Fermions coupled to the Palatini action in ndimensions Jorge Romero and Merced Montesinos

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Fermions coupled to the Palatini action in ndimensions

Jorge Romero ∗and Merced Montesinos †

Departamento de F´ısica, Centro de Investigaci´on y de Estudios Avanzados del

Instituto Polit´ecnico Nacional, Avenida Instituto Polit´ecnico Nacional 2508,

San Pedro Zacatenco, 07360 Gustavo Adolfo Madero, Ciudad de M´exico, Mexico

Ricardo Escobedo ‡

Departamento de F´ısica, Centro Universitario de Ciencias Exactas e Ingenier´ıas,

Universidad de Guadalajara, Avenida Revoluci´on 1500,

Colonia Ol´ımpica, 44430, Guadalajara, Jalisco, Mexico

(Dated: October 21, 2022)

We study minimal and nonminimal couplings of fermions to the Palatini action in ndimensions

(n≥3) from the Lagrangian and Hamiltonian viewpoints. The Lagrangian action considered is not,

in general, equivalent to the Einstein-Dirac action principle. However, by choosing properly the

coupling parameters, it is possible to give a ﬁrst-order action fully equivalent to the Einstein-Dirac

theory in a spacetime of dimension four. By using a suitable parametrization of the vielbein and the

connection, the Hamiltonian analysis of the general Lagrangian is given, which involves manifestly

Lorentz-covariant phase-space variables, a real noncanonical symplectic structure, and only ﬁrst-

class constraints. Additional Hamiltonian formulations are obtained via symplectomorphisms, one

of them involving half-densitized fermions. To confront our results with previous approaches, the

time gauge is imposed.

I. INTRODUCTION

General relativity in ndimensions, in the ﬁrst-order

formalism, is given by the Palatini action principle, which

depends functionally on the vielbein eIand the Lorentz

connection ωIJ, which are the fundamental independent

variables of the theory. This framework is the natural

arena to make the coupling of fermions to gravity, which

is not possible in the metric formalism of general relativ-

ity. When there are no matter ﬁelds coupled to gravity,

the equation of motion for the connection ωIJcan be

solved to yield ωIJas a function of the vielbein and its

derivatives, and substituting it into the Palatini action

leads to an equivalent second-order action principle for

general relativity, which depends only on the vielbein eI.

On the other hand, when a fermion ﬁeld is minimally

coupled to the Palatini action, the theory is not equiva-

lent to the Einstein-Dirac theory because of the coupling

of the Lorentz connection to the fermion ﬁeld (see, for

instance, Refs. [1,2] for a spacetime of dimension four).

In the context of an n-dimensional spacetime, the

Hamiltonian analysis of fermions minimally coupled to

gravity in the ﬁrst-order formalism has been studied in

Ref. [3]. The Hamiltonian formulation derived there re-

lies on the time gauge. Such a gauge ﬁxing simpliﬁes

the handling of the second-class constraints that emerge

during the usual Hamiltonian analysis, but it breaks the

local Lorentz symmetry in the process. Since the local

Lorentz symmetry is one of the fundamental symmetries

∗ljromero@ﬁs.cinvestav.mx

†Corresponding author

merced@ﬁs.cinvestav.mx

‡ricardo.escobedo@academicos.udg.mx

of nature that is also required to make the coupling of

fermions to gravity at the Lagrangian level, it is essential

to maintain it during the Hamiltonian analysis to get a

deeper understanding of the gravity-fermion interaction.

Therefore, in this work, we study the coupling of

fermions to the n-dimensional Palatini action (n≥3)

in the Hamiltonian formalism without spoiling the local

Lorentz invariance. Moreover, to avoid the introduction

of second-class constraints in the Hamiltonian analysis—

and the complications they imply [4]—we follow the

method presented in Ref. [5] where authors get the

Hamiltonian formulation of the n-dimensional Palatini

action from scratch by making a suitable parametrization

of the vielbein eIand the connection ωIJ(see also Ref. [6]

where the Hamiltonian analysis of the Holst action is per-

formed following the same procedure). An advantage of

the approach of Refs. [5,6] is that it naturally allows us

to identify the manifestly Lorentz-covariant phase-space

variables of the theory and, after eliminating the auxil-

iary ﬁelds from the action using their own equations of

motion, the Hamiltonian formulation formed solely by

ﬁrst-class constraints easily follows, which simpliﬁes con-

siderably the analysis. This approach has also been used

to study the coupling of fermions to the Holst action [7].

