q ,Q̇ × R. It indicates that the system is always
Frobenius integrable as mentioned in Section 3.6 and that two integral constants exist. In
fact, LT is nothing but describing the uniform motion in a straight line. The equation of
motion is given as follows:−2Q̈ = 0 .

(3.115)This equation has a unique solution: Q(t) = At + B, where A and B are integral constants
and the boundary conditions uniquely determine these constants. The fundamental maps
are given as follows: κ|σ2 (t) : T Q|Q ,Q̇ → T Q|Q ,P ; Q̇ 7→ P and O|σ2 (t) : Xt = Q(∂/∂ Q̇) +Q̇(∂/∂Q) + (∂/∂t) 7→ ∗ X t = (∂HT /∂P )(∂/∂Q) − (∂HT /∂Q)(∂/∂P ) + (∂/∂t), where HT =87

P 2 /4+constant; these are introduced in a well-defined manner and invertible. The invertible
map ι is ι|σ3 (t) : Q|Q [t1 ] × Q|Q [t2 ] → C|σ3 (t) ; (Q(t1 ) , Q(t2 )) 7→ (A , B) with A = (Q(t1 −Q(t2 ))/(t1 − t2 ) and B = (t1 Q(t2 ) − t2 Q(t1 ))/(t1 − t2 ).
The second example is the Lagrangian L2 in Eq (3.1):
1
L2 = q̇ 1 q̇ 3 + q 2 (q 3 )2 .
2

(3.116)This model has no propagating degrees of freedom but is historically crucial; it was proposed
as a counter-example for the Dirac conjecture [Dirac (1950)]. The kinetic matrix K is
0 0 1


,
K=
0
0
0




1 0 0

(3.117)

where the following canonical momentum variables are used: p1 = q̇ 3 , p2 = 0 , p3 = q̇ 1 . Thereis a primary constraint due to rank K = 2:
ϕ(1) = p2 := 0 .

(3.118)The total Hamiltonian is derived as follows:
HT = H0 + ζϕ(1) ,1
H0 = p1 p3 − q 2 (q 3 )2 .
2

88

(3.119)

The Dirac-Bergmann procedure generates a secondary and a tertiary constraint as follows:1ϕ̇(1) = {ϕ(1) , HT }P.b. ≈ (q 3 )2
2
∴ ϕ(2) = q 3 := 0 ,
(3)

ϕ

(2)

= ϕ̇

(3.120)(2)= {ϕ, HT }P.b. ≈ p1

∴ ϕ(3) = p1 := 0 .
ϕ̇(3) ≈ 0 is automatically satisfied. All ϕ(1) , ϕ(2) , and ϕ(3) are classified into first-class constraint: all Poisson brackets among them vanish. The Shanmugadhasan-Maskawa-Nakajima
theorem indicate that this system has a canonical transformation from the original coordinates to the ones such that a part of their canonical momenta themselves are the first-classconstraints. In fact, the symplectic 2-form of the system is computed as follows:(3.121)Ω = dq i ∧ dpi = dΞα ∧ dΨα ,where Ξα and Ψα are defined as follows:Ξ1 = q 1 ,(3)

Ψ1 = ϕΞ2 = q 2 ,,Ξ3 = −p3(1)

Ψ2 = ϕ,(3.122)(2)

Ψ3 = ϕ.

Notice that the second equality in the equation of Ω is the strong equality, not weak equality.
The total Hamiltonian is transformed as follows:1
HT = −Ψ1 Ξ3 − Ξ2 (Ψ3 )2 + ζΨ2 .
2(3.123)This system is always Frobenius integrable as mentioned in the section 3.3. In fact, Eq (3.41)

89

can be computed as follows:



∂H
∂
dt
XH + ζXΨ2 +∗ θ α (∗ XT ) = dΨα +
∂Ξα
∂t
∂H
=dΨα (XH ) + dΨα (ζXΨ2 ) +∂Ξα∂H≈{Ψα , H} + ζ{Ψα , Ψ2 } +∂Ξα(3.124)=0 ,∴ ∗ θα (∗ XT ) ≈ 0 .
Therefore, the system has six integral constants, of which three integral constants are occupied by the consistency conditions of the constraints. Therefore, the remaining three independent integral constants, which originated from the three equations: Ξ̇1 = −Ξ3 , Ξ̇2 = ζ,and Ξ̇3 = −Ξ2 Ψ3 , have to be fixed by imposing boundary conditions in the variational
principle. There are a possible quasi-canonical embedding σ̃ 1 (t) and a possible canonicalembedding σ1 (t), which are defined in Sections 3.6 and 3.7, respectively.
We derive the pullback of total Lagrangian LT by σ̃1 (t) and compute the first-ordervariation of the action functional for LT . LT is defined as follows:
LT := σ̃1∗ (t)hαΨα Ξ̇ − HT

i


d 
1 2
2
α′
Ψα′ Ξ
,
∴ LT = Ψ1 Ξ + Ξ (Ψ3 ) +2
dt(3.125)3which is defined in the space T Q|Ξ ,Ξ̇ × R. Here, we abbreviated the pullback operator σ̃1∗ (t),α′ ∈ {1 , 3}, and we used σ̃1∗ (t)Ψ2 = 0. Then the effective first-order variation is given as
follows:
δeffective (σ̃1∗ (t)I) = Ψ1

Z t2
t1


t
dtδΞ3 + Ψ1 δΞ1 + Ψ3 δΞ3 t21 .

90(3.126)

The appropriate boundary conditions are set as follows:′δΞα (t1 ) = 0 ,(3.127)′then δΞα (t2 ) = 0 are automatically satisfied since each solution of the equations of motion′for Ξα is only one integral constant, respectively. That is, each solution gives a definite
value at the boundary t = t2 . Then the variational principle for the effective first-ordervariation, δeffective (σ̃1∗ (t)I) := 0, leads to Ψ1 = 0. In addition, δ(σ̃1∗ (t)I) := 0 derives Ψ3 = 0.
Here, we abbreviated the pullback operator σ̃1∗ (t). Remark that the Frobenius integrability
condition (3.43) in Section 3.3 restricted to σ̃1∗ (t)(T Q × R) is also satisfied, and there occurs
no phase space reduction. It indicates that the six integral constants hold, of which thethree constants are occupied by the consistency conditions, or equivalently the embeddingσ̃1∗ (t). This fact can be also led by that Eq (3.41) given in Section 3.3 is always satisfied
in this system by virtue of a zero-eigenvector τ i = (0 , τ 2 , 0) and η i ≈ (0 , η 2 , 0) whereτ 2 and η 2 are arbitrary function in the space T Q|Ξ ,Ξ̇ × R. Then the invertible map ιis ι|σ̃3 : Q|Ξ [t1 ] → C|σ̃3 ; (Ξ1 (t1 ) , Ξ3 (t1 )) 7→ (c1 , c2 ). In fact, we have the two equations:Ξ̇1 = −Ξ3 and Ξ̇3 = 0. Therefore, c1 = Ξ1 (t1 ) − Ξ3 (t1 )t1 , c2 = Ξ3 (t1 ). Remark that κ|σ̃1 (t)and O|σ̃1 (t) do not exist in this case since this system does not have any dynamics. The
remaining one integral constant is assigned for Ξ̇2 = ζ; this constant is not determined until
some gauge fixing conditions are imposed.
Next, let us impose the following gauge fixing conditions to all Ξα : Ξ̇1 = −Ξ3 := 0,Ξ̇2 = ζ := 0, and Ξ̇3 = −Ξ2 Ψ3 := 0. The first and the third equations are automaticallysatisfied by virtue of Ξα ≈ 0 (α ∈ {1 , 2 , 3}). The second equation is satisfied if and only if
ζ := 0. Remark that all these ingredients are derived in the entire space T ∗ Q|Ψ,Ξ × R: thetarget space of the embedding σ1 (t). Then the pullback of total Lagrangian LT by σ1 (t) is91

introduced as follows:hiLT = σ1∗ (t) Ψα Ξ̇α − HT = constant ,(3.128)which is defined in the null subspace {0} × R. That is, this system does not describe any
dynamics. Therefore, of course, κ|σ̃1 (t) and O|σ̃1 (t) do not exist. Map ι is in the same situation:
all the integral constants, which are implied by Eq (3.41) given in Section 3.3 by virtue ofthe same reason as the previous paragraph, are occupied by the consistency conditions forΨα and Ξα , or equivalently fixing the embedding σ1 (t).
The third example is the Lagrangian L3 in Eq (3.1):
L3 = q 1 q̇ 2 − q 2 q̇ 1 − (q 1 )2 − (q 2 )2 .

(3.129)This model is an imitation of the Dirac system for spin 1/2-particles in field theory. The
kinetic matrix K is



0 0
K=
,
0 0

(3.130)where the following canonical momentum variables are used: p1 = −q 2 , p2 = q 1 . There are
two primary constraints due to rank K = 0:(1)

ϕ1 = p1 + q 2 := 0 ,
(1)

(1)

ϕ2 = p2 − q 1 = 0 .

(3.131)(1)

The Poisson bracket is {ϕ1 , ϕ2 }P.b. = 2 and otherwise vanish. The total Hamiltonian isderived as follows:
HT = H0 + ζ α ϕ(1)α ,H0 = (q 1 )2 + (q 2 )2 .

(3.132)

The Dirac procedure determines all Lagrange multipliers:ζ 1 ≈ −q 2 ,92

ζ 2 ≈ q1 .

(3.133)

Then the consistency conditions for ϕ(1) , ϕ(2) are satisfied: ϕ̇(1) ≈ 0 , ϕ̇(2) ≈ 0. That is, allconstraints are classified into second-class constraints and the propagating degrees of freedomof the system is (2 × 2 − 2)/2 = 1. The Shanmugadhasan-Maskawa-Nakajima theorem
indicates that the system has a canonical transformation from the original coordinates tothe ones such that a part of the entire canonical coordinates are represented by the linear
combination of the second-class constraints. In fact, the symplectic 2-form is computed asfollows:Ω = dq i ∧ dpi = dQ ∧ dP + dΘ1 ∧ dΘ1 ,(3.134)

where Θ1 , Θ1 , Q and P are defined as follows:
1 (1)
1 (1)
Θ1 = √ ϕ1 , Θ1 = √ ϕ2 ,22
1
1
Q = √ (q 1 + p2 ) , P = √ (p1 − q 2 ) .
2
2(3.135)

Notice that the second equality in the equation of Ω is the strong equality, not weak equality.
The total Hamiltonian is transformed as follows:1111
HT = P 2 + Q2 − (Θ1 )2 − (Θ1 )2 .
2
2
2
2

(3.136)

The pullback of total Lagrangian by σ2 (t) is derived as follows:LT =σ2∗ (t) Θ1 Θ̇1 + P Q̇ − HT1111=P Q̇ − Q2 − P 2 + (Θ1 )2 + (Θ1 )2 ,2
2
2
2
1 2 1 2∴ LT = P Q̇ − Q − P + constant ,22(3.137)

which is defined in the subspace T Q|Q ,Q̇ × T ∗ Q|Q ,P × R. Here, we abbreviated the pullback
operator σ2∗ (t), which is defined in Section 3.5, in the second and the last line. Remark

93

that the pullback of HT by σ2 (t) above is defined in the symplectic submanifold (T ∗ Q|Q ,P ×
R , ΩQ ,P = dQ ∧ dP ). This system is Frobenius integrable as mentioned in Section 3.6. In
fact, it can be shown that this system has a unique solution.
The first-order variation of the action functional of LT is computed as follows:δ (σ2∗ (t)I) =Z t2 hihi−Ṗ − Q δQ dt + Q̇ − P δP dt + [P δQ]tt21 .

(3.138)t1

The appropriate boundary conditions are set as follows:δQ(t2 ) = δQ(t1 ) := 0 .

(3.139)Then the variational principle leads to the following equations:−Ṗ − Q = 0 ,Q̇ − P = 0 .

(3.140)The second equation gives the explicit form for the canonical momentum variable P . Therefore, LT is rewritten as follows:1
1
LT = (Q̇)2 − Q2 + constant ,22(3.141)

which is now defined in the subspace T Q|Q ,Q̇ × R. It indicates that the system is always
Frobenius integrable as mentioned in Section 3.6 and that two integral constants exist. In
fact, LT is nothing but describing the one-dimensional harmonic oscillator. The equation of
motion is as follows:−Q̈ − Q = 0 .

(3.142)This equation has the unique solution Q(t) = A exp(+it) + B exp(−it) and the boundary
condition uniquely determines the integral constant A , B. The fundamental maps are given94

as follows: κ|σ2 (t) : T Q|Q ,Q̇ → T Q|Q ,P ; Q̇ 7→ P and O|σ2 (t) : Xt = Q(∂/∂ Q̇) + Q̇(∂/∂Q) +(∂/∂t) 7→ ∗ X t = (∂HT /∂P )(∂/∂Q)−(∂HT /∂Q)(∂/∂P )+(∂/∂t), where HT = Q2 /2+P 2 /2+
constant; these are introduced in a well-defined manner and invertible. The invertible mapι is ι|σ3 (t) : Q|Q [t1 ] × Q|Q [t2 ] → C|σ3 (t) ; (Q(t1 ) , Q(t2 )) 7→ (A , B) with A = (Q(t1 )exp(it2 ) −Q(t2 )exp(it1 ))/2i sin(t2 − t1 ) and B = (Q(t2 )exp(−it1 ) − Q(t1 )exp(−it2 ))/2i sin(t2 − t1 ).
The final example is the Lagrangian L4 in Eq (3.1):
1
1
L4 = − q q̈ − q 2 .
22(3.143)For this system, applying the consideration given in Section 3.2, there exists the counter-term
W given as follows:
1
W = q q̇ + C(q) ,2(3.144)

where C(q) is an arbitrary function of q. Then L4 becomes as follows:
L′4 = L4 +

dW
1
1
∂C
= (q̇)2 − q 2 +
q̇ .
dt
2
2
∂q

(3.145)This is none other than the Lagrangian for a one-dimensional harmonic oscillator. In fact,
the first-order variation of L′4 :
Z t2

d
δI =
(−q̈ − q)δqdt +
dt
t1
′


 t2
∂C
q̇ +
δq
∂q
t1

(3.146)and the well-posed variational principle under the Dirichlet boundary condition δq(t2 ) =δq(t1 ) = 0 leads to an equation: −q̈ − q = 0. The solution is, of course, q(t) = A exp(+it) +B exp(−it), where A , B are integral constants that are determined by the boundary conditions. For this trivial system, we apply the two different methodologies which are introducedin Section 2.5, to formulate the well-defined variational principle without any counter-term.

95

That is, (i) Ostrogradsky’s original formulation and (ii) Pons’ modified formulation.
Let us consider applying (i) Ostrogradsky’s original formulation. The configuration space,denoted by Q, in this method is coordinated by Q(1) = q and Q(2) = q̇. Then the corresponding canonical momenta are given as follows: P (1) = ∂L4 /∂Q(2) − (d/dt)(∂L4 /Q̇(2) ) =Q(2) /2 and P (2) = ∂L4 /Q̇(2) = −Q(1) /2. Therefore, the rank of the kinetic matrix K (2) =∂P (2) /∂ Q̇(2) is zero, and there is a primary constraint: ϕ(1) = P (2) + Q(1) /2 := 0. Then the
total Hamiltonian is HT = P (1) Q(2) + P (2) Q̇(2) − L4 + ζϕ(1) = P (1) Q(2) + (Q(1) )2 /2 + ζϕ(1) ,where ζ is a Lagrange multiplier. The consistency condition for ϕ(1) generates a secondary
constraint: ϕ(2) = −P (1) + Q(2) /2 := 0, but it expected from the definition of P (1) . The
consistency condition for ϕ(2) determines the Lagrange multiplier ζ as Q(1) in the weakequality. Therefore, the procedure stops here. The Poisson brackets between ϕ(1) and ϕ(2)is {ϕ(1) , ϕ(2) }P.b. = −1 and otherwise vanish. This system has two second-class constraints,
and it indicates that the propagating degrees of freedom of the system is (2 × 2 − 2)/2 = 1
and that we have to use the canonical embedding σ2 (t) given in Section 3.5 to introduce thewell-posed variational principle.
The symplectic 2-form is computed as follows: Ω = dQ(i) ∧ dP (i) = dΘ1 ∧ dΘ1 + dQ ∧ dP ,where Θ1 = ϕ(2) , Θ1 = ϕ(1) , Q = P (1) + Q(2) /2, and P = P (2) − Q(1) /2. The Poisson brackets
are {Θ1 , Θ1 }P.b. = 1, {Q, P }P.b. = 1, and otherwise vanish. The original variables Q(1) , Q(2) ,P (1) , and P(2) are expressed by using the transformed variables as follows: Q(1) = Θ1 + Q,
Q(2) = Θ1 − P , P (1) = (Q − Θ1 )/2, and P (2) = (P + Θ1 )/2. Then the total Hamiltonianis transformed as follows: HT = P 2 /2 + Q2 /2 − 2Θ1 P − (Θ1 )2 /2 + 3(Θ1 )2 /2 and is definedin the symplectic manifold (T ∗ Q|Q ,P × T ∗ Q|Θ × R , Ω). The pullback of HT by σ2 (t) is
σ2∗ (t)HT = P 2 /2 + Q2 /2 − 2Θ1 P + constant in the symplectic submanifold (T ∗ Q|Q ,P ×
R , σ2∗ (t)Ω = dQ ∧ dP ), and this is a Frobenius integrable system. That is, there are two
integral constants. The pullback of total Lagrangian is LT := σ2∗ (t)[P Q̇ + Θ1 Θ̇1 − HT ] =P Q̇ − P 2 /2 − Q2 /2 − 2Θ1 P + constant in the subspace T Q|Q ,Q̇ × T ∗ Q|Q ,P × R. Therefore,
96the first-order variation of the action functional is given as follows:δ(σ2∗ (t)I) =Z t2 ht1ihi−Ṗ − Q δQdt + Q̇ − P − 2Θ1 δP dt + [P δQ]tt21 .

(3.147)

Under the boundary conditions:δQ(t2 ) = δQ(t1 ) = 0 ,(3.148)the well-posed variational principle δ(σ2∗ (t)I) := 0 leads to the equations of motion: −Ṗ −Q =0 and Q̇ − P − 2Θ1 = 0. That is, using σ2∗ (t)Θ1 = constant, −Q̈ − Q = 0; this is none otherthan the equation of a one-dimensional harmonic oscillator. Remark, here, that LT is now
defined in the subspace T Q|Q ,Q̇ × T ∗ Q|Q ,P × R ≃ T Q|Q ,Q̇ × R and Frobenius integrable.
In fact, the solution is Q(t) = A exp(+it) + B exp(−it), and this includes the two integral
constants: A and B. The boundary conditions fix these constants.
Finally, the maps κ, O, and ι are introduced as follows: κ|σ2 (t) : T Q|Q ,Q̇ → T Q|Q ,P ; Q̇ 7→
P and O|σ2 (t) : O[T (T Q|Q ,Q̇ × R)] → O[T (T ∗ Q|Q ,P × R)]; Xt = Q(∂/∂ Q̇) + Q̇(∂/∂Q) +(∂/∂t) 7→ ∗ X t = (∂HT /∂P )(∂/∂Q) − (∂HT /∂Q)(∂/∂P ) + (∂/∂t) are introduced in a welldefined manner and invertible, where HT = P 2 /2 + Q2 /2 − 2Θ1 P + constant in T ∗ Q|Q ,P ×
T Q|Ξ ,Ψ × R. ι is ι|σ2 (t) : Q|Q [t1 ] × Q|Q [t2 ] → C|σ3 (t) ; (Q(t1 ) , Q(t2 )) 7→ (A , B) with A =(Q(t1 )exp(it2 )−Q(t2 )exp(it1 ))/2i sin(t2 − t1 ) and B = (Q(t2 )exp(−it1 )−Q(t1 )exp(−it2 ))/2i sin(t2 − t1 ).
Let us consider applying (ii) Pons’ modified formulation. The original Lagrangian L4 isrepresented by using a Lagrange multiplier λ as follows:
1
1
L′′4 = − q ẋ − q 2 + λ(x − q̇) .
2
2(3.149)Regarding λ also as a position coordinate of the configuration space, the canonical momenta
are derived as follows: p = −λ, y = −q/2, and π = 0. Therefore, the rank of the kinetic97

matrix Kij = ∂pi /∂ q̇ j (where p1 = p , p2 = y , p3 = π , q 1 = q , q 2 = x , q 3 = λ) is zero,
(1)

(1)

and there are three primary constraints: ϕ1= p + λ :≈ 0, ϕ2= y + q/2 := 0, and

(1)

(1)

ϕ3 = π := 0. The total Hamiltonian is computed as follows: HT = q 2 /2−λx+ζ α ϕα , where(1)

ζ α are Lagrange multipliers and α ∈ {1, 2, 3}. The consistency conditions for ϕα become
(1)

(1)

(1)

as follows: ϕ̇1 ≈ −q − ζ 2 /2 + ζ 3 := 0, ϕ̇2 ≈ λ + ζ 1 /2 := 0 and ϕ̇3 ≈ x − ζ 1 := 0; there
is a secondary constraint: ϕ(2) = λ + x/2 where we used ζ 1 ≈ x. The consistency condition
for ϕ(2) restricts the relation between ζ 2 and ζ 3 , and this determines all the multipliers as
follows: ζ 2 ≈ −q and ζ 3 ≈ q/2. The Poisson brackets among the constraints are computed as
(1)

(1)

(1)

(1)

(1)

(1)

follows: {ϕ1 , ϕ2 }P.b. = −1/2, {ϕ1 , ϕ3 }P.b. = 1, {ϕ(2) , ϕ2 }P.b. = 1/2, {ϕ(2) , ϕ3 }P.b. = 1,
and otherwise vanish. This system has four second-class constraints, and this indicates thatthe propagating degrees of freedom of the system is (3 × 2 − 4)/2 = 1 and that we have to use
the canonical embedding σ2 (t) given in Section 3.5 to introduce the well-posed variational
principle.
The symplectic 2-form is computed as follows: Ω = dq ∧ dp + dx ∧ dy + dλ ∧ dπ = dΘ1 ∧
(1)

(1)

(1)

(1)

dΘ1 +dΘ2 ∧dΘ2 +dQ∧dP , where Θ1 = ϕ(2) −ϕ1 , Θ1 = ϕ2 , Θ2 = (ϕ(2) +ϕ1 )/2, Θ2 = ϕ3 ,Q = q/2 − y + π/2, and P = p + x/2. Then the total Hamiltonian is transformed as follows:
HT = Q2 /2 + P 2 /2 − (Θ1 )2 /2 in the symplectic manifold (T ∗ Q|Q ,P × T ∗ Q|Θ × R, Ω). The
pullback of HT by σ2 (t) is σ2∗ (t)HT = Q2 /2 + P 2 /2 + constant in the symplectic submanifold(T ∗ Q|Q ,P × R , σ2∗ (t)Ω = dQ ∧ dP ), and this is a Frobenius integrable system. That is, there
are two integral constants. The pullback of total Lagrangian is LT = σ2∗ (t)[P Q̇ + Θa Θ̇a −HT ] = P Q̇ − Q2 /2 − P 2 /2 + constant in the subspace T Q|Q ,Q̇ × T ∗ Q|Q ,P × R. Therefore,
the first-order variation of the action functional is given as follows:δ(σ2∗ (t)I) =Z t2 hihi−Ṗ − Q δQdt + Q̇ − P δP + [P δQ]tt21 .
t1

98(3.150)Under the boundary conditions

δQ(t2 ) = δQ(t1 ) := 0 ,(3.151)

the well-posed variational principle δ(σ2∗ (t)I) := 0 leads to the equations of motion, resolving as an equation −Q̈ − Q = 0; this is none other than the equation of a onedimensional harmonic oscillator. Remark, here, that LT is now defined in the subspace
T Q|Q ,Q̇ × T ∗ Q|Q ,P × R ≃ T Q|Q ,Q̇ × R and Frobenius integrable. In fact, the solution isQ(t) = A exp(+it) + B exp(−it), and this includes the two integral constants: A and B. The
boundary conditions fix these constants. The maps κ, O, and ι are the same as the previous
case by just replacing HT = P 2 /2+Q2 /2−2Θ1 P +constant by HT = P 2 /2+Q2 /2+constant.

3.9. Summary
In this chapter, a methodology to make the variational principle well-posed in degenerate
point particle systems was constructed, although that of field theories is under development.
The guiding principle is stated as “as many integral constants (/initial conditions) in thesolutions of a given system as possible should be uniquely determined through the boundary
conditions”. The resulting methodology can be summarized into three steps as follows:(1). For a given system, just performing Hamilton-Dirac analysis, reveal the constraint structure.
(2). Construct constraint coordinates. (Refer to the proof of Theorem 1 (ShanmugadhasanMaskawa-Nakajima theorem) ( or Lemma 1 and/or 2) in [Tomonari (2023b)].) Then, computing the symplectic 2-form, find a new canonical coordinate system which is indicated by
Theorem 1 (Shanmugadhasan-Maskawa-Nakajima theorem). Then select a suitable embedding (see Table 3.1).
(3). Consider the pullback of the Legendre transformation of the total Hamiltonian by theselected embedding: the pullback of the total Lagrangian by the selected embedding.
99

Table 3.1: Embeddings for each type of system. “1st-class system” means a system with
1st-class constraint(s). Others are defined in the same manner.
Gauge fixingYes
No (or no gauge d.o.f)

1st-class system
σ1 (t)σ̃1 (t)2nd-class system
σ2 (t)

1st- and 2nd-class system
σ3 (t)σ̃3 (t)-(i) If one uses στ (t) (τ = 1 , 2 , 3), take the first-order variation of its action functional,
and just fix the emerged configurations in the boundary term at both end-points. Then thevariational principle becomes well-posed.
-(ii) If one uses σ̃τ̃ (t) (τ̃ = 1 , 3), take the first-order variation of its action functional, under
the assumption that the pullback of primary first-class constraint coordinates by σ̃τ̃ (t) is set tobe zero in advance. Then fix the configurations for the propagating degrees of freedom at both
end-points and the configurations which correspond to higher-order (more than secondary)first-class constraint coordinates at either end-point. Then the variational principle becomeswell-posed.
The embeddings in Table 3.1 have those proper constant parameters, and these parameters correspond to the existence of the constraints of a given system. In the case of systems
with only existing second-class constraints, these constant parameters form a set of conjugate
pairs and drop out from the symplectic 2-form; the symplectic manifold reduces to its physical submanifold not depending on the values of these parameters. This indicates that theseparameters can be set as zeros as a formality. In the case of systems with existing first-classconstraints, however, the situation gets changed. The conjugate configurations with respect
to (the linear combinations of) the first-class constraints are not constraints; these variables
are also dynamical although not physical. Therefore, in general, the symplectic manifold
does not reduce to its physical submanifold. This indicates that these conjugate configurations need those initial conditions via the boundary conditions of the variational principle.
While the existence of the undetermined Lagrange multipliers forbids fixing the conjugate100configurations of the primary first-class constraints on the boundaries; the primary first-classconstraints have to be set as zeros in advance. In other words, these constraints cannot betaken as coordinates to be fixed for introducing the embeddings. Taking into account thisregard, the configurations only of the higher-order first-class constraints should be fixed on
the boundaries.
If the guiding principle is neglected, it is also possible to demand consistency only with theconstraint structure of a given system. In this case, in order to make the variational principle
well-posed, just evaluating the symplectic potential around the constraint surface: ω =PI δQI +Θa δΘa +Ψα δΞα ≈ PI δQI , it reveals that fixing the configurations of the propagating
degrees of freedom QI on the boundaries provides appropriate boundary conditions for thewell-posed variational principle [Izumi et al. (2023)]. Therefore, depending on whether or
not the guiding principle is assumed, there occurs a case that the well-posed variational
principle needs additional surface terms. In the case of systems with only existing secondclass constraints, the assumption does not change the situation: ω = PI δQI + Θa δΘa ≈PI δQI . However, in the case of systems with existing first-class constraints, the situation
gets changed; if the boundary terms appear as ω = PI δQI + Ξα δΨα , (i) When not assuming
the guiding principle, the term Ξα δΨα have to be switched in Ξα and Ψα by introducing aC(K)counter-term W = −Ξα Ψα : Ξα δΨα − δW = −Ψα δΞα ≈ 0 as a necessary and sufficientcondition; (ii) When assuming the guiding principle, the term Ξα δΨα vanishes since all Ψα arearbitrary constants in the restriction to an embedded phase subspace, excepting the primary
first-class constraint coordinates: Ψα1 = 0, i .e ., these are not taken as arbitrary. That is,
it is possible to add some counter-terms in the completely same manner as Case (i) only
for the primary first-class constraint coordinates. That is, in Case (ii), counter-terms have′C(1)a different meaning: Ξα δΨα − δW = −Ψα δΞα = −Ψα1 δΞα1 − Ψα′ δΞα = −Ψα1 δΞα1 ≈ 0
as a necessary and sufficient condition due to the boundary conditions either Eq (3.88) or
Eq (3.89). In this case, the imposition of full constraints is a sufficient condition.
101

The well-posedness discussed in [Tomonari (2023b); Izumi et al. (2023)] is considered
under the consistency with C(K) . As mentioned in Section 3.1, the Dirac-Bergmann analysis itself is valid independently from the well-posedness of the variational principle. Thisimplies the possibility of arguing the well-posedness in each constraint surface: C(k) (k =1 , 2 , · · · , K). Taking into account the consideration in the previous paragraph, hereinafterin the present thesis, let us treat the well-posedness under the consistency only with C(1)and call it the “weak” well-posedness of the variational principle. Otherwise, let us call thewell-posedness under the consistency with full constraints that of “strong” one.

102

Chapter 4
General theory of relativity
Chapter abstract
In this chapter, the general theory of relativity, which is the most successful classical theory ofgravity in modern physics, is established. First of all, the fundamental concepts of the theory
such as spacetime, and weak/Einstein/strong equivalence principles are introduced. Based onthese concepts, the geodesic equations and the Einstein field equations are derived from both
viewpoints of physical considerations (bottom-up approach) and the variational principle(top-down approach). As mentioned in Section 3.9, we assume that the well-posednessof the variational principle demands consistency with the primary constraint(s): C(1) . In
particular, before deriving the Einstein field equations by using the variational principle, theADM-foliation/decomposition of the spacetime is introduced. This technique is mandatory
not only to apply the variational principle in a well-posed manner being consistent with C(1)but also to perform the Dirac-Bergmann analysis. Finally, performing the Dirac-Bergmannanalysis, the propagating and gauge degrees of freedom of the theory are identified.

4.1. Spacetime and fundamental principles
In this section, the concept of spacetime and fundamental principles for establishing thegeneral theory of relativity, that is, the principle of equivalence and of general relativity/covariance, are introduced. First of all, let us formulate Minkowski spacetime.
In the absence of gravity, the special theory of relativity describes the motions of nature,and the theory is established upon the following two fundamental principles: the principle ofconstant speed of light, stated as the speed of light does not depend on inertial frames, and theprinciple of special relativity/Lorentz covariance, stated as the physical laws do not depend103

on inertial frames. Then the first principle formulates Minkowski spacetime. Let S be an
inertial frame equipped with coordinates (ct, xi ) (i ∈ {1 , 2 , · · · , n}), where c is the speedof light, and an arbitrary particle and a photon are emitted from the origin O, coordinatedby (0, 0), of the system S. After passing time ∆t, let the particle and the photon exist ata point Q coordinated by (c∆t, xiQ ) and a point R coordinated by (c∆t, xiR ), respectively.
Then an observer can consider the subtraction of OR, which is the square of the length of thephoton path, from the square of the Euclidean distance of QR in the n-dimensional space asfollows:(∆s)2 = −c2 (∆t)2 +nX(∆xi )2 ,

(4.1)

i=1

where ∆xi = xiR − xiQ . If the time ∆t is infinitesimal, i .e ., ∆t → dt then the above
equation inserts a distance of the (n + 1)-dimensional space which is coordinated by (ct, xi ).
Introducing the metric tensor ηµν = diag (−1 , +1 , · · · , +1), the distance can be expressed
as follows:
ds2 = ηµν dxµ dxν .

(4.2)

This space is nothing but Minkowski spacetime. In the special theory of relativity, all the lawsof physics have to be described in Minkowski spacetime. The fundamental property of the
distance ds2 is that it does not depend on inertial frames. That is, for another inertial frame
S ′ , the distance is invariant: (ds′ )2 = ds2 , and it can be shown that this crucial property isa conclusion of the first principle. This property introduces the Lorentz symmetry, in termsof group theories, SO(1, n), and all physical laws have to satisfy this symmetry. Explicitly,
the Lorentz symmetry is expressed as follows:ηµν Λµ ρ Λν σ = ηρσ ,(4.3)

where Λµ ρ is originated from the coordinate transformation ϕ : S → S ′ ; xµ 7→ x′µ =104Λµ ρ xρ . The coordinate transformation ϕ which satisfies the Lorentz symmetry is the socalled Lorentz transformation. Furthermore, more generically, the symmetry of Minkowski
spacetime can be extended into the Poincaré symmetry, i .e ., P n+1 = T n+1 ⋊ SO(1, n)
with the well-defined operation (a1 , Λ1 ) · (a2 , Λ2 ) = (a1 + Λ1 · a2 , Λ1 · Λ2 ), which is nothing
but an affine transformation: xµ 7→ x′µ = Λµ ν xν + aµ for a constant (n + 1)-vector aµ in
Minkowski spacetime. Therefore, the second principle is a requirement under the Lorentz
transformation, and it is achieved by expressing all quantities and laws of physics in termsof tensor fields in Minkowski spacetime.
The special theory of relativity can describe physics only in inertial frames, but the realworld requires the description of physics in accelerating frames. The principle of equivalence
makes it possible to realize the latter requirement of nature and has three different statements: Weak Equivalence Principle (WEP), Einstein’s Equivalence Principle (EEP), andStrong Equivalence Principle (SEP). The essential point of the principle is the reconsideration of the meaning of the accelerating frames when subjecting only to gravity.
Let us consider a freely falling test particle. The second law of Newtonian mechanics
describes this motion in an equation as follows:

mi a = −mg ∇Φ ,(4.4)

where a, mi , and mg are the acceleration vector, the inertial mass, and the gravitational
mass of the test particle, respectively. Φ is a Newtonian gravitational field. The WEP statesthat the two types of mass are equivalent: mi = mg , and it leads toa = −∇Φ .

(4.5)

That is, the acceleration does not depend on the sort of the test particle; only depends on thegravitational field. Experimentally, the Eötvös experiment and its modern successors verify
105this statement in high precision. Further considerations of this result give insight into a
new understanding of the acceleration; the gravitational effect cannot be distinguished from
uniform acceleration in a small enough region. That is, the WEP states more generically thatin small enough regions of spacetime, the motion of freely falling particles in a gravitational
field and of particles in a uniform accelerating frame are not distinguishable. Here, spacetimeis the generalized concept of Minkowski spacetime: a space, or more generically, a pseudoRiemannian manifold, equipped with a metric tensor gµν (x), not restricting to ηµν .
The statement of the WEP restricts its application only to freely falling particles, but
A. Einstein extended the statement to any other experiments that obey special relativity,
which is nothing but the EEP, stated as in small enough regions of spacetime, the laws ofphysics restore those of special relativity. This means that any local experiments cannot
detect a gravitational field. Remark here that the EEP is the statement only for the nongravitational laws of physics due to its own property. Including the gravitational laws ofphysics, the statement is distinguished as the SEP.
In the special theory of relativity, on one hand, the laws of physics are invariant under
the Poincaré transformation. On the other hand, in the presence of gravity, it demands theinvariance of that under the general coordinate transformations, or more generically, diffeomorphisms, xµ 7→ x′µ = (∂x′µ /∂xν )xν , of spacetime. Remark that in this case ∂x′µ /∂xν
is not a constant matrix like Λµ ν in the Poincaré transformation but a function matrix, or
equivalently, a Jacobian matrix; a system of coordinate functions spans only a local regionof the entire spacetime. This statement is nothing but the principle of general relativity/covariance and is easily realized by expressing all quantities and laws of physics in termsof tensor fields in spacetime; it is a straightforward generalization of the principle of special
relativity/covariance. Explicitly, let ϕ : M → M and T be a diffeomorphism of spacetime M and a tensor field of some physical quantity on M. Then the principle claims the106satisfaction of the following condition:
ϕ∗ T = T ,or equivalently Lξ T = 0 ,(4.6)

where ξ is a vector field which generates the diffeomorphism ϕ and Lξ is the Lie derivative
generated by ξ. Expressing it in a component form, the first condition becomes as follows:∗µ1 ···(ϕ T )


∂xµ1
∂x′σ1
···
···
T ρ1 ··· σ1 ··· (x′ ) ,ν1 ··· (x) =
∂x′ρ1
∂xν1
x′ =ϕ(x)(4.7)

as desired.
So far the main concepts and principles have been introduced: (Minkowski) spacetime,the EEP, and the principle of general relativity/covariance. Based on these ingredients, thegeneral theory of relativity is established. In particular, the EEP leads to an interpretationthat a freely falling frame is equivalent to the state of a frame in un-accelerated, the so-called
local inertial frame, if subjecting only to gravity, and it gives an implication of describing
the trajectory of a particle in a gravitational field as geodesics.

4.2. Geodesic equations
In this section, geodesic equations are introduced in two ways: a bottom-up approach focusedon the motion of particles for grasping the meaning of physics and a top-down approach basedon the variational principle for formulating the geodesics in a mathematical manner. Finally,
the geometrical interpretation of gravity is presented. Let us start first with the bottom-up
approach for introducing geodesics. Hereinafter, the speed of light c is set as unity: c = 1.
In the absence of forces including gravity, although it will be revealed that gravity is nota force but the bending of spacetime, a test particle (of which mass is unity in the system107

of units) satisfies the following equations of motion:aµ =d 2 xµ
= 0,
dτ 2

(4.8)

where xµ = xµ (τ ) and τ are the position (n+1)-vector and the proper time of the test particle,
respectively. Performing a general coordinate transformation xµ 7→ x′µ = (∂x′µ /∂xν )xν ofwhich Jacobian matrix being non-singular, the above equation becomes as follows:aρ = 0 7→ a′ρ =d2 x′ρ
dx′µ dx′ν
′ρ
= 0,
+
Γ
µν
dτ 2
dτ dτ

Γ′ρ µν =∂x′ρ ∂ 2 xσ.
∂xσ ∂x′µ ∂x′ν

(4.9)

This equation is nothing but the geodesic equation in the pseudo-Riemannian geometry, if◦

replacing Γ′ρ µν by the so-called Christoffel symbols/Levi-Civita connection Γρ µν , given as
follows:◦1Γρ µν = g ρσ (∂µ gνσ + ∂ν gµσ − ∂σ gµν ) ,2(4.10)◦and the connection Γρ µν transforms by the general coordinate transformation as follows:◦Γ′ρ µν =∂x′ρ ∂ 2 xσ∂xα ∂xβ ∂x′ρ ◦ γ
Γ+.
αβ∂x′µ ∂x′ν ∂xγ∂xσ ∂x′µ ∂x′ν

(4.11)

Then the equation describes the trajectory of the test particle as a geodesic. If tµ = dxµ /dτ
is a tangent vector of the geodesic then the equation can be expressed as follows:◦◦aµ = tν ∇ν tµ = 0 .

