# Symplectic Approach of Three-Dimensional Palatini Theory Plus a Chern-Simons Term

Authors: Alberto Escalante; Prihel Cavildo-Sánchez
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

Using the symplectic framework of Faddeev-Jackiw, the three-dimensional Palatini theory plus a Chern-Simons term [P-CS] is analyzed. We report the complete set of Faddeev-Jackiw constraints and we identify the corresponding generalized Faddeev-Jackiw brackets. With these results, we show that although P-CS produces Einstein’s equations, the generalized brackets depend on a Barbero-Immirzi-like parameter. In addition, we compare our results with those found in the canonical analysis showing that both formalisms lead to the same results.


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## Body

## 1. Introduction

It is well known that the core of canonical gravity is based on the Hamiltonian formalism for singular systems developed by Dirac [1, 2]. In fact, the Dirac formalism is a powerful approach for studying gauge systems and relevant information is obtained from it. For instance, it is possible to obtain the gauge symmetry, the counting of the physical degrees of freedom, and the identification of the quantum brackets, the so-called Dirac brackets. The modern starting point for studying the canonical approach of gravity is based on the Holst action, which depends on a connexion valued on the SO(3,1) group, a tetrad 1-form, and a parameter called the Barbero-Immirzi (the so-called γ parameter) [3–5]. In fact, because of the presence of the γ parameter, the Holst action provides a set of actions classically equivalent to Einstein’s theory and its contribution can be appreciated at the classical level in the structure of the constraints. On the other hand, in the coupling of fermionic matter with gravity, γ determines the coupling constant of a four-fermion interaction [5]. It is important to comment that Holst’s action also allows us to reproduce the different classical formulations of canonical gravity reported in the literature; for instance, if γ is real, then the Barbero formulation is obtained [6]; moreover, if the parameter is complex, then the Ashtekar approach is reproduced [7]. However, at the moment, the γ parameter is controversial because its value is unknown. In fact, in the quantum version of the Barbero formulation, there is a contribution of γ in the area operator [8, 9]. Thus, if we were able to develop an experiment for calculating quanta of area, then we could calculate the value of γ.On the other hand, in the three-dimensional case, there is a model with the same characteristics of the Holst action, the so-called Bonzom-Livine action [BL] [10]. In fact, the BL theory turns out to be classically equivalent to three-dimensional Palatini theory. It provides a set of actions sharing the classical equations of motion with the Palatini theory, and its symplectic structure depends on a γ-like parameter. In this respect, it is also possible to obtain similar results from an action principle with a more simple structure than BL theory: the Palatini theory plus 1/γ times the Chern-Simons term [P-CS]. In fact, the P-CS theory also provides a classical set of actions sharing the equations of motion with Einstein’s theory and their symplectic structure depends on the γ parameter. It is important to comment that the [P-CS] has been analyzed from the Lagrangian and Hamiltonian points of view [10]; however, all the fundamental steps were ignored; that is, the complete structure of the constraints and the symmetries of the theory were not reported.In this manner, with the antecedents mentioned above, in this paper, we will study the P-CS theory from a symplectic point of view. For this aim, we will use the symplectic formalism of Faddeev-Jackiw [FJ] [11–25], due basically to the fact that the FJ approach is more economical than Dirac’s method. In fact, the FJ is a symplectic description where all relevant information of the theory can be obtained through a symplectic tensor, which is constructed from the symplectic variables that are identified from the Lagrangian. If the theory is singular, then there will be constraints and the FJ method avoids the classification of the constraints in primary, secondary, first class, or second class, as it is done in Dirac’s method. Furthermore, from the components of the symplectic tensor, it is possible to identify the FJ generalized brackets; at the end of the calculations, the Dirac and the FJ generalized brackets are equivalent. Furthermore, in order to complete our analysis, we have added as an appendix the Dirac approach of P-CS, where we report the complete structure of the constraints and at the end we compare the results obtained in both formulations.The paper is organized as follows. In Section2, the symplectic analysis is developed; we report the complete FJ constraints and a symplectic tensor is constructed. From that symplectic tensor, the generalized FJ brackets are identified and we compare the FJ results with those obtained in the Dirac method. In Section 3, we present prospects and conclusions. Finally, an appendix is added resuming the canonical analysis.