We begin our analysis in Sec. II, where we present the

ﬁrst-order action principle for a fermion ﬁeld coupled to

the Palatini action in ndimensions used throughout the

manuscript. The coupling of the fermion ﬁeld is generi-

cally nonminimal, but it also includes the minimal cou-

pling as a particular case. We eliminate ωIJfrom the

action principle using its equation of motion and ob-

tain the equivalent second-order action principle, which

turns out to be diﬀerent from the Einstein-Dirac theory

in the generic case. However, we show that a particular

choice of the coupling parameters in the ﬁrst-order La-

arXiv:2210.07378v2 [gr-qc] 19 Oct 2022

grangian action in four dimensions is equivalent to the

Einstein-Dirac action principle plus a boundary term.

Next, in Sec. III, the Hamiltonian analysis of the general

Lagrangian is performed straightforwardly. In Sec. IV,

we present two additional Hamiltonian formulations; one

of which is obtained through a symplectomorphism while

the other employs half-densitized fermions, which simpli-

ﬁes even more the constraints. For the sake of complete-

ness, we impose the gauge ﬁxing known as time gauge

in Sec. V, and compare some of our results with those

obtained in Ref. [3]. We ﬁnish the paper by making

some remarks in Sec. VI. Our notation and conventions

are collected in the Appendices A–C. Further details of

the Hamiltonian formulations when the spacetime has di-

mensions three and four are given in the Appendices D

and E, respectively.

II. LAGRANGIAN ANALYSIS

A. The action principle

The gravitational ﬁeld is given by the n-dimensional

Palatini—also known as Einstein-Cartan—action

SP[e, ω] = κZM?eI∧eJ∧FIJ −2Λρ,(1)

where κ= (16πG)−1modulates the strength of gravity,

Gis Newton’s gravitational constant, FIJ:= dωIJ+

ωIK∧ωKJis the curvature of the SO(n−1,1) connection

ωIJ,ρ:= (1/n!)I1...IneI1∧···∧eInis the volume form, Λ

is the cosmological constant, and ?stands for the Hodge

dual (see Appendix Afor more details).

The fermion ﬁeld ψ, coupled to gravity, is given by the

action

SF[e, ω, ψ, ¯

ψ] := ZM1

2¯

ψγIEDψ −DψγIE†ψ∧?eI

−m¯

ψψρ,(2)

where ¯

ψ= iψ†γ0,γIare the Dirac matrices, mis the

mass of ψ,Dstands for the covariant derivative with re-

spect to ωIJ[see (A5a) and (A5b)], and Eis the coupling

matrix deﬁned by

E:= (1 + iθ)1−iξΓ,if nis even

(1 + iθ)1,if nis odd ,(3)

with θand ξbeing dimensionless real parameters and Γ

being the chirality matrix (A10). The coupling matrix E,

E+E†= 21, involves minimal and nonminimal couplings

depending on the values of the parameters. The minimal

coupling is when E=1, which amounts to set θ=ξ= 0.

Note that if nis odd, then Γ is proportional to 1, and

thus it is already considered in E.

It is remarkable that when gravity is turned oﬀ, the ac-

tion principle (2) leads to the Dirac equation with m6= 0

in an n-dimensional Minkowski spacetime for any generic

form of the coupling matrix Egiven by (3) (see Ap-

pendix B). Thus, the action (2) has the correct limit when

there is no gravity.

In this paper we are interested in the coupling of

fermions to general relativity. Therefore, the theory we

are going to study is given by the action principle

S[e, ω, ψ, ¯

ψ] := SP[e, ω] + SF[e, ω, ψ, ¯

ψ],(4)

which generalizes the one considered in Ref. [3], where

authors study only the minimal coupling (E=1).

B. Second-order action

Before performing the Hamiltonian analysis of the

ﬁrst-order action (4) and to better understand the na-

ture of the coupling of fermions to gravity, we eliminate

the connection ωIJfrom the action principle (4) using

its equation of motion to get the equivalent second-order

action principle, so we can make some remarks regard-

ing the coupling of the fermion ﬁeld to gravity in both

ﬁrst-order and second-order formalisms.

The variation of the action (4) with respect to the con-

nection ωIJgives the equations of motion

κD ?(eI∧eJ)+1

4ηK[I¯

ψγJ](E−E†)ψ

+¯

ψ{γK, σIJ }ψ? eK= 0,(5)

where we made use of the fact that E+E†= 21and

(A6).

The equation of motion (5) can be rewritten in the

form

DeI:= deI+ωIJ∧eJ=TI,(6)

where TIis the torsion given by

TI:= 1

8κ1

n−2¯

ψγJ(E−E†)ψeI∧eJ

−¯

ψ{γI, σJK }ψeJ∧eK.(7)

The solution for ωIJis

ωIJ= ΩIJ+CIJ,(8)

where ΩIJ=−ΩJIis the torsion-free spin connection

(deI+ ΩIJ∧eJ= 0) and CIJ=−CJIis the contorsion

1-form

CIJ := 1

8κ2

n−2¯

ψγ[J(E−E†)ψeI]+¯

ψ{γK, σIJ }ψeK.