(4.12)

An important insight here is that the first term in Eq (4.11) is absent in Eq (4.9); vanishing◦

the connection Γγ αβ in some coordinate system is an implication of the test particle existingin a local inertial frame. Therefore, taking into account the statement of the EEP, theexistence of gravity has something to do with the Levi-Civita connection. In fact, it will be108

revealed in the last part of this section that the connection corresponds to the gravitational
force itself.
Next, let us derive the geodesic equation from the variational principle and consider its
properties. The action functional is introduced as follows:Z τ2 

µI[x (τ ) , e(τ )] =
τ1


ν
µ
1
e
ρ dx dxgµν (x )
+σ
dτ ,
2e
dτ dτ
2

(4.13)

where σ = +1, 0, −1 for space-like, null, and time-like geodesic, respectively. The auxiliary
variable e = e(τ ) makes it possible to treat the null geodesic, σ = 0, in a unified manner.
This system has a primary constraint with respect to the auxiliary variable. Therefore, thesymplectic potential becomes ω = πe δe + πµ δxµ = πµ δxµ in the Lagrangian formulation,
since πe = 0 is automatically satisfied as an identity, where πe and πµ are the canonicalmomentum variables with respect to e and µ, respectively. (In the Hamiltonian formulation,C(1)it becomes ∗ ω = πe δe + πµ δxµ ≈ πµ δxµ , since πe = 0 have to be imposed: πe := 0). It
indicates that the well-posedness of the variational principle on C(1) demands the boundary
conditions δxµ (τ2 ) = δxµ (τ1 ) = 0 [Tomonari (2023b); Izumi et al. (2023)]. Varying thisaction functional with respect to e, the following equation is derived:
σe2 = gµνdxµ dxν,dτ dτ

(4.14)

on the other hand, with respect to xµ , the following equation is derived:d2 xρ ◦ ρ dxµ dxν
ė dxµ
+
Γ
=
,
µν
dτ 2
dτ dτ
e dτ

(4.15)

under the imposition of the boundary conditions. These two equations are nothing but thegeodesic equations. The action functional (4.13) has a symmetry; it is invariant under the109

following one-dimensional diffeomorphism:
ϕ : (τ1 , τ2 ) → (τ1′ , τ2′ ); τ 7→ τ ′ = ϕ(τ ) .

(4.16)

Then xµ and e are transformed as follows:xµ (τ ) → x′µ = ϕ∗ (xµ (τ )) = xµ (τ ′ = ϕ(τ )) ,e(τ ) → e′ (τ ) = ϕ∗ (e(τ )) = e(τ ′ = ϕ(τ )) ,(4.17)

where ϕ∗ is the pull back operator of ϕ. Therefore, using this diffeomorphism, the parameterτ can be always taken under the satisfaction of the condition(4.18)ė(τ ) = 0 .

In such case, Eq (4.15) results in the geodesic equation (4.12). A parameter under thesatisfaction of the condition (4.18) is called the affine-parameter, and it can be generically
expressed as follows:(4.19)

τ → τ ′ = aτ + b ,where a and b are constants. In this formulation, the proper time is redefined as a parameterτ such that e(τ ) = 1, or equivalently, gµν (dxµ /dτ )(dxν /dτ ) = −1. Notice that the proper
time has a symmetry for τ 7→ τ + constant.
Finally, let us consider the geodesic equation in spacetime which slightly deforms from
Minkowski spacetime. That is, the metric tensor gµν is expressed as follows:gµν = ηµν + hµν ,|hµν | ≪ 1 ,∂0 hµν = 0 .

(4.20)

Then the spatial part of the geodesic equation (4.15) under the imposition of the condi-

110tion (4.18) is approximated as follows:µν
◦d 2 xii dx dx
.
≃
−
Γ
µν
dτ 2
dτ dτ

(4.21)

If dxi /dτ ≪ 1 is satisfied, the above equation is further approximated as follows:◦d 2 xi≃ −Γi00 ,2
dτ(4.22)

where the explicit formula of the Levi-Civita connection (4.10) is used. Comparing thisequation to the equation for the freely falling test particle, Eq (4.5), the following relation
is derived:◦Γi00 ≃ ∂i Φ ,or h00 = −2Φ .

(4.23)

It indicates that the Levi-Civita connection and the metric tensor are nothing but the expression of gravity and of the gravitational potential, respectively. That is, gravity is nothing
but a property of spacetime how it different is from Minkowski spacetime. Therefore, thestatement of the EEP is translated in terms of the Levi-Civita connection into as follows:in small enough regions of spacetime, it exists a coordinate system such that the Levi-Civita
connection vanishes, and this statement is guaranteed by a theorem: It exists a coordinate
system such that Γ̃ρµν = 0 if and only if the lower two indices in the connection is commutative: Γ̃ρµν = Γ̃ρνµ . Here, remark that the connection is not restricted to the Levi-Civita◦

connection Γρνµ . Therefore, the EEP is now expressed only in an equation: Γ̃ρµν = Γ̃ρνµ . The◦

Levi-Civita connection Γρνµ apparently satisfies this condition by virtue of its expression in
Eq (4.10).
Summarizing this section, using the pseudo-Riemannian geometry, the trajectory of aparticle is described by the geodesic equation (4.15) under the imposition of the condition (4.18), or equivalently, Eq (4.12), and the connection of the theory is the Levi-Civita
111connection (4.10). In this geometry, the EEP is automatically satisfied, and gravity and thegravitational potential are expressed by the Levi-Civita connection and the metric tensor,
respectively. Since the geodesic equation is the generalization of the Newton equation for a
freely falling particle, that is, Eq (4.8), therefore, a particle subjecting only to gravity does
continue the uniform motion as a straight line in spacetime, i .e ., a geodesic; gravity is now
engraved into a property of spacetime. Therefore, once the gravitational potential, i .e ., themetric tensor, is determined, the geodesic equations describe how a particle evolves in time.
The Einstein field equations provide a set of field equations to govern the evolution of themetric tensor.

4.3. Einstein field equations
In this section, the Einstein field equations are derived based on two bottom-up approaches
by using the Poisson equation of gravity and the equations of geodesic deviation respectively.
The modern approach based on the variational principle will be presented in Section 4.5.
The speed of light c is set as unity: c = 1. Let us start with the first approach.
The Poisson equation of gravity is given as follows:(4.24)∇2 Φ = 4πGρ ,where G is the Newtonian constant and ρ is the matter density of sources. Using Eq (4.20),
Eq (4.22), and the above equation, the following equation is derived:◦(4.25)∇2 g00 = 8πGT 00 ,◦◦where T 00 = −ρ is the (0, 0)-component of the energy-momentum tensor T µν . Generalizing112this equation in a covariant manner, the field equation should be the following form:◦◦(4.26)

Gµν = 8πGT µν .
◦Therefore, revealing Gµν , the field equations for gravity are obtained, under the satisfaction◦of the following conditions; (i) Gµν is composed of the metric tensor and its derivative up
to second-order; (ii) In the Newtonian limit, Eq (4.26) restores Eq (4.25); (iii) The lower◦◦

two indices of Gµν are symmetric; (iv) Gµν satisfies the covariant conservative property.
◦

◦

◦

◦

Conditions (i) and (iii) allows that Gµν is written as follows: Gµν = ARµν + Bgµν R, where◦◦Rµν and R are the Ricci tensor and the Ricci scalar, A and B are arbitrary constants. Theremay be some leaps to make up this form, but the validity will be able to be confirmed in the◦second approach. Condition (iv) leads to A = −2B. Therefore, Gµν should be the following
formula:◦◦◦1Gµν = Rµν − gµν R .
2(4.27)

This is nothing but the so-called Einstein tensor, which is derived from the Bianchi identity:◦◦∇[ρ| Rµν]σκ = 0, and satisfies Condition (iv). Therefore, the Einstein field equations arederived as follows:◦◦1Rµν − gµν R = 8πGT µν ,
2
◦



◦
◦
1
or Rµν = 8πG T µν − gµν T .
2◦(4.28)

From the second equation above, Condition (ii) leads the Poisson equation of gravity:
Eq (4.25).
The second approach is based on a consideration of the tidal effect. The EEP indicates
that, in small enough regions of spacetime, gravity can be removed by performing appropriate
general coordinate transformation but that, in global regions of spacetime, it is not the case;the tidal effect appears. Let us consider the congruence of geodesics. The two particles in113

geodesics have the relative acceleration given as follows:◦◦◦(4.29)

aµ = tµ ∇µ (tν ∇ν nρ ) ,where nρ are the normal vector of geodesics. Using the geodesic equation (4.12) and the◦◦relation Ln tµ = 0, where Ln is the Lie derivative with respect to the normal vector nµ , theequation of geodesic congruence are derived as follows:◦(4.30)

aµ = Rµ νρσ tν tρ nσ .

This equation describes the tidal effect between two test particles. In the Newtonian case,the following equation holds:(4.31)

ai = −nj ∂j ∂ i Φ .

Now, let us derive the Einstein field equations. Generalizing Eq (4.31) into a covariant form,
it becomes aµ = −nν ∂ν ∂ µ Φ. Therefore, comparing it to Eq (4.30) and using the Poisson
equation of gravity, Eq (4.24), and then tracing out the un-contracted indices, the following
equation holds:◦(4.32)

Rµν tµ tν = κ2 ρ ,◦where κ2 is a constant coefficient. The right-hand side can be generalized as T µν tµ tν in a
covariant form, therefore, the following relation is derived:◦◦Rµν = κ2 T µν

(4.33)

This equation does not satisfy the covariant conservative property but the Bianchi identity:
◦

◦

◦

◦

◦

∇[ρ| Rµν]σκ = 0 leads to ∇µ Gµν = 0, where Gµν is defined in Eq (4.27). Therefore, the above114

equation should be modified as follows:◦◦(4.34)

Gµν = κ2 T µν .

The coefficient κ2 can be determined by considering the Newtonian limit: κ2 = 8πG. This
is nothing but the Einstein field equations (4.28).
◦◦◦

In the first approach, the form of Gµν = ARµν + Bgµν R is assumed and it satisfies
all Conditions (i) - (iv), whereas a term Cgµν satisfies only Conditions (i), (ii), and (iv).
Therefore, if the coefficient C is small enough to be consistent with the Newtonian limit,
then it would be possible to modify the Einstein equation as follows:◦◦Gµν + Λgµν = 8πGT µν .

(4.35)

The constant coefficient, Λ, is the so-called Einstein’s cosmological constant. He was motivated to introduce this term from a philosophy that the universe should be static, althoughmodern observations indicate that the universe is expanding, and it is rather accelerating.
The Einstein field equations are composed of (n + 1)(n + 2)/2 independent equationsin (n + 1)-dimensional spacetime and non-linear partial differential equations. Therefore,
it is generically difficult to solve the equations and to find the true propagating degrees
of freedom. Of course, in order to reveal the latter point, the Dirac-Bergmann analysis
is a powerful tool, but it demands the ADM-foliation of spacetime and the Lagrangian
formulation of the theory.

4.4. ADM-foliation of spacetime and Einstein field equations
In this section, ADM-foliation of spacetime is introduced [Arnowitt et al. (1959, 1960);
Deser et al. (1960); Arnowitt et al. (2008); Baez and Muniain (1995); Nakahara (2003); Carroll(2019); Padmanabhan (2014b)], and it is one of the most important fundamentals of the115

present thesis. The concepts introduced in this section are as follows: ADM-foliation of
spacetime, the metric of the geometry, first fundamental form (projection map), covariant
derivative in a foliation of spacetime, second fundamental form (extrinsic curvature), GaussCodazzi equations, and the ADM-foliation of Einstein field equation.
Let M be a (n + 1)-dimensional manifold of spacetime that has a Cauchy surface. Then
ADM-foliation of spacetime is a diffeomorphism ϕ : M → R × S such that it decomposes
the spacetime M as a disjoint union of hypersurfaces as follows:

M=

[

Σt ,

Σt = {p ∈ M|ϕ∗ τ (p) = t} ,(4.36)

t∈I

where S is a n-dimensional hypersurface, t and τ are a time coordinate of M and of R ×
S, respectively, I is a time interval of M, and ϕ∗ is the pullback operator of ϕ. Eachhypersurface Σt , which is called a leaf, is diffeomorphic to S. The ADM-foliation of spacetime
is denoted as (M , S , ϕ).
Let g = gµν dxµ ⊗ dxν and nµ be a metric tensor of M and a normal vector of a leaf Σt ,respectively. Then the line element ds2 = gµν dxµ dxν is decomposed as follows:
ds2 = ϵN 2 dτ 2 + hij (dxi + N i dτ )(dxj + N j dτ ) ,−1 ∗ µ(4.37)−1 ∗ ν

hij = [(ϕ ) ] i [(ϕ ) ] j gµν ,where ϵ is either −1 or +1 according to either time-like or space-like hypersurface Σt , respectively. N and N i are the lapse function and shift vector, respectively, which is introduced
as follows:(∂t )µ = ϵN nµ + [(ϕ−1 )∗ ]µ i N i .

(4.38)

Therefore, the components of the metric tensor on M are expressed by the spatial metric116

hij , the lapse function N , and the shift vector N i as follows:
g00 = ϵN 2 + hij N i N j ,g0i = gi0 = hij N j ,(4.39)

gij = hij .
The components of the inverse metric tensor on M are derived as follows:g 00 = ϵN −2 ,g 0i = g i0 = −ϵN −2 N i ,(4.40)

g ij = hij + ϵN −2 N i N j .
The normal vector nµ can be expressed in terms of the lapse function and the shift vector
as follows:nµ = ϵN −1 (−1 , N i ) = −N g 0µ ,(4.41)

nµ = (−N , 0) = −N δµ0 .
Remark that a normal vector satisfies the Frobenius theorem: n[µ ∇ν nρ] = 0, where ∇µis a covariant derivative for any geometries/connections, not restricting to the Levi-Civita
connection. Remark also that the expression of nµ above can be extended into a more genericform as follows:nµ = (−N , N Ni ) = −N (δµ0 − Ni δµi )(4.42)

under the condition N i Ni = 0.
In order to describe the geometry of the foliation of spacetime, let us introduce the first117and second fundamental forms. The first fundamental form is defined as follows:Pµν = gµν − ϵnµ nν ,(4.43)or in the components form
P00 = hij N i N j ,P0i = Pi0 = hij N j ,(4.44)

Pij = hij ,or equivalently,
P 00 = 0 ,P 0i = P i0 = 0 ,(4.45)

P ij = hij .
This quantity is also called the projection map by virtue of the following properties:nµ Pµν = 0 ,P µν P ν ρ = P µρ .

(4.46)

Hereinafter, we call it the projection map. Using this quantity, the covariant derivative on aleaf Σt is introduced as follows:Dρ T µ1 µ2 ···µk ν1 ν2 ···νl = P σ ρ P µ1 µ̃1 P µ2 µ̃2 · · · P µ1 µ̃k P ν̃1 ν1 P ν̃2 ν2 · · · P ν̃l νl ∇σ T µ̃1 µ̃2 ···µ̃k ν̃1 ν̃2 ···ν̃l , (4.47)
where T µ̃1 µ̃2 ···µ̃k ν̃1 ν̃2 ···ν̃l is a (k , l)-tensor in the spacetime M and T µ1 µ2 ···µk ν1 ν2 ···νl is a (k , l)tensor in a leaf Σt . Remark that this formula holds for any geometries/connections. In
particular, in a (pseudo-)Riemannian manifold, the projection map satisfies the following
118

properties:◦

Dk Pij = 0 ,1(n) kΓij = P kl (∂i Pjl + ∂j Pil − ∂l Pij ) .
◦(4.48)

2This is nothing but the Levi-Civita connection on a leaf Σt .
The second fundamental form is defined as follows:1Kµν = Ln Pµν ,2(4.49)

where Ln is the Lie derivative with respect to the normal vector nµ . Remark that this quantity is viable in any geometries/connections. This quantity satisfies the following properties:
Kµν = Kνµ ,(4.50)µ

n Kµν = 0 ,and has four equivalent expressions as follows:◦◦1Kµν = P α µ P β ν Ln gαβ ,
2
◦

◦

◦

Kµν = ∇µ nν − ϵnµ aν ,◦

ρ(4.51)◦

Kµν = P µ ∇ρ nν ,◦◦◦Kµν = ∇µ nν + nµ Dν lnN ,where the Frobenius theorem is used. Remark that the properties in Eq (4.50) hold forany geometries/connections. That is, the quantity is symmetric with respect to the lower
two indices and spatial in any geometries/connections. Using these properties, the following
property is derived:
◦

◦

◦

◦

Ln (P α µ P β ν Kαβ ) = P α µ P β ν Ln Kαβ .
119(4.52)

This property shows that the projection map behaves just like the metric tensor in (pseudo)Riemannian geometry regarding the Lie derivative of the second fundamental form withrespect to the normal vector. Notice that these equations excepting the fourth equation in
Eq (4.51) do not depend on a concrete form of the normal vector nµ . If nµ is expressed interms of the lapse function and the shift vector, i .e ., nµ = ϵN −1 (−1, N i ), then the following
important formula is derived:
(n)

◦
◦
◦
ϵ
Kij = N −1 (Di Nj + Dj Ni − ḣij ) .
2(4.53)

This formula plays a crucial role in performing the Dirac-Bergmann analysis of theories ofgravity. The geometrical meaning of the second fundamental form can be interpreted byconsidering a way of embedding a leaf Σt into the spacetime M. Let C be an integral
curve with respect to tµ . Then we can verify the following relation: tµ Dµ tν = P ν ρ tσ ∇σ tρ −ϵnν tρ tσ Kσρ . If the curve C is a geodesic on the leaf then this relation turns into as follows:ϵnν tρ tσ Kσρ = P ν ρ tσ ∇σ tρ . This relation implies that if the curve also is a geodesic of thespacetime M then the second fundamental form vanishes, and vice versa. Therefore, thesecond fundamental form can be interpreted as a quantity that measures the difference in away of freely falling a test particle. That is, let C ′ be a geodesic in the spacetime M andboth C and C ′ path through a common point P . Then, to restrict a test particle on C,
the parallel transformation along with the curve C ′ have to be modified by the amount ofϵnν tρ tσ Kσρ from that of P ν ρ tσ ∇σ tρ . Therefore, the second fundamental form is interpreted
as a sort of curvature. Hereinafter, we call it the extrinsic curvature.
Next, let us introduce the Gauss-Codazzi equations. The Gauss equation is given as
follows:
(n)

◦

◦

◦

◦

(4.54)

Rρ σµν = P ρ γ P δ σ P α µ P β ν Rγ δαβ + 2ϵK ρ [µ Kν]σ .
◦◦◦This relation can be shown by using the relation [Dα , Dβ ]tδ = (n) Rγ δαβ tδ , where tδ is a120

tangent vector on a leaf Σt . The first and second Codazzi equations are given as follows:◦◦◦

P α µ P β ν P σ γ nδ Rγ δαβ = 2D[µ Kν] σ ,αγ

P µn P

β

◦

◦

◦

◦

(4.55)◦◦◦ ◦ρν n Rαγβδ = Kµ Kρν − Ln Kµν + Dµ aν − ϵaµ aν ,δ◦respectively, where aµ = nν ∇ν nµ . These relations can be shown by using the relation
◦

◦

◦

◦

◦

[∇α , ∇β ]nγ = Rγ δαβ nδ . Concerning the terms Dµ aν and/or aµ , it can be replaced by using◦◦

the following relations:◦◦aµ = −nρ nµ Dν lnN ,◦◦Dµ aν = −ϵ(N−1◦(4.56)◦◦◦

Dµ Dν N − aµ aν ) .

From the Gauss equation (4.54), the Ricci tensor and scalar on a leaf are expressed as follows:
(n)
(n)

◦

◦

◦

◦

◦

◦

Rµν = P α µ P β ν P γ δ Rδ αγβ + ϵ(K Kµν − Kµρ K ρ ν ) ,
◦

◦

◦

◦

µ ν

µν

◦

◦

(4.57)

2

R = R − ϵ(2Rµν n n + K Kµν − K ) .

Contracting the first Codazzi equation, it relates the extrinsic curvature to the Ricci tensor
as follows:◦◦◦

2D[µ K µ ν] = nλ P ρ ν Rλρ .

(4.58)

The above equations in this paragraph hold only for (pseudo-)Riemannian geometry. A
generalization of these equations into metric-affine geometries is given in [Ariwahjoedi et al.
(2021a,b)].
Finally, let us decompose the Einstein field equations. The Einstein tensor and the121extrinsic curvature have the following relations:
◦

◦

◦

◦

◦

2Gµν nµ nν = −ϵ (n) R + K 2 − K µν Kµν ,
◦

◦

◦

µ

◦

µ

ν

(4.59)◦

Dµ K ν − Dν K = n P ρ Gµν .
◦◦These relations are shown by using P ρσ P µν Rρµσν = −2ϵnµ nν Gµν . Using these relations, theEinstein field equations (4.28) are decomposed into the following three equations:
(n)

◦

◦

◦

R + K 2 − Kµν K µν = 2κ2 Tµν nµ nν ,

◦

◦

◦

◦

ν

2

ρ

(4.60)

σDµ K − Dν K µ = κ P µ n Tρσ ,and, using also the second equation in Eq (4.55) and the second equation in Eq (4.56),
◦

◦

(n)

Ln Kαβ = −

◦

◦

γ

◦

◦

◦

Rαβ + 2K α Kγβ − K Kαβ + N−1


◦Dα Dβ N + P α P β T µν −
◦

◦

µ

ν


◦
1
gµν T ,
n−1
(4.61)

where ϵ = −1; leaves are taken as space-like hypersurfaces. The third equations (4.61)
describe the time evolution of the spacetime. In fact, using the following formula:◦◦(4.62)

Ln Qµν = N −1 (∂0 Qµν − Ln Qµν ) ,where Qµν is spatial tensor, i .e ., nµ Qµν = 0, then Eq (4.61) is further split into as follows:◦◦∂0 hij − LN hij = −2N Kij ,◦◦∂0 Kij − LN Kij = N

(3)

◦

◦

◦

◦

k

◦

◦

◦

2

µ

ν

Rij + N (K Kij − 2K i Kkj ) − Di Dj N − 2κ P i P j



◦
1
gµν T
T µν −
n−1◦(4.63)
where N µ = (0 , N) = (0 , N i ). These equations imply the possibility of reformulating theEinstein field equations in a Hamiltonian formulation, and the first and second equations
122


N,

would correspond to “ q̇ = ∂H/∂p ” and “ ṗ = −∂H/∂q ”, respectively. In this section, we
have discussed based on the equations of motion. In order to verify this statement, however,it needs to establish the Lagrangian formulation of the general theory of relativity.

4.5. Reformulation based on the well-posed variational principle
In this section, Einstein-Hilbert action is introduced in the Lagrangian formulation, and,based on the well-posed variational principle on the primary constraint phase space C(1) , theEinstein field equations are derived [Einstein (1916)]. This is the first time to consider thewell-posedness of the variational principle in field theories in the current thesis, therefore,
all computations are performed in detail. The Einstein-Hilbert action in the Hamiltonian
formulation is also established, and the Einstein field equations as Hamilton equation are
derived [Padmanabhan (2014b,a); Parattu et al. (2016a,b); Danieli (2020); Jha (2023)].
The Einstein-Hilbert action functional in vacuum is given as follows:1
IGR [gµν ] = 2
2κ

Z

◦

dn+1 x LGR ,LGR = R√

(4.64)−g ,M

where M is a (n + 1)-dimensional spacetime, g is the determinant of the metric tensor gµν
◦of M, R is the Ricci scalar of M, and κ2 = 8πG. In order to apply the variational principle
in a well-posed manner on the primary constraint phase subspace C(1) , let us derive thecanonical momentum variables of the system. Using the second relation of the contracted◦◦◦

Gauss equation (4.57) in the case of ϵ = −1 and the relation: Rµν nµ nν = ∇ρ (aρ − nρ K) −
◦

◦

◦

◦

◦

◦

◦

Kµν K µν + K 2 , which can be derived from [∇α , ∇β ]nγ = Rγ δαβ nδ , the Ricci scalar in theaction functional can be decomposed as follows:
◦

◦

◦

◦

◦

◦

◦

◦

R = (n) R + Kµν K µν − K 2 − 2∇ρ (nρ K − aρ ) .

123(4.65)

Therefore, the Einstein-Hilbert action functional (4.64) becomes as follows:1
IGR [N , N , h ] = 2
2κ
i

ij

Z t2

Z

1
dt
d xLADM + 2
2κ
t1
Σt
n

Z

 √ ◦
dn x +2 hK ,(4.66)∂M

where LADM , the so-called Arnowit-Deser-Misner Lagrangian density, is defined as follows:

√  ◦
◦
◦
◦
LADM = N h (n) R + Kµν K µν − K 2 .

(4.67)

The canonical momentum density variables of the ADM Lagrangian are derived as follows:δLADM (x)δLADM (x)= 0 , πi (x) == 0,
δ Ṅ (y)δ Ṅ i (y)◦◦δLADM (x) pπij (x) == h(x)(K(x)hij (x) − Kij (x))δ (n) (x − y) .
δ ḣij (y)π0 (x) =(4.68)

These results indicates that there are the total number of (n + 1) primary constraints asfollows:(1)∗ ϕ0 = π0 := 0 ,(1)∗ ϕi = πi := 0

(4.69)

in the Hamiltonian formulation. In the Lagrangian formulation, these equations are satisfied
as identities. The symplectic potential in the Lagrangian formulation is given as follows:ω = π0 δN + πi δN i + πij δhij = πij δhij . (In the Hamiltonian formulation, ∗ ω = π0 δN +C(1)πi δN i + πij δhij ≈ πij δhij ). Therefore, the well-posedness of the variational principle on theconstraint phase subspace C(1) demands the imposition of the boundary condition as follows:δhij = 0 on the boundary ∂M [Tomonari (2023b); Izumi et al. (2023)].
Now, let us apply the variational principle to the Einstein-Hilbert action (4.64) under
the imposition of the boundary condition δhij = 0 on ∂M. Varying the Lagrangian density

124

LGR in Eq (4.64), the following equation is derived:◦

δLGR = Gµν δg µν +◦◦√−g ∇µ ∇ν −δg µν + g µν gαβ δg αβ .

(4.70)

The first term is the bulk term and it governs the dynamics of the system when coupling
to matters as discussed later. The second term generates the surface term of the theory.
Integrating out the second term on the spacetime M, it becomes as follows:√dxn ϵ hδV ,Z
δV =(4.71)∂M◦

δV = nµ ∇ν −δgµνµν+ g gαβ δg

αβ



.

Using the relation gµν δg µν = Pµν δP µν + 2ϵnµ δnµ and δnν = nν δg µν − ϵnµ nα nβ δg αβ /2 (, orequivalently δnρ = −ϵnρ nµ nν δg µν ), the density V is rewritten as follows:◦◦◦δV = (∇µ nν )δg µν − Kgµν δg µν + ∇µ δV µ ,µµδV = −nα P β δg

αβµ+ n Pαβ δP◦

αβ(4.72).
◦◦A direct calculation leads to the following property: ∇µ W µ = Dµ W µ + ϵnν ∇ν (nµ W µ ) −ϵW µ aµ . Therefore, using this property, the density V becomes as follows:
◦

√ ◦
◦
◦
◦
1
δV = (Kµν − KPµν )δP µν + √ δ(−2 hK) + Dµ (nα P µ β δg αβ ) .
h(4.73)The third term in the above equation vanishes when integrating it out on ∂M. Therefore,
neglecting this term and using Eq (4.45), Eq (4.73) becomes as follows:
√ ◦
◦
◦
1
δV = (Kij − Khij )δhij + √ δ(−2 hK) ,h125(4.74)

or Eq (4.71) becomes as follows:
Z
δV =

n

ij

√ ◦
dn xδ(−2ϵ hK) ,Z

d x(−ϵ)πij δh +∂M

(4.75)∂M

where the third equation in Eq (4.68) is used. Assembling all results, the first-order variationof the Einstein-Hilbert action (4.64) is calculated as follows:1
δIGR = 22κZ◦Gµν δg
M

µν

1
+ 2
2κ

Z

1
d x(−ϵ)πij δh + 2
2κ
∂M
n

ij

Z

√ ◦
dn xδ(−2ϵ hK) .

(4.76)∂M

The second term vanishes under the imposition of the boundary condition δhij = 0 on ∂M,
but the third term remains. Therefore, let us modify the Einstein-Hilbert action on the
ground of the following equation:1
δ I˜GR = 2κZ◦Gµν δg

µν

M

1
+ 2
2κ

Z

dn x(−ϵ)πij δhij ,(4.77)∂M

where I˜GR is defined as follows:1
I˜GR [N , N i , hij ] = IGR [N , N i , hij ] + 22κZ √ ◦
dn x 2ϵ hK .

(4.78)∂M

The second term is nothing but the so-called Gibbons-York-Hawking counter-term [York(1972, 1986a); Gibbons and Hawking (1977); Hawking and Horowitz (1996); Padmanabhan(2014a); Parattu et al. (2016a,b); Chakraborty et al. (2017); Chakraborty and Padmanabhan(2020)]. At last, the well-posed variational principle on the constraint phase subspace C(1)leads to the vacuum Einstein field equations as follows:◦Gµν = 0 .

(4.79)

Remark that, in this case, the equations are just sufficient conditions to satisfy δ I˜GR := 0126

due to the dependence among δg µν , that is, δg µν are not independent. In fact, the numberof components of gµν is (n + 1)(n + 2)/2 in (n + 1)-dimensional spacetime, but at least(n + 1) components of them depend on other components since there exist at least that
number of primary constraints; the number of (n + 1) components of the metric tensor canbe expressed by the remaining (n + 1)(n + 2)/2 − (n + 1) components. Notice, finally, that,based on Eq (4.78) in the case of ϵ = −1, Eq (4.66) becomes as follows:1
I˜GR [N , N , h ] = 2
2κ
i

ij

Z t2

Z

t1

(4.80)

dn xLADM .

dtΣt

That is, the second term in Eq (4.66) and the second term in Eq (4.78) are canceled out one
another; the canonical momentum variables in Eq (4.68), which were derived by neglecting
the second term in Eq (4.66), are now justified.
Let us introduce matters. Taking into account a matter Lagrangian Lmatter , the net
Lagrangian becomes Lnet = L̃EH + Lmatter . In the case of existing matters, the variational
principle with respect to the matter fields leads to those equations of motion. Assumingthese equations, the first-order variation of the net Lagrangian becomes schematically as
follows:δLmatter =δLmatter
δϕ




δϕ +
g :fixed

δLmatter
δg µν


δg

µν


=

ϕ :fixed

δLmatter
δg µν

δg µν , (4.81)
ϕ :fixed

or, in the form of action functional,
1δImatter = −2Z◦Tµν δg µν dn+1 x .

(4.82)

M

Therefore, the variational principle leads to the Einstein field equations as follows:◦◦Gµν = κ2 Tµν .
127(4.83)

Finally, let us derive the Hamilton equations of motion, although this formulation is
not complete: it needs to take into account that the theory is a degenerate system. The
ADM-Hamiltonian is derived from the Legendre transformation as follows:(4.84)HADM = πij ḣij − LADM .
Using Eq (4.53) in the case of ϵ = −1, the above equation turns into as follows:√ ◦ ◦
◦◦◦◦
HADM = N h Kij K ij − K 2 − (n) R − 2N i Dj πij + 2Di πij N j√  ◦ ◦ ij
◦
◦
◦
T.D.
2
(n)
= N h Kij K − K − R − 2N i Dj πij ,(4.85)

T.D.

where “ = ” denotes the equality under the ignorance of the total divergent term. Varying
HADM , the following formula is obtained:δHADM = Pij δhij + Qij δπij + C0 δN + Ci δN i ,(4.86)

where Pij , Qij , C0 , and Ci are given as follows:√
Pij = − N h◦
1 (n) ◦
Rij − hij R − Dk (πik Nj + πjk Ni − πij Nk )
2






√
◦
◦
◦
◦
N
2N
π2
ππij
k
kl
k
−√
,
+ h Di Dj N − hij Dk D N + √πkl π −
πi πjk −
n−1
n−1
2 h
h
(n)

◦

(4.87)
4N
Q =√
h
ij


π ij −√◦
1
C0 = − h(n) R + √
h

π
hij
n−1


π ij πij −◦Ci = −2Dj πij .

128



1
π2
n−1

(4.88)
,(4.89)(4.90)

Conversely, the inverse Legendre transformation is given as follows:
LADM = πij ḣij − HADM .

(4.91)

Therefore, using Eq (4.86), the first-order variation of LADM in the phase space is calculated
as follows:
T.D.
δLADM = (Pij + π̇ij ) δhij + Qij − ḣij δπij − C0 δN − Ci δN i .

(4.92)The variational principle leads to the following constraint densities:

C0 = 0 ,Ci = 0 ,(4.93)and the following Hamilton equations of motion:ḣij = Qij ,π̇ij = −Pij ,(4.94)

as a set of sufficient conditions. Each equation in Eq (4.94) corresponds to that of Eq (4.63),
respectively. In order to apply the variational principle in a necessary and sufficient mannerfor deriving the Hamilton equations of motion, the methodology that is established in Sections 3.6 and 3.7 is mandatory, although it is hard to find the canonical variables that are
indicated by the Shanmugadhasan-Maskawa-Nakajima theorem. In fact, no one has achieved
it yet at the moment writing this thesis. If this problem is solved, the well-posedness of thevariational principle on the full constraint phase subspace C(K) (K = 2 will be revealed in
the next section) can be established. In this case, it may need to add another counter-term,
which is different from the Gibbons-York-Hawking counter-term, to the action functional.
Anyway, even if one tries to apply the method, the Dirac-Bergmann analysis is necessary to129reveal the constraint structure of the theory and the propagating degrees of freedom.

4.6. Dirac-Bergmann analysis and degrees of freedom
In this section, performing the Dirac-Bergmann analysis, the degrees of freedom of gravity inthe general theory of relativity is investigated [Danieli (2020); Jha (2023); Tomonari and Bahamonde(2023)]. This is the first time to apply the analysis to field theories, therefore, all computations are presented in detail. The analysis is separated into two main steps: (i) The derivationof a set of primary constraint densities and the total Hamiltonian; (ii) The verification of
the consistency conditions for all constraint densities. Let us start with the main step (i):Building the total Hamiltonian.
Based on Eq (4.78), the ADM-Lagrangian (4.67) is derived without any boundary term.
This is the starting point of the analysis. The canonical momentum variables and theprimary constraint densities are already derived: Eq (4.68) and Eq (4.69), respectively. The
fundamental Poisson brackets (PBs) are given as follows:{N (x) , π0 (y)}P.b. = δ (n) (x − y) ,ij{h

{N i (x) , πj (y)}P.b. = δji δ (n) (x − y) ,(i j)(x) , πkl (y)}P.b. = δk δl δ (n) (x − y) .

(4.95)

Under these PB-algebras, the primary constraint densities Eq (4.69) are commutative withrespect to the Poisson brackets (PBs). The Hessian/kinetic matrix given in Eq (2.1) (but infield theories: See Eq (7.6) in Section 7.1 for detail) is computed as follows:

Kab

√



h=




0

01,n

0n,1

0n,n

0n(n+1)/2,1 0n(n+1)/2,n01,n(n+1)/2  (n)0n,n(n+1)/2 
· δ (x−y) ,K(i,j),(k,l)

√ 

h
(k l)
K(i,j),(k,l) =hij hkl − δi δj ,2N
(4.96)

where (i, j) , (k, l) ∈ {(1, 1) , (1, 2) , · · · , (1, n) ; (2, 2) , (2, 3) , · · · , (2, n) ; · · · ; (n, n)}. The rank130

of this matrix is six. The kernel Hamiltonian is calculated as follows:(4.97)

H 0 = N C 0 + N i Ci ,where C and Ci are given in Eqs (4.89) and (4.90). Therefore, the total Hamiltonian isderived as follows:(4.98)HT = H0 + λµ ∗ ϕ(1)µ ,where λµ are the Lagrange multipliers.

The next main step (ii) is the verification of the consistency conditions for all constraint
(1)
(1) C

densities. Those of the primary constraint densities ∗ ϕµ

≈ 0 are calculated as follows:(4.99)(2)∗ ϕµ = Cµ := 0 .

(1)

(2)

Notice that Eq (4.99) is nothing but Eq (4.93). The PBs between ∗ ϕµ and ∗ ϕµ are commutative. The PBs among the secondary constraint densities (4.99) are calculated as follows:
(2)

(2)

(2)

(2)

{∗ ϕi (t , x) , ∗ ϕj (t , x)}P.b. =



(y)
(2)
(x)
(2)
− ∗ ϕi (t , y)∂j
∗ ϕj (t , x)∂i
(2)



C(2)

δ (n) (x − y) ≈ 0 ,

C(2)

(x)

{∗ ϕi (t , x) , ∗ ϕ0 (t , y)}P.b. = ∗ ϕ0 (t , x)∂i δ (n) (x − y) ≈ 0 ,


C(2)
(x)
(2)
(2)
(2)
(2)
ij
ij
{∗ ϕ0 (t , x) , ∗ ϕ0 (t , y)}P.b. = h (t , x)∗ ϕj (t , x) + h (t , y)∗ ϕj (t , y) ∂i δ (n) (x − y) ≈ 0 .
(4.100)
Therefore, all the secondary constraint densities are classified into first-class constraint densities. Based on these algebras, straightforward computations show that the consistency
conditions for the secondary constraint densities are automatically satisfied on the constraint
phase subspace C(2) , and then The Dirac-Bergmann analysis stops here.
Another way of verifying this statement is calculations with the smeared variables. This
calculation method is convenient if the system has only first-class constraints by virtue of131the closed algebra (4.100). The definition of smeared variables is given as follows:Z
CS (N ) =Z

3dx N C0 ,CV (N) =Σt

dx3 N i Ci ,(4.101)Σtwhere N = N i ∂i . Then, first, the following relation holds:(4.102){F (hij , πij ), CV (N)}P.b. = LN F (hij , πij )for arbitrary function F (hij , πij ). Using this, the following algebras are derived:C(2){CV (N1 ) , CV (N2 )}P.b. = CV (LN1 N2 ) ≈ 0 ,C(2){CV (N) , CS (N )}P.b. = CS (LN N ) ≈ 0 .
(4.103)

Neglecting spatial boundary terms, the following algebra holds:{CS (N1 ) , CS (N2 )}P.b.
where “spatial T.D.

=

spatial T.D.

=

CV (N1 ∂ i N2 − N2 ∂ i N1 )∂i C(2)≈ 0,(4.104)” denotes the ignorance of spatial total divergence terms.1 Using these

algebras, it is obvious to verify the satisfaction of the consistency conditions for the secondary
constraint densities. Therefore, the Dirac-Bergmann analysis stops here.
The Dirac-Bergmann analysis reveals that the number of (n + 1) primary and the same
number of secondary constraint densities exist, and all the multipliers remain arbitrary.
1

As discussed in Section 7.1 in detail, this ignorance is restricted to a system such that the diffeomorphism
invariance holds, meaning that Eq (4.100) is satisfied. In this case, imposing appropriate spatial boundary
conditions, i .e ., Ni (t , spatial boundary) := 0, the spatial total divergence terms can be neglected. However,
fortunately, in the case of GR, the boundary terms are composed only of the constraint densities, therefore,
even if this ignorance is not necessary.

132

Therefore, the propagating degrees of freedom (see Section 2.4) is given as follows:(n + 1)(n + 2)
1
1
pDoF = 2 ×− 2 × 0 − 2 × {(n + 1) + (n + 1)} × = (n + 1)(n − 2) .
2
2
2
(4.105)
The gauge degrees of freedom (see Section 2.4) is the number of the primary first-classconstraints as follows:(4.106)

gDoF = n + 1 .

The gauge generator (see Section 2.3) is given as the linear combination of the first-class(k)constraints: ∗ G = ϵµk ∗ ϕµ (k ∈ {1 , 2}). This gDoF inherits the principle of general relativity/covariance.
C(2)The symplectic potential becomes the following form: ∗ ω = PI δQI + Ψα δΞα ≈ PI δQI ,(k)where Ψα are linear combinations of ∗ ϕµ with some function coefficients, QI and PI are
the propagating degrees of freedom, and I ∈ {1 , 2}; α ∈ {1 , 2 , · · · , 8}. Since this system
has only the first-class constraint densities, in order to identify appropriate boundary conditions, the symplectic potential has to be pulled back by the quasi-embedding σ̃1 (t). (See
C(2)

C(2)

Section 3.7.) That is, σ̃1∗ (t)∗ ω ≈ PI (t)δQI (t) + Ψ̃α (t)δΞα (t), where Ψ̃α (t) ≈ σ̃1∗ (t)Ψα :
C(2)

(1)

only the primary first-class constraint densities ∗ ϕµ

≈ 0 in Ψα are set to be zero; other-(2)

wise the secondary first-class constraint densities ∗ ϕµC(2)≈ 0 in Ψα are set to be constantparameters. Therefore, the well-posedness of the variational principle for the effective action(see Section 3.7) on the full constraint phase subspace C(2) demands the boundary conditions δQI = 0 and, in generically, δΞα = 0, on the boundary ∂M. However, the explicit
expressions of these variables are difficult to lead, as mentioned in the final paragraph in
Section 4.5. No one has achieved it yet at the moment writing this thesis.

