# Electrodynamics with violations of Lorentz and U(1) gauge symmetries and
their Hamiltonian structure

Xiu-Peng Yanga,b<EMAIL_ADDRESS>Bao-Fei Lia,b<EMAIL_ADDRESS>Tao
Zhua,b corresponding author<EMAIL_ADDRESS>a Institute for Theoretical
Physics and Cosmology, Zhejiang University of Technology, Hangzhou, 310032,
China
b United Center for Gravitational Wave Physics (UCGWP), Zhejiang University of
Technology, Hangzhou, 310032, China

###### Abstract

This article aims to study the Lorentz/U(1) gauge symmetry-breaking
electrodynamics in the framework of the Standard-Model Extension and analyze
the Hamiltonian structure for the theory with a specific dimension $d\leq 4$
of Lorentz breaking operators. For this purpose, we consider a general
quadratic action of the modified electrodynamics with Lorentz/gauge-breaking
operators and calculate the number of independent components of the operators
at different dimensions in gauge invariance and breaking. With this general
action, we then analyze how the Lorentz/gauge symmetry-breaking can change the
Hamiltonian structure of the theories by considering Lorentz/gauge-breaking
operators with dimension $d\leq 4$ as examples. We show that the Lorentz-
breaking operators with gauge invariance do not change the classes of the
constraints of the theory and the number of the physical degrees of freedom of
the standard Maxwell’s electrodynamics. When the U(1) gauge symmetry-breaking
operators are present, the theories in general lack first-class constraint and
have one additional physical degree of freedom, compared to the standard
Maxwell’s electrodynamics.

##  I Introduction

The Standard Model (SM) successfully describes the fundamental constituents of
matter using quarks, leptons, gauge bosons, and Higgs bosons. It effectively
explains phenomena involving elementary particles and their interactions. The
last predicted elementary particle of the SM, the Higgs boson, was
experimentally confirmed in 2012 Higg1 ; Higg2 , giving a happy ending to the
development of this great theory. General Relativity (GR) links gravity with
the curvature of spacetime and provides a highly successful description of
gravity-related phenomena. Since its inception, it has been recognized as a
great theory and continues to be corroborated through various experiments. Its
basic principles and predictions have been confirmed through multiple
observations and measurements, establishing its foundational position in
modern physics. In 2016, gravitational waves were observed in experiments GR ,
which established the great position of general relativity. These two theories
are expected to be unified at the Planck scale and potentially exhibit
observable quantum gravity effects at accessible low-energy scales. This
signal may be related to Lorentz symmetry breaking and can be described by an
effective field theory EFT1 .

To construct a consistent effective field theory that incorporates both GR and
SM, Kostelecký and colleagues proposed an effective field theory called the
Standard Model Extension (SME) IS1 ; CV5 , which features general Lorentz and
CPT violation. The measured and derived values of coefficients for Lorentz and
CPT violation in the SME can be found in the data table organized by
Kosteleck$\mathrm{\acute{y}}$ IS4 . In recent years, research on the Cosmic
Microwave Background (CMB) radiation has provided new opportunities for
studying the pure photon sector of the SME. One reason is that any prediction
involving the pure photon part that deviates from SM could potentially
indicate Lorentz violation originating from the pure photon sector of the SME.
A large amount of research has been conducted in this area CMB0 ; CMB1 ; CMB2
; CMB3 ; CMB4 ; CMB5 ; CMB6 ; CMB7 . Compared to conventional Maxwell
electrodynamics, the pure-photon sector of the SME includes additional
Lorentz-breaking terms, which can be classified as CPT-even and CPT-odd. The
inclusion of these terms leads to the emergence of new effects, which has
spurred extensive research in this field. Studying the general aspects of the
pure photon sector is a challenging task, Ref. IS3  proposed a general
electrodynamics extension theory with a quadratic action, which can be used to
describe many related studies, including photon interactions PSME1 , optical
activity of media PSME2 , the Lorentz-invariance-Violating term HA1 ; HA2 ,
the Chern-Simons term PSME3 , nonminimal SME PSME4 , and other related
phenomena involving photons PSME5 . Theories with Lorentz-breaking operators
of dimension 4 have received considerable attention, including the Lorentz-
invariance-Violating (LIV) KsLIV , Carroll-Field-Jackiw (CFJ) TX1 , and Proca
electrodynamics Proc , with the first two being U(1) gauge invariant theories
and the last one involving U(1) gauge symmetry breaking. In Refs. HA1 ; HA2 ;
TX1 ; Proc , they conducted Hamiltonian analyses of the aforementioned three
theories and identified the constraint structures of the theories, which
motivates us to study the constraint structure of more general theories with a
specific dimension $d\leq 4$.

The Lagrangian density of quadratic electrodynamics, given in IS3 , is a
quadratic polynomial in the photon field $A_{\alpha_{1}}$ and its higher-order
derivatives $\partial_{\alpha_{3}}\ldots\partial_{\alpha_{d}}A_{\alpha_{2}}$
with $d\geq 2$. Since the constant coefficients
$\mathcal{K}_{(d)}^{\alpha_{1}\alpha_{2}\alpha_{3}\ldots\alpha_{d}}$ which are
contracted with
$A_{\alpha_{1}}\partial_{\alpha_{3}}\ldots\partial_{\alpha_{d}}A_{\alpha_{2}}$
remain invariant under coordinate transformations, it leads to a violation of
the Lorentz symmetry of the theory. These constant coefficients can be
regarded as originating from the vacuum expectation value of an operator in
the underlying theory, or the dominant component of dynamic background fields,
or an averaged effect. Through dimensional analysis, it can be found that the
constants $\mathcal{K}_{(d)}^{\alpha_{1}\alpha_{2}\alpha_{3}\ldots\alpha_{d}}$
associated with
$A_{\alpha_{1}}\partial_{\alpha_{3}}\ldots\partial_{\alpha_{d}}A_{\alpha_{2}}$
which is of dimension $d$ must have dimension $4-d$. Some researchers believe
that the theories with power series that are renormalizable have mass
dimension $d\leq 4$ Colladay:1998fq ; IS3 . Considering this, the study in
this paper will focus on theories with a specific dimension operator in the
range of $d\leq 4$ as this avoids the potential Ostrogradsky instability that
the theory could possess Svanberg:2022kdr ; Woodard:2015zca .

In this work, we will extend the U(1) gauge-invariant Lagrangian density of
quadratic electrodynamics described in IS3  to the one that includes U(1)
gauge symmetry breaking terms. We also perform Hamiltonian analysis for
renormalizable specific dimension cases of $d\leq 4$. Our purpose is to
clarify how the U(1) gauge breaking terms affect the constraint structure and
physical degrees of freedom of the theory.

Our results indicate that the Lagrangian density of general quadratic
electrodynamics is the combination of five terms. One corresponds to the
Lagrangian density of standard electrodynamics, one is U(1) gauge invariant,
one contains both U(1) gauge invariant and breaking, and the remaining two are
U(1) gauge breaking. The Hamiltonian analysis of the general gauge-breaking
system with a specific dimension $d\leq 4$ reveals that the system with $d=4$
requires additional conditions to become a constrained system, while systems
with $d=2$ and $d=3$ do not. It also shows that the Lorentz-breaking operators
with gauge invariance do not change the classes of the constraints of the
theory and the number of the physical degrees of freedom of the standard
Maxwell’s electrodynamics. When the U(1) gauge symmetry-breaking operators are
presented, the theories in general lack first-class constraint and have one
additional physical degree of freedom, compared to the standard Maxwell’s
electrodynamics.

The structure of this paper is as follows. The basic theory is discussed in
Sec. II, where we give the number of independent components of the Lorentz-
breaking operators with U(1) gauge violation and extend it to the general
Lagrangian density containing U(1) gauge-breaking terms. Sec. III is about the
Hamiltonian structure and degrees of freedom of theories with a specific
dimension $d\leq 4$. In Sec. IV, We apply the obtained results to some
specific models and derive the results for LIV, CFJ, and Proca
electrodynamics. Our summary is in Sec. V.

For the sake of clarity and conciseness, we will use two conventions:

  1. 1.

The Greek indices range from 0 to 3 and the Latin indices run from 1 to 3. The
metric of the background spacetime $\eta_{\mu\nu}\equiv(1,-1,-1,-1)$;

  2. 2.

The time argument of the vector field $A_{\mu}$ is suppressed throughout the
manuscript, namely $A_{\mu}(\mathbf{x})\equiv A_{\mu}(t,\mathbf{x})$.

##  II Electrodynamics with violations of Lorentz and U(1) gauge symmetry

In this section, we present a brief introduction of the electrodynamics with
quadratic action in the pure photon sector in the framework of SME with both
Lorentz and U(1) gauge symmetry breaking. This represents an extension of the
Lorentz-violating modified electrodynamics with U(1) gauge invariance in IS3
by including the U(1) gauge symmetry-breaking operators in the quadratic
action. In the construction of this theory, we also analyze the properties of
the coefficients for Lorentz- and U(1) gauge-violating operators.

For our purpose, following the similar construction performed in IS3 , let us
start with the general quadratic action for the Lorentz-violating
electrodynamics in the pure photon sector, which can be written as IS3

$\displaystyle S_{(d)}=\int d^{4}x{\cal L}$ (1)

with

$\displaystyle\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\sum\limits_{d=2}^{\infty}\mathcal{K}_{(d)}^{\alpha_{1}\alpha_{2}\alpha_{3}\ldots\alpha_{d}}A_{\alpha_{1}}\partial_{\alpha_{3}}\ldots\partial_{\alpha_{d}}A_{\alpha_{2}},$
(2)

where $F_{\mu\nu}\equiv\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ and
$\mathcal{K}_{(d)}^{\alpha_{1}\alpha_{2}\alpha_{3}\ldots\alpha_{d}}$ are
constant coefficients with mass dimension $4-d$. One possible explanation for
the coefficients
$\mathcal{K}_{(d)}^{\alpha_{1}\alpha_{2}\alpha_{3}\ldots\alpha_{d}}$
originates from non-zero vacuum expectation values to the Lorentz-tensor
fields. Note that each term associated with the coefficient
$\mathcal{K}_{(d)}^{\alpha_{1}\alpha_{2}\alpha_{3}\ldots\alpha_{d}}$ violates
CPT if $d$ is odd or preserves CPT if $d$ is even.

