Constraints and Conserved Charges for Modified Massive and Massless Abelian
1-Form and 2-Form Theories: A Brief Review

A. K. Rao(a), B. Chauhan(a), R. P. Malik(a,b)
(a) Physics Department, Institute of Science,
Banaras Hindu University, Varanasi - 221 005, (U.P.), India

(b) DST Centre for Interdisciplinary Mathematical Sciences,
Institute of Science, Banaras Hindu University, Varanasi - 221 005, India
e-mails<EMAIL_ADDRESS>bchauhan501gmail.com<EMAIL_ADDRESS>

Abstract: We demonstrate that the generators for the local, continuous and
infinitesimal classical gauge symmetry transformations in the cases of (i) the
St$\ddot{u}$ckelberg-modified massive Abelian 1-form and 2-form theories, and
(ii) the massless Abelian 1-form and 2-form free theories owe their origin to
the first-class constraints (on these theories). We discuss the appearance of
these constraints at the quantum level, within the framework of Becchi-Rouet-
Stora-Tyutin (BRST) formalism, through the physicality criteria w.r.t. the
conserved and nilpotent (anti-)BRST charges. One of the highlights of our
present investigation is the derivation of the nilpotent versions of the
(anti-)BRST charges (from the standard non-nilpotent Noether (anti-)BRST
charges) which lead to the appearance of the operator forms of the first-class
constraints through the physicality criteria at the quantum level in the
context of the modified massive and massless Abelian 2-form theories. We also
comment on (i) the existence of the Curci-Ferrari (CF) type restrictions on
the Abelian 2-form theories (with and without mass), (ii) the modifications in
the St$\ddot{u}$ckelberg-technique for the massive 2D Abelian 1-form and 4D
Abelian 2-form theories and their consequences, and (iii) the off-shell
nilpotent version of the conserved co-BRST charge and its role in the
physicality criteria for the St$\ddot{u}$ckelberg-modified 4D massive Abelian
2-form theory.

PACS numbers: 11.15.-q, 12.20.-m, 03.70.+k

Keywords: Modified massive and massless Abelian 1-form and 2-form theories;
first-class constraints; generators for the local gauge symmetry
transformations; classical Noether conserved charges; (anti-)BRST charges;
modifications in the St$\ddot{u}$ckelberg-technique; fields with negative
kinetic terms; possible candidates for the dark matter/dark energy

##  1 Introduction

The ideas of symmetries in physics have brought together the sophisticated
concepts of mathematics and the physical descriptions of the natural realities
on a platform where the deep insights into the working principle behind a
natural phenomenon (learnt and understood by the theoretical physicists and
mathematicians) have been enriched in a meaningful manner (see, e.g. [1-4] and
references therein). In fact, the principles of the local symmetries dictate
interactions [1]. One such symmetry, in the context of the above, is the gauge
symmetry which has been able to play a decisive role in the precise
theoretical description of the three out of four fundamental interactions of
nature. The most modern definition of a gauge symmetry is elucidated in terms
of the first-class constraints in the terminology of Dirac’s prescription for
the classification scheme of constraints (see, e.g. [5-8] for details). In
other words, those continuous and local symmetry transformations are called as
the gauge symmetry transformations that are generated by the first-class
constraints. In this context, mention can be made of the $U(1)$ Abelian local
gauge symmetry transformations that describe and dictate the electromagnetic
interaction and the $SU(N)$ non-Abelian gauge symmetry transformations of the
Yang-Mills theories that play very important roles in the precise theoretical
description of the weak and strong interactions of nature. The culmination of
the success story of the gauge symmetry transformations is the systematic
construction of the standard model of particle physics (SMPP) where there is
stunning degree of agreements between theory and experiment (see, e.g. [9-13]
for details).