133

4.7. Summary
In this chapter, the general theory of relativity was constructed; the geodesic equations
and the Einstein field equations are derived based on the well-posed variational principle
being consistent with the primary constraint densities. In particular, the latter case showedthat the Gibbons-York-Hawking term is automatically derived and the well-posedness being
consistent with all constraint densities expect to generate new sorts of counter-terms. Every
modified/extended theory of gravity has to be able to restore the Einstein field equations
as a special case under the satisfaction of some specific conditions. Then the dynamics
is described by the geodesic equations of the spacetime of a given theory of gravity. The
ADM-foliation/decomposition introduced in Section 4.4 plays a crucial role in performing
the Dirac-Bergmann analysis of theories of gravity. An example of the analysis for thegeneral theory of relativity was given in Section 4.6. This technique can also apply tomodified/extended theories of gravity and provides one of the fundamental ingredients tothe present thesis.

134Chapter 5
Gauge approach to gravity
Chapter abstract
In this chapter, a mathematical framework for the gauge approach to gravity is constructed.
First of all, three fundamental bundles, i .e ., fiber bundle, G-bundle, and principal G-bundle
are defined. For the principal G-bundle, the Ehresmann connection and its curvature are
introduced. After that, vector bundles together with gauge fields and those curvature (fieldstrengths) are derived. Internal space bundles, which play a crucial role in constructing
the metric-affine gauge theories of gravity, are also derived, in particular, focusing on frame
fields and Weitzenböch connection. Finally, a gauge approach derivation of the Einstein field
equations is described. This chapter gives mathematical basics for the succeeding chapters.

5.1. Fiber bundles and G-bundles
In this section, the basics of fiber bundles for physics are briefly introduced [Kobayashi(1989); Baez and Muniain (1995); Kobayashi and Nomizu (1996); Sharpe (1997); Nakahara(2003)]. First, fiber bundles are defined, and restrictions of the application to physics aredescribed. Second, G-bundles are introduced. Finally, the construction of G-bundles is
explained aiming to clarify geometrical meanings.
A fiber bundle is a structure between two topological spaces, denoted by a total space E
and a base space M, related by a continuous surjective map (, or equivalently, projectionmap,) π : E → M such that, for an open set U , π −1 (U ) is homeomorphic to the direct
product of U and a given topological space F, i .e ., φ : E|U = π −1 (U ) → U × F; (p , ξ) 7→{p} × F = Fp = ϕ(p , ξ). This homeomorphism is called local trivialization. Then the fiber
bundle is denoted as B = (E , M , π , F), and F is called the standard fiber of the fiber
135bundle. On the fiber bundle, a quantity s which is defined on M and valued in E, that iss : M → E, can be introduced as satisfying π ◦ s = id. This map is called a section of E on
M, and the set of all sections is denoted as “ Γ(E) ”.
For applications to physics, the two topological spaces E and M are often assumed assome differential manifolds, so is for the fiber F . Then the projection map π is also assumed
to be differentiable. In particular, M is taken to be a spacetime manifold, and then, ofcourse, coordinate transformations on local regions in M are considered. That is, for anatlas of charts {ϕi : Ui → Rd | i ∈ I}, where U = {Ui }i∈I is an open covering of M and dis the dimension of M, and p ∈ Ui ∩ Uj , a coordinate transformation is given as follows:ϕi ◦ ϕj −1 : Rd → Rd ; (x1 , · · · , xd ) = ϕi (p) 7→ (y 1 , · · · , y d ) = ϕj (p). Then, making a set bygathering all transition functions “ tij = ϕi ◦ϕj −1 ” and inducing the composition of functions“ ◦ ” as a binary operation into the set, denoted by G = (tij , ◦), G forms (a subgroup of) a
general linear group GL(d ; R).
For the total space E, the resemble concepts of a chart and an atlas of charts in a manifold
can be introduced: G-chart(, or equivalently, local trivialization,) and G-atlas, where G isa given Lie group. That is, for Ui ∈ U, a G-chart is a homeomorphism given as follows:
φi : E|Ui = π −1 (Ui ) → Ui × F; (p , ξ) 7→ {p} × F = Fp = ϕi (p , ξ). Then the set of all Gcharts is defined as G-atlas: {φi | i ∈ I}. If a G-atlas is given, for each point p ∈ Ui ∩ Uj , there
exists an element τij (p) ∈ G such that φi ◦ φj −1 : (Ui ∩ Uj ) × F → (Ui ∩ Uj ) × F; (p , ξ) 7→(p , τij (p)η). This map is denoted also as “ τij = φi ◦ φj −1 ” and called a transition function,and it plays the role of a coordinate transformation for each fiber. In this way, the groupstructure of G is implemented into the fiber bundle B as the way of the transformation of
the fiber F. This sort of fiber bundle is called as a G-bundle with a structure group G anddenotesd by G = (E , M , π , F , G).
Conversely, if a set of transition functions τij ∈ G, where G is a Lie group, is provided,
then a G-bundle can be constructed as follows: Let M and {Ui }i∈I be a differential manifold
136

and an open covering of M. Giving a differential manifold F , a set of the direct product`manifolds and its disjoint union are constructed: {Ui × F}i∈I and E ′ = i∈I Ui × F, respectively. Now, using the transition functions τij , an equivalence relation, denoted by “ ∼ ”, inE ′ can be induced in a well-defined manner. That is, for (p , ξ) ∈ Ui × F and (q , η) ∈ Uj × F,
the equivalence relation “ ∼ ” is defined as satisfying the following two conditions: (i) p = q;(ii) ξ = τij η. Let [(p , ξ)] be an equivalent class of the representative element (p , ξ). Thenthe quotient space of E ′ by “ ∼ ”, i .e ., E = E ′ / ∼, becomes G-bundle. In fact, the projection
map and the local trivialization are given as π : E|U = U × F → M; [(p , ξ)] 7→ p andφ : π −1 (U ) → U × F; [(p , ξ)] 7→ (p , ξ), respectively: G = (E , M , π , F , G).
Famous examples of G-bundles are a cylinder and a Möbius strip. Let B = (S 1 ×
F , S 1 , π , F) be a fiber bundle, where S 1 and F are a circle and an interval [−1 , 1]. Then,if a group Gcylinder = {e} is given then the G-bundle, G = (S 1 × F , S 1 , π , F , Gcylinder ),becomes a cylinder. If a group GMoebius = {e , −e} is given then the G-bundle, G = (S 1 ×
F , S 1 , π , F , GMoebius ), becomes a Möbius strip.
For applications to physics, G-bundles need more additional structure. In particular,
connection/covariant derivative and curvature/field strength are mandatory. In order to formulate these concepts, a right action of the group G to a G-bundle is introduced, (the left
action of the group G to G-bundle is nothing but a generalization of coordinate transformations) and it invokes a new geometric structure: the principal bundles.

5.2. Principal G-bundles, Ehresmann connections, and curvatures
In this section, making a relation between the fiber, F, and the structure group, G, to a Gbundle, G = (E , M , π , F , G), as a group action, a principal G-bundle is introduced [Kobayashi(1989); Kobayashi and Nomizu (1996); Sharpe (1997)]. For the principal G-bundle, a connection so-called Ehresmann connection is formulated, and its fundamental properties arederived [Kobayashi (1989); Kobayashi and Nomizu (1996)]. Let us start by introducing a137

principal G-bundle.
A principal G-bundle is a G-bundle with a right action of the group G, i .e ., R : P ×G →P; (u , g) 7→ u · g, acting on the total space P in the following manner: (u · g)g ′ = u · (gg ′ )and u · g = u, and it satisfies the following conditions; (i) π(u · g) = π(u); (ii) π(u) = π(u′ )implies the existence of an element g ∈ G as follows u′ = u · g (simply transitivity). In
addition, assume that (iii) all sections σi : Ui → π −1 (Ui ) = Ui × G are differentiable, whereUi ∈ U and U is an open covering of the base space M. “ · ” denotes the right actionof the group G. Conditions (i) and (ii) identify the fiber F with the structure group G.
Therefore, the principal G-bundle is denoted as P = (P , M , π , G ; R). Condition (iii) leads
to a local trivialization φi : P|Ui → Ui × G; σi (p) 7→ (p , g) for Ui ∈ U, where the relation,i .e ., σi (p) = σi (p) · e = σi (p) · (g −1 g) = (σi (p) · g −1 ) · g = σi′ (p) · g for g ∈ G, is used. The
transition functions are defined by using the local trivializations, {φi }i∈I , with respect to theright action of the structure group G as usual: {τij = φ−1i ◦ φj }.
A famous example of a principal G-bundle is the Hopf fibration. Let G = (S 3 , S 2 , π , F , U (1))
be a U (1)-bundle with a fiber F , where U (1) is the 1-rank unitary group. Introducing a rightaction of U (1) into G, for the projection map π : (x1 , x2 , x3 , x4 ) 7→ (ξ 1 , ξ 2 , ξ 3 ) such that
ξ 1 = 2(x1 x3 + x2 x4 ) , ξ 2 = 2(x2 x3 − x1 x4 ) , ξ 3 = (x1 )2 + (x2 )2 − (x3 )2 − (x4 )2 , becomes a
principal U (1)-bundle: P = (S 3 , S 2 , π , U (1) ; R). Notice that, since the elements in U (1)can be expressed as exp(iθ) for θ ∈ (−π , π), the right action is not distinguished from theleft action of the group.
Next, let us introduce the Ehresmann connection. The definition is given as follows:(i) ω(A∗ ) = A (A ∈ g) ,
∗

(5.1)
−1

(ii) Rg ω(X) = Adg (ω) = g ωg ,where g is the Lie algebra of the structure group G, ω is a g-valued 1-form, ω ∈ g ⊗ T ∗ P,
Rg ∗ is the pull back operator of the right action with respect to an element g ∈ G, and Adg138

is the adjoint representation with respect to g ∈ G. A∗ is the so-called fundamental vector
field defined as follows:A∗ u f (u) =d[f (ug(t))]
,
dt
t=0

(5.2)

where u ∈ P , g(t) = exp(tA) ∈ G, and f is an arbitrary function on P. The right action,
R : P × G → P, induces the differential map dR : T P × T G → T P; it means that theright action of g = Te G to P is well-defined. Therefore, the specific form of the fundamental
vector field can be written as follows:A∗ u = uA ,(5.3)

and it of course belongs to Tu P.
The Ehresmann connection has two different expressions: the geometric and algebraic
formulations. The geometric formulation is given as follows: a g-valued 1-form ω on P is theEhresmann connection if and only if the following conditions are satisfied:(i) Tu P = Vu P ⊕ Hu P ,(5.4)(ii) Rg ∗ (Hu P) = Hug P ,where Vu P and Hu P are the vertical and horizontal subspaces, respectively, defined as follows:
Vu P = {X ∈ Tu P | π∗ (X) = 0} ,Hu P = {X ∈ Tu P | ω(X) = 0} ,(5.5)

where π∗ is the push forward operator of the projection map π. In particular, the verticalsubspace is expressed equivalently as follows:
Vu P = {A∗u = uA ∈ Tu P | A ∈ g} .

139

(5.6)

The equivalence of the two definitions (5.1) and (5.4) is obvious from the definitions of
the horizontal and vertical subspaces. Since the Principal G-bundle P is expressed as adirect product π −1 (U ) = U × G in a local region, the above properties give a geometricalinterpretation that U and G represent the horizontal and vertical direction on π −1 (U ) ⊂ P,
respectively.
The curvature/field strength of the Ehresmann connection in the geometrical representation (5.4) is introduced as follows:(5.7)Ω(X , Y ) = dω(X H , Y H ) ,where Ω is a g-valued 2-form on P, X and Y belong to T P, and X H and Y H belong to HP.
Expanding this, the following equation is derived:1Ω(X , Y ) = − ω [X H , Y H ] .
2

(5.8)

Therefore, the flatness, which is of course expressed as Ω = 0, of the principal G-bundle P
is equivalent to satisfying the following condition:(5.9)[X H , Y H ] = 0 .

This condition means that the horizontal subspace HP =u∈P Hu P is Frobenius integrable,Fand it restores the entire of the total space P; in other words, if Ω ̸= 0, the Frobenius theorem
restores only a part of the total space P as a submanifold. Notice that the vertical subspaceF
V P = u∈P Vu P is always Frobenius integrable, and it restores the total space P.
Finally, let us introduce the algebraic formulation of the Ehresmann connection. This
formulation plays the main role in establishing gauge theories. The algebraic formulation is140

given as follows: for the Ehresmann connection ω in Eq (5.1),
ωi = σi∗ ω ,(5.10)

where Ui ∈ U, σi∗ is the pull back operator of σi : Ui → π −1 (Ui ). Then the following relation
is satisfied:ωj = τij −1 ωi τij + τij −1 dτij ,(5.11)

where τij ∈ G is the transition function on Ui ∪ Uj for Ui , Uj ∈ U. Conversely, if therelations (5.11) are given then the Ehressmann connection in Eq (5.1) is derived. The proofof this statement is not trivial but easy to give as follows: Eq (5.10) is transformed from
Ui to Uj by using the transition function τij as ωj = ω((σi τij )∗ X), where X ∈ T P. Here,(σi τij )∗ X is calculated as follows: (σi τij )∗ X = σi∗ (u)(X)τij (u) + A∗σj (u) (X), where xt is a
curve in P such that x0 = u and (dxt /dt)|t=0 = X. Eq (5.3) is also used. Therefore,
ωj = [τij −1 (σi∗ ω)τij + τij −1 dτij ](X), i .e ., ωj = τij −1 ωi τij + τij −1 dτij , where Eq (5.1) are used.
Conversely, if Eq (5.11) holds then, using the fact that τij −1 dτij is Maurer-Cartan form (leftinvariant g-valued 1-form with respect to the group action of L : G × P → P) [Kobayashi(1989); Kobayashi and Nomizu (1996)], it is shown that ω̃i = τij −1 (π ∗ ωi )τij + τij −1 dτij is also
valid on π −1 (Ui ∩ Uj ); ω̃i is defined on the entire region of P: ω = τij −1 (π ∗ ωi )τij + τij −1 dτij .
Taking into account ωi (A∗ ) = 0, ω derives the first condition in Eq (5.1). (The satisfactionof the second condition in Eq (5.1) is obvious.) The pull back of ω by a section σi leads toωi = σi ∗ (ω).
The curvature/field strength of the Ehresmann connection in the algebraic representation (5.11) is introduced as follows:1Ωi = dωi + [ωi , ωi ] .
2

141(5.12)

This expression can be derived by calculating the pull back of the definition (5.7) by a section
σi and is now defined on g-valued 2-form on M. For the curvature Ωj on Uj , the following
relation is easily verified:Ωj = Adτij (Ωi ) = τij −1 Ωi τij ,(5.13)

where τij is the transition function from Ui to Uj . These quantities also live as g-valued
2-forms on M.
For applications to physics, sections in a principal G-bundle, P, have to express the fields
that describe physical phenomena. In order to achieve this, representations of the structuregroup G and those actions to the total space P are introduced, and then a new sort of fiber
bundle is invoked: the vector bundles.

5.3. Vector bundles, gauge fields, and curvatures
In this section, introducing a representation of the structure group G of a principal G-bundle
P = (P , M , π , G ; R), denoted by ρ : G → End(V ) for a r-dimensional representation space
V , an associated vector bundle is introduced [Kobayashi (1989); Kobayashi and Nomizu(1996); Sharpe (1997); Nakahara (2003)]. Inheriting its properties, the generic definition ofvector bundles is also given. Then gauge fields/connection 1-form in covariant exterior derivative and its curvature are introduced, respectively [Kobayashi (1989); Kobayashi and Nomizu(1996)]. Sections of a vector bundle are nothing but fields that describe physical phenomena.
Therefore, these concepts play a central role in constructing a theory of physics. First of all,let us introduce an associated vector bundle.
An associated vector bundle is derived from a quotient space of P × V with an equivalentrelation, denoted by “ ∼ ”, of the structure group G such that, for (ξ , u) and (η , v) in P × V ,these two elements are equivalent if and only if the following two relations hold: (i) ξ = Rg η;(ii) u = ρ−1 (g)v. The quotient space is Vρ = P ×ρ V = (P × V )/ ∼ and an equivalentclass is denoted as [(ξ , v)]. Then the structure V = (Vρ , M , π , V ; R , ρ) has a one-to-one
142correspondence on a local region U ⊂ M as follows: P|U × V → U × G × V ; (σ(p) , v) 7→(p , e , v), where p ∈ U , σ is a section of P on U , and e is the unit element of G, since thegroup action R is simply transitive. Therefore, the quotient space V can be identified in a
local region U with Vρ |U = U × V . That is, the representation space V is nothing but thestandard fiber of the associated vector bundle Vρ . This property also indicates the existenceof local trivializations as follows: φi : π −1 (Ui ) → Ui × V , where Ui ∈ U and the projection
map is given as follows: π : Vρ → M ; [(σi (p) , v)] 7→ p, where σi is a section of P on
Ui ∈ U. Based on these properties, it would be interpreted that an associated vector bundle
is nothing but a result of replacing the structure group G by a representation space V usinga group representation ρ in a principal G-bundle.
Based on the properties of an associated vector bundle described above, the genericdefinition of vector bundles is defined as follows: a fiber bundle V = (V , M , π , V ) suchthat (i) the fiber V is a vector space; (ii) for each Ui ∈ U, a local trivialization φi : V|Ui =π −1 (Ui ) → U × V exists. Conversely, if a vector bundle is given then a principal G-bundle
exits such that its associated vector bundle by the representation ρ = id coincides with the′original vector bundle: “ V = ∃ Vρ=id” [Kobayashi and Nomizu (1996)]. Therefore, in orderto reveal the properties of vector bundles, it is enough to consider that of associated vector
bundles.
Next, let us introduce gauge fields/connection 1-form of an associated vector bundle. In
order to define the operation of covariant exterior derivative, it needs to prepare V -valued
˜ on P as a section on V ⊗ Vp T ∗ P as follows: for X1 , · · · , Xp ∈ Γ(T P),
p-form, denoted by ξ,
it satisfies
˜ 1 , · · · , Xp ) = u−1 (ξ(π∗ X1 , · · · , π∗ Xp )) ,ξ(X
where ξ ∈ Γ(Vρ ⊗Vp(5.14)T ∗ M). If p = 0 then ξ˜ is a V -valued function given as follows:ξ˜ : P → V ; u 7→ u−1 (ξ(π(u))) .
143(5.15)

This V -valued p-from ξ˜ on P obviously satisfies the following two properties:(i) Rg∗ ξ˜ = ρ−1 (g)ξ˜ ,(ii) X1 ∈ HP

˜ 1 , · · · , Xp ) = 0 .
then ξ(X(5.16)

V
These two properties lead to a one-to-one correspondence between ξ˜ ∈ Γ(V ⊗ p T ∗ P) andVξ ∈ Γ(Vρ ⊗ p T ∗ M) in Eq (5.14). Then the covariant exterior derivative, denoted by dD ,Vof ξ˜ ∈ Γ(V ⊗ p T ∗ P) is introduced in a well-defined manner as follows:
˜ 1 , · · · , Xp ) = dξ(X˜ H , · · · , XH) ,dD ξ(X1p(5.17)

where X1 , · · · , Xp ∈ Γ(T P) and X1H , · · · , XpH ∈ Γ(HP). In terms of the Ehresmann connection, the above equation can be expressed as follows:dD ξ˜ = dξ˜ + dρ(ω) ∧ ξ˜ ,(5.18)

where dρ is the representation of the Lie algebra g induced by ρ. Then the properties inVVV
Eq (5.16) lead also to dD ξ˜ ∈ Γ(V ⊗ p+1 T ∗ P). For α ∈ Γ( p T M) and ξ˜ ∈ Γ(V ⊗ q T ∗ P),
the following formula holds:
dD (ξ˜ ∧ α) = dD ξ˜ ∧ α + (−1)q ξ˜ ∧ d(π ∗ α) .

(5.19)VV
The one-to-one correspondence between ξ˜ ∈ Γ(V ⊗ p T ∗ P) and ξ ∈ Γ(Vρ ⊗ p T ∗ M) leadsVto the covariant exterior derivative in Γ(V ⊗ p T ∗ P); just removing “ ˜ ”, “ H ”, “ dρ ”,
and “ π ∗ ”, from Eqs (5.17), (5.18) and (5.19), and just replacing T P by T M, HP by T M,
V
and V by Vρ , the covariant exterior derivative dD in Γ(Vρ ⊗ p T ∗ M) is introduced in awell-defined manner. Notice that in this case the Ehresmann connection becomes a section144

of End(V) × T∗ M: ω = ω i jµ ei ⊗ej ⊗dxµ . This Ehresmann connection is, in particular, called′gauge fields. Since the theorem “ V = ∃ Vρ=id” [Kobayashi and Nomizu (1996)] as mentioned

in the previous paragraph guarantees that a vector bundle has its associated principal Gbundle, all formulae above, i .e ., Eqs (5.18) and (5.19), can apply also to any given vector
bundle.
For a vector bundle V = (V , M , π , V ), the covariant derivative of a section of V can bederived by considering a special case of the covariant exterior derivative; let p in Eq (5.18) set
zero. Then dD becomes a linear operator as follows: dD : Γ(V) → Γ(V ⊗ T ∗ M ) ; ξ 7→ dD ξ.
In particular, for a function f on M, it satisfies dD (f ξ) = df ⊗ ξ + f dD ξ. The image of ξby dD is defined as the covariant derivative of ξ and denoted by dD ξ = ∇ξ. Explicitly, it isexpressed as follows:dD ξ = ∇ξ = (d + ω)ξ .

(5.20)

From the definition of dD , the covariant derivative ∇ can be interpreted as a linear operator
as follows: for a vector field X ∈ Γ(T M), ∇X : Γ(V) → Γ(V) ; ξ 7→ ∇X ξ such that (i)∇f X ξ = f ∇X ξ; (ii) ∇X+Y ξ = ∇X ξ + ∇Y ξ, where f is a function on M, X , Y ∈ Γ(T M),
and ξ ∈ Γ(V). In other words, ∇X ∈ End(Γ(V)). Then Eq (5.20) becomes as follows:∇X ξ = X(ξ) + ω(X)ξ .

(5.21)

Since the fiber V of the vector bundle V is vector space, V|U = π −1 (U ) = U × V has a basisof Γ(V): eI = eI (p) for p ∈ U , where I ∈ {1 , · · · , r} and r is the dimension of V . Therefore,
Eqs (5.20) and (5.21) can be rewritten as follows:∇ξ = (dξ I + ω I J ξ J )eI ,∇µ ξ = (∂µ ξ I + ω I Jµ ξ J )eI ,145(5.22)

respectively, where X = ∂µ . Defining the components of ∇µ ξ as (∇µ ξ)I , i .e ., ∇µ ξ =(∇µ ξ)I eI , the second equation becomes as follows:(5.23)(∇µ ξ)I = ∂µ ξ I + ω I Jµ ξ J .

This is a familiar form of covariant derivative in physics. For a component ξ I the second
formula in Eq (5.22) becomes more familiar form: ∇µ ξ I = ∂µ ξ I + ω I Jµ ξ J .
Finally, let us introduce the curvature (2-form on M) of the covariant derivative ∇ givenin (5.20). Based on Eq (5.7), the following equation is derived:R(X , Y )ξ =

1
∇X ∇Y − ∇Y ∇X − ∇[X ,Y ] ξ ,2(5.24)

where R is End(V)-valued 2-form on M, X and Y are vector fields on M, and ξ is a sectionof the vector bundle V. This formula is called the Ricci identity. Calculating it for ξ = eI ,the following equations are derived:
ReI = ΩI J ⊗ eJ ,J

J

(5.25)JΩI = dωI + ωK ∧ ωI

K

J

= dD ω I ,where Eq (5.18) is used. The first equation and second relation are the so-called structure
equation and curvature form of the covariant derivative ∇, respectively. Notice that thecurvature form coincides with Eq (5.12). That is, it is nothing but a relation replaced ωi ,which lives as a g-valued 1-form on M, by ωI J , which lives as a End(V)-valued 1-form on
M. Therefore, for the gauge transformation of Eq (5.11), the curvature 2-form changes as
Eq (5.13).
So far, it is revealed that the connection 1-form ω and curvature 2-form Ω are the firstand second-order covariant exterior derivative of a basis eI of the fiber V , respectively.

146

Considering the third-order covariant exterior derivative of eI , or more generically, ξ ∈ Γ(V),the following identity is derived
dD R = 0 .

(5.26)

This is nothing but the so-called Bianchi identity.
Depending on the structure group G and its representation, the mathematical theory
presented above can describe a specific physics. For instance, for G = U (1), Maxwell’s electromagnetic interaction is obtained [Weyl (1929); O’Raifeartaigh and Straumann (1998)].
The structure group G = U (1) × SU (2) and G = SU (3) describe the electroweak interaction [Weinberg (1967); Salam (1968)] and the quantum chromodynamics/strong interaction [Greenberg (1964); Han and Nambu (1965)], as non-abelian gauge Yang-Mills gauge
theory [Yang and Mills (1954)], respectively. In addition, identifying irreducible representations for each theory, gauge fields/particles are taken into account. In the same manner, it
may be expected that a theory of gravity is also obtained from the proper structure groupand its representation [Utiyama (1956)]. This is the motivation for gauge theories of gravity. Before introducing these theories, let us relate the Ehresmann connection to the affine
connection.

5.4. Internal space bundles, frame fields, and Weitzenböch connections
In this section, the main concepts for gauge theories of gravity, and internal space bundles,
in particular, focusing on frame fields/vielbein, are introduced. Relating it, an important
formula, i .e ., the Weitzenböch connection, between the so-called spin connection and theaffine connection is derived. This relation plays a crucial role in formulating the metric-affine
gauge theories of gravity as presented in the next Chapter 6.
Frame fields/vielbein are the components of a bundle morphism, denoted by e, from a vector bundle V = (V , M , π) to the tangent bundle of a spacetime manifold T = (T M , M , ι).
147

That is, for any open set U of an open covering of M and a smooth function f : M → M ,the bundle map e = π −1 ◦ f ◦ ι : V → T M ; V|U 7→ e(V|U ) = T M|U defines a set of
sections as follows:eI = e(ξI ) = eI µ ∂µ ,(5.27)

where ξI and ∂µ are a basis of sections of V|U and the local coordinate/holonomic basisof T M|U , respectively. Then the set of components eI µ , or equivalently, eI , and the vector
bundle V are called frame fields/vielbein with respect to the vector bundle V and the internal
space bundle, respectively. Remark that both eI and eI µ are sections of the tangent bundle
T M, although the indices represent that of the internal space bundle V. Therefore, thebundle morphism can be written also as follows:e = e I µ ∂µ ⊗ ξ I ,(5.28)

where ξ I are the dual basis of ξI . This is a section of the vector bundle of the tensor product,i .e ., T M ⊗ V ∗ . The projection map of the constructed bundle is obtained just by taking
the tensor product of each projection map in the order; so are the local trivializations.
The inverse map of e can be introduced in a well-defined manner by virtue of its definitione = π −1 ◦ f ◦ ι. That is, e−1 = ι−1 ◦ f −1 ◦ π : T M → V ; T M|U 7→ e−1 (T M|U ) = V|U .
Using this map, the dual basis of ξI , i .e ., ξ I , are pulled back to T ∗ M as follows:eI = (e−1 )∗ (ξ I ) = eI µ dxµ ,(5.29)

where dxµ are the dual basis of ∂µ . Then the components eI µ , or equivalently, eI , are called
co-frame fields/co-vielbein with respect to the internal space bundle V. Remark that botheI and eI µ are sections of the co-tangent vector bundle T ∗ M. Therefore, the inverse map

148can be expressed as follows:(e−1 )∗ = eI µ dxµ ⊗ ξI .

(5.30)

This is a section of the dual vector bundle T ∗ M ⊗ V of T M ⊗ V ∗ . The projection map andlocal trivializations are also introduced in a dual manner.
The frame fields/co-frame fields satisfy the following relations:eI µ eJ µ = δIJ ,eI µ eI ν = δνµ .

(5.31)For a metric tensor of the tangent space g = gµν dxµ ⊗ dxν and a metric of the internal spacebundle g = gIJ ξ I ⊗ ξ J , pulling back it by e and (e−1 )∗ , the following relation are obtained:gIJ = eI µ eJ ν gµν ,gµν = eI µ eJ ν gIJ ,(5.32)

respectively. The total number of independent variables in the tangent bundle is, on one
hand, (n+1)(n+2)/2 for the metric tensor gµν . On the other hand, that in the internal spacebundle is the total number of (n + 1)(n + 1) for the frame fields eI µ and of (n + 1)(n + 2)/2 forthe metric tensor gIJ , respectively. Therefore, in order to make the correspondence betweengµν , and, eI µ and gIJ in a one-to-one manner, the internal space variables have to be imposedsome conditions. In many applications in theories of gravity, the internal space bundle istaken to be a trivial vector bundle M × Rn+1 . This indicates that all components of themetric gIJ are constant; it reduces the number of the independent variables in the internal
space bundle into the number of the frame fields. The remaining extra degrees of freedom,i .e ., (n + 1)(n + 2)/2 − (n + 1)(n + 1) = n(n + 1)/2, can be erased by imposing a gauge
symmetry of the (co-)frame fields. In the case of the trivial bundle M × Rn+1 , the structuregroup G is sometimes set to be the Lorentz group: G = SO(n , 1). Then, since an element of
G = SO(n , 1) has the number of n(n + 1)/2 independent components, the (co-)frame fields
149

are restricted by the group via the transition functions. Therefore, in this case, the numberof independent variables on each bundle coincides with one another.
The covariant derivative of sections of the internal space bundle is also given by Eq (5.20)
or Eq (5.22) in the previous section 5.3. Since the frame field e can be expressed as Eq (5.28),
i .e ., a section of the tensor product bundle T ∗ M ⊗ V, the covariant derivative of e, denotedby De, becomes as follows:
De = (deI µ − Γ̃ρνµ eI ρ dxν + ω I Jµ eJ ν dxν )dxµ ⊗ ξI ,

(5.33)

where Γρµν and ω I Jµ are the affine connection of the tangent bundle and the spin connectionof the internal space bundle. In component form, the above formula is expressed as follows:
Dµ eI ν = ∂µ eI ν − Γ̃ρµν eI ρ + ω I Jµ eJ ν .

(5.34)In the same way, Dµ eI ν is calculated as follows:
Dµ eI ν = ∂µ eI ν + Γ̃νµρ eI ρ − ω J Iµ eJ ν .

(5.35)

These formulae are the result from V|U ≃ T M|U on a local region U in M; this local relation
implies that the affine connection and the spin connection can be identified in the local region
U . The transition functions τij = φ−1i ◦ φj of the internal space bundle V form its structuregroup G. Therefore, for Λ ∈ G, Eq (5.11) leads to the transformation of the spin connection
as follows:ω I Jµ → ω ′I Jµ = (Λ−1 )I K ∂µ ΛK J + (Λ−1 )I K ω K Lµ ΛL J .

(5.36)

Then the co-frame fields eI µ and those covariant derivatives Dµ eI ν are transformed as eI µ →e′I µ = ΛI J eJ µ and Dµ e′I ν = ΛI J Dµ eJ ν , respectively. Dµ eI ν transforms in the same manner150

under the gauge transformation (5.36): Dµ e′I ν = (Λ−1 )J I Dµ eJ ν .
Finally, let us derive the Weitzenböch connection. A section, denoted by X, of thetangent bundle T M is of course expressed as X = X µ ∂µ . Pulling back X µ to the internal
space bundle by using co-frame fields eI µ , the components of a section on the internal space
are obtained: X̃ I = eI µ X µ , or equivalently, X µ = eµ I X̃ I . Therefore, X is equivalent toX̃ = X̃ I ξI . The covariant derivative of X and X̃ become as follows:∇X = (∂ν X µ + Γ̃µνρ X ρ )∂µ ⊗ dxν

(5.37)

DX̃ = (eI ν ∂µ X ν + X ν ∂µ eI ν + ω I Jµ eJ ν X ν )ξI ⊗ dxµ ,(5.38)

andrespectively. Using the pull back of ∂µ by the frame fields eI µ , ξI = eI µ ∂µ , the left-hand sideof the second formula becomes as follows:DX̃ = (∂ν X µ + X ρ eI µ ∂ν eI ρ + ω I Jν eI µ eJ ρ X ρ )∂µ ⊗ dxν .

(5.39)

This formula is of course the same as the first one; the covariant derivatives ∇X and DX̃
are also equivalent. Therefore, the following relation is derived:Γ̃ρµν = eI ρ ∂µ eI ν + ω I Jµ eI ρ eJ ν .

(5.40)

This is the so-called Weitzenböch connection, which relates the spin connection to the affine
connection via the (co-)frame fields. Notice that Eq (5.40) implies Dµ eI ν = 0, or equivalently,
Dµ eI ν = 0. These properties are sometimes called vielbein postulate but, as shown in theabove, always satisfied. That is, it is not a postulate. In addition, taking into account thegauge invariant property of them, it guarantees that these properties hold in the global region151

of M.
The metric-vielbein relation (5.32) and the Weitzenböch connection (5.40) play a fundamental role of gauge theories of gravity. That is, assume that an action of gravy is given as
follows:
I[gµν , Γ̃ρµν ] =Z√
L(gµν , ∂ρ gµν , · · · ; Γ̃ρµν , ∂λ Γ̃ρµν , · · · ) −g dn+1 x .

(5.41)

M

Then, using the metric-vielbein relation and the Weitzenböch connection, the above action
is rewritten in terms of the metric of the internal space, the (co-)frame fields, and the spin
connection as follows:
I

Z

I

L(gIJ , ∂ρ gIJ , · · · ; eI µ , ∂ρ eI µ , · · · ; ω I Jµ , ∂λ ω I Jµ , · · · ) det(e−1 ) dn+1 x ,I[gIJ , e µ , ω Jµ ] =M(5.42)
where det(e−1 ) denotes the determinant of the co-frame fields. This quantity is well-definedby virtue of V|U ≃ T M|U . Remark that the theory given in Eq (5.42) does not always
restore that in Eq (5.41). Conversely, starting with constructing an internal space bundle
V = (V , M , π), a generic theory which is described by the action functional (5.42) isobtained. This theory is much richer than the original theory given by Eq (5.41) since
impositions of proper gauge conditions would give rise to a lot of theories of gravity. As a
special case, for the trivial internal space bundle V = M × Rn+1 |U , Eq (5.42) becomes asfollows:
I

I

Z

I[e µ , ω Jµ ] =L(eI µ , ∂ρ eI µ , · · · ; ω I Jµ , ∂λ ω I Jµ , · · · ) det(e−1 ) dn+1 x .

(5.43)

M

The metric tensor gIJ has those components as constant and defined on the global region of
M × Rn+1 . This theory is always equivalent to that given by Eq (5.41).

152

5.5. A derivation of Einstein field equations based on gauge approachto gravity
In this section, the Einstein field equations (4.28) are derived in the way of the gauge theories
of gravity that is provided in the previous section 5.4 [Palatini (1919); Utiyama (1956);
Baez and Muniain (1995)].
The action functional given as Eq (5.43) of the internal space bundle V = (V = M ×
Rn+1 , M , π) realize this purpose. First, in order to specify the metric gIJ , the following
metric-preserving condition is imposed:vg(ξ , ξ ′ ) = g(Dv ξ , ξ ′ ) + g(ξ , Dv ξ ′ ) ,(5.44)

where v is a vector field on M, and, ξ and ξ ′ are sections of the internal space M × Rn+1 .
Considering it with respect to the basis of sections of the internal space M × Rn+1 , i .e ., ξI ,the connection ω I Jµ satisfies the following property:

ωIJµ = −ωJIµ .

(5.45)

Therefore, under the metric-preserving condition, this connection is nothing but the Lorentz
connection: ω (L) IJµ ∈ so(n , 1). This satisfies the demand from the EEP. Second, a specific
Lagrangian in Eq (5.43) is given as follows:
I

I

Z

−1

L det(e ) dx

I[e µ , ω̃ Jµ ] =
M

n+1

Z
=

eµ I eν J Ω̃IJ µν det(e−1 ) dxn+1 ,(5.46)

M

where Ω̃IJ µν is the curvature form given in Eq (5.25). Explicitly, Ω̃IJ µν is written as follows:
˜ [µ ω̃ IJ ν] + ω̃ I K[µ ω̃ KJ ν] ,Ω̃IJ µν = ∇153(5.47)

˜ µ is the covariant derivative of the Weitzenböch connection Γ̃ρ given in Eq (5.40).
where ∇µν
To apply the variational principle in a well-posed manner with respect to the constraint
phase space C(1) , let us clarify appropriate boundary conditions. The action functional (5.46)
can be rewritten as follows [Lagraa et al. (2017); Lagraa and Lagraa (2018)]:
I

Z

I

I[e µ , ω̃ Jµ ] =hi
˜ 0 ω̃IJi − ∇˜ i ω̃IJ0 ) − AiIjJ Ω̃IJij det(e−1 ) dxn+1 ,AiI0J (∇

(5.48)

M

where AµIνJ is defined as follows:(5.49)AµIνJ = eIµ eJν − eIν eJµ .

Canonical momentum variables are calculated as follows:
π µI =

δL= 0,
˜ 0 eIµ )δ(∇

P iIJ =δL
= AiI0J ,˜ 0 ω̃IJi )δ(∇

P 0IJ =

δL
= A0I0J = 0 .
˜ 0 ω̃IJ0 )δ(∇(5.50)All of these variables become primary constraint densities, and these constraint densities
specify the phase subspace C(1) . Then the symplectic potential is given as follows: θ =π µI δeIµ + P iIJ δ ω̃IJi + P 0IJ δ ω̃IJ0 = P iIJ δ ω̃IJi . Therefore, the variational principle becomeswell-posed on the constraint phase subspace C(1) under the imposition of boundary conditions
δ ω̃IJi = 0 on ∂M.
Varying Eq (5.46) with respect to the frame fields eI α , the following equation is derived:1δe I = 2R̃µν − gµν R̃ η IJ eI µ δ(eJ β )det(e−1 )dxn+1 .
2MZ(5.51)

Note that there are no boundary terms: this is consistent with the appropriate boundary
conditions that are derived in the previous paragraph. Therefore, the variational principle

154

leads to the following equations in a well-posed manner:1G̃µν = R̃µν − gµν R̃ = 0 .
2(5.52)I

Varying Eq (5.46) with respect to the connection ω I Jµ around the Lorentz connection ω (L) Jµ :Iω̃ I Jµ = ω (L) Jµ + ω̃ ′ I Jµ , the following equation is derived:
Z

β
γ
β
α
α
β
β
α
) det(e−1 )dxn+1
δCµα
δCβα
+ Cβν
δCβν
− Cµν
+ Cγµ
δCµν
g µν (−Cαβ

δIω =
M

Z
+

(nν g

µν

(5.53)

ρρδCρµ− g µν nρ δCµν
) det(e−1 )dxn ,∂Mνρ= 0 that are= eI ρ eJ µ ω̃ ′ I Jν . Applying the appropriate boundary conditions δCµI
where Cµν

derived in the previous paragraph, the above equation becomes as follows:ZδIω = · · · +ρρ(nν g 0ν δCρ0
− g µ0 nρ δCµ0
) det(e−1 )dxn .

(5.54)∂M

Remark that the frame fields are fixed in this variation. To erase this boundary term, let us
impose a gauge condition as follows:Iω̃ ′ J0 = 0 ,µor equivalently, Cν0= 0.

(5.55)

This is the so-called temporal gauge condition. Then the boundary term is removed fromthe first-order variation of the action as follows:
Z
δIω =
M

γ
β
α
β
β
α
α
β
g µν (−Cαβ
δCµν
+ Cγµ
δCβν
− Cµν
δCβα
+ Cβν
δCµα
) det(e−1 )dxn+1 .