The symmetry among indices $\\{\alpha_{3},\ldots,\alpha_{d}\\}$ of tensor
$\partial_{\alpha_{3}}\ldots\partial_{\alpha_{d}}$ and the use of integration
by parts results in two properties of the coefficients
$\mathcal{K}_{(d)}^{\alpha_{1}\alpha_{2}\alpha_{3}\ldots\alpha_{d}}$. One is
the total symmetry in the $d-2$ indices $\\{\alpha_{3},\cdots,\alpha_{d}\\}$,
another is the symmetry of the two indices $\\{\alpha_{1},\alpha_{2}\\}$ when
$d$ is even and antisymmetry when $d$ is odd. Depending on the specific
intrinsic symmetries of the Lorentz-violating operators, the coefficients
$\mathcal{K}_{(d)}^{\alpha_{1}\alpha_{2}\alpha_{3}\ldots\alpha_{d}}$ can be
decomposed into five representations IS3 . When one imposes the conditions of
the $U(1)$ gauge invariance on the Lorentz-violating operators, these five
representations are reduced into two representations, one corresponds to the
CPT-even coefficient and the other corresponds to CPT-odd coefficient IS3 .

###  II.1 Lorentz-violating electrodynamics with $U(1)$ gauge invariance

Let us first consider the Lorentz-violating electrodynamics with $U(1)$ gauge
invariance IS3 . The $U(1)$ gauge invariance is a symmetry of the theory under
the $U(1)$ gauge transformation

$\displaystyle A_{\mu}\to\tilde{A}_{\mu}=A_{\mu}+\partial_{\mu}\Lambda,$ (3)

where $\Lambda$ is an arbitrary function. The variation of the action (1)
under this gauge transformation reads

$\displaystyle\delta_{g}S_{(d)}$ $\displaystyle=$
$\displaystyle-\int\differential^{4}x\mathcal{K}_{(d)}^{\alpha_{1}\alpha_{2}\alpha_{3}\cdots\alpha_{d}}\Lambda$
$\displaystyle\times\partial_{\alpha_{3}}\cdots\partial_{\alpha_{d}}\quantity(\partial_{[\alpha_{1}}A_{\alpha_{2}]{\pm}}+\frac{1}{2}\partial_{[\alpha_{1}}\partial_{\alpha_{2}]\pm}\Lambda)=0,$

where $\pm$ correspond to odd $d$ and even $d$, respectively, and the brackets
$[]_{\pm}$ indicate symmetrization and antisymmetrization. Specifically, the
terms in the big bracket of the above expression can be written as

$\displaystyle\partial_{[\alpha_{1}}A_{\alpha_{2}]{\pm}}+\frac{1}{2}\partial_{[\alpha_{1}}\partial_{\alpha_{2}]\pm}\Lambda$
$\displaystyle~{}~{}~{}~{}=\begin{cases}\partial_{\alpha_{1}}A_{\alpha_{2}}+\partial_{\alpha_{2}}A_{\alpha_{1}}+\partial_{\alpha_{1}}\partial_{\alpha_{2}}\Lambda,&d\;\mathrm{is\;even},\\\
\partial_{\alpha_{1}}A_{\alpha_{2}}-\partial_{\alpha_{2}}A_{\alpha_{1}},&d\;\mathrm{is\;odd}.\end{cases}$
(5)

The $U(1)$ gauge invariance requires the variation of the action in
(LABEL:equ7) vanishes. When $d$ is even, since the two first indices
$\alpha_{1}$ and $\alpha_{2}$ are symmetric, in order to make (LABEL:equ7)
vanishes, one has to require both indices $\alpha_{1}$ and $\alpha_{2}$ are
antisymmetric with one of $\\{\alpha_{3},\alpha_{4},\cdots,\alpha_{d}\\}$ IS3
. With these properties, it is straightforward to infer that all the CPT-even
operators with $d=2$ are gauge-violating and the gauge invariance requires
$d\geq 4$. By using these properties, one can also count the number of
independent components of the CPT-even operators, which leads to

$\displaystyle N_{F}=(d+1)d(d-3).$ (6)

When $d$ is odd, similarly, the first two indices $\alpha_{1}$ and
$\alpha_{2}$ are antisymmetric, and the vanishing of (LABEL:equ7) require
$\alpha_{1}$ and $\alpha_{2}$ are antisymmetric with one of
$\\{\alpha_{3},\alpha_{4},\cdots,\alpha_{d}\\}$. For this case, only operators
with $d\geq 3$ are allowed and the number of independent components of the
CPT-odd operators is

$\displaystyle N_{AF}=\frac{1}{2}(d+1)(d-1)(d-2).$ (7)

Note that in Table. 1, we summarize the number of independent components for
the CPT-odd and CPT-even operators.

###  II.2 Lorentz-violating electromagnetics with the violations of $U(1)$
gauge symmetry

Now let us turn to consider the case with the breaking of the $U(1)$ gauge
symmetry. When the $U(1)$ gauge symmetry is violated, one does not require the
variation of the action in (LABEL:equ7) vanishing. For this case, one does not
need to impose extra conditions on the coefficients
$\mathcal{K}_{(d)}^{\alpha_{1}\alpha_{2}\alpha_{3}\ldots\alpha_{d}}$ to ensure
the gauge invariance of the theory. For CPT-even operators without gauge
invariance, as we mentioned before, while the indices
$\\{\alpha_{3},\alpha_{4},\cdots,\alpha_{d}\\}$ in the CPT-even coefficients
$\mathcal{K}_{(d)}^{\alpha_{1}\alpha_{2}\alpha_{3}\ldots\alpha_{d}}$ are
symmetric, first two indices $\alpha_{1}$ and $\alpha_{2}$ are also symmetric.
Similarly, for CPT-odd operators without gauge invariance, the indices
$\\{\alpha_{3},\alpha_{4},\cdots,\alpha_{d}\\}$ are symmetric, while the first
two indices $\alpha_{1}$ and $\alpha_{2}$ are antisymmetric. One can then
count the number of independent components of the CPT-even and CPT-odd
operators with a specific dimension $d$, which are summarized in Table. 1.

Table 1: The number of independent components of the Lorentz-violating operators with/without $U(1)$ gauge invariance. $d$ | CPT | Gauge | Number of
---|---|---|---
invariance | independent components
2 | even | no | 10
even, $\geq 4$ | even | yes | $(d+1)d(d-3)$
no | $\frac{2}{3}(d+2)(d+1)d$
odd, $\geq 3$ | odd | yes | $\frac{1}{2}(d+1)(d-1)(d-2)$
no | $\frac{1}{2}(d+2)(d+1)(d-1)$

For later convenience, one can decompose the Lagrangian density (2) into two
parts, the $U(1)$ gauge invariance part and the gauge-violating part.
Specifically, this involves rewriting the Lorentz-breaking coefficients in
five different forms with distinct symmetries. The specific representation
decomposition of these five forms can be found in IS3 . Then, the Lagrangian
density (2) can be rewritten as

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}F_{\alpha\nu}F^{\alpha\nu}$
$\displaystyle+\sum_{{\rm
even}\;d=2}^{\infty}\mathcal{K}_{(d)}^{(1)\alpha_{1}\alpha_{2}\alpha_{3}\ldots\alpha_{d}}A_{\alpha_{1}}\partial_{\alpha_{3}}\ldots\partial_{\alpha_{d}}A_{\alpha_{2}}$
$\displaystyle+\sum_{d=3}^{\infty}\mathcal{K}_{(d)}^{(2)\alpha_{1}\mu\nu\alpha_{2}\ldots\alpha_{d-2}}A_{\alpha_{1}}\partial_{\nu}\partial_{\alpha_{2}}\ldots\partial_{\alpha_{d-2}}A_{\mu}$
$\displaystyle+\sum_{d=3}^{\infty}\mathcal{K}_{(d)}^{(3)\mu\alpha_{1}\nu\alpha_{2}\ldots\alpha_{d-2}}A_{\mu}\partial_{\nu}\partial_{\alpha_{2}}\ldots\partial_{\alpha_{d-2}}A_{\alpha_{1}}$
$\displaystyle+\sum_{d=4}^{\infty}\mathcal{K}_{(d)}^{(4)\mu\nu\rho\sigma\alpha_{1}\ldots\alpha_{d-4}}A_{\mu}\partial_{\rho}\partial_{\sigma}\partial_{\alpha_{1}}\ldots\partial_{\alpha_{d-4}}A_{\nu}$
$\displaystyle+\sum_{{\rm
odd}\;d=3}^{\infty}\mathcal{K}_{(d)}^{(5)\mu\nu\rho\alpha_{1}\ldots\alpha_{d-3}}A_{\mu}\partial_{\rho}\partial_{\alpha_{1}}\ldots\partial_{\alpha_{d-3}}A_{\nu},$

where the five coefficients ${\cal
K}_{(d)}^{(i)\alpha_{1}\alpha_{2}\alpha_{3}\cdots\alpha_{d}}$ with
$i=1,2,3,4,5$ are distinguished by their distinct symmetries among the indices
$\\{\alpha_{1},\alpha_{2},\alpha_{3},\cdots,\alpha_{d}\\}$. The coefficients
${\cal K}_{(d)}^{(1)\alpha_{1}\alpha_{2}\alpha_{3}\cdots\alpha_{d}}$ are all
CPT-even coefficients with $d\geq 2$, while their indices
$\\{\alpha_{1},\alpha_{2},\alpha_{3},\cdots,\alpha_{d}\\}$ are all symmetric.
It is obvious to infer from (LABEL:equ7) that these coefficients violate the
$U(1)$ gauge symmetry of the theory. For the coefficients ${\cal
K}_{(d)}^{(2)\alpha_{1}\mu\nu\alpha_{2}\cdots\alpha_{d-2}}$, they are be
either CPT-even or CPT-odd with $d\geq 3$. Except the two indices $\mu$ and
$\nu$ are antisymmetric, the rest indices
$\\{\alpha_{1},\alpha_{2},\cdots,\alpha_{d-2}\\}$ are all symmetric. Note that
these coefficients are also gauge-violating. Similarly, for ${\cal
K}_{(d)}^{(3)\mu\alpha_{1}\nu\alpha_{2}\cdots\alpha_{d-2}}$, they can also be
either CPT-even or CPT-odd with $d\geq 3$. The two indices of this coefficient
$\mu$ and $\nu$ are antisymmetric and the rest indices
$\\{\alpha_{1},\alpha_{2},\cdots,\alpha_{d-2}\\}$ are all symmetric. One can
check that these coefficients also break the $U(1)$ gauge symmetry of the
theory. Then for the indices in the coefficients ${\cal
K}_{(d)}^{(4)\mu\nu\rho\sigma\alpha_{1}\ldots\alpha_{d-4}}$, the two indices
$\mu$ and $\rho$, and the two indices $\nu$ and $\sigma$, are both
antisymmetric. We also note that these coefficients are symmetric upon
interchange of the two pairs of indices $(\mu,\rho)$ and $(\nu,\sigma)$. By
inspecting the variation of the action (LABEL:equ7) with these coefficients,
one can infer that the CPT-odd operators with ${\cal
K}_{(d)}^{(4)\mu\nu\rho\alpha_{1}\cdots\alpha_{d-4}}$ violates the gauge
invariance, while the CPT-even ones are gauge invariance. The last
coefficients $\mathcal{K}_{(d)}^{(5)\mu\nu\rho\alpha_{1}\ldots\alpha_{d-3}}$
are CPT-odd coefficient with $d\geq 3$, and their three indices
$\\{\mu,\nu,\rho\\}$ are antisymmetric. These coefficients preserve the $U(1)$
gauge symmetry of the theory.