Despite many successes, the standard model of particle physics (based on the
interacting non-Abelian 1-form gauge theory) is not a complete theory due to
the experimental evidence of the masses of all the three varieties of
neutrinos. Hence, one has to go beyond the realm of the validity of the SMPP.
One of the most promising candidates, for such an endeavor in theoretical high
energy physics, is the idea of superstring theories (see, e.g. [14-18] and
references therein) whose quantum excitations lead to the existence of the
higher $p$-form $(p=2,3,...)$ gauge fields. Thus, the study of the higher
$p$-form gauge theories is important and essential. In our present endeavor,
we have concentrated on the study of the modified massive and massless Abelian
2-form gauge theories which are the modest examples as well as the lowest
rungs***The idea of an antisymmetric tensor Abelian 2-form (i.e.
$B^{(2)}=[(d\,x^{\mu}\wedge d\,x^{\nu})/{2!}]\,B_{\mu\nu}$) gauge field was
introduced by V. I. Ogievetsky and I. V. Palubarinov: Sov. J. Nucl. Phys.
(Yad. Fiz.) 4, 156 (1967) who coined the nomenclature “notoph” (i.e. the
opposite of “photon”) for this field whose Dirac’s quantization scheme and
constraint structures were studied by R. K. Kaul: Phys. Rev. D 18, 1127
(1978). This Abelian 2-form theory, in modern times, is popularly known as the
Kalb-Ramond theory. of the tower of the higher $p$-form $(p=2,3,...)$ gauge
fields that appear in the quantum excitations of the (super)string theories.
We have purposely discussed the modified massive and massless Abelian $p=1,2$
form gauge theories together so that the decisive differences and striking
similarities between the two types of theories, at the classical and quantum
levels, could become clear. As far as the discussions on the above two types
of theories at the quantum level is concerned, we have chosen the
mathematically rich and physically very intuitive theoretical method of BRST
formalism [19-22] where the sacrosanct requirements of unitarity and quantum
gauge [i.e. (anti-)BRST] invariance are respected together at any arbitrary
order of perturbative computations for a given physical process that is
allowed by the (anti-)BRST invariant theory (see, e.g. [23-26, 8]).

In our present endeavor, we have laid a great deal of emphasis on the
existence of the first-class constraints and the associated gauge as well as
the (anti-)BRST symmetry transformations and established a deep connection
between the first-class constraints and the Noether conserved charges. The
latter are derived from (i) the infinitesimal, local and continuous gauge
symmetry transformations that are generated by the first-class constraints at
the classical level, and (ii) the (anti-)BRST symmetry transformations which
are the generalizations of the classical local gauge symmetry transformations
to the quantum level. The appearance of the operator forms of the constraints
of the classical gauge theories (with mass and without mass), at the quantum
level, has been established by the requirement of the physicality criteria
where we demand that the physical states (in the total Hilbert space of
quantum states) are those that are annihilated by the conserved and off-shell
nilpotent (anti-)BRST charges. Furthermore, it has been pointed out that the
2D modified Proca and 4D modified massive Abelian 2-form theories are field-
theoretic models of Hodge theory [27-30] because the physical states are
annihilated together by the (i) the first-class constraints, and (ii) the dual
versions of the first-class constraints of the classical gauge theories (of
the modified massive and massless varieties) within the framework of BRST
formalism. The above statements are correct because the physical states are
annihilated by the conserved and off-shell nilpotent versions of the
(anti-)BRST and (anti-)co-BRST charges together. Hence, these specific
theories in 2D and 4D are very special in some sense because the ideas from
the topological field theories [31, 32] also play some crucial roles in the
massless versions of these Abelian 1-form and 2-form gauge theories [33, 34].

Our present brief review article has been spurred by the following key
motivating factors. First, we have discussed various aspects of the massless
and modified massive versions of the Abelian 3-form gauge theories in our
earlier works within the purview of BRST [35, 36] and related superfield
formalism [37]. Hence, it has been a challenge for us to assimilate our
insights into the various aspects of the massless and modified massive Abelian
1-form and 2-form theories [38, 39, 27-30, 33, 34] in a coherent fashion at
one place and discuss the deep connections that exist between the conserved
charges and the first-class constraints of these theories at the classical and
quantum levels. Second, the 2D modified massive Abelian 1-form and 4D modified
massive Abelian 2-form theories have been very special in our studies over the
years [27-30, 33, 34, 38, 39]. It has been a challenge for us to state
something worthwhile about these theories coherently in a single article at
one place. We have accomplished this goal in our present endeavor. Third, in
our earlier work [40], we have systematically developed a theoretical method
to obtain the nilpotent versions of the (anti-)BRST charges from the standard
conserved but non-nilpotent versions of the Noether conserved (anti-)BRST
charges. However, we have not discussed anything about the (anti-)co-BRST
charges that exist for the field-theoretic models of Hodge theory (see, e.g.
[27-30]). We have accomplished this goal, too, in our present endeavor.
Finally, in all our previous works on the Abelian $p=1,2,3$ form theories of
massless and modified massive varieties, we have not concentrated on the
constraints. In our present endeavor, we have given a great deal of importance
to the first-class constraints.