(5.56)

Therefore, the variational principle leads to the following conditions are derived:
µ
Cνi
= 0,

I

or equivalently, ω̃ ′ Ji = 0 .
155(5.57)

Taking into account Eqs (5.55) and (5.57), the Weitzenböch connection (5.40) becomes asfollows:IΓ̃ρµν = eI ρ ∂µ eI ν + ω (L) Jµ eI ρ eJ ν .

(5.58)

This is equivalent to the Christoffel/Levi-Civita connection given in Eq (4.10). That is,
Eq (5.52) becomes the Einstein field equations in vacuum. In the case of coupling matterwith this theory, using the covariant derivative of spinor fields ψiIJDµ = ∂µ − ω (L) µ σIJ ,4i
σIJ = [γI , γJ ] ,2(5.59)

where γI are Dirac matrices, and pull-backed Dirac matrices γ µ (x) = eI µ (x)γ I , the Lagrangian of matters are constructed as “ iψ̄γ µ Dµ ψ − mψ̄ψ ” on a generic spacetime M.
Other fields are also coupled with gravity in the same manner provided in Section 4.5.

5.6. Summary
In this chapter, the mathematical basics for the gauge approach to gravity were introduced.
The gauge approach to gravity provides a method to construct gravity theories based on a
local symmetry just like the Yang-Mills gauge field theories in the elementary particle physics.
In particular, the ingredients introduced in Sections 5.2 and 5.4 are repetitively utilized
for constructing the metric-affine gauge theories of gravity in the next chapter. A local
symmetry is introduced into the internal space and leads to possible geometric quantities.
The Weitzenböch connection bridges these geometric quantities on a spacetime manifold toones on an internal space via (co-)frame fields, and vice versa. This dual structure plays acrucial role in constructing the metric-affine gauge theories of gravity.

156

Chapter 6
Metric-affine gauge theories of gravity
Chapter abstract
In this chapter, the metric-affine gauge theories of gravity are constructed. First, based on
Sections 5.2 and 5.4, three fundamental geometric quantities, i .e ., curvature, torsion, andnon-metricity are introduced. Second, using the so-called Möbius representation of a gauge
group, the metric-affine geometries are classified into eight subclasses: Riemann-Cartan,
teleparallel, Riemann, Minkowski, Weyl-Cartan, Weyl, metric-affine, and symmetric teleparallel geometries. In this formulation, each geometric quantity is given by the covariant exterior derivative of each corresponding potential. Third, The geometrical trinity of gravity,
which is composed of the three subclasses: Riemann (general relativity), symmetric teleparallel, and torsion teleparallel, is established. These three subclasses are equivalent up toboundary terms in the meaning of action functional. Finally, two extensions of the geometrical trinity of gravity, that is, quadratic extension and non-linear extension, are introduced.
The latter extension plays a main role in the next Chapter 7.

6.1. Curvatures, torsions, and non-metricities
In this section, based on the tangent bundle T M of a spacetime manifold M, three fundamental geometric quantities, i .e ., curvature, torsion, and non-metricitie are introduced [Kobayashi(1989); Kobayashi and Nomizu (1996); Nakahara (2003)]. Throughout those introductions,the positioning of the affine connection in these quantities is also clarified. Let us first reformulate curvature and define torsion. To do this, it needs to introduce an associated principal
bundle from the tangent bundle.
Each fiber Tp M of the tangent bundle T M has a basis {ei }, where p ∈ M and i ∈157

{1 , 2 , · · · , n + 1}. n + 1 is the dimension of each fiber, or equivalently, that of the spacetime
manifold. Then an isomorphism from Rn+1 to Tp M, i .e ., u : Rn+1 → ⟨e1 , e2 , · · · , en+1 ⟩ =Tp M always exists, where ⟨· · · ⟩ denotes a vector space spanned by the basis “ · · · ”. Gathering
all such u, denotes the set as L(M)p and then construct a disjoint union as follows: L(M) =Fp∈M L(M)p . For each L(M)p , a right action R of GL(n + 1 ; R) is defined in a well-definedmanner as follows: Rg : L(M)p ×GL(n+1 ; R) → L(M)p ; (u , g) 7→ u◦g since Rg (u) = u ◦ g
is nothing but a basis transformation of Tp M. Then a manifold structure is inserted into
the disjoint union L(M) by using the inverse map of a local trivialization φU : L(M)|U →U × GL(n + 1 ; R) ; σU (p) ◦ g → (p , g), where σU is a set of a basis of Tp M at point p ∈ U .
Based on these structures, the disjoint union L(M) becomes a principal GL(n + 1; R)bundle associated to the tangent bundle T M: L = (L(M) , M , ι , GL(n + 1 ; R) ; R), where
the projection map is given as follows: ι : L(M) → M ; L(M)|p 7→ p.
From Chapter 5, in particular, Sections 5.2 and 5.3, the associated principal bundle L(M)has, on one hand, the Ehresmann connection, ω ∈ Γ (gl(n + 1 ; R) ⊗ T ∗ L(M)), defined as
Eq (5.1). These 1-forms ω lives in the dual space of the vertical space V L(M). On the other
hand, the associated principal bundle L(M) has its proper Rn+1 -valued 1-forms on L(M),denoted by θi , defined as follows:
θi (X) = σU−1 (p)(ι∗ X) ,(6.1)

where X is a vector field on L(M) and σU is a section of L(M). Then the pull back of θi
becomes the dual basis of a basis of a tangent space Tp M: σU∗ θi (ej ) = δji , where {ei } denotes
a basis of Tp M. Therefore, these 1-forms θi lives in the dual space of the horizontal space
HL(M). Remark that the existence of such 1-forms is a proper characteristic of associatedprinciple bundles since generic principle bundles do not have any basis of those fibers.
For the Ehresmann connection ω, the curvature 2-form is defined by Eq (5.7). Similarly,158

for the 1-forms θi , Rr -valued 2-form θi is defined as follows:
T (X , Y ) = dθ(X H , Y H ) ,(6.2)

where X , Y ∈ Γ(T L(M)), θ ∈ ⟨θi ⟩, and X H , Y H ∈ HL(M). This quantity is called thetorsion tensor of the spacetime manifold M. In the same manner as the curvature 2-form
given in Eq (5.12), the torsion 2-form has the algebraic expression given as follows:
1
σU∗ T = dσU∗ θ + [σU∗ ω , σU∗ θ] .
2

(6.3)

This relation holds on the spacetime manifold M. Here, U is an open set of M. Since thecurvature 2-form Ω given in Eq (5.7) and the torsion 2-form given in the above vanish for
X , Y ∈ V L(M), these quantities can be expressed in algebraic forms as follows:Ω = Ωi j ei ⊗ θj

(6.4)

T = Θi ⊗ ei ,

(6.5)

and

respectively, where Ωi j and Θi are defined as follows:1Ωi j = Ri jkl θk ∧ θl
2

(6.6)

1
Θi = T i jk θj ∧ θk ,
2

(6.7)

and

respectively. Eqs (6.4) and (6.5) are a gl(n+1 ; R)-valued 2-form on L(M) and a T M-valued
2-form on L(M), respectively.

159

Now, let us apply the framework given in Section 5.3 to the associated principle bundle
L(M) and its geometrical quantities Ω and T . For the representation ρ = id, the principal
bundle L(M) turns back into the original vector bundle T M. Then the Ehresmann connection ω becomes a section of End(T M)-valued 1-form on M; this connection ω is nothing
but the affine connection. That is, a connection of the tangent bundle is defined as an affine
connection. Then, for the curvature 2-form, on one hand, it becomes a section of End(T M)valued 2-form on M, and the Ricci identity (5.24) and the structure equation (5.25) hold.
On the other hand, for the torsion 2-form, it becomes T M-valued 2-form on M, and theresemble formula to the curvature 2-form is derived as follows:

T̃ (X , Y ) =1 ˜˜ Y X − [X , Y ] ,∇X Y − ∇2(6.8)

where X and Y are vector fields on M, i .e ., sections of T M. Hereinafter, we denote thequantities in the affine connection by putting “ ˜ ” over those notations. Calculating it for abasis {ei } of T M|U , where U is an open set of M, the following equations are derived:T̃ (ei , ej ) = dθk + ω k l ∧ θl (ei , ej ) ⊗ ek ,(6.9)

where θi are the dual basis of ei ; these θi correspond to Eq (6.1). Comparing the above
equation with Eq (6.5), the components Θi are determined as follows:Θ̃i = dθi + ωj i ∧ θj = dD̃ θi ,(6.10)

where ωi j are the components of the Ehresmann connection ω and dD is the covariant exterior
derivative of the connection ω. This equation is called Cartan’s first structure equation. The

160

equation of the curvature 2-form, i .e .,Ω̃i j = dωi j + ωk j ∧ ωi k = dD̃ ωi j ,(6.11)

is called Cartan’s second structure equation.
In the existence of the torsion, in addition to the Bianchi identity given in Eq (5.26),
another identity that relates the curvature to the torsion exists:
dD̃ T̃ = R̃ei ∧ θi .
This identity is defined on T M ⊗V3(6.12)T ∗ M. Acting it to three vector fields X, Y , and Z,

the above identity is expressed as follows:

R̃(X , Y )Z + R̃(Y , Z)X + R̃(Z , X)Y = 2(dD̃ T̃ )(X , Y , Z) .

(6.13)

These identities (6.12) and (6.13) are called Bianchi’s first identity. The Bianchi identity
is given in Eq (5.26) with replacing dD by dD̃ is then called the Bianchi’s second identity,
distinguishing from the first one.
Next, let us introduce the non-metricity tensor. Assume that a metric tensor g = gij θi ⊗θjof the tangent bundle is given. Then the metric tensor generically satisfies the following
relation:˜
˜ , Y ) + g(X , ∇Y˜ ),dg(X , Y ) = (∇g)(X, Y ) + g(∇X

(6.14)

˜ Z g)(X , Y ) + g(∇
˜ Z X , Y ) + g(X , ∇˜ ZY ) ,Zdg(X , Y ) = (∇(6.15)

or equivalently,

˜ or equivalently, ∇˜ X g iswhere X, Y , and Z are vector fields on M. Then the quantity ∇g,161

defined as the non-metricity tensor and denoted by Q̃, or equivalently, Q̃X . For the basis{ei }, the non-metricity tensor is expressed as follows:Q̃ = dD̃ g ,(6.16)

or equivalently,
˜ ij = dgij − ωi k gkj − ωj k gik ,Q̃ij = ∇g
˜ k gij = dgij (ek ) − ωik l glj − ωjk l gil .
Q̃kij = ∇(6.17)

Both the curvature tensor and the torsion tensor have those Bianchi identities. The nonmetricity tensor is the same as these quantities. Considering the exterior covariant derivativeof it, the following identity is derived:dD̃ Q̃ = R̃θi ∧ gei .

(6.18)

This identity is called the zero-th Bianchi identity.
Finally, let us express these geometrical quantities, i .e ., curvature R̃, torsion T̃ , andnon-metricity Q̃, in terms of the standard coordinate basis, i .e ., {xµ } and those partial
derivatives {∂µ }. In this basis, the affine connection ω is denoted as Γ̃, and the italic indicesi , j , k , · · · are replaced by the greek indices µ , ν , ρ , · · · . Then the Ricci identity (5.24) of
the curvature tensor is expressed as follows:R̃σ µνρ = 2∂[ν Γ̃σρ]µ + 2Γ̃σλ[ν Γ̃λρ]µ = ∂ν Γ̃σρµ − ∂ρ Γ̃σνµ + Γ̃σλν Γ̃λρµ − Γ̃σλρ Γ̃λνµ .

(6.19)For the torsion tensor, using Eq (6.8), the following expression is derived:T̃ ρ µν = 2Γ̃ρ[µν] = Γ̃ρµν − Γ̃ρνµ .
162(6.20)

These expressions can be also derived from Eqs (6.6) and (6.7), respectively. The nonmetricity tensor is expressed as follows:Q̃ρµν = ∂ρ gµν − 2Γ̃λρ(µ g|λ|ν) = ∂ρ gµν − Γ̃λρµ gλν − Γ̃λρν gλµ .

(6.21)

Remark that the order of the lower two indices in Γ̃ has to be as this order.
Conversely, using the torsion and non-metricity tensors, the affine connection is decomposed as follows [Schouten (1954); Hehl et al. (1989, 1995); Heisenberg (2019); Beltrán Jiménez et al.
(2019)]:◦(6.22)

Γ̃ρ µν = Γρ µν + K ρ µν + Lρ µν ,where the contorsion tensor K ρ µν and the disformation tensor Lρ µν are defined as follows:1
K ρ µν = T ρ µν + T(µ ρ ν)2

(6.23)

1Lρ µν = Qρ µν − Q(µρν)
2(6.24)

andrespectively. This decomposition can be shown by a straightforward calculation. According
to this decomposition, the curvature tensor (6.19) is also decomposed into as follows:◦◦R̃σ µνρ = Rσ µνρ + 2∇[ν N σ ρ]µ + 2N σ [ν|λ| N λ ρ]µ ,◦(6.25)◦

where Rσ µνρ is the curvature tensor of the Levi-Civita connection Γρµν , N ρ µν is the distorsion
tensor defined as follows:◦N ρ µν = Γ̃ρ µν − Γρ µν .

(6.26)

That is, the distorsion tensor is nothing but the difference between the affine connection andthe Levi-Civita connection and can be taken as the contorsion tensor and/or the disformation163

tensor.
In Section 5.4, a frame field e is nothing but a bundle map from an internal space bundle
V = (V , M , π) to the tangent bundle T = (T M , M , ι). Notice that the isomorphism “u : Rn+1 → ⟨e1 , e2 , · · · , en+1 ⟩ = Tp M ” is possible to play the role of a frame field byreplacing the basis {ei } by {∂µ }. Therefore, all statements for the frame fields in Section 5.4
hold also for the sections of the associated principle bundle L(M). This means also thatthe italic indices i , j , k , · · · are identified as the capital indices I , J , K , · · · . In particular,
the affine connection (6.22) is equivalent to the Weitzenböch connection (5.40). That is, thefollowing important equation holds:◦eI ρ ∂µ eI ν + ω I Jµ eI ρ eJ ν = Γρ µν + K ρ µν + Lρ µν .

(6.27)

This equation relates the gauge structure of the internal space to the metric-affine geometry.

6.2. Metric-affine gauge approach to gravity and its subclass
In this section, based on the following gauge groups, i .e ., affine group A(n + 1 ; R) =T (n + 1 ; R) ⋊ GL(n + 1 ; R) [Hayashi and Shirafuji (1980); Baekler et al. (2011)], Weyl group
W (n+1 ; R) = D(n+1 ; R)⋊A(n+1 ; R) [Charap and Tait (1974); Kopczyński et al. (1988);
Hehl et al. (1989); Blagojevic (2001)], and metric-affine group M A(n+1 ; R) = S(n+1 ; R)⋊
W (n + 1 ; R) [Hehl et al. (1995, 1989); Tomonari (2023a)], the full theory is classified into
eight subclasses: (i) Riemann-Cartan geometry: Un+1 , (ii) Teleparallel geometry: Tn+1 ,(iii) Riemann geometry: Vn+1 , (iv) Minkowski geometry: Mn+1 , (v) Weyl-Cartan geometry:Yn+1 , (vi) Weyl geometry: Wn+1 , (vii) metric-affine geometry: Ln+1 , and (viii) symmetricteleparallel geometry: Sn+1 . Throughout these considerations, the relations between eachgauge group structure and the relevant geometries are clarified. Further, the so-called spin,
dilation, and shear currents are introduced, and the relations between each gauge groupstructure and these currents are briefly discussed. First of all, let us reconstruct the metric164

affine geometry from the viewpoint of the gauge group structure in a bottom-up approach.
Let us first consider the case of the affine group: A(n + 1 ; R) = T (n + 1 ; R) ⋊ GL(n +1 ; R). This group forms an internal space bundle introduced in Section 5.4. The group
multiplication of A(n + 1 ; R), denoted by “ ◦ ”, is defined as follows:(6.28)(s1 , t1 ) ◦ (s2 , t2 ) = (s1 · s2 , s1 t2 + t1 ) ,where s1 = s1 (p) , s2 = s2 (p) ∈ GL(d ; R), t1 = t1 (p) , t2 = t2 (p) ∈ T (d ; R) for p ∈ M, and“ · ” is the group multiplication of GL(n + 1 ; R). This law of multiplication is the common
one for the semi-direct products. The Lie algebra of this group, denoted by a(n + 1 ; R) =t(n + 1 ; R) ⋊ gl(n + 1 ; R), is given as follows:[Pi , Pj ] = 0

il[Li j , Pk ] = C (A) jk Pl ,[Li j , Lk l ] = C (A)

ikn

jlm L

m

n,

(6.29)

where Pi and Lij are the generator of the Lie groups T (n+1 ; R) and GL(n+1 ; R), respectively.
il

C (A) jk and C (A)iknjlm are structure constants. In this case, C(A) ili l(A) ikn
jk = δ k δ j and C
jlm =δ i l δ k m δ n j −δ i m δ k j δ n l . The affine group A(d ; R) can be regarded as a subgroup of GL(n+2 ; R)as follows:(
AMobius (n + 1 ; R) =



)
g(p)
τ
(p)


 g(p) ∈ GL(n + 1 ; R) , τ (p) ∈ T (n + 1 ; R) . (6.30)
01This representation is called the Möbius representation [Kobayashi (1972); Mielke (1987);
Hehl et al. (1995)]. Then the ordinary matrix product of the elements of AMobius (n + 1 ; R)
restores the semi-direct product defined in Eq (6.28) in as the first row of the result of
the matrix computations. In addition, acting an element Λ(p) ∈ AMobius on T (x , 1), wherex = x(p) is the coordinates of the point p ∈ M, it is revealed that the first component of165

the result is nothing but the affine transformation of the point p as follows: Λ(p) T (x , 1) =T(s(p)x(p) + τ (p) , 1). Therefore, the action of the affine group A(n + 1 ; R) leaves the (n + 1)-

dimensional hyperplane Rn+1 invariant.
In this representation, the affine connection is given as follows [Kobayashi and Nomizu(1996)]:


 

(E)
(T )
(E) i
j
(T ) iω  ω
⊗ Pi 
j ⊗L i ωωω (A) = 
=
,
0
0
0
0

(6.31)

where ω (E) and ω (T ) are a gl(n + 1 ; R)-valued 1-form on M and a t(n + 1 ; R)-valued 1-form
on M, respectively. Then, for a frame transformation ei → e′i = Λi j ej , a(n + 1 ; R)-valued
1-form ω (A) on M is transformed as follows:(6.32)

ω (A) → ω ′(A) = Λ−1 dΛ + Λ−1 ω (A) Λ ,where Λ ∈ AMebius (n + 1 ; R), or equivalently,
ω (E) → ω ′(E) = s−1 ds + s−1 ω (E) s ,
ω

(T )

→ω

′(T )

−1

−1

= s dD t + s ω

(T )

,

(6.33)
dD t = dt + dρ(ω

(E)

)t

where s ∈ GL(n + 1 ; R) and t ∈ T (n + 1 ; R). The first transformation law in Eq (6.33)
indicates that ω (E) is nothing but the Ehresmann connection of the principal GL(n + 1 , ; R)bundle. Therefore, dD t = dt + dρ(ω (E) )t is actually the covariant exterior derivative of t withrespect to the connection ω (E) , where dρ is a representation of the Lie algebra gl(n + 1 ; R).
Similarly, the curvature 2-form is given as follows:

 

(E)
(T )
(E)
(E)
(E)
(T )
(E)
(T )
Ω  dω + ω ∧ ωdω + ω ∧ ω ΩΩ(A) = dω (A) + ω (A) ∧ ω (A) = 
=
.
0
0
0
0
(6.34)

166

Then Ω(A) is a gl(n+1 ; R)-valued 2-form on M. For a frame transformation ei → e′i = Λi j ej ,Ω(A) is transformed as follows:Ω(A) → Ω′(A) = Λ−1 Ω(A) Λ .

(6.35)

This is the same only in the form as Eq (5.13). That is, ω (T ) is not an Ehresmann connectionof the principal T (d ; R)-bundle; the above law is not a gauge transformation law.
Eq (6.3), or equivalently, Eq (6.10) implies the meaning of Ω(T ) = dω (T ) + ω (E) ∧ ω (T ) .
That is, if the ω (T ) is equivalent to the coframe field e−1 , Ω(T ) is nothing but the torsion of
the spacetime manifold M. To confirm this speculation, let us introduce an auxiliary vector
field ξ = ξ i Pi . Then the coframe field e−1 can be expressed as follows:
e−1 = ω (T ) + dD ξ ,(6.36)

where e−1 = ei ⊗ Pi = ei µ dxµ ⊗ Pi and ω (T ) = ω (T ) i ⊗ Pi = ω (T ) i µ dxµ ⊗ Pi . Therefore, if thefollowing condition holds then the statement is shown:dD ξ := 0 .

(6.37)

Under the imposition of the above condition, the affine connection (6.31) and curvature
2-form become as follows [Kobayashi (1972)]:


(E)
−1
e 
ω
ω (C) = 
,
0
0



(E)
T,
Ω
Ω(C) = dω (C) + ω (C) ∧ ω (C) = ,00(6.38)

where Ω(E) and T are givens as follows:
T = de−1 + ω (E) ∧ e−1 = dD e−1 ,Ω(E) = dω (E) + ω (E) ∧ ω (E) = dD ω (E) .
167(6.39)

These two equations are nothing but Cartan’s first and second structure equations. Thisconnection ω (C) is the so-called Cartan connection, although only the part “ ω (E) ” in ω (C)is an Ehresmann connection; the remaining part “ e−1 ” has nothing to do with Ehresmannconnections, therefore, strictly speaking, this use “ connection ” is just an abuse of jargon.
Note that, for a frame transformation ei → e′i = Λi j ej , Ω(E) and T transform as follows:Ω(E) → Ω′(E) = Λ−1 Ω(E) Λ ,T → T ′ = Λ−1 T ,(6.40)

respectively. Remark also that this decomposition is valid only in the satisfaction of Eq (6.37).
The geometry based on the affine group A(n + 1 ; R) = T (n + 1 ; R) ⋊ GL(n + 1 ; R)is, therefore, the so-called (i) Riemann-Cartan geometry, which includes not only curvature
but also torsion. The Riemann-Cartan geometry is denoted by “ Un+1 ”, where n + 1 isthe dimension of the geometry. Un+1 has four subclasses depending on whether or not thecurvature and/or torsion vanishes; (ii-a) “ Tn+1 ”: The curvature vanishes: Ω(E) = 0 then thegeometry turns into teleparallel geometry; (iii) “ Vn+1 ”: The torsion vanishes: T = 0 then thegeometry turns into Riemann geometry; (iv) “ Mn+1 ”: The curvature and torsion vanish:Ω(E) = 0 and T = 0 then the geometry turns into Minkowski geometry. Therefore, theRiemann-Cartan geometry provides geometrical extensions of the general theory of relativity.
In particular, the existence of torsion gives rise to a new subclass, i .e ., Tn+1 , for theories ofgravity.
Next, let us extend the affine group to the so-called Weyl group: W (n + 1 ; R) = D(n +1 ; R) ⋊ A(n + 1 ; R). The Lie algebra of W (n + 1 ; R), i .e ., w(n + 1 ; R) = d(n + 1 ; R) ⋊a(n + 1 ; R), is identified by the affine algebra (6.29) and the following algebra:j[D , Pi ] = C (W ) i Pj ,j[D , Li j ] = 0 ,j(6.41)

where C (W ) i is structure constant. In this case, C (W ) i = δ j i [Charap and Tait (1974)]. The
168

Möbius representation of the Weyl group is given as follows [Tomonari (2023a)]:¯ ( g(p) τ (p) d(p))
WMobius (n+1 ; R) = 
1
0 
 0
 g(p) ∈ GL(n+1 ; R) , τ (p) ∈ T (n+1 ; R) , d¯ ∈ D(n+1 ; R) .


0
0
1
(6.42)
This is of course a subgroup of GL(n + 3 ; R). Then the Weyl connection is defined asfollows [Tomonari (2023a)]: (T ) i
(D)
j
(E) i
(E)
(T )
(D)
⊗ Pi ω̃ ⊗ Dωω  ω
j ⊗L i ω
ω




,
=
ω (A) = 
0
0
0
0
0
0

 


 

0
0
0
0
0
0

(6.43)whereω̃ (D) =1ln(−det(g))e−1 .
n+1(6.44)The same consideration as the case of the affine group leads the following transformation
law [Tomonari (2023a)]:ω (E) → ω ′(E) = s−1 ds + s−1 ω (E) s ,ω (T ) → ω ′(T ) = s−1 dD t + s−1 ω (T ) ,dD t = dt + dρ(ω (E) )t ,

ω (D) → ω ′(D) = s−1 dD d¯ + s−1 ω (D) ,dD d¯ = dd¯ + dρ(ω (E) )d¯(6.45)

for the frame transformation ei → e′i = Λi j ej . Therefore, the d(n + 1 ; R)-valued 1-form ω (D)is not an Ehresmann connection. Alternatively, this valued 1-form becomes a potential togenerate the following geometric quantity [Tomonari (2023a)]:∆ = d∇ ω (D) −111ln(|det(g)|) T =Tr ( d∇ g ) e−1 =Tr ( Q ) e−1 .
n+1
n+1
n+1
169

(6.46)

where “ Tr ” is the trace operator and Q = d∇ g is the non-metricity 2-form defined inEq (6.16). Remark that the commutativity of the generators D and E I J ensures ω (E) ∧ω (D) = 0. This quantity ∆ is the so-called dilation 2-form. Notice that if the torsion 2-form
vanishes: T = 0, Eq (6.46) turn to simply to be [Tomonari (2023a)]∆ = d∇ ω (D) =1
1
Tr ( d∇ g ) e−1 =Tr ( Q ) e−1 ,n+1
n+1(6.47)The torsion-free condition can be realized by taking the auxiliary field ζ in Eq (6.36) as
follows [Tomonari (2023a)]:d∇ ω (T ) + d∇ 2 ζ := 0 .

(6.48)

Remark that d∇ 2 = d∇ d∇ does not vanish unlike the ordinary exterior derivative: d2 = dd =0. Based on the classification of the Riemann-Cartan geometry, such geometry can containthe subclasses Vn+1 and En+1 (or Mn+1 ).
The geometry based on the Weyl group W (n+1 ; R) = D(n+1 ; R)⋊A(n+1 ; R) is richer
than the Riemann-Cartan geomrtry by virtue of the exixtence of the dilation. This geometryis called (v) the Weyl-Cartan geometry and denoted as “ Yn+1 ”. On one hand, Yn+1 contains
Un+1 as a special case of vanishing the dilation: ∆ = 0. On the other hand, Yn+1 gives new
subclasses; (vi-a) “ Wn+1 ”: The torsion vanishes: T = 0 then the geometry turns into the
Weyl geometry; (ii-b) Tn+1 : The curvature vanishes: Ω(E) = 0 then the geometry turns into
one of the teleparallel geometry with the dilation.
Further generalization of the Weyl geometry is possible; the dilation given in Eq (6.46)
implies the existence of a geometry such that the non-metricity tensor is generated from a
covariant exterior derivative of some potential quantity. Consider the metric-affine group:
M A(n + 1 ; R) = S(n + 1 ; R) ⋊ W (n + 1 ; R). The Lie algebra of M A(n + 1 ; R), ma(n +1 ; R) = s(n + 1 ; R) ⋊ w(n + 1 ; R), is identified by the affine algebra (6.29) and the following

170

algebra [Tomonari (2023a)]:il[Di j , Pk ] = C (M A) jk Pl ,ilwhere C (M A) jk and C (M A)
and C (M A)

ikn

jlm = 0.

ikm

[Di j , Lk l ] = 0 ,[Di j , Dk l ] = C (M A)

jln are structure constants.

ikm

jln D

m

n,

(6.49)

il

In this case, C (M A) jk = δ i k δ l j

Notice that this algebra is nothing but a generalization of the Weyl

algebra (6.41); in the case of its dimension being one, Eq (6.49) results in Eq (6.41). The
Möbius representation of the metric-affine group is given as follows [Tomonari (2023a)]:¯
g(p) τ (p) d(p)
(
 0
1
0

M A Mobius (n + 1 ; R) = 
 0
0
1


0
0
0


s(p)
p 

0 

 g(p) ∈ GL(n + 1 ; R) , τ (p) ∈ T (n + 1 ; R) ,
0 


1
)
d¯ ∈ D(n + 1 ; R) , sp ∈ S(n + 1 ; R) .
(6.50)

This is a subgroup of GL((n + 2)(n + 3)/2 + 1 ; R). Then the metric-affine connection is
introduced as follows [Tomonari (2023a)]:

 

(E)
(T )
(D)
(S)
(E) i
j
(T ) i
(D)
(S)i
j
ω
ω
ω  ω
⊗ Pi ω ⊗ D ω j ⊗ D i j ⊗L i ω
ω

 

 0
 

0
0
0
0
0
0
0

 

ω (A) = 
=
 
,
 0



0
0
0  
0
0
0
0



 

0
0
0
0
0
0
0
0
(6.51)

171

whereω (S)i j = ω̃ (S) δ i j ,


1
(S)
ω̃ = g −ln(−det(g)) e−1 .
n+1(6.52)For a frame transformation, ω (S) obeys the same rule as ω (T ) , ω (D) in Eq (6.45). Thenthe s(n + 1 ; R)-valued 1-form ω (S) becomes a potential to generate the following geometricquantity [Tomonari (2023a)]:↗
∆ = d∇ ω

(S)


− g−




1
1
−1
ln(|det(g)|) T = Q e − ∆ = Q −Tr ( Q ) e−1 . (6.53)
n+1
n+1

This quantity ↗
∆ is the so-called shear 2-form. Remark that the commutativity of the generators D and E I J ensures ω (E) ∧ ω (S) = 0. Notice that the case of vanishing torsion turns
Eq (6.53) to be [Tomonari (2023a)]↗
∆ = d∇ ω

(S)

= Qe

−1


−∆= Q−

1
Tr ( Q ) e−1 .
n+1(6.54)Taking into account the dilation 2-form, the non-metricity 2-form is restored as follows;

Q=↗
Q+

1
Tr (Q)
n+1(6.55)

as a 1-form on M, where ↗
Q is defined as ↗
∆ = ↗
Q e−1 under Eq (6.53). Therefore, theexistence of both the dilation and shear 2-forms is equivalent to that of the non-metricity
2-form, and vice versa. Notice, finally, that Eq (6.55) holds not depending on whether or
not Condition (6.48) is satisfied.
The geometry based on the metric-affine group M A(n+1 ; R) = S(n+1 ; R)⋊W (n+1 ; R)is the most generic geometry by virtue of the existence not only the dilation but also the172shear. This geometry is nothing but (vii) the metric-affine geometry and denoted as “ Ln+1”. On one hand, Ln+1 contains Yn+1 as a special case of vanishing the shear: ↗
∆ = 0. On theother hand, Ln+1 gives new subclasses: (vi-b) Wn+1 : The torsion vanishes: T = 0 then thegeometry turns into the most generic subclass of the Weyl geometry with the shear; (viii)“ Sn+1 ”: The torsion and the curvature vanish: T = 0 and Ω(E) = 0 then the geometry
turns into the symmetric telepalallel; (ii-c) Tn+1 : The curvature vanishes: Ω(E) = 0 then thegeometry turns into the most generic subclass of the teleparallel geometry with the dilationand the shear: the non-metricity.
Finally, let us consider the relation between the above classification and the field equations. The action functional is given as follows [Hehl et al. (1989)]:
Z

Z
L vol =IMAG =M

LMAG (gij , ei , ω (E) i j , Qij , T i , Ri j ) vol + Lmatter (gij , ei , Φ , DΦ) vol ,M(6.56)
where ei is the coframe fields, Φ denotes a set of matter fields, and D is the covariant
derivative given in Eq (5.59). “ vol ” is the volume form on M. Varying this action
functional, the following field equations are derived:

∂LMAG
∂LMAG
+2
= −σ ij ,
D 2
∂Qij
∂gij


∂LMAG
∂LMAG
= −Σi ,
D
+
∂T i
∂ei


∂LMAG
∂LMAG
∂LMAG
D 2
+ ei ∧+ 2gjk= −Πi j ,
j
j
∂Ri
∂T
∂Qki


(6.57)

where σ ij , Σi , and Πi j are the metric stress-energy tensor, the canonical energy-momentum
tensor, and the hypermomentum, respectively, given as follows:σ ij = 2

δLmatter
,
δgij

Σi =

δLmatter
,
δei

173

Πi j =

δLmatter.
δω (E) i j(6.58)

The equations of motion of the matter fields Φ are given by the Euler-Lagrange equations as
usual. The connection decomposition (, strictly speaking, the pull back of it by the coframe
field,) in Eq (6.22) implies that the hypermomentum is decomposed as follows:Πij = τ ij +

1 ij
g Π+↗
Πij ,
n+1

(6.59)

where the spin current τ ij and shear current ↗
Πij are given as follows:τ ij =

δL
,
δKij

↗
Πij =

δL
1 ij
= Π(ij) −
g Π,
δ↗
Qij
n+1

(6.60)The dilation current Π is the trace of the hypermomentum, which is also given as follows:

Π=

1 δL
.
2 δQ

(6.61)

Therefore, the gauge structures are engraved also into the hypermomentum structure. Thatis, the metric-affine geometry Ln+1 includes all components: the spin, the dilation, and theshear current. the Weyl-Cartan geometry Yn+1 contains the spin and the dilation current.
The affine geometry Un+1 has only the spin current. In the Riemann geometry Vn+1 , theteleparallel geometry Tn+1 , and the Minkowski geometry Mn+1 , the hypermomentum vanishes.
Summarizing, the most generic class is the metric affine geometry denoted as Ln+1 ofwhich the gauge group is the metric-affine group M A(n + 1 ; R) = S(n + 1 ; R) ⋊ W (n +1 ; R). Ln+1 is described by the three geometric quantities: the curvature R, the torsion T ,and the non-metricity Q and has two subclasses: the Weyl-Cartan geometry Yn+1 , which
is described by the curvature R, the torsion T , and the dilation ∆, and the symmetricteleparallel geometry Sn+1 , which is described by the non-metricity Q. In the viewpointof the equations of motion, the entire class Ln+1 has all the three currents: the spin, the174

dilation, and the shear current. The subclass Yn+1 has its gauge group as the Weyl group
W (n+1 ; R) = D(n+1 ; R)⋊A(n+1 ; R). The possible subclasses of Yn+1 are Weyl geometry
Wn+1 , which is described by the curvature R and the dilation ∆, and the Riemann-Cartan
geometry Un+1 , which is described by the curvature R and the torsion T . In the viewpointof the equations of motion, this subclass Yn+1 has the spin and dilation currents. The
latter geometry Un+1 has its gauge group as the affine group A(n + 1 ; R) = T (n + 1 ; R) ⋊GL(n + 1 ; R) and the three subclasses: the Riemann geometry Vn+1 , which is describedby the curvature R, the teleparallel geometry Tn+1 , which is described by the torsion T ,and the Minkowski geometry Mn+1 , which is described by vanishing all the three geometricquantities. From the viewpoint of the equations of motion, this subclass Un+1 has only thespin current. The subclasses Vn+1 , Tn+1 , and Mn+1 do not have any hypermomentum.
Remark that the geometric classification based on whether or not each of the threegeometric quantities vanish(es) does not take into account the gauge group structure of thetheory. However, notice that the four geometries Vn+1 , Tn+1 , Sn+1 , and Mn+1 are the common
subclasses in the algebraic classification and the geometrical classification. In addition, Vn+1
and Mn+1 are nothing but the geometry of the general and the special theory of relativity,
respectively. Therefore, let us focus on these two subclasses in the next chapter: gravity
theories based on the teleparallel geometry Tn+1 and the symmetric teleparallel geometry
Sn+1 : the so-called geometrical trinity of gravity/general relativity.
Notice, finally, that, if it is possible to further extend the metric-affine gauge group using
a semi-direct product then a new geometric quantity would be discovered, and then it would
also be possible to construct a new subclass of gravity. The investigation of this perspective
is for an intriguing future work including its realizability.2
2

See Appendix C.

175

6.3. Geometrical trinity of gravity
In this section, the so-called geometrical trinity of gravity, of which geometries are taken in aset of subclasses, i .e ., Vn+1 , Tn+1 , Sn+1 , in the metric-affine gauge theories of gravity, is established [Beltrán Jiménez et al. (2019)]. Since these subclasses satisfy the common condition ofvanishing curvature, let us start by introducing the most generic class of the teleparallel gravity: (ii-c) [Beltrán Jiménez et al. (2020a); Gomes et al. (2023b); Bahamonde et al. (2023a);
Heisenberg (2023)]. In order to construct the Lagrangian of the theory, it needs to unveil
the relation between the Ricci scalar of the Levi-Civita connection in the (pseudo-)Riemanngeometry and the geometric quantities in the metric-affine geometry.
The decomposition of the curvature tensor given in Eq (6.25) provides the fundamental
relation to establishing the metric-affine gauge theories of gravity. Contracting Eq (6.25),
the following relation is derived:◦˜ µ (Qµ − Q̄µ + 2T µ ) ,R̃ = R + T + Q + T ρµν Qµνρ − T µ Qµ + T µ Q̄µ + ∇(6.62)

where Qµ = g νρ Qµνρ , Q̄ρ = g µν Qµνρ , T µ = g µν T ρ νρ , and T and Q are defined as follows:
1
T=
2




1
1
Tµνρ + Tνµρ − gµν Tρ T µνρ
42(6.63)and1
1
1
1
Q = Qµνρ Qµνρ − Qµνρ Qνµρ − Qµ Qµ + Qµ Q̄µ ,42
42(6.64)

respectively. Here, we abbreviated “ ˜” for each geometric quantity excepting the Ricci scalar
˜ ”. The latter scalar quantity is also expressed in terms ofand the covariant derivative “ ∇the disformation tensor given in Eq (6.24) as follows:
Q = g µν Lρ ρλ Lλ µν − Lρ λµ Lλ νρ .
176

(6.65)

Notice that the decomposition (6.62) can represent the Ricci scalar in terms of the geometricquantities in the metric-affine gauge theories of gravity. Therefore, generalizing the pseudoRiemannian geometry of the Einstein-Hilbert action functional given in Eq (4.64) into themetric-affine geometry and replacing the Ricci scalar in the action functional by the quantities
in the metric-affine geometry, the following action functional is derived:
IMAG [gµν , Γ̃ρµν , Φ] =

1
2κ2

Z

[R̃ − T − Q − T ρµν Qµνρ + T µ Qµ − T µ Q̄µM˜ µ (Qµ − Q̄µ + 2T µ )] √−g dn+1 x + Imatter [Φ , DΦ] .
−∇(6.66)

This action functional is defined in the tangent bundle of the spacetime manifold. In orderto move to the internal space bundle, it is just to replace the variables gµν and Γ̃ρµν by eI µand ω I Jµ making use of Eq (5.32) and Eq (5.40). In the case of the teleparallel gravity (ii-c),
the vanishing curvature is assumed: Rσ µνρ = 0. Therefore, the action functional becomes asfollows:
IGT EGR [gµν , Γρµν , Φ] =

1
2κ2

Z

[ − T − Q − T ρµν Qµνρ − T µ Qµ − T µ Q̄µM˜ µ (Qµ − Q̄µ + 2T µ )] √−g dn+1 x + Imatter [Φ , DΦ] .
−∇(6.67)
This theory is called the General Teleparallel Equivalnt to General Relativity (GTEGR) andprovides the basics of the geometrical trinity of gravity. Here, notice that the assumption ofvanishing curvature tensor, this condition is called teleparallel condition/teleparallelism inthe metric-affine gauge theories of gravity, leads to the following connection:Γ′ρµν = 0 ,(6.68)

in some coordinate system. Conversely, this connection trivially satisfies the teleparallel177

condition. In addition, for a general coordinate transformation: xµ → x′µ = x′µ (xν ), recall
that the affine connection obeys the following transformation law:Γρµν =∂x′ρ ∂xκ ∂xσ ′λ∂xρ ∂ 2 x′λΓ
+.
∂xλ ∂x′µ ∂x′ν κσ ∂x′λ ∂xµ ∂xν(6.69)◦

This law holds by virtue of the transformation law for the Levi-Civita connection Γρµν andthat both the contorsion and the disformation in Eq (6.22) are tensorial quantities. Under
the imposition of Eq (6.68), the affine connection with the teleparallel condition in a genericcoordinate system is, on one hand, expressed as follows:Γρµν = (Λ−1 )ρ λ ∂µ Λλ ν ,(6.70)

where Λµ ν = ∂xµ /∂x′ν . This connection also satisfies the teleparallel condition. On the other
hand, the relation given in Eq (5.40) always has the following form in some local region:Γρµν = eI ρ ∂µ eI ν

(6.71)

by virtue that any vector bundle has the so-called standard flat connection in some local
region: ω I Jµ = 0. In the case of restricting such local region, the matrix componentsΛµ ν can be replaced by the coframe field components eI µ . In particular, the case that theinternal space bundle is M × Rn+1 makes it possible to identify the index “ λ ” in Eq (6.70)
with the index “ I ” in Eq (6.71) and the matrix components Λµ ν with the coframe fieldcomponents eI µ . In this special case, eI µ , or equivalently, Λµ ν , belongs to the general linear
group GL(n + 1 ; R), which is nothing but the structure group of the internal space bundle.
Hereinafter, we assume this special case.
Next, let us consider the case of vanishing non-metricity tensor: Qρ µν = 0. The GTEGR

178

action functional turns into as follows:
ITEGR [gµν , Γρµν ] =

1
2κ2

Z

[−T − 2∇µ T µ ]√−g dn+1 x + Imatter [Φ , DΦ] .