To simplify the later handling of the Hamiltonian analysis of the theory, one
can rewrite the Lagrange density of the theory in a compact form of

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}F_{\alpha\nu}F^{\alpha\nu}+A_{\alpha_{1}}\hat{\mathcal{K}}^{(1)\alpha_{1}\alpha_{2}}A_{\alpha_{2}}+\frac{1}{2}A_{\alpha_{1}}\hat{\mathcal{K}}^{(2,3)\mu\nu\alpha_{1}}F_{\nu\mu}$
(9)
$\displaystyle-\frac{1}{4}F_{\mu\rho}\hat{\mathcal{K}}^{(4)\mu\rho\nu\sigma}F_{\nu\sigma}+\frac{1}{2}\epsilon^{\kappa\mu\nu\rho}A_{\mu}\hat{\mathcal{K}}^{(5)}_{\kappa}F_{\nu\rho},$

where

$\displaystyle\hat{\mathcal{K}}^{(1)\alpha_{1}\alpha_{2}}$
$\displaystyle\equiv$
$\displaystyle\sum_{d=\mathrm{even}}\mathcal{K}_{(d)}^{(1)\alpha_{1}\alpha_{2}\alpha_{3}\ldots\alpha_{d}}\partial_{\alpha_{3}}\ldots\partial_{\alpha_{d}},$
(10) $\displaystyle\hat{\mathcal{K}}^{(4)\mu\rho\nu\sigma}$
$\displaystyle\equiv$
$\displaystyle\sum_{d=4}^{\infty}\mathcal{K}_{(d)}^{(4)\mu\nu\rho\sigma\alpha_{1}\ldots\alpha_{d-4}}\partial_{\alpha_{1}}\ldots\partial_{\alpha_{d-4}},$
(11) $\displaystyle\hat{\mathcal{K}}^{(5)}_{\kappa}$ $\displaystyle\equiv$
$\displaystyle\frac{\epsilon_{\kappa\mu\nu\rho}}{6}\sum_{d=\mathrm{odd}}\mathcal{K}_{(d)}^{(5)\mu\nu\rho\alpha_{1}\ldots\alpha_{d-3}}\partial_{\alpha_{1}}\ldots\partial_{\alpha_{d-3}},$

and

$\displaystyle\hat{\mathcal{K}}^{(2,3)\mu\nu\alpha_{1}}\equiv$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\sum_{d=3}^{\infty}\left[\mathcal{K}_{(d)}^{(2)\alpha_{1}\mu\nu\alpha_{2}\ldots\alpha_{d-2}}+(-1)^{d}\mathcal{K}_{(d)}^{(3)\mu\alpha_{1}\nu\alpha_{2}\ldots\alpha_{d-2}}\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\partial_{\alpha_{2}}\ldots\partial_{\alpha_{d-2}}.$
(13)

The indice symmetries of the operators in (9) are shown in Table. 2. The
indices enclosed in the same braces $\\{\\}$ of the second and third columns
represent symmetry and antisymmetry between them, respectively.
$\\{\mu\rho,\nu\sigma\\}$ in the second column indicates that the
corresponding operators are symmetry when the two pairs $(\mu,\rho)$ and
$(\nu,\sigma)$ are exchanged. The fourth column displays the conditions under
which each class of operators appears.

Table 2:  Symmetries of the indices of the coefficients of operators in (9). Coefficient | Symmetry | Antisymmetry | $d$
---|---|---|---
$\hat{\mathcal{K}}^{(1)\alpha_{1}\alpha_{2}}$ | $\\{\alpha_{1},\alpha_{2}\\}$ | $\cdots$ | even,$\;\geq 2$
$\hat{\mathcal{K}}^{(2,3)\mu\nu\alpha_{1}}$ | $\cdots$ | $\\{\mu,\nu\\}$ | $\geq 3$
$\hat{\mathcal{K}}^{(4)\mu\rho\nu\sigma}$ | $\\{\mu\rho,\nu\sigma\\}$ | $\\{\mu,\rho\\},\\{\nu,\sigma\\}$ | $\geq 4$
$\hat{\mathcal{K}}_{\kappa}^{(5)}$ | $\cdots$ | $\cdots$ | odd,$\;\geq 3$

##  III Hamiltonian structure of the theory

In this section, we perform Hamiltonian analysis on the Lorentz-violating
electromagnetic with and without $U(1)$ gauge invariance by using the Dirac-
Bergmann procedure Berg1 ; Berg2 ; Dirac1 ; Dirac2 ; Dirac3 . For simplicity,
we will focus on the theory with Lagrangian densities with a specific
dimension $d\leq 4$ of Lorentz/gauge-breaking operators.

###  III.1 $d=4$

We start the Hamiltonian analysis for $d=4$. For this case, the Lagrangian
density of the theory reduces to

$\displaystyle{\cal L}_{(4)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}{U}^{\mu\nu\rho\sigma}\partial_{\mu}A_{\nu}\partial_{\rho}A_{\sigma}$
(14)
$\displaystyle-\frac{1}{2}V^{\mu\nu\rho\sigma}F_{\nu\mu}\partial_{\sigma}A_{\rho}-\frac{1}{4}W^{\mu\rho\nu\sigma}F_{\mu\rho}F_{\nu\sigma},$

where

$\displaystyle U^{\mu\nu\rho\sigma}$ $\displaystyle=$ $\displaystyle
2\mathcal{K}_{(4)}^{(1)\mu\nu\rho\sigma},$ (15) $\displaystyle
V^{\mu\nu\rho\sigma}$ $\displaystyle=$
$\displaystyle\mathcal{K}_{(4)}^{(2)\rho\mu\nu\sigma}+\mathcal{K}_{(4)}^{(3)\mu\rho\nu\sigma},$
(16) $\displaystyle W^{\mu\rho\nu\sigma}$ $\displaystyle=$
$\displaystyle\mathcal{K}_{(4)}^{(4)\mu\nu\rho\sigma}.$ (17)

The terms in ${\cal L}_{(4)}$ with coefficients $U^{\mu\nu\rho\sigma}$ and
$V^{\mu\nu\rho\sigma}$ break the U(1) gauge symmetry. The four indices
$\\{\mu,\nu,\rho,\sigma\\}$ in $U^{\mu\nu\rho\sigma}$ are total symmetric,
while $V^{\mu\nu\rho\sigma}$ are antisymmetric in $\\{\mu,\nu\\}$ and
symmetric in $\\{\rho,\sigma\\}$. The terms with coefficients
$W^{\mu\rho\nu\sigma}$ are gauge invariant and the corresponding indices of
the coefficients $W^{\mu\rho\nu\sigma}$ have the same symmetric properties of
the Riemann tensor, i.e. the first pair $\\{\mu\rho\\}$ and the last pair
$\\{\nu\sigma\\}$ of $W^{\mu\rho\nu\sigma}$ are both antisymmetric, and
symmetric upon interchange of the two pairs. Variation the action (1) with the
above Lagrangian with respect to the field $A_{\mu}$, one obtains the equation
of motion of the electromagnetic, i.e.,

$\displaystyle\partial_{\mu}\Big{(}F^{\mu\nu}+U^{\mu\nu\rho\sigma}\partial_{\rho}A_{\sigma}+V^{\nu\mu\rho\sigma}\partial_{\sigma}A_{\rho}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{1}{2}V^{\rho\sigma\nu\mu}F_{\sigma\rho}+W^{\rho\sigma\mu\nu}F_{\rho\sigma}\Big{)}=0.$
(18)

Now to perform the Hamiltonian analysis, it is convenient to define the
conjugate momentum

$\displaystyle\pi^{\mu}\equiv\partialderivative{\mathcal{L}_{(4)}}{\dot{A}_{\mu}}$
$\displaystyle=$
$\displaystyle-F^{0\mu}-U^{0\mu\nu\rho}\partial_{\nu}A_{\rho}-V^{\mu
0\nu\rho}\partial_{\rho}A_{\nu}$ (19) $\displaystyle-\frac{1}{2}V^{\nu\rho\mu
0}F_{\rho\nu}-W^{0\mu\nu\rho}F_{\nu\rho},$

which makes the fundamental Poisson brackets (PB) as

$\displaystyle\\{A_{\mu}(\mathbf{x}),\pi^{\nu}(\mathbf{y})\\}=\delta^{\nu}_{\mu}\delta(\mathbf{x}-\mathbf{y}).$
(20)

From the conjugate momentum, the canonical Hamiltonian of the system can be
expressed as

$\displaystyle{\cal H}_{(4)}\equiv\dot{A}_{\mu}\pi^{\mu}-{\cal L}_{(4)}.$ (21)

In Hamiltonian analysis, a significant portion of the work involves computing
the Poisson bracket between the Hamiltonian and functions in phase space. In
this context, it is important to express the Hamiltonian as a function of the
conjugate momenta and coordinates. To do so, let’s write the time and spatial
components of the conjugate momentum for (19),