## 2. Symplectic Analysis

The theory that we will analyze along this paper is described by the following action:(1)Se,A=2∫Mei∧FiA+1γ∫MAi∧dAi+23ϵijkAi∧Aj∧Ak,where M is a three-dimensional manifold without boundary, F=dA+A∧A is the curvature of the connexion A, e’s are the triad fields, i,j,k,…=0,1,2 are internal SU(2) indices, and γ is an arbitrary parameter like a Barbero-Immirzi parameter [10]. From the variation of the action, the following equations of motion arise: (2)De+1γF=0,F=0, which are classically equivalent to the three-dimensional Einstein equations. Along the paper, we will use the notation μ,ν,…=0,1,2 for space-time indices and the alphabet letters a,b,c for space indices. Moreover, we will suppose that the manifold has topology M=Σ×R, where Σ is a Cauchy surface and R is an evolution parameter. With these considerations at hand, we perform the 2+1 decomposition, and action (1) takes the following form:(3)Se,A=∫ϵ0abebi+12γAbiA˙ai+12e0iFabi+A0iDaebi+12γFabid3x,where (4)Fabi=∂aAbi-∂bAai+ϵijkAajAbk,Daebi=∂aebi+ϵijkAajebk. From (3), we can identify that the Lagrangian density has the following symplectic form [11–25]:(5)L0=ϵ0abebi+12γAbiA˙ai+12ϵ0abe0iFabi+ϵ0abA0iDaebi+12γFabi=ϵ0abebi+12γAbiA˙ai--12ϵ0abe0iFabi-ϵ0abA0iDaebi+12γFabi=ai0ξ˙0i-V0,whereξ(0)i are the set of symplectic variables,(6)ξ0i=e0i,eai,A0i,Aai,a(0)’s are the corresponding 1-forms(7)ai0=0,0,0,ϵ0abebi+12γAbi,and in addition V(0) is the so-called symplectic potential:(8)V0=-12ϵ0abe0iFabi-ϵ0abA0iDaebi+12γFabi. In the FJ context, the equations of motion are expressed as fij(0)ξ˙(0)j=∂V(0)/∂ξ(0)i, where fij(0) is the so-called symplectic matrix and it is given by fij(0)=∂aj(0)/∂ξ(0)i-∂ai(0)/∂ξ(0)j [11–25]. In this manner, using the symplectic variables and the 1-forms given above, the symplectic matrix acquires the form(9)fij0=0000000ϵ0abδil00000-ϵ0abδil01γϵ0abδilδ2x-y. Furthermore, we observe that fij(0) is a singular matrix; hence, this means that the Lagrangian describes a constrained theory; therefore, there will be constraints. In order to identify these FJ constraints, we calculate the zero modes of fij(0): (10)V10=Ve0i,0,0,0,V20=0,0,VA0i,0, where Ve0i and VA0i are arbitrary functions. In this manner, the FJ constraints are found from the following contractions [11–25]: (11)Ω10=∫d3xV10iδδξ0i∫d3yV0ξ=∫d3xVe0i∫δV0xδe0iyd2y=Ve0i-12ϵ0abFabi=0.Ω20=∫d3xV20iδδξ0i∫d3yV0ξ=∫d3xVA0i∫δV0xδA0iyd2y=-VA0iϵ0abDaebi+12γFabi=0,In this manner, the FJ constraints read(12)Ω10i=12ϵ0abFabi=0,Ω20i=ϵ0abDaebi+12γFabi=0. Now, we will determine whether there are more FJ constraints; for this aim, we require consistency of the constraints [11–25]. This means that the following system is satisfied: (13)f¯ij0ξ˙j=Zi, where (14)f¯ij0=fij0∂Ω0∂ξj,Zi=∂V0∂ξi0. In this manner, the matrix f¯ij(0) has the explicit form(15)
f
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0
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0
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1
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ϵ
0
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j
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0
a
b
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j
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b
+
ϵ
i
j
k
A
b
k
0
ϵ
0
a
b
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i
j
∂
b
+
ϵ
i
j
k
A
b
k
0
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0
a
b
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j
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1