(6.72)

M

This theory is called Teleparallel Equivalent to General Relativity (TEGR) and implies
the equivalence to the general theory of relativity up to boundary terms [Einstein (1928);
Unzicker and Case (2005)]. Varying this, the following equations of motion are derived:1Tµν − Tgµν = κ2 Tµν , Tµν = (∇ρ + Tρ ) S(µν) ρ + tµν ,2h√iρ(∇ρ + Tρ ) −gS[µ ν] = 0 ,(6.73)

where the symmetric tensors Sρ µν , which is called the torsion conjugate tensor, and tµν aredefined as follows:∂T11= Tρ µν + T [µρ ν] − δρ [µ T ν] ,
ρ
∂T µν
4
2
1
∂T
tµν = µν = S(µ| λκ Tν)λκ − T λκ (µ Sλκ|ν) .
∂g
2Sρ µν =

(6.74)

The first equations in Eq (6.73) has the same form as the Einstein field equations.
The TEGR action functional is also written in terms of (co)frame fields. The assumptionof vanishing non-metricity tensor leads to the following equation:
2ei ρ ∂β ei (µ gν)ρ = ∂β gµν .

(6.75)

This equation is easily solved in terms of coframe field components as follows:gµν = eI µ eJ ν cIJ ,(6.76)

where cIJ is arbitrary constant symmetric tensor. This solution includes the case of Eq (5.32);
179

cIJ is nothing but the internal space metric ηIJ . The torsion tensor is also expressed asfollows:
T ρ µν = 2eρ I ∂[µ eI ν] .

(6.77)

Of course, this relation corresponds to Eq (5.40). Using Eq (5.32), or more generically
Eq (6.76), and Eq (5.40), or equivalently, Eq (6.77), Eq (6.72) becomes as follows:
1
ITEGR [e µ ] = 2
2κ
I

Z




−T − 2DI T I det(e−1 ) dn+1 x + Imatter [Φ , DΦ] .

(6.78)

M

The action functionals Eq (6.72) and Eq (6.78) correspond to each other in a one-to-one
manner. (See Section 5.4 in detail.)
Finally, let us consider the case of vanishing torsion tensor: T ρ µν = 0. The GTEGR
action functional turns into as follows:
ISTEGR [gµν , Γρµν ] =

1
2κ2

Z


√
−g dn+1 x + Imatter [Φ , DΦ] .
−Q − ∇µ (Qµ − Q̄µ )(6.79)

M

This is called Symmetric Teleparallel Equivalent to General Relativity (STEGR) [Nester and Yo(1999)] and implies the equivalence to the general relativity up to boundary terms. Varying
this, the following equations of motion are derived:√11Qµν − Qgµν = κ2 Tµν , Qµν = √ ∇ρ −gP ρ µν + qµν ,2−g
√−gP µν ρ = 0 .
∇µ ∇ν

(6.80)

where the symmetric tensors P ρ µν , which is called the non-metricity conjugate tensor, and180

qµν are defined as follows:

1
1
1
1 ∂Q
1 ρ
ρ
ρ
ρ
ρ
Q
+
Q
+
g
Q
−
g
Q̄
+
δ
Q
=
−
µν
µν
µν
(µ
ν)
2 ∂Qρ µν
42 (µ ν) 44∂Qqµν = µν = P(µ|λκ Qν) λκ − 2P λκ (ν Qλκ|µ) .
∂g
P ρ µν =(6.81)

Note that the non-metricity scalar Q can be expressed by using the non-metricity conjugate
tensor P ρ µν as follows:
Q = Pρµν Qρµν .

(6.82)

The first equations in Eq (6.80) has the same form as the Einstein field equations.
The STEGR action functional is of course written in terms of (co)frame fields by using
Eqs (5.32) and (5.40). This is the same formulation as the TEGR case. Whereas, in the
STEGR case, a specific formulation exists; the theory can be expressed in terms of theso-called Stükelberg fields. To see this, let us focus on a proper solution of the following
equation:∂µ eI ν = 0 .

(6.83)

This equation is derived from the assumption of vanishing torsion tensor and easily solvedin terms of the so-called Stükelberg fields, denoted by ξ I = ξ I (x), as follows:eI µ =∂ξ I.
∂xµ(6.84)

Then the affine connection becomes as follows:∂xρρΓµν = I ∂µ ∂ν ξ I .
∂ξ

181(6.85)

Therefore, the STEGR action functional given in Eq (6.79) becomes as follows:
1
ISTEGR [ξ , e µ ] = 2
2κ
I

I

Z




−Q − DI (QI − Q̄I ) det(e−1 ) dn+1 x + Imatter [Φ , DΦ] .

(6.86)

M

Notice that the above action functional contains the second-order derivative terms of the
Stükelberg fields ξ µ . This is the reason why the action functional includes the variables
eI µ = ∂µ ξ I ; higher-order terms have to be treated as independent variables in its own
right. Such higher-order terms could cause the Ostrogradski ghost modes as explained in
Sections 2.5 and 2.6. However, the work given in [Hu et al. (2023)] showed that the theory
does not suffer from such ghost modes. Notice also that, under the condition of fixing
the internal space metric, the use of the variables eI µ is equivalent to that of the variablesgµν : ISTEGR [ξ I , eI µ ] = ISTEGR [ξ I , gµν ]. Since the original action functional (6.79) includes
the derivatives of the metric tensor only up to first-order, this alternative expression is an
implication of the absence of the Ostrogradski ghost modes.
The STEGR subclass has a special case known as the so-called Coincident General Relativity (CGR) [Beltrán Jiménez et al. (2018)]. This special theory can be derived by imposing
the so-called coincident gauge condition to the STEGR subclass [Beltrán Jiménez and Koivisto(2022)]. In order to clarify this gauge condition, let us go back to Eq (6.85). This equation
is apparently invariant under the following global transformation:
ξ I (x) → ξ ′I (x) = M I J ξ J (x) + ζ I ,(6.87)

where M I J ∈ GL(n + 1 ; R) and ζ I are arbitrary constant (n + 1)-vector components. Since
the internal space bundle is locally isomorphic to U × R, a section on the bundle can be
identified with that on the tangent bundle T M by virtue of the local relation U ×R ≃ T M|U ,where U is an open set of M. This indicates that the internal space indices I , J , K , · · · canbe replaced by the spacetime indices µ , ν , ρ , · · · . Therefore, Eq (6.87) gives the following
182

equation in a local region:
ξ I (x) = M I µ xµ + ζ I .

(6.88)

This relation is equivalent to the case just replacing ξ I (x) by xµ in Eq (6.87) as follows:ξ I (x) = δ I µ xµ + AI ,(6.89)

where AI are arbitrary constant (n + 1)-vector components. That is, the coincident gauge
is nothing but a condition on the Stükelberg fields such that it restricts to those that are
obtained only from an affine transformation of a spacetime point. This indicates also that,based on Eq (6.84), the co-frame fields are obtained from a given spacetime coordinates.
Notice that in this gauge Eq (6.84) turns into eI µ = δ I µ , and it is nothing but the coefficientsof “ xµ ” in Eq (6.89). This is the another signal of the local relation: “ U × R ≃ T M|U” [Tomonari and Bahamonde (2023)]. Under the imposition of the coincident gauge, i .e .,Eq (6.88), Eq (6.85) becomes as follows:Γρµν = 0 .

(6.90)

This indicates that the coincident gauge condition changes the covariant derivative “ ∇ ” tothe partial derivative “ ∂ ”.
The CGR action functional is of course directly obtained from Eq (6.79) just replacing “∇ ” in the action functional by “ ∂ ”. However, this naive replacement unfortunately blinds
the essence of the theory. Therefore, using Eq (6.65), let us express the STEGR action
functional and impose the coincident gauge condition.
ISTEGR [gµν , Γρµν ] =

1
2κ2

Z

 µν ρ √−gL ρλ Lλ µν − Lρ λµ Lλ νρ−g dn+1 x + Imatter [Φ , DΦ] ,M(6.91)

183

where the boundary terms are neglected. Imposing the coincident gauge condition (6.88) andusing the decomposition (6.22), the disformation tensor can be expressed by the Levi-Civita
connection as follows:◦Lρ µν = −Γρµν .

(6.92)

Therefore, the CGR action functional is given as follows:
1
ICGR [gµν ] = 2
2κ

Z

h
◦
i √
◦
◦
◦
−g dn+1 x + Imatter [Φ , DΦ] .
g µν Γρ ρλ Γλ µν − Γρ λµ Γλ νρ

(6.93)

M

Notice that this is nothing but the Einstein-Hilbert action functional without boundary terms.
That is, the dynamics of CGR is equivalent to that of the general relativity.
The theories discussed in this section, i .e ., the geometrical trinity of gravity, so far are
just equivalent subclasses to the general theory of relativity from the viewpoint of the action
functional up to boundary terms. That is, there is no departure from the general theoryof relativity. However, by extending these theories in a proper manner, the situations getchanged; the theories may acquire the possibility to provide explanations for dark energy
and dark matter and also to generate new propagating degrees of freedom and symmetry
breaking.

6.4. Modified theories of geometrical trinity of gravity
In this section, two ways of extension/modification of the geometrical trinity of gravity, i .e .,quadratic extensions and non-linear extensions, are presented. Let us start by investigating
the quadratic extension of TEGR and STEGR.
The TEGR action functional (6.72) (, or equivalently, Eq (6.78),) contains terms up toquadratic with respect to the torsion tensor T ρ µν and its contraction T ρ under neglecting
boundary terms. Therefore, the possible quadratic Lagrangian with respect to these quan-184

tities are given as follows:
1
IqTEGR = − 22κT̂ = c1 Tρµν T

ρµν

Z

√
T̂ −g dn+1 x + Imatter [Φ , DΦ] ,M+ c2 Tµρν Tρµν(6.94)

ρ+ c3 Tρ T ,where c1 , c2 , and c3 are arbitrary constant parameters. Therefore, the extension of TEGR in
this manner forms the theory of a three-parameters family. These three parameters have to
do with the emergence of new degrees of freedom of the theory [Blixt et al. (2019b,a, 2021);
D’ambrosio and Heisenberg (2021); Heisenberg (2023)]. However, a complete Hamiltonian
analysis has not been performed yet at the moment of writing this thesis. Remark that in thecase of c1 = 1/4 , c2 = 1/2 , c3 = −1 the theory results in the TEGR action functional (6.72)(, or equivalently, Eq (6.78)). Varying Eq (6.94), the following field equations are derived:1T̂µν − T̂gµν = κ2 Tµν , T̂µν = (∇ρ + Tρ ) Ŝ(µν) ρ + t̂µν ,2√ρ(∇ρ + Tρ )−g Ŝ[µ ν] = 0 ,(6.95)

where Ŝρ µν and t̂µν are defined as follows:Ŝρ µν =∂ T̂
= c1 Tρ µν + c2 T [µρ ν] + c3 δρ [µ T ν] ,
ρ
∂T µν

∂ T̂
1
t̂µν = µν = Ŝ(µ| ρκ Tν)ρκ − T ρκ (µ Ŝρκ|ν) .
∂g2(6.96)

Eq (6.95) is the field equations with respect to the metric tensor and to the connection,
respectively. Notice that in the case of c1 = 1/4 , c2 = 1/2 , c3 = −1 the theory results in theequations of motion of TEGR: Eq (6.73) with Eq (6.74).
In the same way as the TEGR case, the STEGR action functional (6.79) (, or equivalently,

185

Eq (6.86),) can be extended into the theory of a five-parameters family as follows:
1
IqSTEGR = − 22κQ̂ = c1 Qρµν Q

ρµν

√

Z
Q̂

−g dn+1 x + Imatter [Φ , DΦ] ,(6.97)

M+ c2 Qµρν Q

ρµν

ρ

ρ

ρ

+ c3 Qρ Q + c4 Q̄ρ Q̄ + c5 Qρ Q̄ ,where c1 , c2 , c3 , c4 , and c5 are arbitrary constant parameters. Notice that the fourth term
“ Q̄ρ Q̄ρ ” is a proper term in this extended theory; the original action functional (6.79)(, or equivalently, Eq (6.86)) does not contain this sort of term. These parameters relate
to new degrees of freedom of the theory [D’ambrosio and Heisenberg (2021); Heisenberg(2023)]. However, a complete Hamiltonian analysis has not been performed yet at the
moment of writing this thesis. Remark that the case of c1 = 1/4 , c2 = −1/2 , c3 = −1/4 , c4 =0 , c5 = 1/2 results in the STEGR action functional (6.79) (, or equivalently (6.86)). Varying
Eq (6.97), the following field equations with respect to the metric tensor and the connection
are derived:
√

ρ
2
1
√ ∇ρ
−g P̂ µν + q̂µν − Q̂gµν = κ2 Tµν ,
−g
2
√

µν
∇µ ∇ν
−g P̂ ρ = 0 ,(6.98)

respectively, where P ρ µν and q̂ µν are defined as follows:ρP̂ µν =q̂ µν =


1
1 ∂ Q̂
ρ= c1 Qρ µν + c2 Q(µρ ν) + c3 gµν Qρ + c4 δ ρ (µ Qν) + c5 gµν Q̄ + δ ρ (µ Qν) ,
µν
2 ∂Qρ
2

∂ Q̂
= P̂(µ|λκ Qν) λκ − 2P̂ λκ (µ Qλκ|ν) .
∂g µν(6.99)The relation “ Q̂ = P̂ ρµν Qρµν ” also holds. Notice that the case of c1 = 1/4 , c2 = −1/2 , c3 =−1/4 , c4 = 0 , c5 = 1/2 results in the STEGR field equations: Eq (6.80) with Eq (6.81).
Finally, let us consider the non-linear extension of TEGR and STEGR. The main idea186

of this extension is the same known as the so-called f (R)-gravity [Buchdahl (1970)]:
1
IfGR [gµν ] = 2
2κ

Z

◦

f (R)

√

(6.100)−g dn+1 x + Imatter [Φ , DΦ] .

M

Varying this, the following field equations are derived:◦◦
1 ◦f ′ (R)Rµν − f (R)gµν + (gµν □ − ∇µ ∇ν )f ′ (R) = κ2 Tµν ,2◦◦(6.101)◦

where f ′ (R) = df (R)/dR. This extension contains the Einstein-Hilbert action functional (4.64)◦◦

as a special case of f (R) = R; this is nothing but a linear case. Further, the case of◦◦f (R) = R − 2Λ leads to the following equations:(6.102)Gµν + Λgµν = κ2 Tµν .

This is nothing but the Einstein field equations with the cosmological constant: Eq (4.35).
◦

◦

◦

◦

Therefore, non-linear extensions need to be f ′′ (R) ̸= 0, where f ′′ (R) = d2 f (R)/dR2 .
In the TEGR case, the extended action functional is given as follows:
1
IfTEGR = − 2
2κ

Z
f (T)

√

−g dn+1 x + Imatter [Φ , DΦ](6.103)

M

Varying this, the field equations with respect to the metric tensor and the connection are
given as follows:
1(∇ρ + Tρ ) f ′ (T)S(µν) ρ + f ′ (T)tµν − f (T)gµν = κ2 Tµν ,2(∇ρ + Tρ ) f ′ (T)S[µρ ν] ,(6.104)

respectively. In the case of f (T) = T, the theory results in the TEGR case: Eq (6.73). The

187

first equations are rewritten by using the Einstein tensor Gµν as follows:f ′ (T)Gµν −1(f (T) − f ′ (T)T) gµν + f ′′ (T)S(µν) ρ ∂ρ T = κ2 Tµν .
2(6.105)The cases of f (T) = T and f (T) = T − 2Λ results in the Einstein field equations (4.28)
and (4.35), respectively. In particular, even if a non-linear case, it is possible to obtain the
Einstein field equations with the cosmological constant (4.35) as an effective theory: let us
consider the case of T = T0 = constant . Then the third term in Eq (6.105) vanishes, andthen the following equations are derived:(6.106)Gµν + Λeff gµν = κ2 Teff µν ,where Λeff and Teff µν are defined as follows:Λeff = −1
2f ′ (T0 )(f (T0 ) − f ′ (T0 )T0 ) ,Teff µν =1
Tµν .
f (T0 )

(6.107)◦

Notice that this sort of effectiveness is absent in the f (R)-gravity. In order to understand thisdifference, the Dirac-Bergmann analysis is available. In fact, the authors in [Blagojević and Nester(2020)] unveiled that the theory has a specific sector (s4) in their notation and the corresponding term emerges caused by the difference of the constraint structure by virtue of theviolation of the local Lorentz symmetry.
In the STEGR case, the extended action functional is given as follows:
1
IfSTEGR = − 22κZ√f (Q) −g dn+1 x + Imatter [Φ , DΦ]M

188(6.108)

Varying this, the following field equations are derived:√12√ ∇ρ −gf ′ (Q)P ρ µν + f ′ (Q)qµν − f (Q)gµν = κTµν ,
−g
2

√
∇µ ∇ν
−gf ′ (Q)P µν ρ = 0 ,(6.109)with respect to the metric tensor and the connection, respectively. The former equations
can be rewritten as follows:f ′ (Q)Gµν −1(f (Q) − f ′ (Q)Q) gµν + 2f ′′ (Q)P ρ µν ∂ρ Q = κ2 Tµν .
2(6.110)

These equations are also resemble structure to the non-linear extension of the TEGR case;(i) The case of f (Q) = Q and f (Q) = Q − 2Λ results in the Einstein field equations (4.28)
and (4.35), respectively; (ii) The case of Q = Q0 = constant leads to an effective field
equations: Eq (6.106) with the following Λeff and Teff µν :Λeff = −1
2f ′ (Q0 )(f (Q0 ) − f ′ (Q0 )Q0 ) ,Teff µν =1
Tµν .
f (Q0 )

(6.111)◦

Notice also that this sort of effectiveness is absent in the f (R)-gravity. Similar to the nonlinear extension of TEGR, the Dirac-Bergmann analysis would unveil the reason for thedifference from the viewpoint of constraint structures.

6.5. Summary
In this chapter, the metric-affine gauge theories of gravity were constructed. There wereeight possible geometries depending on the corresponding Möbius representation of eachgauge group. In particular, the subclasses that are called the geometrical trinity of gravity, i .e ., GR (Riemann), TEGR (Teleparallel Equivalent to GR), and STEGR (SymmetricTeleparalle Equivalent to GR) are important since these subclasses are equivalent to one189another up to boundary terms in the meaning of action functional and include general relativity. These subclasses satisfy the teleparallel condition, or equivalently, the Weitzenböch
gauge condition. STEGR includes a special subclass so-called Coincident GR (CGR) which
satisfies the coincident gauge. In addition, if it is possible further to extend the metric-affine
gauge group using a semi-direct product then a new geometric quantity would be discovered,and then it would also be possible to construct a new subclass of gravity. The investigationof this perspective is for an intriguing future work. In the next chapter, the Dirac-Bergmannanalysis will be performed for each subclass and its non-linear extension to unveil those
propagating degrees of freedom.

190

Chapter 7
Dirac-Bergmann analysis of geometrical trinityof gravity and its extension
Chapter abstract
In this chapter, the Dirac-Bergmann analysis is performed for each subclass of the geometricaltrinity of gravity and its non-linear extension. In order to apply the analysis, which wasintroduced in Section 2.2 but restricted to point particle systems, to field theories, threefundamental issues are clarified. These issues are commonly caused by the spatial derivativesof the Dirac delta function. And then a prescription that circumvents the issues is proposed.
Based on the prescription, the analysis is performed for TEGR, CGR, non-linear extensionsof GR, TEGR, and CGR, and the propagating degrees of freedom of each theory is unveiled.
Throughout these analyses, the three issues in field theories of the Dirac-Bergmann analysis
are concretely clarified for each subclass, and the reason why the prescription is valid forremoving the issues is investigated. I emphasize that these analyses are not a complete
manipulation of the issues; it is just a prescription to circumvent the issues and make itpossible to perform the Dirac-Bergmann analysis. Therefore, the results could be valid only
for some classes of solutions. Finally, this chapter is summarized with the outlook for theseworks.

7.1. Issues of Dirac-Bergmann analysis in field theories and a prescription
In this section, three issues concerning spatial derivative terms in Poisson brackets are clarified. A correct understanding of these issues is important when applying the Dirac-Bergmann
191

analysis to field theories. A prescription that settles these issues is also given. Let us first
scrutinize each step of the extension of the analysis into field theories.
The Dirac-Bergmann analysis of the general theory of relativity, which is one of theanalyses in field theories, was already performed in Section 4.6. The analysis revealed that thetheory has the number of (n+1) gauge degrees of freedom (gDoF) and that of (n+1)(n−2)/2
propagating degrees of freedom (pDoF). The former result is reasonable since this number
is the same as that of the maximal and independent generators of a diffeomorphism. The
validity of the latter result can also be confirmed by using the Einstein field equations of◦◦

motion (4.28) and the Bianchi identity: ∇[ρ| Rµν]σκ = 0. That is, contracting the Bianchi◦◦

identity, ∇µ Gµν = 0 is derived, and it can be expanded as follows: ∂0 G0ν + ∂i Giν + Γiiρ Gρν +◦Γνµρ Gµρ = 0. The second, third, and fourth terms include the time derivative terms up tosecond-order, therefore, the first term should be the same order with respect to the time
derivative. This indicates that G0ν contains the time derivative terms up to first-order; theEinstein field equations with respect to time-space components become constraint densities,
and that number is n + 1. Therefore, the number of pDoF is (n + 1)(n + 2)/2 − (n + 1) −(n + 1) = (n + 1)(n − 2)/2. However, it is difficult to perform this sort of degrees of freedom
counting when the theory gets complicated including the geometrical trinity of gravity andits extensions. In such cases, a systematic method would be a powerful tool to achieve
this purpose. The Dirac-Bergmann analysis provides this convenience although we have tomanage to do with the following issues.
The extension of the Dirac-Bergmann analysis into field theories is achieved in a straightforward manner as usual; it just introduces density variables and those fundamental Poissonbrackets (PBs). That is, integrating field variables φa (t , x), denote it as Φa (t), on a hypersurface Σt in the spacetime manifold, it just regards as variables in point particle systems asfollows:aZΦ (t) =

φa (t , x)

Σt

192

p
|h| dn x ,(7.1)

where a stands for schematic index of the fields and h is the determinant of the metric hijpof Σt . Then, in this formulation, the fields φa (t , x) |h| are, in particular, called density
variables. The Lagrangian density of a given system is composed in terms of these fields likep
L(φa , ∇µ φa , · · · ) |g|, and the action functional is defined as follows:
a

Z t2
p
n
L(Φa (t) , (∇µ Φa )(t) , · · · ) dt , (7.2)
L(φ , ∇µ φ , · · · ) |g| d x dt =Z t2 ZaI[ϕ (t)] =t1

where |g| = N

a

t1

Σt

p
|h| is the determinant of the metric tensor gµν of the spacetime manifold.

Then the canonical momentum density variables are introduced as follows:

πa (t , x) =

δI
.
δ φ̇a

(7.3)

In this formulation, therefore, φa play the role of configuration variables. Then the fundamental PBs are introduced as follows:(7.4){φa (t , x) , πb (t , y)}P.b. = δ a b δ (n) (x − y) .

In the right-hand side, there is no time dependence of φa and πa on the same leaf Σt . Here,the PB itself is defined as follows:aa{F (φ , πb ) , G(φ , πb )}P.b. =

Z X
δF δG
Σt

c

δG δF
− c
c
δφ δπc δφ δπcdn z ,(7.5)

where F and G are functional with respect to the canonical variables φa and πa . Basedon the above formulation, the Dirac-Bergmann analysis is applied to field theories. An
important example is of course the analysis of the general theory of relativity. The variables
φa correspond to the lapse function, N = N (t , x), the shift vector, N i = N i (t , x), and theinduced metric, hij = hij (t , x). The corresponding canonical momentum variables, π0 (t , x),193

πi (t , x), and πij (t , x) are given as Eq (4.68) in Section 4.5. The fundamental PBs are definedas Eq (4.95).
In order to investigate the existence of primary constraint densities, it is necessary to
introduce the Hessian/Kinetic matrixp
Kab (x , y) |h| =δδabδ φ̇ (x) δ φ̇ (y)aZL(φa (z) , ∇µ φa (z) , · · · )p|h| dn z

Σta= Kab (φ (x) , ∇µ φ (x) , · · · ) δ

(n)

(7.6)

(x − y) ,and to scrutinize the rank of the matrix density: Kab (φa (x) , ∇µ φa (x) , · · · ) . If the matrix
density is degenerate and has its rank of n + 1 − r then there exist the number of r primaryconstraint densities. Let us denote these primary constraint densities as follows:
C(1)
(1)
a
∗ ϕα1 (φ (t , x) , πb (t , x)) ≈ 0 ,(7.7)

where α1 ranges 1 , 2 , · · · , r. Then the total Hamiltonian density is introduced as follows:a
HT (φa (t , x) , πb (t , x)) = H0 (φa (t , x) , πb (t , x))+λα1 (φa (t , x) , πb (t , x)) ∗ ϕ(1)α1 (φ (t , x) , πb (t , x)) ,(7.8)
or abbreviating the arguments with respect to the support manifold (the spacetime manifoldM),a
HT (φa , πb ) = H0 (φa , πb ) + λα1 (φa , πb ) ∗ ϕ(1)α1 (φ , πb ) ,(7.9)

where λα1 = λα1 (ϕa , πb ) are Lagrange multipliers. Remark that the variable dependence.
That is, if it was λα1 = λα1 (x) = λα1 (t , x) then these variables are not the Lagrange multipliers; these variables are instead auxiliary field variables, although it also is mathematicallyinappropriate. A great amount of authors mistakes in this regard. Remember over andover again that we are considering the extremum problem in the phase space but not in194the spacetime manifold. That is, for each spacetime point x, we consider the first-ordervariation of the total Hamiltonian density with respect to φa (x) → φa (x) + δφa (x) andπa (x) → πa (x) + δπa (x): not to the spacetime points. As an example of these quantities,
in the case of the general theory of relativity, the Hessian matrix Kab is given by Eq (4.96),
and its rank is six. Therefore, there are four primary constraint densities: Eq (4.69), andthe total Hamiltonian is derived as Eqs (4.98) and (4.97).
So far, every extension of ingredients in the Dirac-Bergmann analysis to field theories has
been straightforwardly achieved, and no problematic situation has occurred. The next stepis, of course, the computations of consistency conditions for the derived constraint densities.
In this step, we will encounter three issues with the analysis of field theories [Sundermeyer(1982); Blagojević and Nester (2020); D’Ambrosio et al. (2023); Tomonari and Bahamonde(2023); Heisenberg (2023)]. Let us consider the consistency conditions of the primary constraint densities as follows:(1)a∗ ϕ̇α1 (φ (t , x) , πb (t , x))=C(1) 

≈(1)aa∗ ϕα1 (φ (t , x) , πb (t , x)) , HT (φ (t , y) , πb (t , y)) P.b.

p

(1)
β1
a
a
a
∗ ϕα1 (φ (t , x) , πb (t , x)) , H0 (φ (t , y) , πb (t , y)) P.b. + λ (φ (t , y) , πb (t , y)) Dα1 β1 (x , y)(7.10)
where “ := ” denotes the equation as an imposition and Dα1 β1 (φa , πb ) is defined as follows:Dα1 β1 (x , y)

p

|h| =

n

o
(1)
(1)
a
a
ϕ
(φ
(t
,
x)
,
π
(t
,
x))
,
ϕ
(φ
(t
,
y)
,
π
(t
,
y))
∗ α1
b
∗ β1
b

P.b.

.

(7.11)

This matrix density is called the Dirac matrix density. The Dirac matrix in point particle
systems is already defined in Eq (2.27). Remark that if λβ1 (φa (t , x) , πb (t , x)) are not LaC(1)grange multipliers but auxiliary fields λβ1 (t , x) then the weak equal “ ≈ ” is just an equal“ = ”. This is easily verified from the definition of Poisson bracket (7.5). Therefore, needless
195|h| := 0 ,to say, Eqs (7.10) and (7.11) are defined on the phase space: not on the support manifold(spacetime manifold) itself. In field theories, Eq (7.11) can be schematically calculated asfollows:
Dα1 β1 (x , y)

p
|h| =

a
(n)
a
(1)
(x − y)f1 (∗ ϕ(1)α1 (φ (x) , πb (x)) , ∂i∗ ϕα1 (φ (x) , πb (x)) , · · · )α1 β1 δ
a
(1)
a
c
a
c
a
+ f2 (∗ ϕ(1)α1 (φ (x) , πb (x)) , ∂i∗ ϕα1 (φ (x) , πb (x)) , · · · ; φ (φ (x) , πb (x)) , ∂i φ (φ (x) , πb (x)) , · · · ;πc (φa (x) , πb (x)) , ∂i πc (φa (x) , πb (x)) , · · · )α1 β1 δ (n) (x − y)
(i)

(x)

a
(1)
a
(n)
+ g1 (∗ ϕ(1)(x − y)α1 (φ (x) , πb (x)) , ∂i∗ ϕα1 (φ (x) , πb (x)) , · · · )α1 β1 ∂i δ
(i)

a
(1)
a
c
a
c
a
+ g2 (∗ ϕ(1)α1 (φ (x) , πb (x)) , ∂i∗ ϕα1 (φ (x) , πb (x)) , · · · ; φ (φ (x) , πb (x)) , ∂i φ (φ (x) , πb (x)) , · · · ;(x)πc (φa (x) , πb (x)) , ∂i πc (φa (x) , πb (x)) , · · · )α1 β1 ∂i δ (n) (x − y)
+ ···
(i)

(x)

a
(1)
(n)
a
+ h1 (∗ ϕ(1)(x − y)α1 (φ (y) , πb (y)) , ∂i∗ ϕα1 (φ (y) , πb (y)) , · · · )α1 β1 ∂i δ
(i)

a
(1)
a
c
a
c
a
+ h2 (∗ ϕ(1)α1 (φ (y) , πb (y)) , ∂i∗ ϕα1 (φ (y) , πb (y)) · · · ; φ (φ (y) , πb (y))) , ∂i φ (φ (y) , πb (y)) , · · · ;(x)πc (φa (y) , πb (y)) , ∂i πc (φa (y) , πb (y)) , · · · )α1 β1 ∂i δ (n) (x − y)(7.12)(i)(i)

where we abbreviated “ (t , x) ” as “ x ”. f1 , g1 , and h1 are functionals that are composed
(i)

(i)

only of the primary constraint densities and those spatial derivatives. f2 , g2 , and h2 arefunctionals that are constituted not only of the primary constraint densities and those spatial
derivatives but also of the canonical variables φa , πb , and those spatial derivatives, such thatnot vanishing under the imposition of the primary constraint densities. In particular, if
(i)

(i)

f1 , g1 , and h1 do not contain any spatial derivatives then these terms vanish under the
(i)

(i)

imposition of the primary constraint densities. The terms including h1 and h2 give rise to
(i)

(i)

total derivative terms. The terms including g1 and g2 give rise to the same sort of terms

196

(i)

(i)

as “ h1 ∂i δ(·) ” and “ h2 ∂i δ(·) ”, respectively, and as “ f1 δ(·) ” and “ f2 δ(·) ”, respectively.
Therefore, Eq (7.12) can be written schematically as follows:Dα1 β1 (x , y)

h
i
p
(i)
|h| = Fα1 β1 δ (n) (x − y) + ∂i Hα1 β1 δ (n) (x − y) ,

(i)

(1)

(7.13)

where Fα1 β1 and Hα1 β1 are specific functionals of ∗ ϕα1 , φa , πa , and those spatial derivatives.
So far, only primary constraint densities have been considered but this manipulation works(i)as well in the same manner for higher-order constraint densities. Remark that if g1 and(i)g2 are composed only of the constraint densities up to those linear combinations then the
manipulation of integral by parts does not need since, just imposing the constraints, thesefunctionals vanish. The PB-algebras (4.99) in Section 4.6 of the general theory of relativity
has this property.
Now, let us clarify the issues of the Dirac-Bergmann analysis in field theories. The first isp
sue relates to the existence of the inverse matrix of the Dirac matrix density Dα1 β1 (x , y) |h|;hi(i)the second term ∂i Hα1 β1 δ (n) (x − y) in Eq (7.13) generically leads to the following equations:ZDα1 γ1 (x , z)p   −1 γ β
p −1  n|h| (D ) 1 1 (z , y) |h|d z(7.14)

Σt

=δ

(n)

(x − y)δα1

β1+ total divergent terms on ∂Σt ,where ∂Σt denotes the boundary of the leaf Σt . That is, the total divergent terms on ∂Σt
may disturb the well-defined calculation of the inverse matrix. That is, the question hereis the method to define the inverse matrix. More fundamentally, “the first issue concerns
the method to define the degeneracy/rank of the matrix under the existence of the boundary
terms.” Since the rank determines the total number of primary constraint densities, therefore,
this issue cannot be ignored. It is also crucial in particular for a system that has second-classconstraint densities since the definition has something to do with determining the multipliers.
197

The second issue relates to the mathematical meaning of the density equations. In orderto obtain the mathematical meaning of Eq (7.13) in a well-defined manner, the density
equations should be integrated over on a leaf Σt with respect to x and y as follows:
Z Z
Σt

p
Dα1 β1 (x , y) |h| dn xdn y =

Z

(1)
a
n
∗ ϕα1 (φ (t , x) , πb (t , x))d x ,

Σt

Σt

Z
=

Z
Σt

F̃α1 β1 dn x +

Σt

h

(i)
ni H̃α1 β1

i

(1)
a
n
∗ ϕβ1 (φ (t , y) , πb (t , y))d y

P.b.

,

∂Σt

(7.15)
where ∂Σt denotes the boundary of the leaf Σt . Here, remark that, when integrating over(i)

Eq (7.13), there may occur the cancellation between the terms from Fα1 β1 and Hα1 β1 . Taking
(i)

(i)

into account this cancellation, Fα1 β1 and Hα1 β1 are redefined as F̃α1 β1 and H̃α1 β1 , respectively.
hi(i)
Excepting the last term ni H̃α1 β1, all other terms are the so-called smeared variables.
∂ΣtThat is, the boundary terms may disturb to make the smeared algebras closed. To makeC(1)matters worse, if Fα1 β1 ≈ 0, the existence of the boundary terms relates directly to whether
C(1)

(1)

or not a primary constraint density, ∗ ϕα1 (φa (t , x) , πb (t , x)) ≈ 0, is classified into first-classconstraint: if the boundary terms are absent then there is a possibility to be the primaryconstraint density as a first-class one. Furthermore, even if non-vanishing boundary terms do
not appear in the consistency conditions for the primary constraint densities, that for higherorder constraint densities would face the same problem. That is, “the second issue is whether
should the boundary terms be also taken into account when classifying a constraint into either
first- or second-class constraint density.” This issue is the most crucial for counting degrees
of freedom. As a special case, we can use the normal vector given in Eq (4.41), and thenthe boundary terms in Eq (7.15) always vanish. In this case, another issue occurs; which
equation, either Eq (7.13) or Eq (7.15), should be applied for the classification? Furthermore,this issue implies that the constraint structure of a given theory depends on the way of taking
a foliation. For a diffeomorphism invariant theory, this property is at a glance pathological.
198

How to resolve this situation? These are also the second issue.
The final issue relates to determining the Lagrange multipliers. Applying the schematic
relation (7.13) to the consistency conditions (7.10), the following schematic equations arederived:⃗ δ (n) (x − y) = ⃗0 ,A(i) ∂i⃗λ + B⃗λ + C(7.16)

or equivalently, integrating over with respect to either x or y on a leaf Σt ,⃗ = ⃗0 ,A(i) ∂i⃗λ + B⃗λ + C(7.17)⃗ are the matrices and the vector that are determined by the given action
where Ai , B, and C
functional (7.2) and composed of the phase space variables φa , πb and those spatial derivatives, respectively. ⃗λ is the column matrix of which components are the Lagrange multipliershi(i)
λα1 (φa (t , x) , πb (t , x)) given in Eq (7.10). Notice here that the terms “ ∂i Hα1 β1 δ (n) (x − y)” in Eq (7.13) give rise to the first term “ Ai ∂i⃗λ ” from the second terms in Eq (7.10) due tothe Leibniz law. Explicitly, Eq (7.17) can be expanded into as follows:
"
A(i)

#
⃗λ
∂⃗λ
∂
⃗ = ⃗0 .
∂i φa +
∂i πb + B⃗λ + C∂φa∂πb(7.18)On one hand, if A(i) = 0 then the above equation becomes an ordinary linear algebraic equation, and the situation is the same as the point particle systems as explained in Section 2.2;
the multipliers λα1 as the same number of the rank of the matrix B are determined, andotherwise secondary constraint densities occur. On the other hand, if the rank of A(i) is
not zero then the situation gets changed; the above equation turns into partial differential
equations of functionals (PDEs of functionals) with respect to the phase space variables φaand πb . When the PDEs are solved, of course, the function forms of the Lagrange multipliers
are determined, and then the equation generically turns into the differential equations with199

respect to the phase space variables themselves. That is, the above equation determines
φa and πb in terms of the coordinate variables x on the leaf Σt . However, the equations of
motion also determine the variables φa and πb in terms of the coordinate variables x (andt). Therefore, “the third issue is whether or not the solved phase space variables φa and πb inthese two different ways are always consistent.” Remark that, hereinafter, we use the term“solvable” for such PDEs in the means that all the Lagrange multipliers are expressed interms of the phase space variables under the satisfaction of this consistency.
These three issues of the Dirac-Bergmann analysis in field theories are commonly caused
hi(i)by the problematic terms “ ∂i Hα1 β1 δ (n) (x − y) ” in Eq (7.13). Therefore, it is mandatory
to unveil a valid treatment of the problematic terms. Furthermore, this problem relates
directly to the well-posedness of PBs. Based on the considerations in the previous threeparagraphs, the most naive treatment to settle these issues is to discard the problematic
terms. That is, if it is possible, Eq (7.14) turns out to give a definition of the inverse matrixof the Dirac matrix density in the usual manner just like Eq (7.6) and then the total numberof first-class constraint densities becomes maximum number (See Section 2.2); Eq (7.15)
plays a role in classifying a constraint into first- or second-class constraint in a usual mannerjust like the point particle cases; Eq (7.17) becomes an ordinary linear algebraic equations.
Since the consistency conditions have to be calculated by using the well-defined PB-algebras
in advance, it makes sense to consider the discard of the problematic terms in the stepof calculating the PB-algebras. Remark, here, that, in the general theory of relativity,
all primary constraint densities are commutative with respect to the PB-algebras; the termshi(i)
Fα1 β1 in Eq (7.13) vanish and the terms “ ∂i Hα1 β1 δ (n) (x − y) ” in Eq (7.13) do not appear.
Therefore, the issues above do not occur.
In order to determine PB-algebras in a well-defined manner, the relations (7.13) should
be considered by integrating over. That is, Eq (7.15) has a key for the well-definedness. Inhi(i)particular, the possibility of vanishing the boundary terms “ ni Hα1 β1” in the equations
∂Σt

200

would relate to whether or not the problematic terms could be discarded. Then the possible
reasons to make the naive treatment of the spatial boundary terms mentioned in the previous(i)paragraph would be that Possibility (i): If the spatial normal vector ni is orthogonal to Hα1 β1on the leaf Σt then the boundary terms vanishes (This include the specific use of Eq (4.41));
Possibility (ii): The boundary terms could be vanished under the imposition of appropriate
spatial boundary conditions to φa and/or πa . In the latter case, the variables that are fixed
on the spatial boundary ∂Σt should be unphysical variables. Possibility (iii): The spacetime
manifold is foliated into leaves with no boundaries. In particular, it seems that Possibility (ii)is the most realizable and applicable to the wide range of theories. Extending this statement
into all other PBs that appear in the Dirac-Bergmann analysis, it is reasonable since thewell-definedness is a mathematical property although this statement needs to be proven, in
this thesis, let us assume that the following prescription [Tomonari and Bahamonde (2023)]:3

Prescription 1.
For functionals F (i) on a phase space which are coordinated by φa (t , x) and πa (t , x), the term“ ∂i F (i) δ (n) (x − y) ” in PB-algebras can be discarded by setting properly spatial boundary
conditions of the “unphysical variables” in φa (t , x) and πa (t , x) on the boundary of a leaf
Σt . Here, the index i runs from 1 to the dimension of a leaf Σt , i .e ., n, and h are thedeterminant of the metric on a leaf Σt .
■Here, “unphysical” means that a variable corresponds to a constraint in the conjugate manner
3
In the original paper [Tomonari and Bahamonde (2023)], the statement was deserted for the terms likep(x)“h(x)A(x)∂i δ (n) (⃗x − ⃗y ) ”. In the current thesis, we already integrated by parts this sort of term in
p
(x)
(x) p
Eq (7.13). That is, if we expand as h(x)A(x)∂i δ (n) (⃗x − ⃗y ) = ∂i ( h(x)A(x)δ (n) (⃗x − ⃗y )) − δ (n) (⃗x −(x) p⃗y )∂i ( h(x)A(x)) then the first term is nothing but ones that have been considered in the current thesis.
(The second term is encapsulated into the first term of Eq (7.13).) The important point here is that these
two statements are equivalent since, in order to evaluate the PBs, we integrate them over twice; every term
above turns into a spatial boundary term and appropriate spatial boundary conditions erase them.