$\displaystyle\pi^{0}$ $\displaystyle=$
$\displaystyle-U^{00\mu\nu}\partial_{\mu}A_{\nu}-\frac{1}{2}V^{\mu\nu
00}F_{\nu\mu},$ (22) $\displaystyle\pi^{k}$ $\displaystyle=$
$\displaystyle{D^{k}}_{i}F^{0i}+N^{k},$ (23)

where the matrices ${D^{k}}_{i}$ and $N^{k}$ are defined as

$\displaystyle{D^{k}}_{i}$ $\displaystyle\equiv$
$\displaystyle-{\delta^{k}}_{i}-{V_{i0}}^{k0}-{V^{k0}}_{0i}-2{W^{0k}}_{0i}-{U^{0k}}_{0i},$
(24) $\displaystyle N^{k}$ $\displaystyle\equiv$
$\displaystyle-\Big{(}\frac{1}{2}V^{jik0}+W^{0kij}\Big{)}F_{ij}-(U^{0k00}+V^{k000})\partial_{0}A_{0}$
$\displaystyle-2(U^{0k0i}+V^{k00i})\partial_{i}A_{0}-(U^{0kij}+V^{k0ij})\partial_{i}A_{j}.$

From the expression of ${D^{k}}_{i}$, it is easy to get that
${D^{k}}_{i}={D_{i}}^{k}$. Further analysis is required to determine the
constraint structure of the system. For this purpose, we assume that
${D^{k}}_{i}$ is a non-degenerate matrix, just like in gauge invariance HA1 ;
HA2 , such that

$\displaystyle F^{0i}=$ $\displaystyle{(D^{-1})^{i}}_{k}(\pi^{k}-N^{k}).$ (26)

After some manipulations, one can finally express the canonical Hamiltonian
density as a function of the field quantity and its conjugate momentum, namely

$\displaystyle\mathcal{H}_{(4)}$ $\displaystyle=$
$\displaystyle\pi^{k}\partial_{k}A_{0}+\frac{1}{2}{(D^{-1})_{jl}}(\pi^{l}-N^{l})\Big{[}\pi^{j}-N^{j}+({V^{0j}}_{0k}-{V_{k0}}^{j0}){(D^{-1})^{k}}_{i}(\pi^{i}-N^{i})-2(U^{0j00}+V^{j000})\partial_{0}A_{0}\Big{]}$
(27)
$\displaystyle+\frac{1}{2}F_{ij}\quantity(\frac{1}{2}F^{ij}+\frac{1}{2}W^{ijkl}F_{kl}+2V^{ji0k}\partial_{k}A_{0}+V^{jikl}\partial_{l}A_{k})+\quantity(\frac{1}{2}U^{ijkl}\partial_{i}A_{j}+2U^{0jkl}\partial_{j}A_{0})\partial_{k}A_{l}$
$\displaystyle+2U^{00kl}\partial_{k}A_{0}\partial_{l}A_{0}-\frac{1}{2}U^{0000}\partial_{0}A_{0}\partial_{0}A_{0}.$

Following the Dirac-Bergmann algorithm, our next step is to write the total
Hamiltonian density of the system, which is composed of the canonical
Hamiltonian density and primary constraints. However, we will see that for the
system in $d=4$, the existence of primary constraints requires certain
additional conditions.

####  III.1.1 Conditions for the existence of the constraints

Let us look at what conditions the system described by (14) contains
constraints. According to the Dirac-Bergmann procedure, the presence of
constraints in the system with Lagrangian density $\mathcal{L}_{(4)}$ can be
determined by checking whether the Hessian matrix
$\partialderivative{\mathcal{L}}{\dot{A}_{\mu}}{\dot{A}_{\nu}}$ is degenerate.
Therefore, for making the system with Lagrangian density $\mathcal{L}_{(4)}$ a
constrained system, it is sufficient to provide the condition that the matrix
$\partialderivative{\mathcal{L}_{(4)}}{\dot{A}_{\mu}}{\dot{A}_{\nu}}$ is
degenerate. The Hessian matrix takes the form,

$\displaystyle\partialderivative{\mathcal{L}_{(4)}}{\dot{A}_{\mu}}{\dot{A}_{\nu}}$
$\displaystyle=$
$\displaystyle-\quantity(\eta^{\mu\nu}-\eta^{0\mu}{\delta_{0}}^{\nu})-2W^{0\mu
0\nu}-U^{0\mu 0\nu}$ (28) $\displaystyle-V^{\mu 0\nu 0}-V^{\nu 0\mu 0}.$

When the Hessian matrix is non-degenerate, the system does not possess any
constraint that is not of our interest. It is easy to observe that for gauge
invariant case since one has

$\displaystyle U^{\mu\nu\rho\sigma}=0=V^{\mu\nu\rho\sigma},$ (29)

the Hessian matrix is identically degenerate. This indicates that the theory
with $U(1)$ gauge symmetry always has constraints. However, when the $U(1)$
gauge symmetry is broken, whether the theory possesses constraints depends on
the specific forms of the gauge-breaking coefficients $U^{\mu\nu\rho\sigma}$
and $V^{\mu\nu\rho\sigma}$.

For our purpose, we intend to consider the case when the Hessian matrix is
degenerate, either with gauge invariance or gauge breaking. Considering the
complexity of the Lagrangian density for $d=4$ case, for simplicity, let us
only focus on the following simple case to make the Hessian matrix degenerate,

$\displaystyle U^{0000}=0,\quad U^{000i}=-V^{i000}.$ (30)

From the above equation, it can be seen that this constraint condition only
restricts the gauge-breaking coefficients and the first row and column of
Hessian matrix that are 0, which allows the subsequent conclusions to be fully
applicable to gauge-invariant systems. For the discussion clarity, similar to
gauge invariance case HA1 ; HA2 , we will assume that the $3\times 3$ matrix
$\partialderivative{\mathcal{L}_{(4)}}{\dot{A}_{i}}{\dot{A}_{j}}$ is non-
degenerate.

Thus, for $d=4$ case, hereafter we will analyze the system with

$\displaystyle\mathcal{L}_{(4)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}{U}^{\mu\nu\rho\sigma}\partial_{\mu}A_{\nu}\partial_{\rho}A_{\sigma}$
(31)
$\displaystyle-\frac{1}{2}V^{\mu\nu\rho\sigma}F_{\nu\mu}\partial_{\sigma}A_{\rho}-\frac{1}{4}W^{\mu\rho\nu\sigma}F_{\mu\rho}F_{\nu\sigma},$

with conditions $U^{0000}=0$ and $U^{000i}=-V^{i000}$.

####  III.1.2 Canonical analysis

After obtaining the constrained system with a Lagrangian density of (31), this
subsection analyzes the constraint structure of the system. Under the
condition of (30), the conjugate momentum in (23) can be written as

$\displaystyle\pi^{0}$ $\displaystyle=$
$\displaystyle-(U^{00ij}+V^{ji00})\partial_{i}A_{j}-2U^{000i}\partial_{i}A_{0},$
(32) $\displaystyle\pi^{k}$ $\displaystyle=$
$\displaystyle{D^{k}}_{i}F^{0i}+N^{k},$ (33)

where

$\displaystyle N^{k}$ $\displaystyle=$
$\displaystyle-(\frac{1}{2}V^{jik0}+W^{0kij})F_{ij}-2(U^{0k0i}+V^{k00i})\partial_{i}A_{0}$
(34) $\displaystyle-(U^{0kij}+V^{k0ij})\partial_{i}A_{j}.$

The rank of matrix
$\partialderivative{\mathcal{L}_{(4)}}{\dot{A}_{\mu}}{\dot{A}_{\nu}}$ is 3,
giving rise to a unique primary constraint

$\displaystyle\phi_{1}^{(4)}=\pi^{0}+(U^{00ij}+V^{ji00})\partial_{i}A_{j}+2U^{000i}\partial_{i}A_{0}\approx
0.$ (35)

The symbol ”$\approx$” is called weak equality symbol which implies the
equation only holds on the constraint surfaces but not throughout phase space.
With the expressions of (30), the canonical Hamiltonian density (27) also
becomes

$\displaystyle\mathcal{H}_{(4)}$ $\displaystyle=$
$\displaystyle\pi^{k}\partial_{k}A_{0}+\frac{1}{2}{(D^{-1})_{jl}}(\pi^{l}-N^{l})\Big{(}\pi^{j}-N^{j}+({V^{0j}}_{0k}-{V_{k0}}^{j0}){(D^{-1})^{k}}_{i}(\pi^{i}-N^{i})\Big{)}$
(36)
$\displaystyle+\frac{1}{2}F_{ij}\Big{(}\frac{1}{2}F^{ij}+\frac{1}{2}W^{ijkl}F_{kl}+2V^{ji0k}\partial_{k}A_{0}+V^{jikl}\partial_{l}A_{k}\Big{)}$
$\displaystyle+\quantity(\frac{1}{2}U^{ijkl}\partial_{i}A_{j}+2U^{0jkl}\partial_{j}A_{0})\partial_{k}A_{l}+2U^{00kl}\partial_{k}A_{0}\partial_{l}A_{0},$

and the total Hamiltonian density can be written as

$\displaystyle\mathcal{H}_{(4)T}=\mathcal{H}_{(4)}+u^{(4)}\phi_{1}^{(4)},$
(37)

which gives the total Hamiltonian in the form of

$\displaystyle H_{(4)T}=\int\mathcal{H}_{(4)T}(\mathbf{x})\differential^{3}x,$
(38)

where $u^{(4)}$ is an arbitrary Lagrangian multiplier. Note that
$\partial_{t}A_{0}(\mathbf{x})=\\{A_{0}(\mathbf{x}),H_{(4)T}\\}=u^{(4)}(\mathbf{x})$,
because the Poisson bracket between $A_{0}(\mathbf{x})$ and
$\mathcal{H}_{(4)}(\mathbf{y})$ is zero, and between $A_{0}(\mathbf{x})$ and
other terms in $\phi_{1}^{(4)}(\mathbf{y})$, except for $\pi(\mathbf{y})$, is
also zero. This gives meaning to the coefficient $u^{1}(\mathbf{x})$: the time
derivative of $A_{0}(\mathbf{x})$.