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.We can observe that this matrix is not square as expected; however, it has null vectors:(16)v1i=0,∂iVa-ϵijkAajVk,0,0,Vi,0,v2i=0,-ϵijkeajVk,0,∂aVi-ϵijkAajVk,0,Vi.On the other hand, the vector Zi is given by(17)Zi=Ω10i-ϵ0ab∂bA0i+ϵijkAbjA0kΩ20iϵ0ab∂ae0i+1γA0i-ϵ0abϵijkA0jeak-ϵ0abϵijke0j+1γA0jAak00. In this manner, the contraction of the null vectors (16) with Zi yields identities. For example, from the contraction of the vector v1i with Zi, one obtains(18)v1iZi=ϵijkViA0jΩ1k0=0, which implies that there are no more FJ constraints. As we have mentioned previously, in the FJ approach, it is not necessary to perform the classification of the constraints in first class, second class, and so forth, but all constraints are at the same level. Hence, the counting of degrees of freedom is carried out in the following form: there are 6 dynamical variables Aai and 6 independent FJ constraints Ω1(0)i, Ω2(0)i; in this manner, the theory is devoid of degrees of freedom, and the theory is topological as expected.Now, we will add to the symplectic Lagrangian the new constraints found above and we define a new symplectic Lagrangian, sayL(1):(19)L1=ai0ξ˙0i-V0Ω1=Ω2=0-Ω1i0α˙i-Ω2i0β˙i,=ϵ0abebi+12γAbiA˙ai-Ω1i0α˙i-Ω2i0β˙i,where we have added the FJ constraints via Lagrange multipliers; namely, e0i≡α˙i and A0i≡β˙i. Note that the symplectic potential vanishes V(0)Ω1=Ω2=0=0; this result is expected because of the general covariance of the theory; all the dynamics of the system are governed by a symmetry. From the symplectic Lagrangian (19), we identify the set of new symplectic variables(20)ξ1i=αi,eai,βi,Aai and the following 1-forms:(21)ai1=-Ω10i,0,-Ω20i,ϵ0abebi+12γAbi. With the help of these symplectic variables, we calculate the new symplectic matrix; namely, fij(1)=∂aj(1)/∂ξ(1)i-∂ai(1)/∂ξ(1)j, which acquires the following explicit form: (22)fij1=000δij∂a+ϵijkAak00δij∂a-ϵijkAakδij0δij∂a+ϵijkAak0ϵijkeak+1γδij∂a+ϵijkAakδij∂a-ϵijkAakδij-ϵijkeak+1γδij∂a-ϵijkAak1γδij×ϵ0abδ2x-y.This matrix is singular, and the corresponding zero modes are the generators of the gauge transformations. The null vectors are given by v~(1)=0,-ϵljkϖjeak,ϖl,∂aϖl-ϵljkϖjAak and v~(2)=-τi,∂aτi-ϵijkτjAak,0,0, where ϖi and τi form a set of infinitesimal parameters characterizing the gauge transformations. In fact, the null vectors generate the following gauge transformations:(23)δαi=-τ˙i,δeai=Daτi+ϵijkeajϖk,δβi=ϖ˙,δAai=Daϖi,where we can see that v~(1) is the generator of rotations and v~(2) of translations. Furthermore, the matrix (22) is still singular; however, we have shown that there are no more constraints and that the theory has a gauge symmetry. In order to construct a symplectic tensor, we need to fix the gauge [11–25], and thus we will fix the temporal gauge, say, e0i=A0i=0, which means that α˙i=0 and β˙i=0. In this manner, the fixing gauge will be added to the symplectic Lagrangian via Lagrange multipliers, Θi and Ξi. Hence, the symplectic Lagrangian which will be called L(2) is(24)L2=ai2ξ˙2i=ϵ0abebi+12γAbiA˙ai-Ω10i+Θiα˙i-Ω20i+Ξiβ˙i, and from this expression the following symplectic variables and the 1-forms are identified: (25)ξ2i=αi,eai,βi,Aai,Θi,Ξi,a2i=-Ω10i+Θi,0,-Ω20i+Ξi,ϵ0abebi+12γAbi,0,0. In this manner, using these sets of symplectic variables and 1-forms, the corresponding symplectic matrix fij(2) is given by (26)fij2=000ϵ0abηij∂b+ϵijkAbkδij000ϵ0abδij∂a-ϵijkAakϵ0abδij000ϵ0abδij∂b+ϵijkAbk0ϵ0abϵijkebk+ϵ0abγηij∂b+ϵijkAbk0δij-ϵ0abδij∂a-ϵijkAak-ϵ0abδij-ϵ0abϵijkeak+1γϵ0abδij∂a-ϵijkAak1γϵ0abδij00-δij0000000-δij000δ2x-y,where we note that fij(2) is not singular; therefore, there exists its inverse. After long calculations, the inverse of fij(2) is given by (27)f2ij-1=0000-δij00-1γϵ0abδij0-ϵ0abδijδij∂b+ϵjikAbk-ϵjikeak00000-δij0ϵ0abδij000-δij∂a-ϵjikAakδij-δij∂b+ϵjikAbk000ϵ0abAaiAbj0ϵjikebkδijδij∂a+ϵjikAak-ϵ0abAaiAbjϵ0abAaiAbj×δ2x-y.This corresponds to a symplectic tensor. From the symplectic tensor (27), we can identify the generalized FJ brackets by means of(28)ξi2x,ξj2yFJ=fij2x,y-1, and hence the FJ brackets are given by(29)eaix,ebjyFJ=-1γϵ0abδijδ2x-y,eaix,AbjyFJ=-ϵ0abδijδ2x-y,Aaix,AbjyFJ=0,and we can observe that these brackets are the same as those obtained using the Dirac method (see the Appendix and [10]). Furthermore, we can observe that there is a γ-contribution in the fundamental brackets, which makes a difference with respect to Palatini theory [26]. In fact, in Palatini theory, the Dirac brackets between the triad fields are commutative, while in P-CS, they are not; however, we observe that P-CS and Palatini are equivalent in the limit when γ goes to infinity.