201and we assume that the classification of constraint densities into either first- or second-classis determined by the use of Eq (7.15). This makes sense since the Dirac delta function has
its mathematical meaning in a well-defined manner when integrated over and since before
performing the calculation for determining multipliers the PB-algebras should be given in awell-defined manner in advance.
The possibilities (i) (excepting the specific use of Eq (4.41)) and (iii) could also be viable
but it would strongly depend on each theory. Therefore, in this thesis, we do not assume
these specific possibilities. Remark, here, that, in the case of the general theory of relativity, the secondary constraint densities given in Eq (4.100) contains the spatial derivative of
the Dirac delta function. However, these terms are not problematic in this meaning; thecoefficients of these terms are composed of the secondary constraint densities, as alreadymentioned. Therefore, the imposition of the secondary constraint densities removes themwithout applying Prescription 1. Notice, finally, that the imposition of spatial boundary
conditions restricts the solution space of a given theory [Heisenberg (2023)]. That is, theDirac-Bergmann analysis based on Prescription 1 gives the propagating degrees of freedom
for the restricted solutions by the spatial boundary condition. For example, let us consider
the Schwarzschild and FLRW spacetime as two solutions of the Einstein field equations. If
we impose the asymptotically flatness as a spatial boundary condition then FLRW spacetime no longer belongs to the solution space of the Einstein field equations. Therefore, incompensation for the use of Prescription 1, the Dirac-Bergmann analysis does not always
unveil all propagating degrees of freedom of a given theory. This is the meaning of the word,“prescription”, and it guarantees the reliable results of the Dirac-Bergmann analysis that do
not suffer from the three issues instead of its generality.
There is an important remark: the specific choice of the normal vector given in Eq (4.41)
always satisfies Prescription 1. For a diffeomorphism theory, on one hand, since taking an
ADM-foliation itself decomposes a spacetime manifold into another one which has a topology
202

R×S in a diffeomorphic manner (See Section 4.4 in detail) and this manipulation is equivalent
to a choice of a coordinate system, without any loss of generality, it means that Prescription
1 can be applied and always provide the full pDoF of a given theory. On the other hand, fora theory with violating its diffeomorphism invariant characteristics, the theory genericallydepends on coordinate systems, or equivalently, the choices of ADM-foliations. It indicatesthat the theory generically has several sectors and each sector generically derives a different
pDoF from that of the other sectors. In this regard, a detailed survey in f (T )-gravity isgiven in [Tomonari and Bahamonde (2023)].
In the succeeding sections, the Dirac-Bergmann analysis of the geometrical trinity ofgravity and its non-linear extension is performed under the imposition of Prescription 1
when facing the three issues. These cases are also good examples for the application of
Prescription 1, or more fundamentally, for making the understanding of the three issues
deeper.

7.2. Degrees of freedom of TEGR
In this section, the Dirac-Bergmann analysis of TEGR is performed [Blagojevic and Nikolic(2000); Blagojevic and Vasilic (2000); Maluf and da Rocha-Neto (2000); Ferraro and Guzmán(2016)], and the propagating and gauge degrees of freedom are clarified. In the case of TEGR,
it is unveiled that the spatial boundary terms appear in PB-algebras but the analysis does
not need Prescription 1; this is just the same situation as the general theory of relativity.
The following derivation particularly obeys the authors of Ref [Ferraro and Guzmán (2016)].
Let us start with the action functional of TEGR given in Eq (6.78) in Section 6.3. The
Lagrangian density is introduced as follows:Z
STEGR =

dn+1 x LTEGR ,LTEGR = −M

1
det (e−1 )T ,2κ2

(7.19)

where we discarded the boundary term in Eq (6.78). Hereinafter, we use the unit 2κ2 = 1
203

and e = det(e−1 ). Expressing in terms of (co)frame fields, Eq (7.19) becomes as follows:
1
LTEGR = − e ∂µ eI ν ∂ρ eJ σ eµ K eν L eρ M eσ N MIJ KLM N ,2(7.20)

where MIJ KLM N is called the supermetric and defined as follows:
[M

L]

[K

N]

MIJ KLM N = 2gIJ g K[M g N ]L − 4δI g N ][K δJ + 8δI g L][M δJ ,(7.21)

where gIJ is the metric of the internal space. Notice that the supermetic is antisymmetricwith respect to the indices (K , L) and (M , N ). The canonical momentum density variables
are computed as follows:πI µ = −e ∂ν eJ ρ e0 K eµ L eν M eρ N MIJ KLM N .

(7.22)

The fundamental PBs are introduced as follows:{eI µ (t , x) , πJ ν (t , y)}P.b. = δJI δµν δ (n) (x − y) .

(7.23)

The antisymmetric property of the supermetric with respect to K and L leads apparently
to the number of (n + 1) primary constraint densities as follows:(1)0∗ ϕI = πI := 0 .

(7.24)

These trivial constraint densities give an insight into finding other remaining primary constraint densities. Eq (7.24) can be deformed as follows:
(1)
0
K
µ J
∗ ϕI = e J δI πK e µ := 0 .

204(7.25)

This implies the following quantity:
GJ I = πI µ eJ µ = −e CIK JL eµ L ∂0 eK µ − e ∂i eL µ e0 K ei M eµ N MIL KJM N .

(7.26)In order to turn these quantities into constraint densities, multiplying V I (X)J to the above
equations
G(X) = V I (X)J GJ I ,(7.27)and then impose the following conditions:
V I (X)J CIK JL := 0 .

(7.28)

I
satisfies Eq (7.28), andHere, the index “ (X) ” is a dummy one. For instance, V I KJ = e0 J δK

then Eq (7.27) restores the primary constraint densities (7.25), or equivalently, Eq (7.24).
In this case, the dummy index “ (X) ” is taken as K. In the same way, let us take V I (X)JI
IgK]L . Then V I KLJ satisfies Eq (7.28), and then the number ofas 2δ[JgK]L : V I KLJ = 2δ[Jn(n + 1)/2 primary constraint densities are derived as follows:
(1)
i K
K
0
i j
∗ ϕIJ = 2gK[J πI] e i + 4 e ∂i e j e [J e I e K] := 0 .

(1)(7.29)(1)

Then the PB-algebras among ∗ ϕI and ∗ ϕIJ are computed as follows:
(1)

(1)

(1)

(1)

{∗ ϕI (t , x) , ∗ ϕJ (t , y)}P.b. = 0 ,{∗ ϕIJ (t , x) , ∗ ϕK (t , y)}P.b. = 0 ,


C(1)
(1)
(1)
(1)
(1)
(1)
(1)
{∗ ϕIJ (t , x) , ∗ ϕKL (t , y)}P.b. = gJL ∗ ϕIK + gIK ∗ ϕJL − gJK ∗ ϕIL − gIL ∗ ϕJK δ (n) (x − y) ≈ 0 ,(7.30)
where we abbreviated the argument “ (t , x) ” on the right hand side in the third PB205

algebras. These PB-algebras do not include the spatial derivative of δ-function. Notice thatthe third PB-algebras are exactly the same as the Lorentz algebra. Therefore, it is expected(1)that the primary constraint densities ∗ ϕIJ (t , x) becomes the generator of the local Lorentz
transformation and are classified into first-class constraint densities.
Next, let us calculate the consistency conditions for the primary constraint densities. The
total Hamiltonian is composed as follows:
(1)

(1)

HT = H0 + λI ∗ ϕI + λIJ ∗ ϕIJ ,(7.31)

where H0 is the kernel Hamiltonian and λI and λIJ are Lagrange multipliers. Remarkthat the multipliers are the functionals of the canonical variables eI µ and πI µ . The kernel
Hamiltonian is derived from the temporal component of the Euler-Lagrange equations under
the primary constraint densities. That is,∂LTEGR C(1) ∂LTEGR ∂LTEGR∂LTEGR∂LTEGR−−≈ ∂i= −∂i πI i −
= 0,
I
I
I
I
∂(∂µ e 0 )∂e 0∂(∂i e 0 )
∂e 0
∂eI 0
∂LTEGR∴ ∂i πI i +
= 0.
∂eI 0

∂µ

(7.32)The second term on the left-hand side is calculated as follows:∂LTEGR
0
J
i
=
e
∂
(e
π
)
−
H
− eJ 0 eI 0 ∂i πJ i + 2eI j ∂[i eJ j] πJ i .
I
i
0
J
0
∂eI 0

(7.33)

Substituting the second formulae into the first equations, the following equations are derived:eJ j eI j ∂i πJ i + eI 0 ∂i (eJ 0 πJ i ) − H0 + 2eI j ∂[i eJ j] πJ i = 0 .

206(7.34)

The temporal component gives the kernel Hamiltonian as follows:(7.35)

H0 = eI 0 eI j eJ j ∂i π i J + ∂i (eI 0 πI i ) .
(1)

The consistency conditions for the primary constraint densities ∗ ϕI

C(1)≈ 0 generate thesecondary constraint densities as follows:
(1)
(2)
(1) C
≈ H0 − ∂i (eI 0 π i I ) := 0 ,∗ ϕ0 = ∗ ϕ̇0(7.36)

(1)
(2)
(1) C
≈ ∂j (eI i πI j ) := 0 .
∗ ϕi = ∗ ϕ̇0

Notice that the first constraint densities of the above equations imply that the total Hamiltonian is composed only of the constraint densities with a total divergent term as follows:
(2)

(1)

C(2)

(1)

HT = ∗ ϕ0 + λI ∗ ϕI + λIJ ∗ ϕIJ + ∂i (eI 0 π i I ) ≈ ∂i (eI 0 π i I ) .

(7.37)

Therefore, taking into account the PB-algebras (7.30), the consistency conditions for the
(1)

C(1)

Lorentz generators ∗ ϕIJ ≈ 0 are automatically satisfied up to boundary terms. The PB(2)(2) C

algebras among ∗ ϕI
(2)

(2)

(2)

(2)

≈ 0 are calculated as follows:{∗ ϕi (t , x) , ∗ ϕj (t , x)}P.b. =



(2)
(x)
(y)
(2)
− ∗ ϕi (t , y)∂j
∗ ϕj (t , x)∂i
(2)



C(2)

δ (n) (x − y) ≈ 0 ,

C(2)

(x)

{∗ ϕi (t , x) , ∗ ϕ0 (t , y)}P.b. = ∗ ϕ0 (t , x)∂i δ (n) (x − y) ≈ 0 ,


C(2)
(2)
(2)
(2)
(2)
(x)
{∗ ϕ0 (t , x) , ∗ ϕ0 (t , y)}P.b. = hij (t , x)∗ ϕj (t , x) + hij (t , y)∗ ϕj (t , y) ∂i δ (n) (x − y) ≈ 0 .
(7.38)
Notice that these PB-algebras are exactly the same as the ones of general relativity given in
Eq (4.100). This fact means that TEGR has the same gauge symmetry as the one of general

207

(2)
(1) C

relativity. The PB-algebras among ∗ ϕI
(1)

(2)

(1)

(2)

(2)
(2) C

≈ 0 and ∗ ϕI

≈ 0 are calculated as follows:{∗ ϕI (t , x) , ∗ ϕI (t , y)}P.b. = 0 ,
(2)

C(2)

(2)

{∗ ϕI (t , x) , ∗ ϕ0 (t , y)}P.b. = (eI 0 ∗ ϕ0 + eI i ∗ ϕi )δ (n) (x − y) ≈ 0 ,(1)(2){∗ ϕIJ (t , x) , ∗ ϕi (t , y)}P.b. = 0 ,
(1)

(2)

(2)

(7.39)C(2){∗ ϕIJ (t , x) , ∗ ϕ0 (t , y)}P.b. = −eK 0 gK[I eJ] 0 ∗ ϕ0 δ (n) (x − y) ≈ 0 .
Using these PB-algebras and the total Hamiltonian (7.37), the consistency condition for the
(2)
(2) C

secondary constraint densities ∗ ϕI

≈ 0 are automatically satisfied up to boundary terms.

Therefore, the Dirac-Bergman analysis stops here. The propagating degrees of freedom of
TEGR in a (n + 1)-dimensional spacetime is given as follows:
pDoF = (n + 1)2 −1(n + 1)(n + 4)
= (n + 1)(n − 2) .
2
2

(7.40)The gauge degrees of freedom is given as follows:

gDoF of Diffeomorphism = n + 1(7.41)

for the diffeomorphism (covariance) symmetry and
1
gDoF of Lorentz = n(n + 1)
2

(7.42)

for the local Lorentz symmetry.
Finally, notice that the PB-algebras (7.30), (7.38), and (7.39) do not need the Prescription
1 since the coefficients of these PB-algebras are composed by the linear combination of the
primary and/or secondary constraint densities. This structure prevents moving from thespatial differential operator to the Lagrange multipliers. That is, TEGR does not suffer
208

from the third issue explained in Section 7.1. In addition, if the constraint densities are
satisfied on the spatial boundary then TEGR is also free from the second issue explainedin Section 7.1. The first issue does not arise since TEGR has only first-class constraint
densities: the Dirac matrix cannot be defined. Therefore, the pDoF given in Eq (7.40) holds
for any solutions of TEGR.

7.3. Degrees of freedom of Coincident GR
In this section, the Dirac-Bergmann analysis of Coincident GR (CGR) is performed [D’Ambrosio et al.
(2020)], and the propagating and gauge degrees of freedom are clarified. In the case of TEGR,
it is unveiled that the spatial boundary terms appear in PB-algebras, but the analysis does
not need Prescription 1; this is just the same situation as general relativity.
Let us start by decomposing the CGR action functional in the ADM-foliation manner.
The action functional of CGR is already derived as Eq (6.93) in Section 6.3. The action
functional can be rewritten as follows:√1
dn+1 x −g M αβσρµν Qαβσ Qρµν ,4
M

Z
SCGR =(7.43)

where 2κ2 = 1 and M αβσρµν is set as follows:M αβσρµν = g αρ g βσ g µν − g αρ g βµ g σν + 2g αν g βµ g σρ − 2g αβ g µν g σρ .

(7.44)

Remark that Qαβγ = ∇α gβγ is now in the coincident gauge: Qαβγ = ∂α gβγ . Applying the
ADM-foliated metric Eq (4.37) and, after performing very long but straightforward algebraic
calculations, the above action can be rewritten as follows [D’Ambrosio et al. (2020)]:ZSCGR =

Z
dt

I

√ 
dn x h N(n)
Q + K ij Kij − K 2 + B1 + B2 + B3 ,

Σt

209

(7.45)

where (n) Q, B1 , B2 , and B3 are set as follows:
(n)

Q=


1  il jm kn−h h h + 2hin hjm hkl + hil hjk hmn − 2hij hmn hkl Qijk Qlmn ,4(7.46)

B1 =hij hkl (Qjkl − Qkjl )∂i N ,∂i N i (∂i N i )(N j ∂j N )(7.47)
−
,
N2
N2

1
(N i ∂j N )(∂i N j ) ∂i N j
i
i mn
2∂
N
+
N
h
Q
+ Ṅ kB3 =−(N hij Qkij − 2∂k N ) .
j
jmn
2
2
N
2N
2N
√
√
√
√
The sum of these boundary terms multiplying h, i .e ., hB1 , hB2 , and hB3 , are calB2 =K∂i N i + Ṅ

culated, neglecting spatial boundary terms, as follows:
√

√ ◦
√
√
i
i
h(B1 + B2 + B3 ) = −N hDi ((n) Q − (n) Q̄ ) + ∂µ N j ∂j ( hnµ ) − ∂i N i ∂µ ( hnµ ) . (7.48)

Remark that this ignorance is irrelevant to the Priscription 1. Therefore, the action becomes
as follows:
Z

Z
dt

SCGR =
I

Σt

h √ n
o
◦
i
i
dn x N h (n) Q + K ij Kij − K 2 − Di ((n) Q − (n) Q̄ )
√ µ i
√ µ
j
−∂i N ∂µ ( hn ) + ∂µ N ∂j ( hn ) .

(7.49)i

This is none other than the ADM-foliation of CGR [D’Ambrosio et al. (2020)]. Notice thatthe derivation of Eq (7.49) neglected only spatial boundary terms. Integrating the sec√√√
ond term in Eq (7.49) by parts: −∂i N i ∂µ ( hnµ ) = −∂i [N i ∂µ ( hnµ )] + ∂µ [N i ∂i ( hnµ )] −√(∂µ N i )(∂i hnµ ), the third term is canceled out with the third term in Eq (7.49). The
remnant terms are only boundary terms; these terms can be neglected by introducing appropriate counter-terms just like the case of general relativity. (See Section 4.5 in detail.)

210

Therefore, we get the final result:Z
SCGR =i√ h◦iidn xN h (n) Q + K ij Kij − K 2 − Di ((n) Q − (n) Q̄ ) .

Z
dt

I

(7.50)

ΣtNote that the above derivation does not need the Gauss equation, unlike the case of thegeneral theory of relativity. (See Sections 4.4 and 4.5.) We just manipulated complicatedalgebraic calculations.
The Dirac-Bergmann analysis is performed for the action functional Eq (7.50). The
canonical momentum variables are calculated as follows:√
π0 = 0 ,πi = 0 ,πij =(7.51)

h(Khij − Kij ) ,and therefore, the primary constraint densities are given as follows:(1)∗ ϕ0 = π0 := 0 ,(1)∗ ϕi = πi := 0 .

(7.52)

These constraint densities restrict the entire phase space to a subspace C(1) . The total
Hamiltonian density is derived as follows:(CGR)

H T = N C0
(CGR)

where C0

(CGR)
C0
=−

(CGR)

and Ci(CGR)+ N i Ci

(7.53)+ λµ ∗ ϕ(1)µ ,are set as follows:
i√ h(n)
◦
1
1
ij
2
(n) i
(n) i
π πij −π,h
Q − Di ( Q − Q̄ ) + √
n−1
h

(CGR)

Ci

◦

= −2Dj πij .
(7.54)

The fundamental PB-algebras are the same as those of the general theory of relativity:
Eq (4.95). Therefore, the consistency conditions for the primary constraint densities Eqs (7.52)

211lead to four secondary constraint densities:
(2)
(CGR)
:= 0 ,∗ ϕ0 = C0

(2)
(CGR)
:= 0 ,∗ ϕi = Ci(7.55)and these constraint densities restrict C(1) to a new subspace C(2) . We can show that thesmeared PB-algebras of Eq (7.55) satisfy the same PB-algebras as those of the general theoryof relativity given in Eq (4.103) and (4.104) [D’Ambrosio et al. (2020)], and it indicates that
CGR has the same gauge symmetry as the general theory of relativity. That is, the analysis
stops here. Therefore, CGR has
1
pDoF = (n + 1)(n − 2)
2

(7.56)

gDoF = n + 1 .

(7.57)

andThat is, CGR is completely equivalent to GR from both viewpoints of dynamics and gauge
symmetry. 4

7.4. Degrees of freedom of non-linear extension of GR
In this section, the Dirac-Bergmann analysis of the non-linear extension of the general theoryof relativity, denoted by “fGR”, is performed [Sotiriou and Faraoni (2010); Deruelle et al.
(2009); Capozziello and De Laurentis (2011); Liang et al. (2017)], and the propagating andgauge degrees of freedom are clarified. In the case of fGR, it is unveiled that the spatial
boundary terms generically appear in PB-algebras due to the existence of the non-linearity,
4

However, the original Lagrangian density given in Eq (6.93) discards the boundary term of the Ricci
scalar in the Einstein-Hilbert action functional. It means that the diffeomorphism invariance is at least
partly violated, (it is also guessed from the imposition of the coincident gauge since this gauge destroys thecovariance of the theory,) and it might imply that the analysis gives rise to some second-class constraint
densities. I would like to leave to scrutinize the validity of the work [D’Ambrosio et al. (2020)] from thisperspective for future work.

212

and the analysis generically needs Prescription 1; this is the departure from the case of thegeneral theory of relativity from the viewpoint of constraint mechanics. That is, the theory
suffers from the second issue, and, caused by it, the first issue may also occur. Further, it
might happen the third issue. It means that the result is viable only for the solutions thatsatisfy the spatial boundary condition that is demanded by Prescription 1.
The action functional of the non-linear extension of the general theory of relativity isalready given as Eq (6.100) in Section 6.4:
Z
SfGR =◦√
dn+1 x −gf (R) ,(7.58)

M◦◦◦

where 2κ2 = 1, f ′ (R) = df (R)/dR, and f ′′ ̸= 0. Introducing an auxiliary field φ, Eq (7.58)
is decomposed as follows:Z
SfGR =i◦√ hdn+1 x −g f ′ (φ)R + f (φ) − φf ′ (φ) .

(7.59)

M◦

Using the Gauss equation Eq (4.57) to decompose R, this action becomes as follows:Z
SfGR =dtIi√ h  ◦
dn xN h f ′ (n) R − K 2 + K ij Kij − φ + f
Σt
Z
i
◦
√ h ◦  ◦dn+1 x −g f ′ ∇µ nν ∇ν nµ − nµ ∇ν nν .
−2

Z(7.60)

M

Compared to Eq (4.66), the boundary terms cannot be neglected due to the existence of the
non-linearity of f ′ . Integrating by parts and then neglecting the boundary terms, Eq (7.60)

213becomes as follows [Liang et al. (2017)]:i√ h ′ (n) ◦2ij
R − K + K Kij − φ + f
SfGR = dt
d xN h f
I
Σt


Z
Z

√ 2K  i ◦ ′
◦
◦
n
′′
′ i
d xN h+ dt
N Di f − f φ̇ + 2Di f D lnN .
N
I
Σt
Z

Z

n

(7.61)
The canonical momentum variables are calculated as follows:√
π0 = 0 ,πi = 0 ,

πij =




hij  k ◦ ′′′h f (Khij − Kij ) −N Dk f − f φ̇ ,N′√
πφ = −2K hf ′′ .
(7.62)In the case of f ′ = constant, as expected, Eq (7.62) becomes Eq (4.68). The primaryconstraint densities are given as follows:(1)∗ ϕ0 = π0 := 0 ,(1)∗ ϕi = πi := 0 ,(7.63)and these constraint densities identify the subspace C(1) . The total Hamiltonian density is
calculated as follows:(fGR)

HT = N C0

(fGR)

µ
(1)
+ λµ ∗ ϕ(1)µ = H0 + λ ∗ ϕµ(fGR)

are defined as follows:+ N i Ci

where λµ are Lagrange multipliers, and Cµ



 ◦
i
√ h
√ ◦ ◦
1
1
′ (n)
ij
2
h f +fR−φ + √
π πij −
π + 2 hDi Di f ′n−1
hf ′ ′ 21n−1 f− √ππφ + √
πφ2 ,n hf ′′
n hf ′ f ′′(7.64)

(fGR)
C0
=−

(fGR)

Ci

◦

(7.65)◦=πφ Di φ − 2Dj πij .

In the case of f ′ = constant, Eq (7.65) coincides with Eqs (4.89) and (4.90). The consis214

(1)tency conditions for the primary constraint densities, ∗ ϕµC(1)≈ 0, are gives four secondary

constraint densities as follows:
(2)
(fGR)
:= 0 ,∗ ϕ0 = C0

(2)
(fGR)
:= 0 ,∗ ϕi = Ci(7.66)

under the same fundamental PB-algebras Eq (4.95) and(7.67){φ(x), πφ (y)}P.b. = δ (n) (x − y) .

These constraint densities restrict C(1) to a new subspace C(2) . The consistency conditions
for these secondary constraint densities are computed as follows:
(2)

(2)

(2)

(2)

{∗ ϕi (t , x) , ∗ ϕj (t , y)}P.b. =


C(2)
(2)
(x)
(2)
(y)
− ∗ ϕi (t , y)∂j δ (n) (x − y) ≈ 0 ,∗ ϕj (t , x)∂i{∗ ϕ0 (t , x) , ∗ ϕi (t , y)}P.b. =(2)(x) (n)(x − y) + ∂j
∗ ϕ0 (t , x)∂i δ(2)

hi C(2) hi(j)(j)
Hi δ (n) (x − y) ≈ ∂j Hi δ (n) (x − y) ,(2){∗ ϕ0 (t , x) , ∗ ϕ0 (t , y)}P.b. = C(2)
(2)
(2)
(x)
ij
ij
h (t , x)∗ ϕj (t , x) + h (t , y)∗ ϕj (t , y) ∂i δ (n) (x − y) + ∂i H (i) δ (n) (x − y) ≈∂i H (i) δ (n) (x − y) ,(7.68)(j)where Hiand H (i) are composed of the phase space variables, the secondary constraint

densities, and those spatial derivatives up to second-order. Notice that these functionals(j)appear due to the non-linearlity of f ; if f is a linear functional then Hi and H (i) vanish. The
concrete forms of these functionals are very complicated and may include terms that do notvanish under the imposition of the constraint densities. That is, here, the second issue arises.
215

◦However, it does not matter. Since the f (R)-gravity is a diffeomorphism invariant theory,
without any loss of generality, we can choose an ADM-foliation such that the Prescription
1 demands. Therefore, if we use Prescription 1, then the PB-algebras turn into that of the◦general theory of relativity, and f (R)-gravity has
1
pDoF = (n + 1)(n − 2) + 1
2(7.69)

gDoF = n + 1 .

(7.70)

andComparing to the case of the general theory of relativity, one additional pDoF appears andthe gauge symmetry does not change.
The calculations of the PB-algebras among the secondary constraint densities are performed under the imposition of Prescription 1 (or other possibilities (i), (iii) explained in
Section 7.1), and then it is shown that the PB-algebras are the same as the ones of thegeneral theory of relativity. In other words, in general, the analysis performed in this section
is reliable only for the solutions that satisfy the boundary conditions: the shift vector on
the spatial boundary vanishes. This is a good example to scrutinize the influence of thespatial boundary terms on the classification of constraint densities, i .e ., the second issue.
If the spatial boundary terms can be neglected then the pDoF is given by Eq (7.69) andthe shift vector has to vanish at least on the spatial boundary. Otherwise, the pDoF may
generically be changed additionally from (n + 1)(n − 2)/2 + 1 since the secondary constraint
densities are no longer classified into first-class constraint densities. 5 However, since the5

It is known that another sort of extra degrees of freedom is obtained by separating the conformal
mode in the general theory of relativity [Chamseddine and Mukhanov (2013)] and by considering the invertible/disformal transformation of gravity [Jiroušek et al. (2023, 2022)]. If the boundary terms cannot
be ignored, does the additional DoF become some sort of DoF similar to the above? This interpretation
might be a possible story since the Dirac-Bergmann analysis generically counts out the degrees of freedomof a given theory including propagate but unphysical (ghost) modes. However, these statements have to be
proven more rigorously, and this problem is unsolved at the moment of writing this thesis.

216

action functional given in Eq (7.61) indicates that the theory is diffeomorphism invariant,the change of the constraint structure depending on the way of taking an ADM-foliation
would be a contradiction. As already mentioned in the final remark in Section 7.1, the resultof the analysis on a diffeomorphism invariant theory should be independent of the choice of
ADM-foliation. Therefore, it makes sense to neglect the boundary terms, and it should hold◦even for the well-definedness of all PBs in the analysis of f (R)-gravity.

7.5. Degrees of freedom of non-linear extension of TEGR
In this section, the Dirac-Bergmann analysis of the non-linear extension of TEGR, denoted by“fTEGR”, is performed [Li et al. (2011); Ferraro and Guzmán (2018); Blagojević and Nester(2020)], and the propagating and gauge degrees of freedom are clarified. In the case offTEGR, it is unveiled that the spatial boundary terms appear in PB-algebras and the analysis
generically needs Prescription 1; this is the departure from the case of the general theory of
relativity from the viewpoint of constraint mechanics. In addition, differing from the fGRcase, it is also unveiled that fTEGR suffers from the second issue and third issue. The second
issue may also cause the first issue. This means that the result is viable only for the solutions
that satisfy the spatial boundary condition that is demanded by Prescription 1.
◦

We performed the Dirac-Bergmann analysis of GR, TEGR, CGR, f (R)-gravity in a(n+1)-dimensional spacetime. However, since the non-linear extension of TEGR is describedby (co-)frame fields, it makes the theory far more complicated than the metric formalism.
Therefore, first of all, the analysis is performed in a 4-dimensional spacetime, and then, thecase of a generic dimension is estimated.
The Lagrangian density of TEGR is already given as Eq (7.19) in Section 7.2. The
Lagrangian density of the theory is given as follows:Z
SfTEGR =d4 x LfTEGR ,LfTEGR = −

M

217

1
det(e−1 ) f (T)2κ2(7.71)

Hereinafter, we use the unit 2κ2 = 1 and e = det(e−1 ). Using an auxiliary field φ, the above
Lagrangian density is decomposed as follows [Blagojević and Nester (2020)]:(7.72)

LfTEGR = e [φT − V (φ)]
under the imposition of the following condition:
δLfTEGR:= 0 ,
δφ(7.73)

where V = V (φ) is a function such that V ′ (φ) is invertible and V ′′ (φ) ̸= 0. This method◦of decomposing the Lagrangian density is different from that of f (R)-gravity, but it has
the advantage of simplifying the analysis of a non-linear theory in terms of the vielbein
formalism. That is, the explicit form of the torsion tensor
T I µν = eI ρ T ρ µν = ∂µ eI ν + ω I Jµ eJ ν − ∂ν eI µ + ω I Jν eJ µ = ∂µ eI ν − ∂ν eI µ ,(7.74)

indicates that T I 0µ can be used equivalently as velocity variables of the co-frame fields,
where the Weitzenböch gauge (, or equivalently, the teleparallel condition,) is applied. In
this formalism, canonical momentum variables of the decomposed Lagrangian density (7.72)
with respect to the velocity T I 0µ are computed as follows: First, we compute
δLfTEGR= eφ
HIJK =
δT IJK1
1
TIJK + T[KJ]I − gI[J TK]
4
2


.

(7.75)

Then the canonical momentum variables are given as follows:πI µ =

δLfTEGR= HI 0µ = e0 J eµ K HI JK .
I
δT 0µ

218

(7.76)

The canonical momentum variable with respect to the auxiliary field φ is given as follows:πφ =δLfTEGR= 0.
δ(∂0 φ)(7.77)

Therefore, primary constraint densities are derived as follows:(1)0∗ ϕI = πI := 0(1)
∗ ϕφ = πφ := 0 .

(7.78)Taking into account the considerations in Section 7.2, it expects that there exist further
primary constraint densities corresponding to the generators of local Lorentz transformation.
In order to find the remaining other primary constraint densities, the (1+3)-decompositionof a vector is introduced as follows:V⊥ = nI VI

VI¯ = VI − nI V⊥ ,(7.79)

where V = V I ξI is a vector of the internal space and n = nI ξI is a normal vector to a leaf Σt .
ξI are a basis of sections of the internal space. Using this decomposition and recalling theprocess of deriving the canonical momentum variables (7.29) from Eq (7.26), the following
quantities
π̂I J¯ = πI µ eJµ =

φ
H ¯,
N I⊥J

(7.80)

HI⊥J¯ = TI⊥J¯ + (TJ⊥I− T⊥JI¯¯ ) − 2(nI VJ¯ − gI J¯V⊥ )lead to the remaining primary constraint densities as follows:(1)∗ ϕIJ = HIJ + φBIJ := 0 ,219(7.81)

where HIJ and BIJ are defined as follows:
HIJ = π̂I J¯ − π̂JI
¯ ,(7.82)

2
BIJ = − (T ⊥ I¯J¯ − nI T K̄ K̄ J¯ + nJ T K̄ K̄ I¯) .
NTherefore, fTEGR has the number of (4 + 1) + 6 = 11 primary constraint densities. Thenthe total Hamiltonian is derived as follows:
1 IJ (1)(1)
i I
HT = N H⊥ + N i Hi + λI ∗ ϕI + λφ ∗ ϕ(1)φ + λ ∗ ϕIJ + ∂i (πI e 0 ) ,2(7.83)

where H⊥ and Hi are defined as follows:¯H⊥ = π̂I J T I ⊥J¯ −jI1
Lf T EGR − nI ∂j πI j ,NI(7.84)j

Hi = πI T ij − e i ∂j πI ,and λI , λφ , and λIJ are Lagrange multipliers. The PB-algebras are computed as follows:
(1)

(1)

(1)

(1)

(1)

(1)

{∗ ϕI (t , x) , ∗ ϕJ (t , y)}P.b. = 0 ,{∗ ϕI (t , x) , ∗ ϕJK (t , y)}P.b. = 0 ,{∗ ϕIJ (t , x) , ∗ ϕKL (t , y)}P.b.


4
(1)
(1)
(1)
(1)
⊥M̄ N (3)= gJL ∗ ϕIK + gIK ∗ ϕJL − gJK ∗ ϕIL − gIL ∗ ϕJK δ (3) (x − y) + φM̄ gN L δIJK
δ (x − y) ,N
(7.85)
where φI¯ = ∂I¯φ. We abbreviated the argument “ (t , x) ” on the right-hand side in the
(1)

C(1)

third PB-algebras. Notice that the third PB-algebras implies that some of ∗ ϕIJ ≈ 0 areclassified into second-class constraint densities; there occurs a departure from the TEGR case.
(1)(Compare to the PB-algebras (7.30) in Section 7.2.) The PB-algebras between ∗ ϕφ

220

C(1)

≈ 0

and others are computed as follows:(1){∗ ϕ(1)φ (t , x) , ∗ ϕI (t , y)}P.b. = 0 ,(7.86)(1)(3){∗ ϕ(1)φ (t , x) , ∗ ϕIJ (t , y)}P.b. = −BIJ (t , x)δ (x − y) .

(1)

C(1)

C(1)

(1)

The second PB-algebras imply that all the constraint densities ∗ ϕIJ ≈ 0 and ∗ ϕφ

≈ 0

are classified into second-class constraint densities. Therefore, fTEGR does not have the full
local Lorentz symmetry.
(1)
(1) C

Next, let us investigate the consistency conditions for these primary constraints: ∗ ϕI
(1)

C(1)

(1)

0, ∗ ϕIJ ≈ 0, and ∗ ϕφ

≈

C(1)

≈ 0. From the PB-algebras (7.85) and (7.86), it is easy to confirm

the first case relative to the second and third ones. The conditions
(1)

C(1)

(1)

i
∗ ϕ̇I = {∗ ϕI , HT }P.b. ≈ NI H⊥ + eI Hi := 0(7.87)

lead to the following secondary constraints:(2)∗ ϕ̄0 = H⊥ := 0 ,(2)∗ ϕ̄i = Hi := 0 .

(7.88)In order to construct a closed PB-algebra, the above constraints can be modified as follows [Mitrić (2019)]:
C(2)
(2)
(2)
(1)
∗ ϕ0 = ∗ ϕ̄0 + ∗ ϕφ ∂⊥ φ ≈ 0 ,

C(2)
(2)
(2)
(1)
∗ ϕi = ∗ ϕ̄i + ∗ ϕφ ∂i φ ≈ 0 .

221(7.89)

Then these modified PB-algebras form the following PB-algebras:
(2)

(2)

C(2)

(2)

(2)

C(2)

{∗ ϕi (t , x) , ∗ ϕj (t , y)}P.b. ≈


C(2)
(2)
(x)
(2)
(y)
(n)
ϕ
(t
,
x)∂
−
ϕ
(t
,
y)∂
δ
(x
−
y)
≈ 0,
∗ j
∗ i
i
j
{∗ ϕ0 (t , x) , ∗ ϕi (t , y)}P.b. ≈(2)(x) (n)(x − y) + ∂j
∗ ϕ0 (t , x)∂i δ(2)

hi C(2) hi(j)(j)
Hi δ (n) (x − y) ≈ ∂j Hi δ (n) (x − y) ,(2){∗ ϕ0 (t , x) , ∗ ϕ0 (t , y)}P.b. = C(2)(2)(2)(x)
hij (t , x)∗ ϕj (t , x) + hij (t , y)∗ ϕj (t , y) ∂i δ (n) (x − y) + ∂i H (i) δ (n) (x − y) ≈∂i H (i) δ (n) (x − y) ,(7.90)(I)

where HJ and H (I) are composed of the phase space variables, the secondary constraint densities, and those spatial derivatives up to second-order, which arise due to the non-linearity
f , or equivalently, V . Remark that the above algebras hold only in the weak equality on
(1)
(1) C

the constraint phase subspace restricted at least by ∗ ϕφ

(1)
(1) C

(2)
(2) C

≈ 0, and ∗ ϕI

≈ 0, ∗ ϕI≈ 0.

Here, notice that we face the second issue. If the total divergent terms in the second and(2)third PB-algebras remain then the secondary constraint densities ∗ ϕIC(2)≈ 0 are classified

into second-class constraint densities, and then the third issue occurs. Therefore, in order torestrict our consideration to algebraically solvable cases, let us apply Prescription 1. Then
(2)
(2) C

the PB-algebras of the secondary constraint densities ∗ ϕI

≈ 0 becomes commutative ones.