(9)

The contorsion and the torsion are related by TI=CIJ∧

eJ.

Due to the fact ωIJhas been solved using its equation

of motion, it is an auxiliary ﬁeld [8]. Next, we substitute

the solution for the connection (8) into the action (4) and

obtain, using (A7) and after some algebra, the equivalent

second-order action principle

Seﬀ[e, ψ, ¯

ψ] := κZM?eI∧eJ∧RIJ −2Λρ

+ZM1

2¯

ψγIDΩψ−DΩψγIψ∧?eI

−m¯

ψψρ+Sint[e, ψ, ¯

ψ]

−1

4(n−2) Z∂M

ψγI(E−E†)ψ ? eI,

(10)

where RIJis the curvature of ΩIJ,RIJ=dΩIJ+ΩIK∧

ΩKJ, and the covariant derivatives of ψand ¯

ψare given

DΩψ:= dψ +1

2ΩIJ σIJ ψ, (11a)

DΩψ:= d¯

ψ−1

2ΩIJ ¯

ψσIJ .(11b)

A relevant aspect of the second-order Lagrangian for-

mulation (10) is the presence of the interaction term

Sint[e, ψ, ¯

ψ] := 1

64κZMn−1

n−2¯

ψγI(E−E†)ψ¯

ψγI(E−E†)ψ+¯

ψ{γI, σJK }ψ¯

ψ{γI, σJK }ψρ. (12)

Therefore, due to the interaction term Sint, the resulting

second-order action (10) is generically diﬀerent from the

Einstein-Dirac theory, unless the interaction term van-

ishes. Note that the last term in (12) corresponds to the

well-known interaction term predicted by the Einstein-

Cartan theory (see, for instance, Ref. [2]).

However, in a four-dimensional spacetime it is possi-

ble to choose the coupling parameters in the ﬁrst-order

action (4) in such a way that the resulting second-order

action (10) is precisely the Einstein-Dirac theory. This is

shown next.

1. Four-dimensional spacetime

If n= 4, then we have the result for the anticommu-

tator (see Appendix A)

{γI, σJK }= iIJKLΓγL.(13)

Using this, the fact that E−E†= 2i (θ1−ξΓ), and

taking into account the deﬁnition of the real vector VI

and axial AIcurrents given by

VI:= i ¯

ψγIψ, (14a)

AI:= i ¯

ψΓγIψ, (14b)

the interaction term (12) acquires the form

Sint =3

32κZMθ2VIVI+ 2θξVIAI+ξ2−1AIAIρ.

(15)

It is clear that the interaction term is not invariant under

the parity transformation due to the middle term in (15).

However, for the couplings when θ= 0 or ξ= 0, the mid-

dle term vanishes, and the interaction term is invariant

under parity transformations.1Note that any of these

two choices is not the Einstein-Dirac theory.

Furthermore, even if we take both θ= 0 = ξ, the

resulting theory is also not the Einstein-Dirac theory be-

cause of the presence of the axial-axial term in (15), i.e.,

the minimal coupling (E=1) in the ﬁrst-order formal-

ism (4) is not equivalent to the Einstein-Dirac theory.

Nevertheless, if we consider the particular choice θ= 0

and ξ=±1 =: τ, the interaction term vanishes

Sint = 0.(16)

Thus, in a four-dimensional spacetime, the ﬁrst-order ac-

tion (4) with nonminimal coupling matrix E=1−τiΓ is

—eliminating the connection ωIJfrom (4) using its equa-

tion of motion— equivalent to the Einstein-Dirac action

plus a boundary term

Seﬀ =κZM?eI∧eJ∧RIJ −2Λρ

+ZM1

2¯

ψγIDΩψ−DΩψγIψ∧?eI−m¯

ψψρ

−τ

4Z∂M

AI? eI.

(17)

Therefore, the usual belief that the ﬁrst-order formalism

of fermions coupled to gravity is intrinsically diﬀerent

from the second-order formalism given by the Einstein-

Dirac theory is not true. As we have shown, it is possible

1The same holds for any even dimension. This conclusion comes

from writing (12) in terms of the axial and vector currents for

even dimensions.

to make them equivalent to each other by choosing a

particular nonminimal coupling in the ﬁrst-order formal-

ism2.

