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

Category(English)

Doctoral Thesis

種別(和文)

論文要旨

Type(English)

Summary

Powered by T2R2 (Tokyo Institute Research Repository)

（博士課程）Doctoral Program論 文 要 旨ＴＨＥＳＩＳ ＳＵＭＭＡＲＹ

系・コース：
Department of, Graduate major in

物理学
物理学

学生氏名：
Student’s Name系
コース

友成 恭介

申請学位（専攻分野）：

博士

Academic Degree Requested

Doctor of

審査員主査：
Chief Examiner（ 理学 ）須山 輝明

要旨（英文 800 語程度）
Thesis Summary （approx.800 English Words ）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 say thattheories of constraint systems are a fundamental ground of cosmology.
A wide range of theories of modern physics is constructed upon the variational principle, andso 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 Dirichletboundary conditions are imposed as usual. For degenerate systems, however, the existence ofconstraints 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 thesolutions. In particular, since the theories of gravity have the property of 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 thestudy of establishing the methodology.
Metric-affine gauge theories of gravity are one of the modified/extended theories of generalrelativity 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 latetime accelerated expansion of the universe (, or equivalently, dark energy problem, cosmologicalconstant problem), the origin of inflation from the viewpoint of low-energy effective theories ofquantum gravity based on the UV-competition, 𝐻! - and/or 𝜎" -tension problems. These theories
are classified into eight subclasses from viewpoints of gauge 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 of General Relativity (GR), Teleparallel
Equivalent to GR (TEGR), and Symmetric TEGR (STEGR), have gathered attention in recent
days since the non-linear extensions of these subclasses 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 the study 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 the existenceof the primary constraints, and all constraints satisfy the so-called consistency conditions, whichdemands all constraints to be static. For point particle systems, this analysis is mathematicallywell-formulated, meaning that the Poisson brackets are always well-defined. For field theories,
which include, of course, theories of gravity, however, the well-definedness should bereconsidered since the existence of the Dirac delta function causes problematic situations when
determining the Lagrange multipliers such as contradictions in the consistency conditions to theproperties of the given system that are derived from its action functional. In this thesis, the issuesof 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, the analysis is performed for each subclass in the geometrical trinityof gravity and its non-linear extension.
Finally, the uncompleted works and future directions of this thesis are discussed.

備考 : 論文要旨は、和文 2000 字と英文 300 語を 1 部ずつ提出するか、もしくは英文 800 語を 1 部提出してください。
Note : Thesis Summary should be submitted in either a copy of2000 Japanese Characters and 300 Words (Engｌish) or 1copy of 800

Words (English).
注意：論文要旨は、東工大リサーチリポジトリ（T2R2）にてインターネット公表されますので、公表可能な範囲の内容で作成してください。
Attention: Thesis Summary will be published on Tokyo Tech Research Repository

Website (T2R2).