In addition, applying the Castellani algorithm [Castellani (1982)] to the primary constraint
(2)
(1) C

densities ∗ ϕI

(2)
(2) C

≈ 0 and ∗ ϕI

≈ 0, taking into account the second PB-algebras in Eq (7.85)

and the first PB-algebras in Eq (7.85), at least these constraint densities form a closed PBalgebra up to linear combinations of these constraint densities. Therefore, under Prescription
(2)
(1) C

1, ∗ ϕI(2)(2) C≈ 0 and ∗ ϕI

≈ 0 are classified into first-class constraint densities. The satisfac-

tion of the consistency conditions for these first-class constraint densities are now trivial; the
222

(2)
(1) C

Dirac-Bergmann procedure on the sector ∗ ϕI

(2)
(2) C

≈ 0 and ∗ ϕI

≈ 0 stops here. Remark thatthe action functional given in Eq (7.72) implies that the theory is diffeomorphism invariant.
Therefore, the application of Prescription 1 does not change the generality of the result of
the analysis, as already mentioned in Section 7.1.
(1)
(1) C

(1)
(1) C

The consistency condition for the primary constraint densities ∗ ϕIJ ≈ 0 and ∗ ϕφ

≈ 0determine some of the Lagrange multipliers λI , λφ , and λIJ by virtue of the PB-algebras (7.85)
and (7.86). The secondary constraint densities with respect to these constrain densities arecomputed as follows:
1 MN(2)BM N := 0 ,∗ ϕφ = −N ∂φ H̃⊥ − λ2(7.91)(2)K∗ ϕIJ = GIJ ∂K φ := 0 ,where H̃⊥ denotes H⊥ evaluated on shell and GIJ K are determined from the following relations:
GIJ ⊥ = N BIJ ,
GIJ

K̄


2 ⊥K̄ N̄ M
K̄ M̄ N̄ R̄
= δIJM
λ N̄ − N φ−1 g K̄ M̄ nI π̂(M̄ J)¯ − nJ π̂(M̄ I)¯ − δI¯J¯R̄ T M̄ N̄ .
N

(7.92)

These secondary constraint densities lead to the following equations:
(2)
1
¯
¯¯ C
λ⊥J¯B ⊥J + λI¯J¯B I J ≈ −N ∂φ Ĥ⊥ ,2C(2)2⊥K̄ N̄ M
BIJ λ̄φ + φK̄ δIJM
λ N ≈ XIJ ,N

223(7.93)

respectively, where λ̄φ , φI¯, and XIJ are defined as follows:λ̄φ = λφ − N I ∂I φ ,(7.94)

φI¯ = ∂I¯φihK̄ M̄ N̄ R̄
XIJ = φK̄ N φ−1 g K̄ M̄ nI π̂(M̄ J)¯ − nJ π̂(M̄ I)¯ + δI¯J¯R̄ T M̄ N̄ .
Performing (1 + 3)-decomposition, the following three equation are derived:
C(2)B ⊥N̄ λ⊥N̄ + B N̄ λN̄ ≈ −N ∂φ H̃⊥ ,C(2)(7.95)B ⊥M̄ λ̄φ − ZK ϵK̄ M̄ N̄ λN̄ ≈ X ⊥M̄ ,C(2)B M̄ λ̄φ − ZK ϵK̄ M̄ N̄ λ⊥N̄ ≈ X M̄ ,where ZI¯, FI¯, and λI¯ are defined as follows:

ZI¯ =

2
φ¯
N I

1
BI¯ = ϵI¯M̄ N̄ B M̄ N̄
2

1
λI¯ = ϵI¯M̄ N̄ λM̄ N̄
2¯¯(7.96)X I = φK̄ ϵK̄ M̄ N̄ T I M̄ N̄ .

Let us consider the case of φI¯ ̸= 0. Combining the contraction of the second and third
equations in Eq (7.95) with φM̄ , the following equation is derived:


φM̄ B

⊥M̄

2



+ φM̄ B

M̄

2 

C(2)

λ̄φ ≈






φM̄ B ⊥M̄ φN̄ X ⊥N̄ + φM̄ B M̄ φN̄ X N̄ .

(7.97)

2This equation with the first relation in Eq (7.94) determines the multiplier λφ when φM̄ F ⊥M̄ +2
φM̄ B M̄ ̸= 0. Then a new secondary constraint density arises as follows:



 

(2)
⊥M̄
N̄
M̄
⊥N̄
ϕ
=
φ
B
φ
X
−
φ
B
φ
X
:= 0 .
∗
M̄
N̄
M̄
N̄

(7.98)The emergence of the additional constraint density implies that some of the remaining mul(2)tipliers λI and λIJ are determined since the PB-algebras between ∗ ϕ(2) and ∗ ϕIJ are not224commutative under the imposition of all constraint densities [Blagojević and Nester (2020)].
The transformed multipliers λ⊥I¯ and λI¯ can be decomposed into parallel and orthogonalcomponents as follows:λ⊥I¯ = λ⊥ φI¯ + λ̂⊥I¯ ,λI¯ = λφI¯ + λ̂I¯ ,¯λ̂⊥I¯φI = 0 ,(7.99)I¯λ̂I¯φ = 0 .

Combining these decomposition with Eq (7.94), the following equation are derived:




¯
¯
¯
¯
λ⊥ B ⊥I φI¯ + λ B I φI¯ = −N ∂φ Ĥ⊥ − B ⊥I λ̂⊥I¯ − B I λ̂I¯ .

(7.100)C(1)In addition, the consistency condition for the secondary constraint density ∗ ϕ(2) ≈ 0 leads
to the following equation:
1 M̄ N̄ K̄
(1)
(1)
I¯
(2)
(2)
λ⊥ φ {∗ ϕ⊥I¯ , ∗ ϕ }P.b. + λϵφK̄ {∗ ϕM N , ∗ ϕ }P.b.
2C(2)(7.101)≈ multiplier independent and known terms .

Therefore, the following equation is derived:λ(x)D(x , x′ ) = G(x′ ) ,(7.102)where D(x , x′ ) and G(x′ ) are defined as follows:′D(x , x ) = B

⊥I¯

 1

φI¯

2

J¯K̄ L̄ϵ(1)φL̄ {∗ ϕJ¯K̄ , ∗ ϕ(2) }P.b.






¯
¯
(1)
− B I φI¯ φJ {∗ ϕ⊥J¯ , ∗ ϕ(2) }P.b. ,G(x′ ) = all known terms .
(7.103)Here, the term D(x , x′ ) contains the spatial derivatives of the Dirac delta function: schemat225

(x)ically, “ A(x)∂i δ (3) (x − y) ”. Integrating Eq (7.102) by parts, the following PDE is derived:(7.104)Ai ∂i u + αu = G ,schematically. This is nothing but Eq (7.17); there gives rise to the third issue. However,
(2)
(2) C

applying Prescription 1 (we already imposed it to make the constraint densities ∗ ϕI

≈ 0first-class constraint densities), the first term in the left-hand side does not appear since all(x)terms such like “ A(x)∂i δ (3) (x − y) ” in the PB-brackets are erased by the imposition of thespatial boundary condition Ni (t , ∂Σt ) = 0. As mentioned in the final remark in Section 7.1,
since f (T )-gravity is a diffeomorphism invariant theory, the imposition of Prescription 1
holds the generality of the analysis. Therefore, Eq (7.102) becomes just a linear algebraic
equation, and it is always solvable as long as D ̸= 0; the Dirac-Bergmann procedure stops
here. The pDoF of f (T )-gravity is

pDoF =

1[(16 + 1) − 8 × 2 − 8] = 5 ,2(7.105)the gDoF is
gDoF = 4 .

(7.106)The local Lorentz gauge symmetry is broken due to the third PB-algebras in Eq (7.85) andthe second PB-algebras in Eq (7.86). In general, for a (n + 1)-dimensional spacetime, thepDoF and gDoF are given as follows [Blagojević and Nester (2020)]:
1
pDoF = (n + 1)(n − 2) + n
2(7.107)

gDoF = n + 1 .

(7.108)

and

226

The local Lorentz gauge symmetry is of course broken. Remark that the authors [Li et al.
(2011)] also derived the same result but overlooked the third issue. In the work [Blagojević and Nester(2020)], other three possible cases are mentioned; (i) If φI¯ ̸= 0 and D = 0 then pDoF isC(2)N/A; (ii) If φI¯ ̸= 0 and ∗ ϕ̇(2) ≈ 0 then pDoF = 4; (iii) If φI¯ = 0 then pDoF = 2. Here,
we used Prescription 1. (The original work in [Blagojević and Nester (2020)] takes into
account the spatial derivatives of the Dirac delta function.) There is a remark: the workin [Blagojević and Nester (2020)] overlooked a generic sector of (iii), which is indicated by
the work [Ferraro and Guzmán (2018)]. In this work, the authors find out (iii)’ If φI¯ = 0 thenpDoF = 3. The difference between (iii) and (iii)’ is that Sector (iii) implicitly demands also
φ = constant. That is, φ = φ(t) in Sector (iii)’ implies that one propagates but unphysicalDoF exists. In this regard, a detailed consideration is given in [Tomonari and Bahamonde(2023)].
An important lesson from the analysis of f (T )-gravity is that the violation of the local
Lorentz symmetry provided several sectors and each sector has a different pDoF from theother sectors. As mentioned in the final remark in Section 7.1, for a diffeomorphism invariant
theory, without any loss of generality, Prescription 1 can be applicable. Therefore, the above
analysis unveiled the full pDoF of each sector. However, for a theory with violating its
diffeomorphism invariance, the result of the analysis depends on the choice of an ADMfoliation. That is, the analysis under the use of Prescription 1 does generically not provide thefull pDoF of the theory. Nevertheless, this sort of analysis has an advantage for circumventing
the three issues. In other words, as long as an analysis suffers from the three issues, any
result cannot be beyond a speculation. The use of Prescription 1 would unveil the pDoF
only of a special sector, which is not the full pDoF of the most generic sector, but the results
are exact, meaning that it is not a speculation since no pathological PDE(s) of Lagrange
multiplier(s) emerges, and, therefore, reliable. This situation happens rather in the analysisof the coincident f (Q)-gravity.
227

7.6. Degrees of freedom of non-linear extension of CGR
In this section, the Dirac-Bergmann analysis of the non-linear extension of CGR, denotedby “ fCGR ”, is performed [Hu et al. (2022); Tomonari and Bahamonde (2023)], and the
propagating and gauge degrees of freedom are clarified. In the case of fCGR, it is unveiled
that the spatial boundary terms appear in PB-algebras and the analysis generically needs
Prescription 1; this is the departure from the case of the general theory of relativity from theviewpoint of constraint mechanics. In addition, differing from the fGR and fTEGR cases,
it is also unveiled that fCGR suffers from the first issue and third issue. This means thatthe result is viable only for the solutions that satisfy the spatial boundary condition that isdemanded by Prescription 1.
◦

We performed the Dirac-Bergmann analysis of GR, TEGR, CGR, f (R)-gravity in a(n + 1)-dimensional spacetime. As we will see, however, the existence of the second-classconstraint densities makes it difficult to understand whether the consistency conditions determine the multipliers or derive new constraint densities since the size of the Dirac matrix
becomes bigger. Therefore, in this section, for simplicity, we perform the analysis for f (Q)in a (3 + 1)-dimensional spacetime and then estimate the general case of the dimension of(n + 1). A general proof would be completed by applying the mathematical induction.
◦

In the same manner as the case of f (R)-gravity, the CGR can be extended non-linearly
into as follows:√
d4 x −gf (Q) ,Z
SfCGR =

(7.109)M

where f is an arbitrary function of the nonmetricity scalar. By introducing an auxiliary
variable we obtain:Z
SfCGR =√
d4 x −g [f ′ Q + f − φf ′ ] ,(7.110)M

where f is an arbitrary function of the auxiliary variable φ and f ′′ = d2 f /dφ2 ̸= 0. From

228

Section 7.3, using the ADM-foliation of Q given in Eq (7.49), Eq (7.110) can be decomposed
as follows:
Z

Z
dt

SfCGR =
I

h √
n

o
√
◦
i
i
d3 x N hf ′ (3) Q + K ij Kij − K 2 − Di (3) Q − (3) Q̄
+ N h (f − φf ′ )

Σt

n
√

√
o √
n 
oi
◦
i
i
+f ′ ∂µ N i ∂ihnµ − ∂i N i ∂µhnµ+ hf ′ Di N (3) Q − (3) Q̄.
(7.111)

Remark that this foliation takes all boundary terms into account. Integrating by parts andneglecting the spatial boundary terms by imposing appropriate spatial boundary conditionson ∂Σt , we get
Z
SfCGR =

Z
dt

I

oo
o
h √ n
n 

◦
i
i
+ f − φf ′d3 x N h f ′ (3) Q + K ij Kij − K 2 − Di f ′ (3) Q − (3) Q̄

Σt


√
oi
n
√
hnµ − ∂i N i ∂µhnµ.
+f ′ ∂µ N i ∂i(7.112)Further, integrating by parts the first term of and the second term of the boundary term withrespect to the spatial derivative and the spacetime derivative, respectively, and neglecting
each the boundary term on ∂Σt and ∂M, respectively, we obtain the following formula:
h √ n
oo
o
n 

◦
′ (3)
ij
2
′
′ (3) i(3) i
SfCGR = dt
dx N h f
Q + K Kij − K − Di f
+ f − φfQ − Q̄
I
Σt
#
√
√
√


h
h
h
+
N i ∂j N j − N j ∂j N i ∂i f ′ −(∂i f ′ ) Ṅ i +f ′′ ∂i N i φ̇ .
N
N
N
Z

Z

3

(7.113)That is, the non-linearity of f changes the constraint structure of CGR. This action was first
derived in [Hu et al. (2022)] by a different method that resembles the GR case. At first glance,
the Lagrangian density is not a scalar, and it would indicate that this theory generically has
only second-class constraint densities. That is, this theory has no diffeomorphism invariance.
229

It would be expected from the ignorance of the surface terms like “ ∂Γ ” in Eq (6.93) for thecomparison to the Ricci scalar and indicates that the diffeomorphism symmetry is at least
partly broken. A related consideration is shown in Ref [Mukhopadhyay and Padmanabhan(2006)].
The canonical momentum variables are computed as follows:
π0 =δSfCGR (t , x)
= 0,
δ Ṅ (t , y)√h ′′
δSfCGR (t , x)f ∂i φδ (3) (x − y),
πi =
=−
i
N
δ Ṅ (t , y)
δSfCGR (t , x) √ ′πij == hf (Kij − Khij ) δ (3) (x − y),δ ḣij (t , y)√h ′′
δSfCGR (t , x)=f ∂i N i δ (3) (x − y) ,πφ =δ φ̇(t , y)N

(7.114)The canonical momentum variables with respect to the shift vectors depart from the ordinary
CGR case; it depends on the lapse function, the 3-metric hIJ , and the non-linearity part byf ′′ . The canonical momentum with respect to the auxiliary variable φ generates a constraint◦

density, which is different from the case of f (R)-gravity. That is, in coincident f (Q)-gravity,◦

the auxiliary variable φ does not have any physical feature unlike f (R)-gravity.
The Hessian matrix of the system has the size of 11 × 11 - components only being nonvanishing components with respect to the canonical momenta πij . All other components ofthe matrix vanish. Therefore, the rank of the Hessian matrix is six, and it implies that thesystem has five primary constraint densities given as follows:(1)∗ ϕ0 =π0 := 0 ,√h ′′f ∂i φ := 0 ,
N
√
h ′′
(1)
f ∂i N i := 0 .
∗ ϕφ =πφ −N(1)∗ ϕi =πi +230(7.115)

This feature also implies that the physical DoF of the system is up to six. These constraint
densities restrict the entire phase space to the subspace C(1) . The PB-algebras among theseprimary constraint densities are computed as follows:
1 √ ′′
hf ∂i φδ (3) (x − y) = Ai δ (3) (x − y) ,N2
1 √ ′′(1){∗ ϕ0 (t , x) , ∗ ϕ(1)hf ∂i N i δ (3) (x − y) = Bδ (3) (x − y) ,φ (t , y)}P.b. = −
2
N
√
1
(1)
{∗ ϕi (t , x) , ∗ ϕ(1)hf ′′′ ∂i φδ (3) (x − y) = Ci δ (3) (x − y) ,φ (t , y)}P.b. =
N
(1)

(1)

{∗ ϕ0 (t , x) , ∗ ϕi (t , y)}P.b. =(7.116)

where we used Prescription 1 proposed in Section 7.1. Explicitly, the problematic term
occurs in the third PB given in Eq (7.116) as follows:(1){∗ ϕi (t , x) , ∗ ϕ(1)φ (t , y)}P.b.
 √
!
h
(x)
f ′′
∂i +
= Ci δ (3) (x − y) + 
N
x

(1)


!
√
h ′′
(y)
f
∂i  δ (n) (x − y) .
N

(7.117)y(1)

Taking into account that ϕI (x) and ϕφ (y) are density variables, in order to reveal themeaning of the above equations in mathematically correct manner, we have to integrate itout with respect to dn x and dn y on a leaf Σt . Then the first term in Eq (7.117) turns into
the smeared one, and then the second terms become generically as follows:!
#
Z Z " √h ′′(x) (n)f∂i δ (x − y) dxn dy n
N
Σt Σt
x
!
!
#
√
√
Z Z "
h
h
(x)
(x)
=
∂i
f ′′ δ (n) (x − y) − ∂if ′′ δ (3) (x − y) dxn dy n
N
N
Σt Σt
√
= − Ni hf ′′ (1 − D(x ⇄ y)),(7.118)∂Σtwhere we assumed nµ = (−N , N Ni ) with N i Ni = 0 and D(x ⇄ y) denotes the Lebesgue
integration of the second term. If the naive replacement of “ y ” in the spatial derivative231

by “ x ” is possible then D(x ⇄ y) = 1, and Eq (7.118) vanishes. Therefore, in thiscase, if we fix the shift vectors Ni = 0 as the spatial boundary condition, that is, applying
Prescription 1, the problematic term is removed from the analysis, and it is always consistent
with the coincident gauge since the gauge restricts only to the Stükelberg fields via the affine
transformation of a given coordinate system. This is reasonable since the shift vectors are
just unphysical variables. In fact, the action functional (7.113) contains only the first-ordertime derivative of Ni , and it indicates that Ni cannot be any physical variable; it affects only
the constraint structure of the system. It would also be possible to impose f ′′ = 0 on ∂Σt .
(If φ2 ∈ f ′′ then we modify the term φ2 as αφ2 with α → 0 on ∂Σt ). Hereinafter, we use thisprescription also for PBs, which is formulated in detail in Section 7.1. Therefore, these five
primary constraint densities are classified into second-class constraint densities, and, as we
will see, the Dirac matrix has its rank two. Note, here, that the theory is free from the second
issue at least for the primary constraint densities. Otherwise, only one additional secondary
second-class constraint density appears and the Dirac procedure stops, and then the pDoFof the theory turns out to be eight [Hu et al. (2022)], although the normal vector should betaken as Eq (4.41) with Eq (4.42) and Ni ̸= 0 at least on the spatial boundary. Since thecoincident f (Q)-gravity does not have diffeomorphism invariance, this difference of the shift
vectors indicates that we perform the analysis for another sector from the sector analyzed
by the authors in [Hu et al. (2022)]. This discrepancy of pDoF is caused by the differencein the rank of the Dirac matrix density, and this problematic situation has something to dowith the first issue. In addition, strictly speaking, in the work [Hu et al. (2022)], the third
issue explained in Section 7.1 is overlooked in the calculation of the consistency conditions
for the primary constraint densities, therefore, a more exact analysis is mandatory.
The Legendre transformation of the coincident f (Q)-gravity is calculated as follows:(fCGR)

H0 = N C0(fCGR)+ N i Ci

232,

(7.119)(fCGR)

where C0(fCGR)and Ci√are defined as follows:



n 
o
1
1 2
′ (3) i
(3) i
′
ij
,
h f Q − Di f
Q − Q̄
+ f − φf − ′ π πij − π
hf
2
√

◦
h ′′
(fCGR)jCi= − 2D πij −f ∂j N j ∂i φ − ∂i N j ∂j φ .
N(7.120)(fCGR)
C0
=−



◦

′(3)

Therefore, the total Hamiltonian density of the system is introduced as follows:
(1)

(1)

(7.121)HT = H0 + λ0∗ ϕ0 + λi ∗ ϕi + λφ∗ ϕ(1)φ .

The PB-algebras among the primary constraint densities and the density H0 are given in
Appendix A.
(1)

The consistency conditions for the primary constraint densities ∗ ϕα are given as follows:
(1)
(1)
(1)
(1)
β
(1)
∗ ϕ̇α = {∗ ϕα , HT }P.b. ≈ {∗ ϕα , H0 }P.b. + λ {∗ ϕα , ∗ ϕβ }P.b. := 0 ,(7.122)where α, β run in the range of α, β ∈ {0, 1, 2, 3, φ}. Since all these primary constraint
densities are classified into second-class constraint densities, it is necessary to investigate the
(1)

(1)

(1)

rank of the Dirac matrix Dαβ δ (3) (x − y) := {∗ ϕα , ∗ ϕβ }P.b. :


0

A1

A2

A3

B







−A1
0
0
0
C1 




(1)

D =
−A
0
0
0
C
2
2 ,





0
0
0
C3 
−A3


−B −C1 −C2 −C3 0(7.123)where Ai , Ci , and B are defined by Eq (7.116). Applying the fundamental matrix transfor-

233

mations to Eq (7.123), we get the following matrix:




0
0
0
0 B




 0

0
A
A
0
12
13




′(1)
(1) (1) (1)
D = P D Q =  0 −A120
A23 0 , 0 −A13 −A23 0 0 


−B
0
0
0 0

(7.124)

where we set Aij := 2A[i Cj] . P (1) and Q(1) are set as follows:


1


 C1
−
 B

C2
P (1) = 
− B

 C3
− B

0





0 0 0

0

1 0 0

− AB1 

0 1 0 − AB2  ,
0 0 1 − AB3 

0 0 0
1

1 − CB1

− CB2



0
1
0


Q(1) = 
0
1
0


0
0
0

0 − AB1 − AB2

− CB3 0


0
0


0
0
.


1
0

A3
−B 1

(7.125)The straightforward computations lead to Aij = 0. Therefore, The Dirac matrix D(1) has
a rank of two. It indicates that two multipliers are determined, and then three secondary
constrain densities appear. Using these fundamental matrices P (1) and Q(1) , and the Diracmatrix D(1) , the consistency conditions Eq (7.122) becomes as follows:
′(1)

(1)

β

(7.126)ββλ .

(7.127)

P (1) α β {∗ ϕβ , H0 }P.b. + Dαβ λ(1) δ (3) (x − y) := 0 ,where we setαλ(1) = Q(1)

234

−1 α

(1)

(1)

For α = φ and α = 0, the corresponding multipliers λφ and λ0 are determined as follows:
1
(1)
λ(1)
φ = − {∗ ϕ0 , H0 }P.b. ,
B

(1)

λ0 =

1
{∗ ϕ(1)
φ , H0 }P.b. .
B(7.128)The explicit formulae of these multipliers are derived by using the formulae given in Appendix
iα

A. The multipliers λ(1) remain arbitrary. Converting λ(1) into the original multipliers λα ,we get1
Ci (1) iλ0 = {ϕ(1)λ ,φ , H0 }P.b. −
B
B
Ai
1 (1)
i
λφ = − {ϕ0 , H0 }P.b. − λ(1) ,B
B(7.129)iλi =λ(1) .
The secondary constraint densities are derived in the correspondence to the undeterminedi

multipliers λi = λ(1) (i ∈ {1, 2, 3}):
Ci
Ai
(2)
(1)
(1)
(1) α
(1)
{∗ ϕ0 , H0 }P.b. + {∗ ϕi , H0 }P.b. − {∗ ϕ(1)∗ ϕi = Pi {∗ ϕα , H0 }P.b. = −φ , H0 }P.b. := 0 .
B

B(7.130)The explicit formulae of these secondary constraint densities can be derived by using theformulae given in Appendix A and it reveals that all the secondary constraint densities areclassified into second-class constraint densities. These constraint densities restrict C(1) to the
new subspace C(2) .
Using Eq (7.129), the multipliers λ0 and λφ in the total Hamiltonian density Eq (7.121)
are replaced by λi :
(2)

(2)

HT = H0 + λi ∗ Φi ,235

(7.131)(2)(2)

where H0 and ∗ Φiare set as follows:

1
1
(1)
(1)
{∗ ϕ0 , H0 }P.b.∗ ϕ(1){∗ ϕ(1)φ +φ , H0 }P.b.∗ ϕ0 ,B
B
Ai (1) Ci (1)(2)(1)− ∗ ϕ0 ≈ 0 .
∗ϕ∗ Φi =∗ ϕi −
B φ
B
(2)

H0 =H0 −(7.132)In the next section, we calculate the tertiary constraint densities.
(2)

The consistency conditions for the secondary constraint densities ∗ ϕi are given as follows:
(2)
(2)
(2)
(2)
(2)
j
∗ ϕ̇i = {∗ ϕi , H0 }P.b. + λ {∗ ϕi , ∗ Φj }P.b. := 0 .

(7.133)(2)

The existence of tertiary constraint densities depends on the rank of the matrix Dij δ (3) (x −
(2)

(2)

y) = {∗ ϕi , ∗ Φj }P.b. . Using the formulae in Appendix A, we get the following result:(2)

Dij = ∂i φ(α∂j φ + ∆j ) ,(7.134)

where ∆i := βi j ∂j φ and the explicit formulae of α and βi j are given in Appendix B. In
matrix form, D(2) is expressed as follows:
φ1 (αφ1 + ∆1 ) φ1 (αφ2 + ∆2 ) φ1 (αφ3 + ∆3 )

D(2) = φ2 (αφ1 + ∆1 ) φ2 (αφ2 + ∆2 ) φ2 (αφ3 + ∆3 ) ,φ3 (αφ1 + ∆1 ) φ3 (αφ2 + ∆2 ) φ3 (αφ3 + ∆3 )(7.135)where φi := ∂i φ. Applying the fundamental matrix transformations to Eq (7.135), we getφ1 (αφ1 + ∆1 ) 0 0,D′(2) = P (2) D(2) Q(2) = 
0
0
0




0
0 0

236

(7.136)

where P (2) and Q(2) are set as follows:
αφ2 +∆2
αφ3 +∆3
1 − αφ1 +∆1 − αφ1 +∆1 .
Q(2) = 
0
1
0




0
0
1



0 0
 1


 ,
P (2) = 
−φ
φ
0
1
 2



−φ3 0 φ1(7.137)

Therefore, the matrix D(2) has its rank of one. This indicates that one multiplier is determined and then two tertiary constraint densities appear. The consistency conditions in
Eq (7.133) can be rewritten as follows:
(2)

j

′(2)

(2)

(7.138)P (2) i j {∗ ϕj , H0 }P.b. + Dij λ(2) δ (3) (x − y) := 0 ,iwhere λ(2) is set as follows:iλ(2) = Q(2)

−1 i

jλ

j

(7.139).

1

Therefore, the multiplier to the x-component, λ(2) , is determined as follows:11

λ(2) = −1i

φ1 (αφ1 + ∆1 )

(2)

(2)

P (2) {∗ ϕi , H0 }P.b. = −1(2)

φ1 (αφ1 + ∆1 )(2){∗ ϕ1 , H0 }P.b. ,(7.140)where we used P (2) in Eq (7.137). The explicit formula of Eq (7.140) can be derived byusing Eq (7.130), the first formula in Eq (7.132), and the formulae given in Appendix A. Thei′

i

multipliers λ(2) (i′ ∈ {2, 3}) remain arbitrary. Converting λ(2) into the original multipliers
λi , we get
λ1 = −i′λ =λ1
φ1 (αφ1 + ∆1 )

(2) i

(2)

(2)

{∗ ϕ1 , H0 }P.b. −αφ2 + ∆2 (2) 2 αφ3 + ∆3 (2) 3−,λλ
αφ1 + ∆1
αφ1 + ∆1′

(7.141).

The tertiary constraint densities are derived in the correspondence to the undetermined

237

′

i′

multipliers λi = λ(2) :
(3)
(2)
(2)
(2) j
∗ ϕi′ := Pi′ {∗ ϕj , H0 }P.b. := 0 ,(7.142)that is,
(3)
(2)
(2)
(2)
(2)
∗ ϕ2 := − φ2 {∗ ϕ1 , H0 }P.b. + φ1 {∗ ϕ2 , H0 }P.b. := 0 ,(7.143)

(3)
(2)
(2)
(2)
(2)
∗ ϕ3 := − φ3 {∗ ϕ1 , H }P.b. + φ1 {∗ ϕ3 , H0 }P.b. := 0 .

The explicit formulae can be derived by using Eq (7.130), the first formula in Eq (7.132),
and the formulae given in Appendix A and it reveals that all the tertiary constraint densities
are classified into second-class constraint densities. These constraint densities restrict C(2)to the new subspace C(3) .
Using Eq (7.142), the total Hamiltonian density Eq (7.131) is rewritten as follows:
(3)

′

(3)

(7.144)

HT = H0 + λi ∗ Φi′ ,
(3)

(3)

where H0 and ∗ Φi′ are set as follows:
(3)

(2)

H0 =H0 −

1

(2)

(2)

(2)

{∗ ϕ1 , H0 }P.b.∗ Φ1 ,φ1 (αφ1 + ∆1 )αφi′ + ∆i′ (2)
(3)
(2)
.
∗ Φi′ =∗ Φi′ −∗Φ
αφ1 + ∆1 1

(7.145)(3)

The consistency conditions for the tertiary constraint densities ∗ ϕi′ are given as follows:
(3)
(3)
(3)
(3)
(3)
j′
∗ ϕ̇i′ = {∗ ϕi′ , H0 }P.b. + λ {∗ ϕi′ , ∗ Φj ′ }P.b. := 0 .

(7.146)(3)

The existence of quaternary constraints depends on the rank of the matrix Di′ j ′ δ (3) (x−y) :=
(3)

(3)

{∗ ϕi′ , ∗ Φj ′ }P.b. . It is easy to confirm that the rank of D(3) is two (full-rank; its determinantdoes not vanish) since the spatial boundary terms no longer vanish without accidental cases
238due to that the primed indices run the range only of 2, 3, although it is very tedious to leadto the explicit formula of D(3) and its determinant. Therefore, the remaining multipliers are
determined and then the procedure stops here excepting accidental cases. Since there are
five primary, three secondary, and two tertiary constraint densities and all the constraint
densities are classified into second-class constraint densities, therefore, the pDoF and the
gDoF of coincident f (Q)-gravity are

pDoF =

1× (22 − 5 − 3 − 2) = 6 ,2

and gDoF = 0 .

(7.147)For simplicity, so far, we considered that the spacetime manifold has (3 + 1) dimensions,however, it would be possible to extend this result for any spacetime dimension. This analysis
would also give an implication of pDoF and gDoF. Let M be a (n+1)-dimensional spacetime
manifold. To do this, we have to consider the two cases depending on whether n is odd oreven. If n is an odd number then we might get n + 2 primary, n secondary, and n − 1 tertiary
constraint densities and all these constraint densities would be classified into second-classconstraint densities. Therefore, when n is an odd number, we have1
pDoF = (n2 + 3) ,2

and gDoF = 0 ,(7.148)respectively. If n is an even number then we might get n + 2 primary, n secondary, n − 1
tertiary, and 1 quaternary constraint densities, and all these constraint densities would beclassified into second-class constraint densities. Therefore, when n is an even number, we
have
1
pDoF = n2 + 1 ,2

and gDoF = 0 .

(7.149)These pDoFs are just an estimation based on the result of the analysis in the case of a
(3 + 1)-dimensional spacetime. However, it would be possible to strictly prove these results239

by applying mathematical induction.
There is a remark: If the Lebesgue measure in Eq (7.118) is unity, i .e ., D(x ⇄ y) = 1,(x)then the terms like “ A(x)∂i δ (3) (x − y) ” in Eq (7.117) automatically vanish. In this case,another scenario is possible: the case of the absence of Prescription 1. Then the PB-algebras
in the second term in Eq (7.133) contain the spatial boundary terms, and then the matrix
given in Eq (7.134) could be non-singular (full-rank). This means that the tertiary constraints
do not appear and the procedure stops here with the pathological PDEs of the Lagrange
multipliers (third issue): this is a resemble situation to the fTEGR case. The same issue
would arise also in Eqs (7.122) and (7.123). Therefore, if all of these PDEs are solvable, thepDoF becomes seven [Tomonari and Bahamonde (2023)]. Notice that this is another sector
from the sector we have considered in the previous paragraphs so far and this perspective
would be consistent with the recent works of the cosmological perturbation [Gomes et al.
(2023a); Heisenberg et al. (2023)] in the non-trivial branch I [Gomes et al. (2023b)]. Where
the non-trivial branch I is one of five branches, which are the classification of the Stükelberg
fields according to the possible Möbius representation in the teleparallel theories of gravity.
A detailed description is given in [Gomes et al. (2023b)].

7.7. Summary
In this chapter, the Dirac-Bergmann analysis performed for each subclass of the geometrical trinity of gravity, i .e ., GR, TEGR, and CGR, and for each non-linear extension.
Before performing the analysis, the three issues in consistency conditions were revealed.
These issues were commonly caused by the existence of the total divergent terms like “hi(i)∂i Hα1 β1 δ (n) (x − y) ”. Then Prescription 1 was introduced, and it demanded appropriate
spatial boundary conditions if it is necessary. Since all computations of consistency conditions have to be based on the well-defined PB-algebras, Prescription 1 should be imposed atevery step of evaluating PB-algebras when emerging new constraints. As the points to be240careful, for a theory with the characteristics of diffeomorphism invariance, without any lossof generality, Prescription 1 can be applicable. If this is not the case, however, the application of Prescription 1 provides the result of a special sector of the full theory. Nevertheless,this prescription circumvents the three issues, therefore, the result is exact and reliable; it is
not a speculation.
For the analyses of GR and TEGR, on one hand, the analysis leads to two pDoF foreach theory without Prescription 1. For that of CGR, there is a question as mentioned infootnote 4. The investigation of this question is a future work. On the other hand, for the
analyses of non-linear extensions of these theories, we assumed the imposition of Prescription 1. This prescription generically demanded the spatial boundary condition of vanishing
the shift vector on the spatial boundary: Ni (t , ∂Σt ) := 0. Otherwise, for fGR and fTEGR,
some additional degrees of freedom appeared due to the breaking of the closed PB-algebras
of first-class constraints (second issue), and then, caused by it, the Dirac matrix arises. It
might cause a reduction of its rank (first issue). fCGR had the same issue as the latter case.
fTEGR and fCGR suffered from the pathological PDEs of the multipliers (third issue; See
Eq (7.17) in Section 7.1). fGR might also had such concern. The status of each subclass tothese issues is summarized as 7.1 as follows:

Table 7.1: Presence of issues in each non-linear extension of the geometrical trinity of gravity.
“ ⃝ ” denotes the theory is free from the issue, “ × ” denotes the theory suffers from the issue,and Prescription 1 would be necessary. “ △ ” denotes the necessity of further investigating.
Subclass
1st issue
2nd issue
3rd issue

GR
⃝
⃝
⃝

TEGR
⃝
⃝
⃝

CGR
△
△
△

fGR
△
×
△

fTEGR
△
×
×

fCGR
×
⃝
×

Prescription 1 removed these issues from the non-linear extended theories. For diffeomor◦

phism theories, i .e ., f (R)- and f (T )-gravity, the generality of the results were remained.
241

However, the coincident f (Q)-gravity, the result was restricted to a special sector. An analysis of the most generic sector is already performed in [Hu et al. (2022)], but the analysis
suffered from the first and third issues. In this regard, the most detailed investigation is
mandatory.
In particular, in the case of the non-linear extension of CGR, there are three possible
cases. The first criterion is the satisfaction of D(x ⇄ y) = 1. Since this is a mathematicalfact, it should be proved whether or not this statement holds, and the fact is common
for all subclasses once the truth is unveiled. The second criterion is the ignorance of thehi(i)total divergent term ∂i Hα1 β1 δ (n) (x − y) under the imposition of some spatial boundary
condition. In this thesis, we have imposed Ni (t , x) = 0 on the spatial boundaries commonly
to all subclasses or taken a specific foliation characterized by the normal vector given in
Eq (4.41). The current status is given in the following table 7.2:

Table 7.2: Three possibilities of pDoF in coincident f (Q)-gravity. “ ⃝ ” and “ × ” denotethe satisfaction and non-satisfaction of the condition, respectively. “BS” and “SB” denotethe bulk space and spatial boundaries, respectively.
h
i
(i)
(n)
Condition D(x ⇄ y) = 1 Ni (t , x) = 0 (Ignorance of ∂i Hα1 β1 δ (x − y) ) pDoF
Case 1Case 2
Case 3×⃝
Either ⃝ or ×× (Both on BS and SB)× (Only on SB)⃝ (Only on SB)

8
7
6

The most reasonable approach to settle this controversy is to clarify whether or not D(x ⇄y) = 1 holds. If D(x ⇄ y) = 1 then Case 2 or 3 are possible cases since the problematic
term in the PB-algebras among the primary constraints vanish without applying Prescription
1 [Tomonari and Bahamonde (2023)]. In this case, there appear at least the number ofthree secondary constraint densities. In these cases, the most generic case is Case 2 withno restriction to the shift vector although this case suffers form the first issue and third
issue; the latter is the same situation as that f (T )-gravity faced in Eq (7.104). Applying
242

Prescription 1 to Case 2, two tertiary constraints arise, and Case 3 is resulted; this situation
is different from the f (T )-gravity case. Otherwise, only Case 1 is possible case [Hu et al.
(2022)] although it suffers from the first issue and third issue. Case 1 and/or Case 2 iscompatible with the recent works of the cosmological perturbation [Gomes et al. (2023b);
Heisenberg et al. (2023)] in the non-trivial branch I [Gomes et al. (2023a)]. Case 3 is still
alive for the trivial branch and non-trivial branch II, in which two tensor modes and up to
two scalar modes exist as propagating modes. A detailed description of these branches isgiven in [Gomes et al. (2023b)].
It would be also viable either to alternate the method to count out the pDoF or to generalize CGR itself into a theory without the coincident gauge: restoring STEGR. Concerning
the former case, the authors in [Errasti Díez et al. (2020, 2024)] proposed a new method of
degrees of freedom counting for first-order field theories. In this methodology, all procedures
proceed in Lagrange formulation; we do not make use of the problematic PB-algebras. Fortunately, the method is applicable to CGR and fCGR since these subclasses are composed of
up to first-order derivatives. Concerning the latter case, when the diffeomorphism invarianceis restored, the analysis would be performed being analogous to the case of f (T )-gravity interms of vielbein formalism. Otherwise, an alternative formulation proposed in [Hu et al.
(2023)] is also viable. These considerations are important for future work.
Finally, we summarize the current status of the Dirac-Bergmann analysis of the geometrical trinity of gravity and its non-linear extension. The analysis can be performed in thefollowing two formalisms: (i) Metric formalism and (ii) Vielbein formalism. In (i), the analysis can unveil the pDoF and the presence of diffeomorphism symmetry from the viewpointof gauge theory. In (ii), the analysis can unveil the pDoF and the presence not only ofdiffeomorphism symmetry but also of internal space symmetry from the viewpoint of gauge
theory. Therefore, theoretically speaking, it is enough to analyze the latter case, althoughcomputations are far more complicated relative to the former case. In this perspective, we243could summarize the current status of our theories as follows:

Table 7.3: “MF” and “VF” denote the Dirac-Bergmann analysis in the metric formalism andthe vielbein formalism, respectively. “ ⃝ ” denotes the analysis is performed and has acommon understanding. “ × ” denotes the analysis is not performed. “ △ ” denotes theanalysis is performed, but there is no common understanding.
Subclass
MF
VF

GR
⃝
⃝

TEGR
×
⃝

CGR
△
×

fGR
⃝
×

fTEGR
×
⃝

fCGR
△
×

The analysis of GR in the vielbein formalism is performed in the works [Deser and Isham(1976); Henneaux (1983); Charap and Nelson (1983a,b); D’Adda et al. (1985); Charap (1987);
Charap et al. (1988)]. Fulfilling this table by ⃝ is a great future work.