The theoretical contents of our present endeavor are organized as follows.
First of all, as a warm-up exercise, we consider the case of the modified
massive and massless Abelian 1-form theories (i) to discuss their first-class
constraints and associated generators, (ii) to establish connections of the
latter with the Noether conserved charges, and (iii) to show the existence of
these constraints at the quantum level within the framework of BRST formalism
in our Sec. 2. Our Sec. 3 is devoted to the discussion on the existence of the
first-class constraints for the St$\ddot{u}$ckelberg-modified massive Abelian
2-form theory whose massless limit is valid for the free Abelian 2-form gauge
theory. In Sec. 4, we concentrate on the construction of the generator, in
terms of the first-class constraints, for the modified massive Abelian 2-form
theory whose massless limit is true for the Abelian 2-form gauge theory. Our
Sec. 5 deals with the derivations of the Noether conserved currents and
charges for the above Abelian 2-form theories (without mass and with mass)
within the framework of BRST formalism. In Sec. 6, we discuss the
modifications in the St$\ddot{u}$ckelberg-technique for the 2D Proca and 4D
massive Abelian 2-form theories. Finally, we make some concluding remarks and
point out a few future directions for further investigations in our Sec. 7.

Our Appendix A is devoted to the derivation of the proper 2D Lagrangian
density for the modified  2D Proca theory where only the covariant notations
have been used. In our Appendix B, we comment briefly on the existence of the
CF-type restrictions for the massless and modified massive Abelian 2-form
theories from the symmetry considerations.

Convention and Notations: We adopt the convention of the left-derivative
w.r.t. all the fermionic fields of our theories. The background D-dimensional
flat Minkowskian spacetime manifold is endowed with the metric tensor
$\eta_{\mu\nu}=$ diag $(+1,-1,-1,...)$ so that the dot product between two
non-null Lorentz vectors $S_{\mu}$ and $T_{\mu}$ is defined as: $S\cdot
T=\eta_{\mu\nu}\,S^{\mu}\,T^{\nu}=S_{0}\,T_{0}-S_{i}\,T_{i}$ where the Greek
indices $\mu,\,\nu,\,\lambda,...=0,1,2,...,D-1$ denote the time and space
directions and the Latin indices $i,\,j,\,k...=1,2,...D-1$ stand for the space
directions only. Einstein’s summation convention has been adopted in the whole
body of our text. We always denote the (anti-)BRST symmetry transformations by
the symbols $s_{(a)b}$. We adopt the notation $\delta_{g}$ for the
infinitesimal, continuous and local classical gauge symmetry transformations
in the entire text. The over-dot on a generic field $\Psi$ denotes the partial
time derivative as: $\dot{\Psi}=(\partial\,\Psi/\partial\,t)$.

##  2 St$\ddot{u}$ckelberg Modified Massive and Massless Abelian 1-Form
Theories: A Bird’s Eye View

Our present section is divided into three parts. In subsection 2.1, we discuss
the constraint structures of the St$\ddot{u}$ckelberg-modified Proca (i.e.
massive Abelian 1-form) and massless Abelian 1-form gauge theories and
establish that they belong to the first-class variety (see, e.g. [5-8]). These
constraints are present in the standard expressions for the generators of the
gauge transformations. In subsection 2.2, we derive the Noether conserved
charges from the local, infinitesimal and continuous classical gauge symmetry
transformations and establish their connections with the classical symmetry
generators (cf. Subsec. 2.1). Finally, in subsection 2.3, we demand the
physicality criteria w.r.t the conserved and off-shell nilpotent (anti-)BRST
charges and demonstrate the existence of the first-class constraints of the
original St$\ddot{u}$ckelberg-modified massive and massless gauge theories (in
their operator forms) at the quantum level within the framework of BRST
formalism which are found to be consistent with Dirac’s quantization
conditions for the systems endowed with any kinds of constraints (see, e.g.
[6, 8] for details).