Following the standard Dirac-Bergmann procedure, we then analyze the
requirement for the preservation of the primary constraint. Such a requirement
is also known as the consistency condition of the primary constraint, which
requires that the time derivative of this constraint also vanishes. The
consistency condition of the primary constraint, i.e.,

$\displaystyle\dot{\phi}_{1}^{(4)}(\mathbf{x})=\\{\phi_{1}^{(4)}(\mathbf{x}),{H}_{(4)T}\\}\approx
0,$ (39)

gives rise to a secondary constraint of the system,

$\displaystyle\phi_{2}^{(4)}$ $\displaystyle=$
$\displaystyle\partial_{k}\pi^{k}+{M^{m}}_{i}(\partial_{m}\pi^{i}-\partial_{m}N^{i})$
(40) $\displaystyle+(5U^{00ij}+V^{ji00})\partial_{i}\partial_{j}A_{0}$
$\displaystyle+V^{ji0k}\partial_{k}F_{ij}+2U^{0jkl}\partial_{j}\partial_{k}A_{l}\approx
0,$

where

$\displaystyle{M^{m}}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}{(D^{-1})_{ij}}+{(D^{-1})_{ji}}$ (41)
$\displaystyle~{}~{}~{}~{}+2({V^{0l}}_{0k}-{V_{k0}}^{l0})(D^{-1})_{li}{(D^{-1})^{k}}_{j}\Big{]}$
$\displaystyle\times\quantity(3U^{0j0m}+2V^{j00m}+V^{jm00}),$

which indicates that the structure of Gauss’s law is influenced by the gauge-
breaking term.

Similar to the primary constraint, the secondary constraint $\phi_{2}\approx
0$, being a constraint itself, also has a corresponding consistency condition,
which is

$\displaystyle\dot{\phi}_{2}^{(4)}(\mathbf{x})=\\{\phi_{2}^{(4)}(\mathbf{x}),{H}_{(4)T}\\}\approx
0.$ (42)

This condition then leads to

$\displaystyle{O_{1}^{ik}}_{j}\partial_{i}\partial_{k}(\pi^{j}-N^{j})+{O_{2}^{ijk}}\partial_{i}\partial_{j}\partial_{k}A_{0}+O_{3}^{jilk}\partial_{j}\partial_{i}\partial_{l}A_{k}$
$\displaystyle~{}~{}~{}+O_{4}^{klij}\partial_{k}\partial_{l}F_{ij}+T^{ij}\partial_{i}\partial_{j}u^{(4)}\approx
0,$ (43)

where

$\displaystyle O_{3}^{jilk}$ $\displaystyle\equiv$
$\displaystyle({\delta^{j}}_{m}+{M^{j}}_{m})(\frac{1}{2}V^{mikl}+U^{imlk}),$
(44) $\displaystyle O_{4}^{klij}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\quantity({\delta^{k}}_{m}+{M^{k}}_{m})(\eta^{li}\eta^{mj}+W^{ijlm}+V^{jiml}),$
(45) $\displaystyle{O_{1}^{ik}}_{j}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\Big{[}({\delta^{i}}_{m}+{M^{i}}_{m})\big{(}V^{mkl0}+2W^{0lkm}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+U^{0lkm}+V^{l0km}\big{)}$
$\displaystyle~{}~{}+{M^{i}}_{m}(V^{lkm0}+2W^{0mkl}+U^{0mkl}+V^{m0kl})$
$\displaystyle~{}~{}+2(V^{li0k}+U^{0ikl})\Big{]}$
$\displaystyle\times\Big{[}{(D^{-1})_{jl}}+{(D^{-1})_{lj}}$
$\displaystyle~{}~{}~{}~{}~{}~{}+2({V^{0{i_{1}}}}_{0{k_{1}}}-{V_{{k_{1}}0}}^{{i_{1}}0})(D^{-1})_{{i_{1}}j}{(D^{-1})^{{k_{1}}}}_{l}\Big{]},$
(46) $\displaystyle{O_{2}^{ijk}}$ $\displaystyle\equiv$
$\displaystyle({\delta^{i}}_{l}+{M^{i}}_{l})(V^{lj0k}+2U^{0jkl})$ (47)
$\displaystyle+{M^{i}}_{l}(U^{0ljk}+V^{l0jk})+2U^{0ijk},$

and

$\displaystyle T^{ij}\equiv
6U^{00ij}+(3U^{00jk}+V^{kj00}+2V^{k00j}){M^{i}}_{k}.$ (48)

For gauge invariance case, one has $T^{ij}=0$, one can check that the
condition (III.1.2) satisfies identically. When the $U(1)$ gauge symmetry of
the theory is violated, $T^{ij}$ are in general nonzeros, then (III.1.2)
represents a restriction on the Lagrange multiplier $u^{(4)}$. One special
case is that $T^{ij}=0$ for some specific combination of the gauge breaking
coefficients, for which a new constraint arises, which is

$\displaystyle\phi_{3}^{(4)}$ $\displaystyle=$
$\displaystyle{O_{1}^{ik}}_{j}(\partial_{i}\partial_{k}\pi^{j}-\partial_{i}\partial_{k}N^{j})+{O_{2}^{ijk}}\partial_{i}\partial_{j}\partial_{k}A_{0}$
(49)
$\displaystyle+O_{3}^{jilk}\partial_{j}\partial_{i}\partial_{l}A_{k}+O_{4}^{klij}\partial_{k}\partial_{l}F_{ij}\approx
0.$

With this constraint, repeating the above procedure, its consistency condition
may produce more constraints under certain conditions. According to the
analysis in HDOF6 ; HDOF5 , such constraint may also lead to the unphysical
half degree of freedom. However, this requires a very special choice of gauge-
breaking coefficients. For simplicity, we will not explore these specific
cases in detail in this paper.

Before we go further, we would like to summarize the main results we have
gotten so far for the $d=4$ case. For both the gauge invariance case and gauge
breaking case with the degenerate condition (30), the theory can have one
primary constraint $\phi_{1}^{(2)}$ and one secondary constraint
$\phi_{2}^{(4)}$.

####  III.1.3 Counting the degrees of freedom

After obtaining all the primary and secondary constraints of the system, let
us turn to identify the first- and second-class constraints of the system by
analyzing the Poisson bracket of the constraints. In general, the first-class
constraints are associated with the gauge symmetry of the theory, they are
gauge generators, which generate gauge transformations that don’t alter the
physical state. The second-class constraints can not generate gauge
transformations since the transformation generated by a second-class does not
preserve all the constraints, which violates the consistency condition, but
are important in the definition of Dirac bracket, which plays a key role in
the transition from classical theory to quantum theory Dirac2 . Specifically,
the first-class constraints are those constraints whose Poisson bracket with
every constraint vanishes weakly, otherwise are the second-class of
constraint.

The Poisson bracket of the primary constraint $\phi_{1}^{(4)}$ and the
secondary constraint $\phi_{2}^{(4)}$ gives

$\displaystyle\\{\phi_{2}^{(4)}(y),\phi_{1}^{(4)}(x)\\}$ $\displaystyle=$
$\displaystyle(C_{1})^{ij}{\partial^{\prime}}_{i}\partial_{j}\delta(x-y)$ (50)
$\displaystyle+(C_{2})^{ij}{\partial^{\prime}}_{i}{\partial^{\prime}}_{j}\delta(x-y).$

where $\partial^{\prime}$ denotes the partial derivative with respect to $y$,
and

$\displaystyle(C_{1})^{ij}$ $\displaystyle=$
$\displaystyle-\quantity(U^{00ji}+(U^{00jk}+V^{kj00}){M^{i}}_{k}),$ (51)
$\displaystyle(C_{2})^{ij}$ $\displaystyle=$ $\displaystyle
2{M^{i}}_{k}(U^{0k0j}+V^{k00j})+5U^{00ij}.$ (52)

It is easy to see that for the gauge invariance case,
$C_{1}^{ij}=0=C_{2}^{ij}$, therefore, $\phi_{1}^{(4)}$ and $\phi_{2}^{(4)}$
are both first-class constraints. For the case without the gauge symmetry, the
gauge breaking coefficients $U^{\mu\nu\rho\sigma}$ and $V^{\mu\nu\rho\sigma}$
are in general nonzero, thus the theory does not possess any first-class
constraint. In this case, $\phi_{1}^{(4)}$ and $\phi_{2}^{(4)}$ are both
second-class constraints. This result is also what we expect since the first-
class constraint can only exist when the theory has gauge symmetry, and thus a
theory without gauge symmetry should not have first-class constraint.

Getting all first- and second-class constraints, one can count the number of
physical degrees of freedom ($N_{\rm DOF}$) by using the following formula b1
,

$\displaystyle N_{\rm
DOF}=\frac{1}{2}(N_{\mathrm{var}}-2\mathrm{NOF}-\mathrm{NOS}),$ (53)

where “$N_{\mathrm{var}}$” means the total number of canonical variables,
“NOF” is the number of constraints of the first class, and “NOS” represents
the number of second-class constraints. Thus, for the gauge invariance case,
the number of degrees of freedom is

$\displaystyle N_{\rm DOF}=\frac{1}{2}(8-2\times 2)=2,$ (54)

which is the same as that in the standard Maxwell’s electrodynamics, while in
the case of gauge violation, one has

$\displaystyle N_{\rm DOF}=\frac{1}{2}(8-2)=3.$ (55)

Therefore, the violation of the $U(1)$ gauge symmetry induces one additional
physical degree of freedom, compared to the standard Maxwell’s electrodynamics
and the theory with gauge invariance.