244

Chapter 8
Conclusions
This thesis has quested the following three pillars: the well-posedness of variational principle, the classification of the metric-affine gauge theories of gravity based on the unified
understanding of the geometric quantities in the metric-affine geometry, i.e., curvature, torsion, and non-metricity in the Möbius representations, and the Hamiltonian/Dirac-Bergmannanalysis of the geometrical trinity of gravity and its non-linear extension.
The summary and future directions of the first pillar are given as follows: In Chapter 2,a theory of analytical mechanics for theories of gravity, focusing on degenerate and higherorder derivative point particle systems, was established. The Hamiltonian/Dirac-Bergmannanalysis unveiled the constraint structures such as the gauge symmetries and the possible
propagating degrees of freedom of a given degenerate system. In Chapter 3, a methodology
to make the variational principle well-posed in degenerate point particle systems was constructed. Throughout the consideration in Summary 3.9, in this thesis, the well-posedness
only with the primary constraints is assumed. This methodology is still developing in thefollowing two points; (i) The elucidation of the difference of treatment between the secondaryor more higher-order first-class constraints and that in the primary first-class constraints;(ii) The extension to field theories in a generic manner. In Chapter 4, the general theory of relativity was reconstructed. In particular, the geodesic equations and the Einsteinfield equations are derived based on the well-posed variational principle being consistent
with the primary constraints. In particular, the latter derivation showed that the GibbonsYork-Hawking term is automatically derived and the well-posedness being consistent withall constraints would generate other sort of counter-terms but specific formulae were notunveiled; this is a future work.
245

The summary and future directions of the second pillar are given as follows: In Chapter 5,a mathematical framework for gauge approach to gravity, i .e ., a bundle theory, is reviewed.
The Ehresmann connection and its curvature on a principle G-bundle was introduced, andusing these ingredients, the internal space bundle and frame fields were introduced together
with the Weitzenböch connection. The gauge structures of the metric-affine gauge theories
of gravity were engraved into the internal space. In Chapter 6, deriving the tangent bundleof a spacetime manifold from a principal G-bundle, curvature tensor, and torsion tensor wereintroduced. The curvature was automatically derived from the Ehresmann connection of thegiven principle G-bundle, but the torsion was deduced as an analogy of the derivation of
the curvature. The non-metricity tensor was introduced separately from the above approachand in an ad hoc manner by considering the covariant derivative. In order to constructthese three quantities in a unified manner and classify the metric-affine gauge theories ofgravity into eight subclasses, the Möbius representations were applied. Then teleparallel
equivalent to general relativity (TEGR) and symmetric teleparallel equivalent to generalrelativity (STEGR) were formulated. In particular, for the STEGR subclass, imposing
the coincident gauge condition, coincident general relativity (CGR) is derived. Further,those quadratic and non-linear extensions were also introduced. Notice, finally, that if it ispossible to further extend the metric-affine gauge group using a semi-direct product then a
new geometric quantity would be discovered, and then it would also be possible to construct
a new subclass of gravity. The investigation of this perspective is for an intriguing futurework.
The summary and future directions of the third pillar are given as follows: In Chapter
7, the Dirac-Bergmann analysis was performed for each subclass of the geometrical trinityof gravity and its non-linear extension. First of all, the three issues on the Dirac-Bergmannanalysis were clarified. The first issue was that “the method to define the degeneracy/rank of
the matrix under the existence of the boundary terms”. This issue concerns the rank of the246Dirac matrix, and, therefore, it determines the total number of second-class constraints; thisissue relates not only to the propagating degrees of freedom counting but also to the determination of Lagrange multipliers. The second issue was that “whether should the boundary
terms be also taken into account when classifying a constraint into either a first- or secondclass constraint”. This issue concerns the gauge symmetry of a given theory, and, therefore,
the constraint structure of the given theory, since first-class constraints form the generator of the symmetry. This issue becomes crucial in particular for diffeomorphism invariant
theories in those consistencies. The third issue was that “whether or not the solved phasespace variables ϕa and πb in the consistency conditions Eq (7.16), or equivalently, Eq (7.17),
and that of the field equations of motion are always consistent”. Where the meaning of thisconsistency is that whether or not the functional forms of the Lagrange multipliers exist suchthat the solutions of Eq (7.16) provide that of the field equations motion. Only when thisconsistency is satisfied, the Dirac-Bergmann analysis can count out the propagating degrees
of freedom of a given theory. All of these issues are caused by the spatial boundary term
h
i
(i)
(x)
like “ ∂i Hα1 β1 δ (n) (x − y) ” in Eq (7.13), or more fundamentally, “ A(x)∂i δ (3) (x − y) ” inEq (7.12). To manipulate the issues and make it possible to perform the Dirac-Bergmannanalysis reliably, Prescription 1 was proposed, stating that appropriate spatial boundary
conditions remove the spatial boundary terms from all PBs. In compensation for the use of
Prescription 1, the solution space of the given theory was restricted; the Dirac-Bergmannanalysis does not always unveil all propagating degrees of freedom of a given theory. In moredetail, for a theory with the characteristics of diffeomorphism invariance, on one hand, theusage did not spoil any generality. On the other hand, for a theory with violating its diffeomorphism invariance, the result of the analysis lost its generality and generically provided
that of a special sector. Finally, based on Prescription 1, the Dirac-Bergmann analysis foreach subclass was performed, and the following results were obtained. (See Table 8.1.)
As for future directions of the third pillar, there are three problems; (i) To reveal the247Table 8.1: Degrees of freedoms (DoF) of each subclass in 4-dimentional spacetime. “pDoF”,“gDoF(Diffeo.)”, and “gDoF(Lorentz)” denote propagating DoF, diffeomorphism symmetry,and local Lorentz symmetry, respectively. All analyses are based on Prescription 1. “ (△)” denotes that further investigation based on the Dirac-Bergmann analysis is necessary (Seealso Table 7.3 in Section 7.7.)
Subclass
pDoF
gDoF(Diffeo.)
gDoF(Lorentz)

GR
2
4
6

TEGR
2
4
6

CGR
2 (△)
4 (△)
6 (△)

fGR
2+1
4
6 (△)

fTEGR
2+3
4
0

fCGR
2+4
0
6 (△)

existence of each issue in each subclass (See Table 7.1 in Section 7.7); (ii) To perform the
Dirac-Bergmann analysis based on Prescription 1 for uncompleted subclasses (See Table 7.3
in Section 7.7); (iii) To settle down the controversies in coincident f (Q)-gravity (See Table 7.2
in Section 7.7.) More essentially, it should be investigated a mathematically correct way tomanipulate the spatial boundary terms. In particular, it is important to unveil whether or
not the consistency in the third issue is always satisfied since this problem relates directly
to the validity of the Dirac-Bergmann analysis.
Finally, as the future direction of this thesis, when completing the above problems (foreach subclass), the cosmological perturbation should be performed. (See Figure 1.1.) In thisstage, we should scrutinize the existence of (infinitely) strong couplings against a backgroundspacetime, which is a discrepancy between the true DoF derived from the Dirac-Bergmannanalysis and/or other alternative methodologies [Kaparulin et al. (2013); Errasti Díez et al.
(2020, 2024)] and the estimated DoF in the cosmological perturbation around a backgroundspacetime. In particular, aiming to the application to cosmology, the investigations of thoseexistences against Minkowski spacetime, the flat and non-flat FLRW spacetime are important. If there exists a (infinitely) strong coupling then the theory cannot be applied to cosmology since the perturbation theory breaks down. For instance, the Dirac-Bergmann analysis
unveiled that f (T )-gravity has five pDoF in a generic sector (See Section 7.5), whereas the linear perturbations around Minkowski spacetime [Golovnev and Guzman (2021); Beltrán Jiménez et al.
248

(2021)], the flat and non-flat FLRW spacetime [Bahamonde et al. (2023b)] lead to only
two pDoF. This situation suggests that f (T )-gravity should not be applied in its current
form to cosmology in a theoretically consistent manner. The recent work also indicatesthat the same pathology may appear in the coincident f (Q)-gravity [Beltrán Jiménez et al.
(2020b); Beltrán Jiménez and Koivisto (2021); Gomes et al. (2023a)]. That is, on one hand,
the linear perturbation theories around Minkowski spacetime and (the flat and non-flat)
FLRW spacetime lead to two, and, four, five, six or seven with respect to its connection dependence, respectively. On the other hand, as shown in Section 7.6, the possible
pDoF of the theory is six in a special sector [Tomonari and Bahamonde (2023)] or, either
seven [Tomonari and Bahamonde (2023)] or eight [Hu et al. (2022)] in a generic sector. The
modified/extended theories of gravity are valid for constructing cosmology only when theissues are cured in some way, and the theory becomes a healthy one. Then, for each healthy
theory, the power spectrum of each ingredient should be derived for comparison to the recent
observations. (See Figure 1.1.) The perturbations and the power spectra of several subclasses
are already calculated. For GR and fGR, see Refs [De Felice and Tsujikawa (2010); Koivisto(2006); Li et al. (2012)], for example. For TEGR and fTEGR, see Refs [Chen et al. (2011);
Izumi and Ong (2013); Bahamonde et al. (2023a,b)], for example. For CGR and fCGR, see
Refs [Beltrán Jiménez et al. (2020b); Gomes et al. (2023b,a); Heisenberg et al. (2023)], forexample. For perturbations of the metric-affine gauge theories of gravity, see Ref [Aoki et al.
(2023)] for example. It would be intriguing to scrutinize the late-time accelerated expansionof the universe (, or equivalently, dark energy problem, cosmological constant problem,) andthe tension problems based on healthy models.

249

Figure 8.1: Modern framework for cosmology (Future direction)

250

Appendix A
PB-algebras of coincident f (Q)-gravity in(n + 1)-dimensional spacetime
In the below calculations, all spatial boundary terms are neglected according to Prescription
1 as discussed in Section 7.1. The assumed spatial boundary conditions are Ni (t , ∂Σt ) := 0for each leaf Σt . In this appendix, the Poisson brackets are denoted as “ {· , ·} ” (abbreviate“ P.b. ”). Also, the argument “ (t , x) ” is abbreviated as “ x ”. We abbreviate the notation of“ ∗ ” for the phase space variables.
(1)

The PB-algebras among the primary constraint densities ϕA (A ∈ {0, i, φ}; i ∈ {1, 2, · · · , n})and the density H0 :√#
h N i ′′f ∂j N j ∂i φ − ∂i N j ∂j φ δ (n) (x − y) ,N N
1 f ′′f (Q)(1)+{ϕi (x) , H0 (y)} = −Ciπ∂i φ δ (n) (x − y) ,n − 1 f′

1 f ′′i(1)′π∂i N δ (n) (x − y) ,{ϕφ (x) , H0 (y)} = −H0 −n − 1 f′"(1)
{ϕ0 (x) , H0 (y)} =

f (Q)
−C0 −

(A.1)

where H0′ is defined by
√

"

(

1
(n)
Q−φ+
h



1
f′

2 

1
H0′ := N h −f ′′π ij πij −
π2
n−1


1 N I ′′′jj−
f ∂j N ∂i φ − ∂i N ∂j φ ,
N N

)

n 
o
◦
i
i
+ Di f ′′ (n) Q − (n) Q̄

(A.2)

251

f (Q)and C0is computed as follows:

f (Q)
C0
:= −

√


h f

′(n)



n 
o
1
1
ij
2
′ (n) i(n) i′.
πQ − Di f
Q − Q̄
+ f − φf − ′ π πij −
hf
n−1
◦

(A.3)
f (Q)

Ci

does not change from Eq (7.120) excepting the range of summations. The PB-algebras(1)among two of the primary constraint densities ϕA (A ∈ {0, i, φ}; i ∈ {1, 2, · · · , n}) and thedensity H0 :√

h N i 2 ′′f ∂j N j ∂i φ − ∂i N j ∂j φ δ (n) (x − y) ,N N N
1 1 f ′′
(1)
(1){{ϕ0 , H0 }(x) , ϕi (y)} =π∂i φδ (n) (x − y) ,n − 1 N f′"(
 2 
)√ ′′ (n)1 1
1 1 f ′′
1
(1)
(1)
2
ij
{{ϕ0 , H0 }(x) , ϕφ (y)} =
hf
Q−φ+
π
−
π∂i N i
π
π
−
ij
′
′
h f
n−1
n−1N f#√h N i ′′′f ∂j N j ∂i φ − ∂i N j ∂j φ δ (n) (x − y) .
−
N N
(1)
(1)
{{ϕ0 , H0 }(x) , ϕ0 (y)} =

(A.4)
√

1 h ′′f ∂j N j ∂i φ − ∂i N j ∂j φ δ (n) (x − y) ,N Nnh (f ′′ )2(1)(1)∂i φ∂j φδ (n) (x − y) ,{{ϕi , H0 }(x) , ϕj (y)} =2(n − 1) N f ′
"√



h ′′′
1 1
f ′′
(1)
(1)
j
j
′′′
{{ϕi , H0 }(x) , ϕφ (y)} =f ∂j N ∂i φ − ∂i N ∂j φ +π∂i φ f − ′Nn − 1 f′f#√nh (f ′′ )2−∂j N j ∂i φ δ (n) (x − y) .
2(n − 1) N f ′(1)(1){{ϕi , H0 }(x) , ϕ0 (y)} = −

(A.5)

252

"
(1)
{{ϕ(1)
φ , H0 }(x) , ϕ0 (y)} =

√

(
hf ′′

1
(n)
Q−φ+
h



1
f′

2 

1
π ij πij −
π2
n−1





f ′′′
i
i
− ′′ ∂i φ (n) Q − (n) Q̄
f√
1 f ′′ 1
2 h f ′′iπ∂i N iπ∂i N +−′′
n−1 f
n−1 f N#√ih N ′′′
f ∂J N j ∂i φ − ∂i N j ∂j φ δ (n) (x − y),
−
N N
"
√
n
h (f ′′ )2
1 f ′′′(1)j(1){{ϕφ , H0 }(x) , ϕi (y)} = −
∂
N
∂
φ
−
∂i φ
j
i
2(n − 1) N f ′n − 1 (f ′ )2
#
√


′′

1
π
h ′′′
f
f ∂j N j ∂i φ − ∂i N j ∂j φ +f ′′′ − ′ ∂i φ δ (n) (x − y),+Nn − 1 f′f"
 ′′ 2


1
f ′′
f
1 1
′′
(1)
(1)
′′′′
i
{{ϕφ , H0 }(x) , ϕφ (y)} = −H0 +f − ′ ∂i N iπ∂i N −
n − 1 f′n − 1 f′f√ (f ′′ )2n+h ′ ∂i N i ∂j N j δ (n) (x − y) ,2(n − 1)f
(A.6)
where H0′′ is defined as follows:
H0′′ := N√" 2 



1 1 1
2
1
2
′′
ij
π
h −f
Q−φ +f −1− ′π πij −
h N f′
f
n−1

o
n 

◦
1 N i ′′′′′′′ (n) i
(n) i
j
j
−
Q − Q̄
+Di f
f∂J N ∂i φ − ∂i N ∂j φ .
N N′′′ (n)

(A.7)

(1)

The PB-algebras among the primary constraint densities ϕA (A ∈ {0, i, φ}; i ∈ {1, 2, · · · , n}),Ai /B, Ci /B, and the density H0 are calculated as follows:

  
Ai
(1)
(y) = 0 ,{ϕA , H0 }(x) ,
B



(1)
{ϕA , H0 }(x) ,



Ci
B




(y) = 0 .

(A.8)

(1)

The PB-algebras among the primary constraint densities ϕA (A ∈ {0, i, φ}; i ∈ {1, 2, · · · , n}),Ai /B, and Ci /B are calculated as follows:


Ai
B





(1)
(x) , ϕ0 (y)


=0,

Ci
B





(1)
(x) , ϕ0 (y)253

1 1 ′′′= − ′′
f ∂i φδ (n) (x − y) . (A.9)
i
f ∂i N

)





Ai
B



Ai
B





(x) , ϕ(1)
φ (y)



(1)
(x) , ϕj (y)


=0,

Ci
B




=0,


(x) , ϕ(1)
φ (y)

254

Ci
B





(1)
(x) , ϕj (y)= 0.

(A.10)
1 1 (f ′′′ )2′′′′=N∂i φδ (n) (x−y) .
−f∂i N i f ′′f ′′(A.11)

Appendix B
The explicit formulae of α and βji
α :=1 n h (f ′′ )2
1 1 f ′′′ 1+π
2 n − 1 N f′n − 1 N f ′ ∂i N i
"√
#
√


′′ 2
′′′
f
h
1
1
n
h
(f
)
1
∂k N k +π f ′′′ − ′ −∂k N k+∂k N k N
n − 1 f′f2(n − 1) N f ′
"
#
√
√


′′
′′
n
f
1
1
f
1
h (f ′′ )21
h−f ′′′ ∂k N k −f ′′′′ − ′ ∂k N k+∂k N k −+∂k N k2(n − 1) N f ′n − 1 (f ′ )2Nn − 1 f′
f
"√
# 
2
f ′′′
h
1
f ′′′′ (1)k−∂N{ϕ , H0 }+N
k∂k N k N∂k N kf ′′ 0
2  ′′′ 22 ′′′
1
f
1
f
(1)
(1)
(1)
+
N {{ϕ0 , H0 }, ϕ(1){{ϕ0 , H0 }, ϕ0 } +
φ }
k
′′
k
′′
∂k N
f
∂k N
f


2 ′′′
2
1
1
f
(1)
(1)
(1)
N {{ϕφ , H0 }, ϕ0 } ++{{ϕ(1)
φ , H0 }, ϕφ }∂k N kf ′′∂k N k(B.1)
βJI := −√1h ′′′f ∂J N I .
K∂K N N

255(B.2)

Appendix C
Inönü-Wigner contraction of metric-affine geometry andits extensions
In this Appendix, based on the unified description of curvature, torsion, and non-metricity(or equivalently, dilation and shear) given Section 6.2 and in Ref [Tomonari (2023a)], therelationships between the algebraic structure and geometric quantities of each geometry
are scrutinized by using the so-called Inönü-Wigner contraction. The ultimate purpose is tobeyond the metric-affine geometry and to pursue a new class of geometry and its gauge theoryof gravity. First, the Inönü-Wigner contraction is introduced. Second, the contractions of
the metric-affine geometry and its extension are performed. Finally, the de Sitter/anti-deSitter cases are discussed.
Inönü-Wigner contraction is a method of parametrizing a Lie algebra to decompose it into
subalgebras [Segal (1951); Inonu and Wigner (1953); Lord (1985); Weimar-Woods (1995);
Khasanov and Kuperstein (2011); Subag et al. (2012)]. The contraction is originally valid
not only for Lie algebras but also for those representations, but, in this paper, we discuss
only the contraction of Lie algebras.
Let G be a Lie group and g be its Lie algebra. Then an element g = g(τi ) ∈ G, whereτi (i = 1 , 2 , · · · , t) are the number of t group parameters, leads to an element of the Liealgebra of G as its derivative with respect to the unit element e ∈ G. Let Ii be such elementsof the Lie algebra. Then the following commutation relations hold:[Ii , Ij ] = C k ij Ik ,(C.1)

where [ · , · ] is the Lie bracket and C k ij are the structure constants of the Lie algebra. For
256

this algebra, a transformation of the elements Ii and the group parameters τi is introduced
as follows:Ii′ = P j i Ij ,(C.2)

τi′ = P j i τj ,where P j i is a parametrization matrix. If the matrix is non-singular then the above transformation is just an automorphism of the Lie algebra. However, if it is singular, the situation
gets changed; a subalgebra can be obtained. In particular, it is an intriguing case that the
singularity is realized as some limits of the parameters.
A simple case is the contraction by the following matrix:







α

0r×(n−r)   ϵ β
0r×(n−r)  1r×r
P ij = +,0(n−r)×r 0(n−r)×(n−r)
0(n−r)×rϵᾱ β̄

(C.3)

where ϵα β ∈ (0 , 1], ϵᾱ β̄ ∈ (0 , 1], α , β ∈ {1 , 2 , · · · , t′ < t}, and ᾱ , β̄ ∈ {t′ + 1 , t′ + 2 , · · · , t}.
Then, on one hand, the elements Ii and those algebras are transformed as follows:Ii′ = (δ α i + δ β i ϵα β )Iα + δ β̄ i ϵᾱ β̄ Iᾱ ,(C.4)[Ii′ , Ij′ ]ϵ = δ α i δ β i C γ αβ Iγ + O(ϵ) ,

(C.5)

and

respectively, where [ · , · ]ϵ denotes the Lie brackets of the transformed elements Ii′ . Let gϵand Gϵ denote a new Lie algebra that is generated by Ii′ with satisfying the above algebra
and the Lie group of gϵ , respectively. Notice that gϵ and Gϵ include g and G as a special
case of ϵα β → +0 and ϵᾱ β̄ → δ ᾱ β̄ , where “ +0 ” means the right-limit to zero in (0 , 1].
Therefore, if the limit of the transformation of the above algebras with respect to ϵα β → +0257

and ϵᾱ β̄ → +0 exists then the following new algebra is obtained:(C.6)[Iα′ , Iβ′ ]ϵ→+0 = C γ αβ Iγ′ .

That is, a new subalgebra g0 = gϵ→+0 = ⟨Iᾱ′ ⟩ is obtained, and g0 generates the Lie group
G0 = Gϵ→+0 . On the other hand, the group parameters τi are contracted as follows:τα′ = τα + ϵβ α τβ → τατᾱ′ = ϵβ̄ ᾱ τβ̄ → 0(ϵβ α → +0) ,

(C.7)

β̄

(ϵ ᾱ → +0) .

Notice that the group parameters τᾱ′ of the group Gϵ converge to zero while remaining thecorresponding parameters τβ̄ of G. Since Gϵ includes not only G but also G0 , this meansthat the parametrization given in Eq (C.2) extended the original group G into a larger groupGϵ as a topological space.
Now, first of all, let us consider the contraction of the metric-affine group. The Liealgebra of M A(n + 1 ; R) is summarized as follows:[PI , PJ ] = 0 ,[E I J , PK ] = δ I K PJ ,[D , D] = 0 ,[D , PI ] = PI ,[S I J , PK ] = δ I K PJ ,[E I J , E K L ] = δ I L E K J − δ K J E I L ,(C.8)[D , E I J ] = 0 ,[S I J , E K L ] = 0 ,[S I J , D] = 0 ,[S I J , S K L ] = 0 .

A parametrization for the Inönü-Wigner contraction of the above algebra can be set as
follows:
PI′ = ϵ(P ) PI ,E ′I J = ϵ(E) E I J ,D′ = ϵ(D) D ,S ′I J = ϵ(S) S I J ,(C.9)

where ϵ(P ) , ϵ(E) , ϵ(D) , and ϵ(S) are a set of parameters and belong to the range (0 , 1]. Hereinafter, let us denote the parameter space spanned by the above parameters as
ϵ type of contraction/geometry in MA = (ϵ(P ) , ϵ(E) , ϵ(D) , ϵ(S) ), where “type of contraction/geometry258

in MA” denotes a geometry which is obtained by performing the contraction. Hereinafter,
we use this notation. Then the above algebras are parameterized as follows:[PI′ , PJ′ ] = 0 ,[E ′I J , PK′ ] = ϵ(E) δ I K PJ′ ,[D′ , D′ ] = 0 ,[D′ , PI′ ] = ϵ(D) PI′ ,[S ′I J , PK′ ] = ϵ(S) δ I K PJ′ ,[E ′I J , E ′K L ] = ϵ(E) (δ I L E ′K J − δ K J E ′I L ) ,[D′ , E ′I J ] = 0 ,[S ′I J , E ′K L ] = 0 ,[S ′I J , D′ ] = 0 ,[S ′I J , S ′K L ] = 0 .
(C.10)

Notice that this algebra does not explicitly contain the parameter ϵ(P ) . Therefore, it isenough to express the parameter space without ϵ(P ) as ϵ type of contraction/geometry in MA =(ϵ(E) , ϵ(D) , ϵ(S) ), and there are eight possible contractions. Of course, if all parameters aretaken to be unity then the above algebra (C.10) results in the original algebra (C.8). In
terms of the parameter space notation, the algebra is expressed as ϵ Ln+1 = (1 , 1 , 1).
The contraction generated by the parameters ϵ Yn+1 = (1 , 1 , +0) derives the following
algebras:[PI′ , PJ′ ] = 0 ,[E ′I J , PK′ ] = δ I K PJ′ ,[D′ , D′ ] = 0 ,[D′ , PI′ ] = PI′ ,[S ′I J , PK′ ] = 0 ,[E ′I J , E ′K L ] = δ I L E ′K J − δ K J E ′I L ,(C.11)[D′ , E ′I J ] = 0 ,[S ′I J , E ′K L ] = 0 ,[S ′I J , D′ ] = 0 ,[S ′I J , S ′K L ] = 0 .

This is nothing but the algebra of the Weyl geometry. In this algebra, the shear given in
Eq (6.53) vanishes, i .e ., ↗
∆ = 0, for the metric tensor of the internal space with satisfying
the following equations:g IK ∂µ gKJ =1 I LMδ J g ∂µ gLM .
n(C.12)

The contraction generated by the parameters ϵ Yn+1 with vanishing dilation = (1 , +0 , 1) leads to259

the following algebras:[PI′ , PJ′ ] = 0 ,[E ′I J , PK′ ] = δ I K PJ′ ,[D′ , D′ ] = 0 ,[D′ , PI′ ] = 0 ,[S ′I J , PK′ ] = δ I K PJ′ ,[E ′I J , E ′K L ] = δ I L E ′K J − δ K J E ′I L ,(C.13)[D′ , E ′I J ] = 0 ,[S ′I J , E ′K L ] = 0 ,[S ′I J , D′ ] = 0 ,[S ′I J , S ′K L ] = 0 .

Then the dilation given in Eq (6.46) vanishes, i .e ., ∆ = 0 for the metric tensor of theinternal space with satisfying the following equations:(C.14)

g IJ ∂µ gIJ := 0 .

Therefore, choosing a constant metric tensor, gIJ = cIJ , is a possible gauge condition forvanishing non-metricity. That is, a flat internal space does not have non-vanishing nonmetricity. Remark that Conditions (C.12) and (C.14) can be independently imposed.
In the same manner, the contraction generated by ϵ Un+1 = (1 , +0 , +0) derives thealgebra of the Riemann-Cartan geometry as follows:[PI′ , PJ′ ] = 0 ,[E ′I J , PK′ ] = δ I K PJ′ ,[D′ , D′ ] = 0 ,[D′ , PI′ ] = 0 ,[S ′I J , PK′ ] = 0 ,[E ′I J , E ′K L ] = δ I L E ′K J − δ K J E ′I L ,(C.15)[D′ , E ′I J ] = 0 ,[S ′I J , E ′K L ] = 0 ,[S ′I J , D′ ] = 0 ,[S ′I J , S ′K L ] = 0 .

The algebras in the first line above are nothing but the affine algebra. In the same manneras the cases of the dilation and the shear, the full contraction, i .e ., ϵ 0 = (+0 , +0 , +0) does
not lead directly to the vanishing curvature and/or torsion as follows:
T = de−1 ,Ω(E) = dω (E) .

(C.16)

That is, to vanish the torsion and/or curvature 2-forms, some additional conditions are260necessary. Using Eq (6.36), the condition of vanishing torsion is given as follows(C.17)

de−1 + d2 ζ = de−1 := 0 ,This is nothing but just imposing T := 0. That is, the theory is consistent. In the case of
the curvature, the so-called Weitzenböch gauge condition
ω (E) := 0 ,Ior equivalently , ω (E) µJ := 0

(C.18)

realizes the vanishing curvature. Under the imposition of the above two conditions, thegeometry becomes either En+1 or Mn+1 . In Section 5.4, in particular, the Weitzenböch
connection is derived as in Eq (5.40). Under the imposition of Eq (C.18), the connection is
given as follows:Γ̃ρµν = eI ρ ∂µ eI ν .

(C.19)

Using Eq (6.8) in a coordinate/holonomic basis, the vanishing torsion is equivalent to theexistence of the so-called Stükelberg fields, denoted by η = η I ξI , as follows:eI µ = ∂µ η I ,or equivalently, e−1 = eI ⊗ ξI = dη I ⊗ ξI .

(C.20)

This, of course, satisfies Eq (C.17). Therefore, even if the flat geometries En+1 or Mn+1 ,the internal space still has the degrees of freedom of the Stükelberg fields. Furthermore,
the imposition of the so-called coincident gauge condition [Beltrán Jiménez et al. (2018);
Beltrán Jiménez and Koivisto (2022)], η I = AI µ xµ + B I , where xµ are the coordinates of apoint in the base space M, AI µ ∈ G(n + 1 ; R) (it is a global symmetry), and B = ξI B I
is a constant vector with respect to the global symmetry, then the Weitzenböch connection
given in Eq (C.19) vanishes.

261

Finally, let us consider the contraction generated by ϵ abelian MA = (+0 , 1 , 1). The algebra
is given as follows:[PI′ , PJ′ ] = 0 ,[E ′I J , PK′ ] = 0 ,[E ′I J , E ′K L ] = 0 ,[D′ , D′ ] = 0 ,[D′ , PI′ ] = PI′ ,[D′ , E ′I J ] = 0 ,[S ′I J , PK′ ] = δ I K PJ′ ,[S ′I J , E ′K L ] = 0 ,(C.21)[S ′I J , D′ ] = 0 ,[S ′I J , S ′K L ] = 0 ,where “abelian” in the subscript of the parameter space means that the algebras in the firstline above, which was the Poincare algebra before performing the contraction, turn now to
be a set of commutative algebras. Hereinafter, we use this notation. Imposing Eqs (C.17)
and (C.18), the curvature and the torsion vanish; the geometry turns into the symmetricteleparallel geometry, Sn+1 , which is described only by the dilation and shear, or equivalently,
the non-metricity. Notice that Eq (C.19) and Eq (C.20) hold for the contraction generatedby ϵ abelian MA = (+0 , 1 , 1). The dilation-free case, ϵ abelian MA with dilation−free = (+0 , +0 , 1),
and the shear-free case, ϵ abelian MA with shear−free = (0+ , 1 , +0), can be constituted in thesame manner.
Second, let us consider an extension of the metric-affine group. An extension of the Liealgebra of M A(n + 1 ; R), let us denote EM A(n + 1 ; R), can be given follows:IL[PI , PJ ] = 0 ,[E I J , PK ] = C (A3)

[D , D] = 0 ,[D , PI ] = C (W 4) I PJ ,[D , E I J ] = C (W 9) J D ,(M A1) ILKI[S J , PK ] = C[S I J , D] = 0 ,If C (A3)IL

JK PL ,JJK PL ,I

[S J , E[E I J , E K L ] = C (A6)

IKN

JLM E

M

N ,

I

L] = C(M A16) IKM

JLN S(C.22)NM ,[S I J , S K L ] = 0 .

I
L(A6) IKN
I
K
N
I
K
N
(W 4) JJ
(M A1) IL
JK = δ K δ J , C
JLM = δ L δ M δ J −δ M δ J δ L , C
I = δ I, C
JK =

δ I K δ L J , and otherwise vanish then the Lie algebra ema(n + 1 ; R) of EM A(n + 1 ; R) turns
into that of M A(n+1 ; R). Therefore, the algebra of Eq (C.22) is an extension of the algebra262

of Eq (C.8). Another important algebra, which includes the Poincare algebra as a subalgebra, is obtained under the following structure constants: C (A3)
C (A6)

JLM
IL

I
L
IL
JK = δ K δ J − g g JK ,J

IKNC (M A1)

IL= g IK g JM δ N L − g KN g JM δ I L − δ K J δ I M δ N L + g KN g JL δ I M , C (W 4) I = δ J I ,LIJK = δ K δ J , and otherwise vanish. Then ema(n ; R) is an extension of the Poincare

algebra with the dilation and the shear. Hereinafter, however, we treat the algebra (C.22)
without specifying the structure constants: after obtaining a contraction, we specify a setof structure constants and then develop a theory of gravity, although constructing physicaltheories is out of scope of the current thesis.
The parametrization given in Eq (C.9) is also taken for the above algebra (C.22) toperform the contraction. Then the parametrized algebra of Eq (C.22) is given as follows:
IL[PI′ , PJ′ ] = 0 ,[E ′I J , PK′ ] = ϵ(E) C (A3)[D′ , D′ ] = 0 ,[D′ , PI′ ] = ϵ(D) C (W 4) I PJ′ ,J[S ′I J , PK′ ] = ϵ(S) C (M A1)[S ′I J , D′ ] = 0 ,IL′
JK PL ,′JK PL ,[E ′I J , E ′K L ] = ϵ(E) C (A6)[D′ , E ′I J ] =[S ′I J , E ′K L ] =

IKN

JLM E

′M

N ,

ϵ(D) ϵ(E) (W 9) IK ′CJ PK ,ϵ(P )ϵ(S) ϵ(E) (M A16) IKM′CJL PM ,ϵ(P )[S ′I J , S ′K L ] = 0 .
(C.23)

Notice that this algebra explicitly contains the parameter ϵ(P ) .

Therefore, we use theparameter space ϵ type of contraction/geometry in EMA = (ϵ(P ) , ϵ(E) , ϵ(D) , ϵ(S) ). This is a different situation from the contraction of the metric-affine algebra.

The geometric quanti-

ties of the contraction generated by ϵ 0 = (1 , +0 , +0 , +0), ϵ abelian EMA = (1 , +0 , 1 , 1),ϵ abelian EMA with dilation−free = (1 , +0 , +0 , 1), and ϵ abelian EMA with shear−free = (1 , +0 , 1 , +0)
are constructed in the same manner as the metric-affine case up to the difference of structure
constants (local symmetry). That is, the departures from the metric-affine case are causedby the non-vanishing parameter ϵ(E) .
Let us consider the geometric quantities in the contraction generated by ϵ EMA = (1 , 1 , 1 , 1).
263

Then the curvature and torsion 2-forms are given as Eq (6.39). The dilation and shear 2forms, which are given as Eq (6.46) and Eq (6.53), respectively, are changed as follows:∆=1
Tr ( Q ) e−1 + ω (E) ∧ ω (D)n(C.24)and
1↗
∆ = Q − Tr ( Q ) e−1 + ω (E) ∧ ω (S) ,n(C.25)

respectively. These changes are caused by the violation of the commutativity of D and
E I J , and, S I J and E I J . Of course, the contraction generated by ϵ EMA with dilation−free =(1 , 1 , +0 , 1) and ϵ EMA with shear−free = (1 , 1 , 1 , +0) leads to ∆ = 0 and ↗
∆ = 0 under theimposition of the gauge conditions given in Eq (C.14) and Eq (C.12), respectively.
The parameter space ϵ type of contraction/geometry in EMA = (0+ , ϵ(E) , ϵ(D) , ϵ(S) ) provides aset of new sort of geometric quantities. The coefficients of the third algebra of the second line and that of the second algebra of the third line in the algebra (C.23) diverge
unless an appropriate set of the parameters ϵ(E) , ϵ(D) , and ϵ(S) converges to zero simultaneously. The case of ϵ abelian EMA = (+0 , +0 , ϵ(D) , ϵ(S) ) is the same situation as that of
ϵ abelian MA = (+0 , 1 , 1) in the metric-affine case up to the difference of structure constants(local symmetry). Therefore, the intriguing cases are those of parameter spaces given byϵ type of contraction/geometry in EMA = (+0 , 1 , ϵ(D) , ϵ(S) ). If the contraction is performed by theparameters ϵ EMA with shear−free = (+0 , 1 , 1 , +0) under the imposition of the gauge conditions (C.12) and (C.14) then the dilation given in Eq (C.24) remains but the shear given in
Eq (C.25) turns into ↗
∆ = ω (E) ∧ ω (S) . In the case of ϵ EMA with dilation−free = (+0 , 1 , +0 , 1)
under the same gauge conditions (C.12) and (C.14), the shear given in Eq (C.25) remains
but the dilation given in Eq (C.24) turns into ∆ = ω (E) ∧ ω (D) . Remark, here, that the limitations of ϵ(D) → +0 and ϵ(S) → +0 are taken in the same order magnitude as ϵ(P ) → +0.
The important point here is that these two algebras provide new geometric quantities, which264

are different from the dilation, the shear, or the non-metricity while holding the richnessof the curvature and torsion 2-forms. This richness together with new geometric quantities
would beyond the metric-affine geometry and yield a new class of gauge theories of gravity.
Finally, let us consider the following extended dS/AdS Lie algebra:[PI , PJ ] = C (A2)KL
IJL E K ,[E I J , E K L ] = C (A6)
[D , D] = 0 ,

IKN

K

JLM E

M

IJL = ϵδK

(A3) ILJK PL ,

N ,(C.26)[D , E J ] = 0 ,[S I J , E K L ] = 0 ,J g IL , C

ILI[D , PI ] = PI ,[S I J , PK ] = δ I K PJ ,where C (A2)[E I J , PK ] = C (A3)[S I J , D] = 0 ,[S I J , S K L ] = 0 ,I
LIL(A6) IKN
IK
g JM δ N L −JK = δ K δ J −g g JK , and C
JLM = gg KN g JM δ I L − δ K J δ I M δ N L + g KN g JL δ I M . If ϵ = +1 , −1 then the algebras of the first line
become that of the de Sitter space and anti-de Sitter space, respectively [de Sitter (1917);
Wise (2010); Blagojevic and Hehl (2012); Blagojević and Hehl (2013)]. The case of ϵ = 0 is
contained as a special case of the extended metric-affine algebra in the previous section 7.3.
In order to perform the contraction, the generators PI , E I J , D, and S I J are parametrized
by Eq (C.9). Then the following parametrized Lie algebras are derived:[PI′ , PJ′ ] =(ϵ(P ) )2 (A2) K′LCIJL E K ,ϵ(E)[E ′I J , E ′K L ] = ϵ(E) C (A6)′′

[D , D ] = 0 ,

IKN

JLM E

[D , PI′ ] = ϵ(D) PI′ ,′[S ′I J , PK′ ] = ϵ(S) δ I K PJ′ ,[E ′I J , PK′ ] = ϵ(E) C (A3)
′M

IL

′
JK PL ,

N ,′(C.27)′I[D , E J ] = 0 ,[S ′I J , E ′K L ] = 0 ,[S ′I J , D] = 0 ,[S ′I J , S ′K L ] = 0 .

The contraction of the above algebra generated by ϵ Poincare in extended dS/AdS = (+0 , 1 , ϵ(D) , ϵ(S) )results in that of the algebra ema(n+1 , R) generated by ϵ Poincare in EMA = (ϵ(P ) , 1 , ϵ(D) , ϵ(S) ),where ϵ(P ) , ϵ(D) , and ϵ(S) can be taken as an arbitrary value in (0 , 1]. So is for the contraction265

of algebras between that generated by ϵ abelian Poincare in extended dS/AdS = (+0 , +0 , ϵ(D) , ϵ(S) )in the extended dS/AdS algebra and that generated by
ϵ abelian Poincare in EMA = (ϵ(P ) , +0 , ϵ(D) , ϵ(S) ) in the extended metric-affine algebra under the
choice of the structure constants as the Poincare algebra. Remark, here, that the contractions
generated by ϵ type of contraction/geometry in extended dS/AdS = (1 , +0 , ϵ(D) , ϵ(S) ) do not exist due
to the divergence of the factor (ϵ(P ) )2 /ϵ(E) of the first algebra in the first line. Therefore, it
is enough to consider the case of ϵ type of contraction/geometry in extended dS/AdS = (1 , 1 , ϵ(D) , ϵ(S) ).
The contraction generated by ϵ type of contraction/geometry in extended dS/AdS = (1 , 1 , ϵ(D) , ϵ(S) )alternates the curvature 2-form given in Eq (6.39) as followsΩ(E) = d∇ ω (E) = dω (E) + ω (E) ∧ ω (E) + e−1 ∧ e−1 .

(C.28)The torsion 2-form given in Eq (6.39) does not change. So are the dilation and shear 2-forms,
which are given in Eq (6.46) and Eq (6.53), respectively. Notice that a new geometricalquantity arises; the above quantity is not the curvature 2-form in the sense of the metricaffine geometry due to the existence of the third term. That is, this departure in Eq (C.28)
from the metric-affine geometry would yield a new class of gauge theories of gravity. In
addition, this geometry is not just the (anti-)de Sitter geometry due to the existence of thedilation and the shear (, or equivalently, the non-metricity).
Finally, it is known that the third term in Eq (C.28) relates to the so-called cosmologicalconstant in general relativity if the torsion, the dilation, and the shear vanish [MacDowell and Mansouri(1977); Wise (2010)]. In order to realize this, of course, it is enough to perform further contraction generated by ϵ dS/AdS = (1 , 1 , +0 , +0) and to impose the torsion-free condition
given in Eq (C.17) and the gauge conditions (C.12) and (C.14). In addition, the contraction
generated by ϵ Vn+1 = (+0 , 1 , +0 , +0) removes out the third term in Eq (C.28), and thegeometry results in the (semi-)Riemann geometry; this is nothing but the contraction of the266metric-affine geometry generated by ϵ Vn+1 = (1 , +0 , +0).

267

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