III. HAMILTONIAN ANALYSIS

Dirac’s approach to Hamiltonian systems calls for the

deﬁnition of the momenta canonically conjugate to all

conﬁguration variables [9], enlarging in this way the

phase space of the theory under consideration, which is

cumbersome most of the times. The method requires us

to also evolve the primary constraints and ﬁnd all the

constraints, which must be classiﬁed into ﬁrst class and

second class. On the other hand, in ﬁrst-order gravity for

n > 4, the issue of the second-class constraints becomes

still more complicated because they are reducible [4],

which must be handled somehow [10]. If, additionally,

the coupling of fermions to general relativity is consid-

ered, it is expected that the analysis becomes worse.

Thus, to avoid these issues, we follow the method de-

veloped in Refs. [5,6], which consists in a three-step algo-

rithm, to neatly arrive at the Hamiltonian formulations

of the n-dimensional Palatini and Holst actions involv-

ing only ﬁrst-class constraints and manifestly Lorentz-

covariant phase-space variables. This method has also

been successfully applied to get the Hamiltonian formu-

lation of fermions coupled to the Holst action [7].

In the ﬁrst step of the approach, we parametrize the

orthonormal frame of 1-forms (vielbein) eI, adapting it

to the geometry of the spacetime foliation. In the second

step, we use the parametrization of the connection ωIJ

naturally induced by the parametrization of the vielbein,

which leads to the phase-space variables of the theory.

Finally, in the third step, we get rid oﬀ the auxiliary

ﬁelds that do not play a dynamical role in the Hamil-

tonian formulation by eliminating them from the action

principle by using their own equations of motion. All of

this is done in what follows.

A. Parametrization of the vielbein

We assume that the spacetime manifold Mis diﬀeo-

morphic to R×Σ, with Σ being a (n−1)-dimensional

spacelike hypersurface without boundary. Then, we foli-

ate the spacetime with hypersurfaces Σtfor every t∈R,

and each Σtis diﬀeomorphic to Σ. Thus, adapted to the

foliation, the local coordinates (xµ)=(t, xa) label the

points on Rand Σ, respectively.

Thus, adapted to the foliation, we write the orthonor-

mal frame of 1-forms and the connection as

eI=e0Idt +eaIdxa,(18a)

ωIJ=ω0IJdt +ωaIJdxa.(18b)

We parametrize the n2components eµIin terms of the

tensor density ˜

ΠaI plus the usual lapse function Nand

the shift vector Naas

e0I=NnI+Nah1

2(n−2)

hab ˜

ΠbI ,(19a)

eaI=h1

2(n−2)

hab ˜

ΠbI ,(19b)

where

nI:= 1

(n−1)!√hIJ1...Jn−1˜

ηa1...an−1˜

Πa1J1··· ˜

Πan−1Jn−1

(20)

is an internal vector orthogonal to Σ that satisﬁes nInI=

−1 and nI˜

ΠaI = 0; ˜

hab is the densitized metric on

Σ whose inverse is given by ˜

hab := ˜

ΠaI ˜

ΠbI, and h:=

det(˜

hab) is a tensor density of weight 2(n−2). The maps

(19a) and (19b) are invertible, see Appendix Cfor the

supplementary maps.

Continuing with the analysis, we use the decomposi-

tion of eIand ωIJgiven in (18a) and (18b) together with

the parametrization (19a) and (19b), and we substitute

these expressions into the action (4) and obtain

S=ZR×Σ

dtdn−1xh−2κ˜

ΠaI nJ˙ωaIJ +1

2h1

2(n−2) nI¯

ψγIE˙

−1

2h1

2(n−2) nI˙

ψγIE†ψ+ω0IJ ˜

GIJ −Na˜

a−˜

N˜

Si,

(21)

where dtdn−1x:= dt ∧dx1∧···∧dxn−1, the dot over the

corresponding ﬁeld denotes ∂t,˜

N:= h−1

2(n−2) N, and

2An analogous situation happens for the nonminimal coupling

of fermions to the Holst action. By making the particular

choice of the parameters in the coupling matrix, θ= 0 and

ξ= (1/γ)(−1±p1 + γ2), where γis the Barbero-Immirzi pa-

rameter, the interaction term, given in Eq. (21) of Ref. [7], van-

ishes Sint = 0.

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FermionscoupledtothePalatiniactioninndimensionsJorgeRomeroandMercedMontesinosyDepartamentodeFsica,CentrodeInvestigacionydeEstudiosAvanzadosdelInstitutoPolitecnicoNacional,AvenidaInstitutoPolitecnicoNacional2508,SanPedroZacatenco,07360GustavoAdolfoMadero,CiudaddeMexico,MexicoRicardoEscobedozDe...

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Fermions coupled to the Palatini action in ndimensions Jorge Romero and Merced Montesinos

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