###  2.1 First-Class Constraints and Generators

In this subsection, we begin with the Lagrangian density for the D-dimensional
Proca (i.e. massive Abelian 1-form) theory and discuss its massless limit in a
concise manner so that the essence of the Maxwell’s theory can also be
captured (as far as the constraint structures are concerned). We begin with
the following Lagrangian density (with rest mass $m$ of the gauge boson) for
the Proca theory (see, e.g. [41, 38, 39] for details)

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

where the 2-form $F^{(2)}=dA^{(1)}=\frac{1}{2!}\,F_{\mu\nu}\;(dx^{\mu}\wedge
dx^{\nu})$ defines the field-strength tensor
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ in terms of the
Abelian 1-form ($A^{(1)}=A_{\mu}\;dx^{\mu}$) massive vector field $A_{\mu}$.
Here the mathematical symbol $d$ stands for the exterior derivative
$d=\partial_{\mu}\;dx^{\mu}$ [with
$d^{2}\equiv\frac{1}{2!}\,(\partial_{\mu}\,\partial_{\nu}-\partial_{\nu}\,\partial_{\mu})\;(dx^{\mu}\;\wedge
dx^{\nu})=0\,\mbox{where}\,\mu,\nu=0,1,2,...D-1$]. The above Lagrangian
density respects no gauge symmetry transformations because it is endowed with
the second-class constraints in the terminology of Dirac’s prescription for
the classification scheme of constraints (see, e.g. [5-8] for details). To
restore the gauge symmetry transformations, we exploit the beauty and
theoretical strength of the St$\ddot{u}$ckelberg-technique of the compensating
field ($\phi$) and take into account the replacement: $A_{\mu}\longrightarrow
A_{\mu}\mp\frac{1}{m}\,\partial_{\mu}\phi$ where the field $\phi$ is a pure
scalar field. The resulting St$\ddot{u}$ckelberg-modified Lagrangian density
[${\cal L}_{(S)}$] for the modified Proca theory is as follows (see, e.g. [41,
38, 39] for details):

$\displaystyle{\cal L}_{(P)}\longrightarrow{\cal
L}_{(S)}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{m^{2}}{2}A_{\mu}\,A^{\mu}\mp
m\,A_{\mu}\partial^{\mu}\phi+\frac{1}{2}\partial_{\mu}\phi\,\partial^{\mu}\phi,$
(2)

The above modified Lagrangian density [${\cal L}_{(S)}$] is endowed with the
first-class constraints. To corroborate this claim, first of all, we compute
the conjugate momenta w.r.t. the basic fields $A_{\mu}$ and $\phi$ (of our
St$\ddot{u}$ckelberg-modified massive Abelian 1-form theory) as:

$\displaystyle\Pi^{\mu}_{(A)}=\frac{\partial\,{\cal
L}_{(S)}}{\partial\,(\partial_{0}\,A_{\mu})}=-\,F^{0\mu}\;\Longrightarrow\;\Pi^{0}_{(A)}=-\,F^{00}\approx
0\quad\Longrightarrow\quad(P.C.),$
$\displaystyle\Pi_{(\phi)}=\frac{\partial\,{\cal
L}_{(S)}}{\partial\,(\partial_{0}\,\phi)}=\mp m\,A^{0}+\partial^{0}\,\phi.$
(3)

In the above, we have adopted Dirac’s notation for the concept of weakly zero
(in $\Pi^{0}_{(A)}=-\,F^{00}\approx 0$) on our theory to define the primary
constraint (P. C.). The time-evolution invariance [24] [i.e.
$({\partial\,\Pi^{0}_{(A)}}/{\partial\,t})\approx 0$] of the naturally present
P. C. (i.e. $\Pi^{0}_{(A)}\approx 0$) on our theory can be computed (see, e.g.
[24]) from the following Euler-Lagrange equation of motion (EL-EoM) that is
derived (w.r.t. the basic field $A_{\mu}$) from ${\cal L}_{(S)}$, namely;

$\displaystyle\partial_{\mu}\,F^{\mu\nu}+m^{2}\,A^{\nu}\mp
m\,\partial^{\nu}\,\phi=0,$ (4)

provided we take into account $\nu=0$, in the above equation, to obtain:

$\displaystyle\partial_{0}\,F^{00}$ $\displaystyle+$
$\displaystyle\partial_{i}\,F^{i0}+m^{2}\,A^{0}\mp m\,\partial^{0}\,\phi=0,$
$\displaystyle\frac{\partial\,\Pi^{0}_{(A)}}{\partial\,t}$ $\displaystyle=$
$\displaystyle\partial_{i}\,E_{i}\mp m\,\Pi_{(\phi)}\approx 0$ (5)
$\displaystyle\equiv$ $\displaystyle\partial_{i}\,\Pi^{i}_{(A)}\mp
m\,\Pi_{\phi}\approx 0\quad\Longrightarrow\quad(S.C.).$