###  III.2 $d=3$

In this subsection, let’s turn to analyze the constraint structure of the
theory with Lorentz/gauge-breaking operators with dimension $d=3$. The
Lagrangian density of the considered theory can be written in the form of

$\displaystyle\mathcal{L}_{(3)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}F_{\alpha\nu}F^{\alpha\nu}+\frac{1}{2}S^{\mu\nu\rho}A_{\rho}F_{\nu\mu}$
(56)
$\displaystyle+\frac{1}{2}\epsilon^{\kappa\mu\nu\rho}{(k_{AF})_{\kappa}}A_{\mu}F_{\nu\rho}.$

where

$\displaystyle{(k_{AF})_{\kappa}}$ $\displaystyle=$
$\displaystyle\frac{1}{3!}\epsilon_{\kappa\mu\nu\rho}{\mathcal{K}}_{(3)}^{(5)\mu\nu\rho},$
(57) $\displaystyle S^{\mu\nu\rho}$ $\displaystyle=$
$\displaystyle\mathcal{K}_{(3)}^{(2)\rho\mu\nu}-\mathcal{K}_{(3)}^{(3)\mu\rho\nu}.$
(58)

Note that the terms with ${(k_{AF})_{\kappa}}$ are gauge-invariant and the
terms with $S^{\mu\nu\rho}$ are gauge-breaking. Variation of the action with
respect to $A_{\mu}$, one obtains field equation

$\displaystyle\epsilon^{\kappa\nu\mu\rho}\left({k}_{AF}\right)_{\kappa}F_{\mu\rho}+\frac{1}{2}S^{\mu\rho\nu}F_{\rho\mu}$
$\displaystyle~{}~{}~{}~{}+\partial_{\mu}\quantity(F^{\mu\nu}-S^{\nu\mu\rho}A_{\rho})=0.$
(59)

The corresponding conjugate momentum of this theory read,

$\displaystyle\pi^{\mu}=$
$\displaystyle\partialderivative{\mathcal{L}_{(3)}}{\dot{A}_{\mu}}=-F^{0\mu}+\quantity(\epsilon^{\beta\nu
0\mu}{k_{AF}}_{\beta}+S^{\mu 0\nu})A_{\nu}.$ (60)

To analyze the constraint structure of the theory, it is convenient to write
down the time and spatial components of the above conjugate momentum (60) as

$\displaystyle\pi^{0}$ $\displaystyle=$ $\displaystyle 0,$ (61)
$\displaystyle\pi^{k}$ $\displaystyle=$
$\displaystyle-F^{0k}+\quantity(\epsilon^{j\mu
0k}{k_{AF}}_{j}+S^{k0\mu})A_{\mu}.$ (62)

One can conclude that in the specific dimension $d=3$, the gauge-breaking and
gauge-invariant Lorentz-breaking electrodynamics have the same form for the
conjugate momentum $\pi^{0}$ of the photon field $A_{0}$. From these
properties, the only primary constraint of the system is

$\displaystyle\phi_{1}^{(3)}=\pi^{0}\approx 0.$ (63)

Then, the canonical Hamiltonian density and total Hamiltonian density are
given by

$\displaystyle\mathcal{H}_{(3)}$ $\displaystyle=$
$\displaystyle\pi^{k}\dot{A}_{k}-\mathcal{L}_{(3)},$ (64)

and

$\displaystyle(\mathcal{H}_{(3)})_{T}$ $\displaystyle=$
$\displaystyle\mathcal{H}_{(3)}+u^{(3)}phi_{1}^{(3)}.$ (65)

Here, similar to the case of $d=4$, the coefficient $u^{(3)}$ is also the time
derivative of $A_{0}$.

Once again, the consistency condition of the primary constraint gives a
secondary constraint

$\displaystyle\phi_{2}^{(3)}$ $\displaystyle=$
$\displaystyle\dot{\phi}_{1}^{(3)}$ (66) $\displaystyle=$
$\displaystyle\partial_{k}\pi^{k}+\frac{1}{2}F_{ik}\quantity(\epsilon^{j0ik}\left({k}_{AF}\right)_{j}+S^{ki0})$
$\displaystyle+{S_{k}}^{00}(\epsilon^{j\mu
0k}\left({k}_{AF}\right)_{j}+S^{k0\mu})A_{\mu}-\pi^{k}{S_{k}}^{00}$
$\displaystyle\approx$ $\displaystyle 0.$

Similarly, this means that the form of Gauss’s law needs to be modified when
considering the presence of the gauge-breaking term. The consistency condition
of $\phi_{2}^{(3)}$ impose a restriction on the $u^{1}$:

$\displaystyle\dot{\phi}_{2}^{(3)}$ $\displaystyle=$
$\displaystyle{S_{l}}^{00}S^{l00}u^{(3)}-\big{(}{S_{i}}^{0k}+{S_{i}}^{k0}\big{)}\partial_{k}\pi^{i}$
(67)
$\displaystyle+{S_{k}}^{00}\big{(}2{\epsilon^{jk0}}_{i}(k_{AF})_{j}+{S_{i}}^{0k}-{S^{k0}}_{i}\big{)}\pi^{i}$
$\displaystyle+{S_{i}}^{00}\big{[}2{\epsilon^{l0i}}_{k}\left({k}_{AF}\right)_{l}+{S^{i0}}_{k}-{S_{k}}^{0i}\big{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\times[{\epsilon^{jn0k}}(k_{AF})_{j}+S^{k0n}]A_{n}$
$\displaystyle+\frac{1}{2}\big{(}-2S^{j00}{\eta^{ki}}+S^{jik})\partial_{k}F_{ij}$
$\displaystyle-\frac{1}{2}(\epsilon^{0kij}\left({k}_{AF}\right)_{0}+S^{jik}){S_{k}}^{00}F_{ij}$
$\displaystyle+2{S_{l}}^{00}(S^{l0k}+S^{lk0})\partial_{k}A_{0}$
$\displaystyle+{S_{i}}^{00}\big{[}2{\epsilon^{l0i}}_{k}\left({k}_{AF}\right)_{l}+{S^{i0}}_{k}-{S_{k}}^{0i}\big{]}S^{k00}A_{0}$
$\displaystyle+\Big{[}{S_{k}}^{00}(\epsilon^{0jik}\left({k}_{AF}\right)_{0}+S^{kij})$
$\displaystyle~{}~{}~{}~{}+({S_{k}}^{0i}+{S_{k}}^{i0})\quantity[{\epsilon^{lj0k}}(k_{AF})_{l}+S^{k0j}]\Big{]}\partial_{i}A_{j}$
$\displaystyle\approx$ $\displaystyle 0.$

For gauge invariance case, all the gauge breaking coefficients
$S^{\mu\nu\rho}=0$, and thus the above consistency condition satisfies
identically. In this case, the theory only has two constraints, the primary
constraint $\phi_{1}^{(3)}$ and the secondary constraint $\phi_{2}^{(3)}$.

When the gauge symmetry is violated, the gauge breaking coefficients
$S^{\mu\nu\rho}$ are in general nonzeros. For this case, the above consistency
condition leads to a specific form of $u^{(2)}$,

$\displaystyle u^{(3)}$ $\displaystyle\approx$
$\displaystyle-\frac{1}{{S_{m}}^{00}S^{m00}}$
$\displaystyle\times\Bigg{[}{S_{i}}^{00}\big{[}2{\epsilon^{l0i}}_{k}\left({k}_{AF}\right)_{l}+{S^{i0}}_{k}-{S_{k}}^{0i}\big{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times[{\epsilon^{jn0k}}(k_{AF})_{j}+S^{k0n}]A_{n}$
$\displaystyle~{}~{}~{}-\big{(}{S_{i}}^{0k}+{S_{i}}^{k0}\big{)}\partial_{k}\pi^{i}$
$\displaystyle~{}~{}~{}+{S_{k}}^{00}\big{(}2{\epsilon^{jk0}}_{i}(k_{AF})_{j}+{S_{i}}^{0k}-{S^{k0}}_{i}\big{)}\pi^{i}$
$\displaystyle~{}~{}~{}+\frac{1}{2}\big{(}-2S^{j00}{\eta^{ki}}+S^{jik})\partial_{k}F_{ij}$
$\displaystyle~{}~{}~{}-\frac{1}{2}(\epsilon^{0kij}\left({k}_{AF}\right)_{0}+S^{jik}){S_{k}}^{00}F_{ij}$
$\displaystyle~{}~{}~{}+2{S_{l}}^{00}(S^{l0k}+S^{lk0})\partial_{k}A_{0}$
$\displaystyle~{}~{}~{}+{S_{i}}^{00}(2{\epsilon^{l0i}}_{k}\left({k}_{AF}\right)_{l}+{S^{i0}}_{k}-{S_{k}}^{0i})S^{k00}A_{0}$
$\displaystyle~{}~{}~{}+\Big{[}{S_{k}}^{00}(\epsilon^{0jik}\left({k}_{AF}\right)_{0}+S^{kij})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+({S_{k}}^{0i}+{S_{k}}^{i0})\quantity[{\epsilon^{lj0k}}(k_{AF})_{l}+S^{k0j}]\Big{]}\partial_{i}A_{j}\Bigg{]}.$
(69)

This indicates that the theory for this case does not have any additional
constraints. Similar to the gauge invariance case, it only has two
constraints, one primary constraint $\phi_{1}^{(3)}$ and one secondary
constraint $\phi_{2}^{(2)}$. Here we would like to mention that, under certain
conditions on the gauge breaking coefficients such that
$S_{l}^{\;\;00}S^{l00}=0$, the theory may produce additional constraints.
However, this requires a very special choice of gauge-breaking coefficients.
For simplicity, we will not explore this specific case in detail in this paper
and focus on the case with $S_{l}^{\;\;00}S^{l00}\neq 0$ when gauge symmetry
is violated.

Then, let us consider the Poisson bracket of the primary constraint
$\phi_{1}^{(3)}$ and the secondary constraint $\phi_{2}^{(3)}$, which is

$\displaystyle\\{\phi_{1}^{(3)}(\mathbf{x}),\phi_{2}^{(3)}(\mathbf{y})\\}=-{S_{k}}^{00}S^{k00}\delta(\mathbf{x}-\mathbf{y}).$
(70)

It is obvious that for gauge invariance case, since $S^{\mu\nu\rho}=0$, the
above Poisson bracket vanishes and the constraints $\phi_{1}^{(3)}$ and
$\phi_{2}^{(3)}$ are both first-class. Thus, the number of the physical
degrees of freedom is

$\displaystyle N_{\rm DOF}=\frac{1}{2}(8-2\times 2)=2,$ (71)

which is the same as that in the standard Maxwell’s electrodynamics.

For the case with gauge violation, since in general ${S_{k}}^{00}S^{k00}\neq
0$, $\phi_{1}^{(3)},\phi_{2}^{(3)}$ are both second-class constraints. Thus,
the number of the physical degrees of freedom is

$\displaystyle N_{\rm DOF}=\frac{1}{2}(8-2)=3.$ (72)

Similar to the case with $d=4$, the violation of the $U(1)$ gauge symmetry
induces one extra physical degree of freedom, compared to the standard
Maxwell’s electrodynamics and the case with gauge invariance.

