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Physics Letters B 769 (2017) 377–384

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

On the origin of Poincaré gauge gravity

J.L. Chkareuli

a,b,∗

Center for Elementary Particle Physics, Ilia State University, 0162 Tbilisi, Georgia

Andronikashvili Institute of Physics, 0177 Tbilisi, Georgia

a r t i c l e i n f o a b s t r a c t

Article history:

Received

8 March 2017

Accepted

6 April 2017

Available

online 10 April 2017

Editor:

M. Cveti

Keywords:

Gravitation

Alternative

gravity theories

Gauge

theories

Spontaneous

symmetry breaking

Sigma

models

Grand

uniﬁed models

We argue that the origin of Poincaré gauge gravity (PGG) may be related to spontaneous violation of

underlying spacetime symmetries involved and appearance of gauge ﬁelds as vector Goldstone bosons.

In essence, we start with an arbitrary theory of some vector and fermion ﬁelds which possesses only

global spacetime symmetries, such as Lorentz and translational invariance, in ﬂat Minkowski space. The

two vector ﬁeld multiplets involved are assumed to belong, respectively, to the adjoint (A

) and vector

) representations of the starting global Lorentz symmetry. We propose that these prototype vector

ﬁelds are covariantly constrained, A

=±M

and e

=±M

, that causes a spontaneous violation

of the accompanying global symmetries (M

A,e

are their presumed violation scales). It then follows that

the only possible theory compatible with these length-preserving constraints is turned out to be the

gauge invariant PGG, while the corresponding massless (pseudo)Goldstone modes are naturally collected

in the emergent gauge ﬁelds of tetrads and spin-connections. In a minimal theory case being linear in a

curvature we unavoidably come to the Einstein–Cartan theory. The extended theories with propagating

spin-connection and tetrad modes are also considered and their possible uniﬁcation with the Standard

Model is brieﬂy discussed.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

One can think that Poincaré gauge gravity (PGG) [1,2] (see also

[3] and references therein) with the underlying vector ﬁelds of

tetrads and spin-connections is perhaps the best theory candi-

date

for gravitation to be uniﬁed with the other three elementary

forces of nature. PGG looks in essence as a gauge ﬁeld theory in

ﬂat Minkowski space which successfully mimics curved space ge-

ometry

when making the transition to the base world space in

terms of general aﬃne connections and metric. Remarkably, there

is some clear analogy between a local frame in PGG and a local

internal symmetry space in conventional quantum ﬁeld theories.

As a result, the vector ﬁelds of the spin-connections gauging the

local frame Lorentz group SO(1, 3)

appear in PGG much as pho-

tons

and gluons appear in the Standard Model. We propose that

such an analogy may follow from their common origin related

to spontaneous breaking of underlying spacetime symmetries in-

volved

(such as relativistic invariance etc.) with all gauge ﬁelds

appearing as massless Nambu–Goldstone bosons [4]. This rather

Correspondence to: Center for Elementary Particle Physics, Ilia State University,

0162 Tbilisi, Georgia.

E-mail

address: j.chkareuli@iliauni.edu.ge.

old idea [5] has gained a further development [6–8] in recent

years.

Here

we will follow the recently introduced emergence conjec-

ture

[9,10] according to which an origin of any gauge symmetry

is basically related to some covariant constraint(s) which, for one

reason or another, is put on a vector ﬁeld system possessing only

some global internal symmetry. As a matter of fact, the simplest

holonomic constraint of this type for vector ﬁeld (or vector ﬁeld

multiplet) A

may be the “length-ﬁxing” condition

C(A) = A

− n

= 0, n

≡ n

=±1(1)

where n

is a properly oriented unit Lorentz vector, while M is

some high mass scale. We will see that gauge invariance appears

unavoidable in the proposed theory, if the equations of motion in-

volved

should have enough freedom to allow a constraint like (1)

be fulﬁlled and preserved over time. Namely, gauge invariance

in such theories has to appear in essence as a response of an in-

teracting

ﬁeld system to putting the covariant constraint (1) on its

dynamics, provided that we allow parameters in the corresponding

Lagrangian density to be adjusted so as to ensure self-consistency

without losing too many degrees of freedom. Otherwise, a given

ﬁeld system could get unphysical in a sense that a superﬂuous

reduction in the number of degrees of freedom would make it

http://dx.doi.org/10.1016/j.physletb.2017.04.012

0370-2693/

SCOAP

378 J.L. Chkareuli / Physics Letters B 769 (2017) 377–384

impossible to set the required initial conditions in an appropriate

Cauchy problem. Furthermore, in quantum theory, to choose self-

consistent

equal-time commutation relations would also become

impossible [11].

see how technically a global internal symmetry may be

converted into a local one, let us consider the question of con-

sistency

of the constraint for vector ﬁeld (1) with its equations

of motion. We propose some arbitrary relativistically invariant La-

grangian

L(A, ψ) which only possesses a global Abelian U (1) sym-

metry,

and, apart from the vector ﬁeld A

, contains the charged

fermion ψ . In the presence of the constraint (1), it follows that the

equations of motion can no longer be independent. This means

that there should be some relationship between all the vector and

matter ﬁeld Eulerians (E

, E

, ...) involved.

Such a relationship

can quite generally be formulated as a functional – but by local-

ity

just a function – of the Eulerians, F (E

, E

), being put equal

to zero at each spacetime point with the conﬁguration space re-

stricted

by the constraint C (A) = 0,

F (C = 0; E

, E

) = 0(2)

for the one matter fermion case proposed.