The above relationship is nothing but the derivation of secondary constraint
(S.C.) on our modified Proca theory (see, e.g. [24]) where
$\Pi^{i}_{(A)}=-\,F^{0i}=E_{i}$ are the components of the electric fields
(i.e. $E_{i}\equiv\vec{E}$) which are the canonical conjugate momenta w.r.t.
the vector field $A_{i}\equiv\vec{A}$. There are no further constraints on the
theory. Since the P. C. and S. C. are expressed in terms of the components of
the conjugate momenta, it is evident that both will commute with each-other
thereby leading to their classification as the first-class constraints in the
terminology of Dirac’s prescription for the classification scheme of
constraints [5-8].

Following the standard formula [42, 43, 26, 8], we can write down the
generator for the local, infinitesimal and continuous classical gauge symmetry
transformations for the St$\ddot{u}$ckelberg-modified Lagrangian density
[${\cal L}_{(S)}$], namely;

$\displaystyle\delta_{g}\,A_{\mu}=\partial_{\mu}\,\chi,\qquad\delta_{g}\,\phi=\pm\,m\,\chi,\qquad\delta_{g}\,F_{\mu\nu}=0,$
(6)

in terms of the first-class constraints as (see, e.g. [42, 43] for details)

$\displaystyle G=\int
d^{D-1}x\,\Big{[}\big{(}\partial_{0}\,\chi\big{)}\,\Pi^{0}_{(A)}-\chi\,\big{(}\partial_{i}\,\Pi^{i}_{(A)}\mp
m\,\Pi_{(\phi)}\big{)}\Big{]},$ (7)

where $\chi\,(\vec{x},t)$ is the physically well-defined infinitesimal,
continuous and local gauge symmetry transformation parameter. Using the Gauss
divergence theorem, the above generator can be re-expressed (in a very
transparent and useful form) as follows:

$\displaystyle G=\int
d^{D-1}x\,\Big{[}(\partial_{0}\,\chi)\,\Pi^{0}_{(A)}+\big{(}\partial_{i}\,\chi\big{)}\,\Pi^{i}_{(A)}\pm
m\,\chi\,\Pi_{(\phi)}\Big{]}.$ (8)

At this juncture, we define the following non-trivial equal-time commutators
for the basic fields $A_{\mu}$ and $\phi$ of our modified Proca theory, in the
natural units ($\hbar=c=1$), as:

$\displaystyle[A_{0}\,(\vec{x},t),\,\Pi^{0}_{(A)}\,(\vec{y},t)]=i\,\delta^{(D-1)}\,(\vec{x}-\vec{y}),$
$\displaystyle[A_{i}\,(\vec{x},t),\,E_{j}\,(\vec{y},t)]=i\,\delta_{ij}\,\delta^{(D-1)}\,(\vec{x}-\vec{y}),$
$\displaystyle[\phi\,(\vec{x},t),\,\Pi_{(\phi)}\,(\vec{y},t)]=i\,\delta^{(D-1)}\,(\vec{x}-\vec{y}).$
(9)

All the rest of the other equal-time brackets are trivially zero. It is
crystal clear now that, we have the following infinitesimal, local and
continuous classical gauge symmetry transformations for the basic fields in
terms of the commutators between the basic fields and the generator $(G)$ of
our St$\ddot{u}$ckelberg-modified Proca theory, namely;

$\displaystyle\delta_{g}\,\phi\,(\vec{x},t)=-\,i\,[\,\phi\,(\vec{x},t),\,G]=\pm
m\ \chi\,(\vec{x},t),$
$\displaystyle\delta_{g}\,A_{0}\,(\vec{x},t)=-\,i\,[A_{0}\,(\vec{x},t),\,G]=\partial_{0}\,\chi\,(\vec{x},t),$
$\displaystyle\delta_{g}\,A_{i}\,(\vec{x},t)=-\,i\,[A_{i}\,(\vec{x},t),\,G]=\partial_{i}\,\chi\,(\vec{x},t).$
(10)

Hence, we have derived the local, infinitesimal and continuous classical gauge
symmetry transformations (6) for the basic fields which, automatically, imply
that $\delta_{g}\,F_{\mu\nu}=0$.