###  III.3 $d=2$

After completing the constraint structure analysis of a specific dimension
$d=3$ and $d=4$, let us turn to analyze the structure of $d=2$, which
Lagrangian density is

$\displaystyle\mathcal{L}_{(2)}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}U^{\mu\nu}A_{\mu}A_{\nu},$
(73)

with

$\displaystyle U^{\mu\nu}$ $\displaystyle=$ $\displaystyle
2\mathcal{K}_{(2)}^{(1)\mu\nu},$ (74)

which are gauge-breaking terms. Variation of the action with respect to
$A_{\mu}$, one obtains

$\displaystyle U^{\nu\mu}A_{\mu}+\partial_{\mu}F^{\mu\nu}=0$ (75)

The conjugate momenta now read

$\displaystyle\pi^{\mu}=\partialderivative{\mathcal{L}_{(2)}}{\dot{A}_{\mu}}=-F^{0\mu}.$
(76)

One difference from $d=3$ and $d=4$ is that at this point the conjugate
momenta $\pi^{\mu}$ are not affected by gauge-breaking coefficients,
$U^{\mu\nu}$, so they are the same as in Maxwell electrodynamics since the
time derivative of the photo field $A_{\mu}$ only appears in the Maxwell term,
the first term in (73). Therefore, the only primary constraint is

$\displaystyle\phi_{1}^{(2)}=\pi^{0}\approx 0.$ (77)

The canonical Hamiltonian density just differs by one term of $U^{\mu\nu}$
from that in Maxwell electrodynamics:

$\displaystyle\mathcal{H}_{(2)}$ $\displaystyle=$
$\displaystyle\pi^{k}\partial_{k}A_{0}-\frac{1}{2}\pi^{k}\pi_{k}+\frac{1}{4}F_{ik}F^{ik}-\frac{1}{2}U^{\mu\nu}A_{\mu}A_{\nu}.$

This gives the total Hamiltonian density:

$\displaystyle\mathcal{H}_{(2)T}=\mathcal{H}_{(2)}+u^{(2)}\phi_{1}^{(2)}.$
(79)

The influence of gauge-breaking terms are reflected in secondary constraint
because it originates from the Poisson bracket between primary constraint and
total Hamiltonian which now is gauge-breaking. It reads

$\displaystyle\phi_{2}^{(2)}=\partial_{k}\pi^{k}+U^{0\mu}A_{\mu}\approx 0.$
(80)

The consistency condition of $\phi_{2}$ is

$\displaystyle\dot{\phi}_{2}^{(2)}=U^{\mu
k}\partial_{k}A_{\mu}+U^{0k}\partial_{k}A_{0}-U^{0k}\pi_{k}+u^{(2)}U^{00}\approx
0,$ (81)

which provides the constraint on $u^{(2)}$ when $U^{00}\neq 0$:

$\displaystyle
u^{(2)}\approx(U^{00})^{-1}\quantity[U^{0k}(\pi_{k}-\partial_{k}A_{0})-U^{\mu
k}\partial_{k}A_{\mu}].$ (82)

Noticing the absence of $\pi^{0}$ in $\mathcal{H}_{(2)}$, we get
$\dot{A}_{0}(\mathbf{x})=\\{A_{0}(\mathbf{x}),H_{(2)T}\\}=u^{1}(\mathbf{x})$.
As a consequence, the meaning of $u^{(2)}$ is the time derivative of $A_{0}$.
Combining (82), the time component $A_{0}$ of the photon field will be
determined by the first-order differential equation.

Finally, the Poisson bracket of $\phi_{1}^{(2)}$ and $\phi_{2}^{(2)}$ is

$\displaystyle\\{\phi_{1}^{(2)}(\mathbf{x}),\phi_{2}^{(2)}(\mathbf{y})\\}=-U^{00}\delta(\mathbf{x}-\mathbf{y}).$
(83)

When the gauge breaking coefficients $U^{\mu\nu}$ are nonzeros,
$\phi_{1}^{(2)},\phi_{2}^{(2)}$ are both second class constraints. Similar to
the cases of $d=4$ and $d=3$, we will not explore in detail the case with
$U^{00}=0$, which may induce more constraint. Then the number of the physical
degree of freedom is

$\displaystyle N_{\rm DOF}=\frac{1}{2}(8-2)=3.$ (84)

Again, compared to the standard Maxwell electrodynamics, since $d=2$ operators
always break the $U(1)$ gauge symmetry, it induces one additional physical
degree of freedom.

##  IV Map to several specific models

The Lorentz-violating electrodynamics presented in this paper provide a
unifying framework for describing possible violations of Lorentz and $U(1)$
gauge symmetries in the electromagnetic interaction. In this section, we
present several specific modified electrodynamics by writing their actions in
the form of (1) and summarize their Hamiltonian structure from our general
analysis. We consider three specific theories, the Lorentz-invariance-
Violating (LIV) electrodynamics, the Carroll-Field-Jackiw (CFJ)
electrodynamics, and the Proca electrodynamics. The first two theories only
break the Lorentz symmetry of the theory, while the last one only breaks
$U(1)$ gauge symmetry.

Table 3: The number of the first-class and second-class constraints, and the number of the physical degrees of freedom for each case with a specific dimension $d=2$, $d=3$, and $d=4$.  $d$ | gauge invariance | number of first-class constraint | number of first-class constraint | number of DOF
---|---|---|---|---
2 | no | 0 | 2 | 3
3 | yes | 2 | 0 | 2
no | 0 | 2 | 3
4 | yes | 2 | 0 | 2
no | 0 | 2 | 3

###  IV.1 LIV electrodynamics

We first consider LIV electrodynamics, which is proposed in HA1 ; HA2  and the
Lagrangian density is given by HA1 ; HA2

$\displaystyle\mathcal{L}_{LIV}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}W^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma},$
(85)

which corresponds to the case in our model where the gauge-breaking
coefficients $U^{\mu\nu\rho\sigma}$ and $V^{\mu\nu\rho\sigma}$ are set to be
zeros for $d=4$. Using the result from (35) and (40), we obtain the constraint
structure in this case:

$\displaystyle(\phi_{1})_{LIV}$ $\displaystyle=$ $\displaystyle\pi^{0}\approx
0,$ (86) $\displaystyle(\phi_{2})_{LIV}$ $\displaystyle=$
$\displaystyle\partial_{k}\pi^{k}\approx 0.$ (87)

And we need verify that the consistency of $(\phi_{2})_{LIV}$ gives no new
constraints. Noticing that $T^{ij}=M^{ij}=0$ by using (41) and (48) since
$V^{\mu\nu\rho\sigma}=U^{\mu\nu\rho\sigma}=0$, then we only need to check
whether $\phi_{3}^{(4)}$ in (49) is identically zero throughout the phase
space, not just on the constraint surface. In this case, the only possible
non-zero coefficients remaining in (49) are ${O_{1}^{ik}}_{j}$ and
$O_{4}^{klij}$. According (33), as a result, $\phi_{3}^{(4)}$ now changes to

$\displaystyle
W^{0lik}((D^{-1})_{jl}+(D^{-1})_{lj})\partial_{k}\partial_{i}(\pi^{j}-N^{j})$
$\displaystyle~{}~{}~{}~{}+\frac{1}{2}(\eta^{li}\eta^{kj}+W^{ijlk})\partial_{k}\partial_{l}F_{ij}.$
(88)

Since the last two indices of $W^{ijkl}$ are antisymmetric, and $i,j$ in
$\eta^{li}\eta^{kj}\partial_{k}\partial_{l}$ are symmetric, (88) is equal to
0. This proves that $(\phi_{2})_{LIV}$ does not give new constraints. In
addition, since the theory has gauge symmetry, both the constraints
$(\phi_{1})_{LIV}$ and $(\phi_{2})_{LIV}$ are first-class, thus the theory has
two degrees of freedom, the same as that in the standard Maxwell
electrodynamics. Our result here is consistent with the results in LIV HA1 ;
HA2 .

###  IV.2 CFJ electrodynamics

For CFJ electrodynamics, its Lagrangian density is given by TX1 :

$\displaystyle\mathcal{L}_{CFJ}=$
$\displaystyle-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}\epsilon^{\kappa\mu\nu\rho}{(k_{AF})_{\kappa}}A_{\mu}F_{\nu\rho},$
(89)

with the caveat that the coefficients ${(k_{AF})_{\kappa}}$ differs by a
factor of $-\frac{1}{2}$ compared to TX1 . The Lagrangian density (89) can be
obtained in our model by setting $S^{\mu\nu\rho}=0$ to zero in $d=3$.
Similarly, according to the results in (63) and (66), we have two constraints
for this theory,

$\displaystyle(\phi_{1})_{CFJ}=\pi^{0}=F^{00}\approx 0,$ (90)
$\displaystyle(\phi_{2})_{CFJ}=\partial_{k}\pi^{k}+\frac{1}{2}F_{ik}\epsilon^{j0ik}\left({k}_{AF}\right)_{j},\approx
0.$ (91)

where $(\phi_{1})_{CFJ}$ is the primary constraint and $(\phi_{2})_{CFJ}$ is
the secondary constraint. According to (67), it can be concluded that in this
case, $(\dot{\phi}_{2})_{CFJ}$ is always zero in phase space, which does not
generate new constraints. This result confirms TX1 , with the additional point
to note that the coefficient ${k}_{AF}$ here differs by a factor of
$-\frac{1}{2}$ compared to TX1 . Note that both the $(\phi_{1})_{CFJ}$ and
$(\phi_{2})_{CFJ}$ are first-class and the number of degrees of freedom is the
same as that in the standard Maxwell electrodynamics.

###  IV.3 The Proca electrodynamics

When $d=2$, if one sets $U^{\mu\nu}=m^{2}\eta^{\mu\nu}$, then the Lagrangian
density $\mathcal{L}_{(2)}$ will return to the case of Proca electrodynamics
Proc :

$\displaystyle\mathcal{L}_{Proca}=-\frac{1}{4}F_{\alpha\nu}F^{\alpha\nu}+\frac{1}{2}m^{2}A_{\mu}A^{\mu},$
(92)

where $m$ is the mass of the photon. Because of this mass term, the Proca
electrodynamics breaks the $U(1)$ gauge symmetry of the theory. Replacing the
constant coefficients $U^{\mu\nu}$ with the product of metric tensor
$\eta^{\mu\nu}$ and $m^{2}$ is the reason why Proca electrodynamics doesn’t
break Lorentz symmetry as the product is also a tensor. Same reason as
$d=3,4$, at this point, the theory has two constraints as well,

$\displaystyle(\phi_{1})_{Proca}=\pi^{0}\approx 0,$ (93)
$\displaystyle(\phi_{2})_{Proca}=\partial_{k}\pi^{k}+m^{2}A^{0}\approx 0.$
(94)

And $(\phi_{2})_{Proca}$ gives no new constraints. They are both second-class
since
$\\{(\phi_{1})_{Proca}(\mathbf{x}),(\phi_{2})_{Proca}(\mathbf{y})\\}=m^{2}\delta(\mathbf{x}-\mathbf{y})$.
Thus this theory propagates three physical degrees of freedom, which is
different from the two degrees of freedom in the standard Maxwell
electrodynamics and the cases with gauge invariance. These results are
consistent with HA3 , with the only difference being that their coefficient
$m^{2}$ differs from ours by a factor of $\frac{1}{2}$.