## 3. Conclusions

In this paper, a symplectic analysis of P-CS has been performed. We reported the complete set of FJ constraints, the gauge symmetry, and the FJ generalized brackets which are not reported in the literature. The advantage for applying the symplectic formalism becomes to be present in a more economical analysis. In fact, it is not necessary to develop the classification of the constraints as in Dirac’s approach (see the Appendix), and the results found in the Dirac method are reproduced in the symplectic scheme. Furthermore, we observed that the generalized brackets depend onγ, and this fact makes the P-CS theory different from Palatini theory. In fact, it is well known that two theories sharing the same equations of motion do not imply that these theories are equivalent at all [27–29]; the difference between the generalized brackets will be relevant in the quantization program; we need to remember that the symplectic structure is an essential ingredient in the quantum canonical approach [30]. It is important to note that the contribution of degrees of freedom just like the addition of matter fields could help us understand the nature of the γ parameter. In fact, it is reported in the literature that, in the description of gravity coupled to fermion fields, the Immirzi parameter is a coupling constant determining the strength of a four-fermion interaction [31]. Hence, we expect that in three-dimensional coupling of P-CS with matter the symplectic structure will not change and therefore we could investigate the relation between γ and matter coupling through their symplectic structure and the constraints. All these ideas are already under study and will be the subject of forthcoming works.



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