Let

us consider a “Taylor expansion” of the function F ex-

pressed

through various combinations of the ﬁelds involved, their

combinations with the Eulerians, as well as the derivatives acting

on them. We basically consider the terms with the lowest mass

dimension 4, corresponding to the Lorentz invariant expressions

∂

)

, A

)

, E

ψ, ψ E

(3)

to eventually have an emergent gauge theory at a renormalizable

level.

All the other terms in the expansion contain ﬁeld combina-

tions

with higher mass dimensions (presumably related to some

Planck mass order scale) and therefore can be neglected.

Now,

together with the constraint (1), which has to be pre-

served

under the time development given by the equations of

motion,

)

= 0(4)

one has in fact the ﬁve equations for the 4-component vector ﬁeld

. This means that not all of the vector ﬁeld Eulerian compo-

nents

can be independent. Therefore, there must be a relationship

of the form given in the emergence equation (2). When being ex-

pressed

as a linear combination of the Lorentz invariant terms (3),

this equation leads to the identity between the vector and matter

ﬁeld Eulerians of the following type

∂

)

= itE

ψ − itψ E

(5)

(where t is some constant) which is in fact identically vanished

when the equations of motion are satisﬁed. This identity imme-

diately

signals about invariance of the basic Lagrangian L( A, ψ)

under vector and fermion ﬁeld local U (1) transformations whose

inﬁnitesimal form is given by

δ A

= ∂

ω,δψ= it ωψ. (6)

Conversely, the identity (5) follows from the invariance of the

physical Lagrangian L(A, ψ) under the transformations (6). In-

deed,

both direct and converse assertions are particular cases of

Noether’s second theorem [12].

So,

we have shown how the constraint (1) enforces the choice

of the parameters in the starting Lagrangian L( A, ψ), so as to

Hereafter, the notation E

stands for the vector ﬁeld Eulerian determined by

the corresponding Lagrangian L(A, ψ) (E

)

≡ ∂L/∂ A

− ∂

[∂ L/∂(∂

)]. We use

similar notations for other ﬁeld Eulerians as well.

convert its global U (1) charge symmetry into a local one, thus

demonstrating an emergence of gauge symmetry (6) that allows

the emerged Lagrangian to be completely determined. For a the-

ory

with renormalizable couplings, it is in fact the conventional

QED Lagrangian supplemented by the constraint (1) imposed on

the vector ﬁeld A

. Interestingly, this type of the QED theory with

the constrained vector potential was considered by Nambu [13]

quite

a long ago.

Let

us make it clearer what does the constraint (1) mean in

the gauge invariant QED framework. This constraint is in fact very

similar to the constraint appearing in the nonlinear σ -model for

pions [14]. It means, in essence, that the vector ﬁeld A

devel-

ops

some constant background value,





M, and the Lorentz

symmetry SO(1, 3) formally breaks down to SO(3) or SO(1, 2) for

the time-like (n

= 1) or space-like (n

=−1) case, respectively.

As a result, the corresponding vector Goldstone mode is produced

which may be associated with a photon. Nonetheless, despite an

evident similarity with the nonlinear σ -model for pions, which

really breaks the corresponding chiral SU(2) × SU(2) symmetry

in hadron physics, the QED theory with the supplementary vector

ﬁeld constraint (1) involved leaves the physical Lorentz invariance

intact. Actually, as was shown in the tree [13] and one-loop [15]

approximations,

there is no physical Lorentz violation in the QED

supplemented by the covariant constraint (1). Later this result was

also conﬁrmed for many other gauge theories with the supple-

mentary

vector ﬁeld constraints, particularly, in the non-Abelian

[16] and supersymmetric theories [10]. So, we conclude with a

remark that in contrast to a spontaneous violation of internal sym-

metries,

a spontaneous Lorentz invariance violation (SLIV) caused by

the length-preserving vector ﬁeld constraint does not necessarily

imply a physical breakdown of Lorentz invariance. Actually, gauge

invariance in QED and other gauge theories always leads to a to-

tal

conversion of SLIV into gauge degrees of freedom of massless

vector Goldstone bosons.

the Section 2 we turn to the construction of an emergent

PGG theory. We start with an arbitrary theory of some vector and

fermion ﬁelds which possesses only global spacetime symmetries,

such as Lorentz and translational invariance, in ﬂat Minkowski

space M

. The two vector ﬁeld multiplets involved are proposed to

belong, respectively, to the adjoint ( A

) and vector (e

) represen-

tations

of the starting global Lorentz symmetry. We show that if

these prototype vector ﬁelds are covariantly constrained then the

only possible theory compatible with these constraints is turned

out to be the standard PGG. In minimal theory case being linear

in curvature we unavoidably come to the Einstein–Cartan theory

that is thoroughly presented in the Section 3. The extended theo-

ries

with propagating spin-connection and tetrad modes and their

possible uniﬁcation with the Standard Model is brieﬂy discussed in

the ﬁnal Section 4, where we also conclude.

2. Towards an emergent Poincaré gravity

Conventionally, we have in PGG the world space (WS) symme-

try

ISO(1, 3)

, which includes translations and the orbital part

of Lorentz transformations, and a local frame (LF) Lorentz sym-

metry

SO(1, 3)

, which only includes the spin part of Lorentz

transformations acting on representation indices. Remarkably, this

duality is in an automatic accordance with the Einstein equiva-

lence

principle which, therefore, need not to be specially pos-

tulated

in PGG as is in the standard GR. We begin with the

entirely global spacetime symmetries, both ISO(1, 3)

and

SO(1, 3)

, and our starting objects are the two vector ﬁeld mul-

tiplets

which are 4-vectors of ISO(1, 3)

and belong, respec-

tively,

to the adjoint (A

) and vector (e

) representations of the

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