##  V Summary and discussion

In this paper, we perform an extended analysis of the modified electrodynamics
with the violations of both Lorentz symmetry and $U(1)$ gauge symmetry in the
framework of the Standard-Model Extension. This represents an extension of the
previous construction of the Lorentz-violating electrodynamics with gauge
invariance IS3 . For our purpose, by following the procedure in IS3 , we
construct the quadratic Lagrangian density of electrodynamics by allowing the
violations of both the Lorentz and $U(1)$ gauge symmetries. The Lorentz- and
gauge-violating effects in the quadratic Lagrangian are represented by the new
operators with specific dimension $d\leq 4$. With the constructed quadratic
Lagrangian, we calculate in detail the number of independent components of the
Lorentz-violating operators at different dimensions with both cases of the
gauge invariance and gauge violation cases.

We then perform the Hamiltonian analysis of the general theory by considering
Lorentz and gauge-breaking operators with dimension $d\leq 4$ as examples.
Specifically, we perform the analysis for $d=4$, $d=3$, and $d=2$,
respectively. It is shown that the Lorentz-breaking operators with gauge
invariance do not change the classes of the constraints of the theory and have
the same number of physical degrees of freedom as that in the standard
Maxwell’s electrodynamics. When the U(1) gauge symmetry-breaking operators are
presented, the theories in general lack first-class constraint and have one
additional physical degree of freedom, compared to the standard Maxwell’s
electrodynamics. The results of the Hamiltonian structure and their
corresponding number of degrees of freedom are presented in Table. 3.

Finally, we also map our general analysis to several specific modified
electrodynamics, including LIV electrodynamics, CFJ electrodynamics, and Proca
electrodynamics. While the former two theories represent two specific examples
of Lorentz-violating theory with gauge invariance, the latter one is a theory
that breaks $U(1)$ gauge symmetry but still keeps the Lorent symmetry. We show
that our general results are consistent with the existing Hamiltonian analysis
in the literature for these specific examples.

## Acknowledgments

This work is supported by the National Key Research and Development Program of
China under Grant No. 2020YFC2201503, the Zhejiang Provincial Natural Science
Foundation of China under Grant No. LR21A050001 and No. LY20A050002, and the
National Natural Science Foundation of China under Grants No. 12275238 and No.
11675143. B.-F. Li is supported by the National Natural Science Foundation of
China (NNSFC) with the grant No. 12005186.

## References

  * (1) G. Aad _et al._ (ATLAS), Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716, 1–29(2012), arXiv:1207.7214 [hep-ex].
  * (2) S. Chatrchyan _et al._ (CMS), Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC, Phys. Lett. B 716, 30–61 (2012), arXiv:1207.7235 [hep-ex].
  * (3) B. P. Abbott _et al._ (LIGO Scientific,Virgo), Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116,061102 (2016), arXiv:1602.03837 [gr-qc].
  * (4) V. A. Kostelecký and R. Potting, CPT, strings, and meson factories, Phys. Rev. D 51, 3923–3935 (1995), arXiv:hep-ph/9501341.
  * (5) D. Colladay and V. A. Kostelecký, $\mathrm{CPT}$ violation and the standard model, Phys. Rev. D 55, 6760–6774 (1997).
  * (6) V. A. Kostelecký, Gravity, lorentz violation, and the standard model, Phys. Rev. D 69, 105009 (2004).
  * (7) V. A. Kostelecký and N. Russell, Data Tables for Lorentz and CPT Violation, Rev. Mod. Phys. 83, 11–31 (2011), arXiv:0801.0287 [hep-ph].
  * (8) V. A. Kostelecký and M. Mewes, Lorentz-violating electrodynamics and the cosmic microwave background, Phys. Rev. Lett. 99, 011601 (2007), arXiv:astro-ph/0702379.
  * (9) V. A. Kostelecký and M. Mewes, Astrophysical Tests of Lorentz and CPT Violation with Photons, Astrophys. J. Lett. 689, L1–L4 (2008), arXiv:0809.2846 [astro-ph].
  * (10) P. Cabella, P. Natoli, and J. Silk, Constraints on CPT violation from WMAP three year polarization data: a wavelet analysis, Phys. Rev. D 76, 123014 (2007), arXiv:0705.0810 [astro-ph].
  * (11) M. Li and X. Zhang, Cosmological CPT violating effect on CMB polarization, Phys. Rev. D 78, 03516 (2008), arXiv:0810.0403 [astro-ph].
  * (12) N. Aghanim _et al._ (Planck), Planck intermediate results. XLIX. Parity-violation constraints from polarization data, Astron. Astrophys. 596, A110 (2016), arXiv:1605.08633 [astro-ph.CO].
  * (13) G. Gubitosi, L. Pagano, G. Amelino-Camelia, A. Melchiorri, and A. Cooray, A Constraint on Planck-scale Modifications to Electrodynamics with CMB polarization data, JCAP 08, 021, arXiv:0904.3201[astro-ph.CO].
  * (14) S. M. Lee, D. W. Kang, J.-O.Gong, D. Jeong, D.-W. Jung, and S. C. Park, Cosmological Consequences of Kinetic Mixing between Photon and Dark Photon, (2023), arXiv:2307.14798 [hep-ph].
  * (15) L. Caloni, S. Giardiello, M. Lembo, M. Gerbino, G. Gubitosi, M. Lattanzi, and L. Pagano, Probing Lorentz-violating electrodynamics with CMB polarization, JCAP 03, 018, arXiv:2212.04867 [astro-ph.CO].
  * (16) V. A. Kostelecký and M. Mewes, Electrodynamics with lorentz-violating operators of arbitrary dimension, Phys. Rev. D 80, 015020 (2009).
  * (17) L. H. C. Borges and A. F. Ferrari, External sources in a minimal and nonminimal CPT-odd Lorentz violating Maxwell electrodynamics, Mod. Phys. Lett. A 37, 2250021 (2022), arXiv:2202.04424 [hep-th].
  * (18) P. D. S. Silva, L. Lisboa-Santos, M. M. Ferreira, Jr., and M. Schreck, Effects of CPT-odd terms of dimensions three and five on electromagnetic propagation in continuous matter, Phys. Rev. D 104, 116023 (2021), arXiv:2109.04659 [hep-th].
  * (19) R. Casana, M. M. Ferreira,Jr,J. S. Rodrigues, and M. R. O. Silva, Finite Temperature CPT-even Electrodynamics of the Standard Model Extension, Phys. Rev. D 80, 085026 (2009), arXiv:0907.1924 [hep-th].
  * (20) C. A. Escobar and M. A. G. Garcia, The Full CPT-Even Photon Sector of the Standard Model Extension at Finite Temperature, Phys. Rev. D 92,025034 (2015), arXiv:1505.00069 [hep-th].
  * (21) R. Avila, J. R. Nascimento, A. Y. Petrov, C. M. Reyes, and M. Schreck, Causality, unitarity, and indefinite metric in Maxwell-Chern-Simons extensions, Phys. Rev. D 101, 055011 (2020), arXiv:1911.12221 [hep-th].
  * (22) M. M. Ferreira, J. a. A. A. S. Reis, and M. Schreck, Dimensional reduction of the electromagnetic sector of the nonminimal standard model extension, Phys. Rev. D 100, 095026 (2019), arXiv:1905.04401 [hep-th].
  * (23) M. A. Anacleto, F. A. Brito, E. Maciel, A. Mohammadi, E. Passos, W. O. Santos, and J. R. L. Santos, Lorentz-violating dimension-five operator contribution to the black body radiation, Phys. Lett. B 785, 191–196 (2018), arXiv:1806.08273 [hep-th].
  * (24) V. A. Kostelecký and M. Mewes, Cosmological constraints on Lorentz violation in electrodynamics, Phys. Rev. Lett. 87, 251304 (2001), arXiv:hep-ph/0111026.
  * (25) S. M. Carroll, G. B. Field, and R. Jackiw, Limits on a Lorentz and Parity Violating Modification of Electrodynamics, Phys. Rev. D 41, 1231 (1990).
  * (26) E. Allys, P. Peter, and Y. Rodriguez, Generalized Proca action for an Abelian vector field, JCAP02, 004, arXiv:1511.03101 [hep-th].
  * (27) D. Colladay and V. A. Kostelecky, Lorentz violating extension of the standard model, Phys. Rev. D 58, 116002 (1998), arXiv:hep-ph/9809521.
  * (28) E. Svanberg, Theories with higher-order time derivatives and the Ostrogradsky ghost,Bachelor thesis (2022), arXiv:2211.14319 [physics.class-ph].
  * (29) R. P. Woodard, Ostrogradsky’s theorem on Hamiltonian instability, Scholarpedia 10,32243 (2015), arXiv:1506.02210 [hep-th].
  * (30) P. G. Bergmann and J. H. M. Brunings, Non-linear field theories II. canonical equations and quantization, Rev. Mod. Phys., 21, 480–487 (1949).
  * (31) J. L. Anderson and P. G. Bergmann, Constraints in covariant field theories, Phys. Rev., 83, 1018–1025 (1951).
  * (32) P. A. M. Dirac, Generalized Hamiltonian Dynamics, Canadian Journal of Mathematics, 2, 129-148 (1950).
  * (33) P. A. M. Dirac, The Hamiltonian Form of Field Dynamics, Canadian Journal of Mathematics, 3, 1-23 (1951).
  * (34) P. A. M. Dirac, Lectures on Quantum Mechanics, Dover Publications (2001).
  * (35) X. Gao and Z.-B. Yao, Spatially covariant gravity theories with two tensorial degrees of freedom: The formalism, Physical Review D 101, 10.1103/physrevd.101.064018 (2020).
  * (36) X. Gao and Z.-B. Yao, Spatially covariant gravity with velocity of the lapse function: the Hamiltonian analysis, JCAP 05, 024, arXiv:1806.02811[gr-qc].
  * (37) M. Henneaux and C. Teitelboim, Quantization of gauge systems, Princeton University Press, chap. 1, pp. 24, 29 (1992).
  * (38) Z. Molaee and A. Shirzad, Constraint structure of the generalized proca model in the lagrangian formalism (2023), arXiv:2307.02384[hep-th].