We conclude this subsection with the following remarks. First, it is clear
that, in the massless limit (i.e. $m=0,\,\phi=0$) of the Proca theory, we
obtain the usual $U(1)$ gauge invariant Maxwell’s theory whose gauge symmetry
generator $(G^{(M)})$ can be obtained from $G$ [cf. Eq. (8), with $m=0$] as
follows

$\displaystyle G^{(M)}$ $\displaystyle=$ $\displaystyle\int
d^{D-1}x\,\Big{[}\big{(}\partial_{0}\,\chi\big{)}\,\Pi^{0}_{(A)}-\big{(}\partial_{i}\,\Pi^{i}_{(A)}\big{)}\,\chi\Big{]}$
(11) $\displaystyle\equiv$ $\displaystyle\int
d^{D-1}x\,\Big{[}\big{(}\partial_{0}\chi\big{)}\,\Pi^{0}_{(A)}+\big{(}\partial_{i}\,\chi\big{)}\,\Pi^{i}_{(A)}\Big{]},$

which leads to the $U(1)$ gauge symmetry transformation:
$\delta_{g}\,A_{\mu}=\partial_{\mu}\,\chi$ if we use the canonical equal-time
commutators (10) appropriately. Second, we note that the gauge theories are
always endowed with the first-class constraints in the terminology of Dirac’s
prescription for classification scheme of constraints (see, e.g. [5-8]).
Third, the first-class constraints are responsible for the appropriate
definitions of the gauge symmetry generators. In other words, the first-class
constraints generate the gauge symmetry transformations which lead to the
sacrosanct (and the most modern) definition of a classical gauge theory.
Fourth, it is the presence of the St$\ddot{u}$ckelberg-compensating field
($\phi$) that has been able to convert the second-class constraints of the
original Proca Lagrangian density [${\cal L}_{(P)}$] into the first-class
constraints of the modified Lagrangian density [${\cal L}_{(S)}$]. Fifth, it
is clear from the P. C. [cf. Eq. (3)] and S. C. [cf. Eq. (5)] that (in the
massless limit) we shall have, once again, the first-class constraints where
$\Pi^{0}_{(A)}\approx 0$ as the P. C. and the S. C. [cf. Eq. (5)] reduces to:
$\partial_{i}\,E_{i}=\vec{\nabla}\cdot\vec{E}\approx 0$ (i.e. the Gauss law of
the source-free Maxwell’s equations). Finally, we note that the modified Proca
theory and Maxwell’s theory present themselves as the field-theoretic examples
of the simple gauge theories as they respect the infinitesimal, continuous and
local classical gauge symmetries that are generated by the first-class
constraints.

###  2.2 Noether Conserved Charge: First-Class Constraints

The purpose of this subsection is to establish a connection between the
Noether conserved charge and the first-class constraints of the
St$\ddot{u}$ckelberg-modified massive Abelian 1-form (i.e. Proca) theory whose
massless limit will be valid for Maxwell’s theory. In this connection, we
would like to point out that the local, continuous and infinitesimal classical
gauge symmetry transformations $(\delta_{g})$ of equation (6) lead to the
derivation of the following Noether current $(J^{\mu}_{(A)})$ from the
modified Lagrangian density [${\cal L}_{(S)}$], namely;

$\displaystyle J^{\mu}_{(A)}=-\,F^{\mu\nu}\,(\partial_{\nu}\,\chi)\pm
m\,(\partial^{\mu}\,\phi\mp m\,A^{\mu})\,\chi,$ (12)

where the superscript $(A)$ on the Noether current denotes the fact that we
are dealing with the massive Abelian 1-form ($A^{(1)}=A_{\mu}\;dx^{\mu}$)
theory where the basic gauge field is $A_{\mu}$. The conservation law
($\partial_{\mu}\,J^{\mu}_{(A)}=0$) can be proven by using the following EL-
EoMs

$\displaystyle\partial_{\mu}\,F^{\mu\nu}+m^{2}\,A^{\nu}\mp
m\,\partial^{\mu}\,\phi=0,$
$\displaystyle\partial_{\mu}\,[\partial^{\mu}\,\phi\mp{m}\,A^{\mu}]=0\;\;\Longrightarrow\;\;\Box\,\phi\mp
m\,(\partial\cdot A)=0,$ (13)

that emerge out from ${\cal L}_{(S)}$ [cf. Eq. (2)] w.r.t. the basic fields
$A_{\mu}$ and $\phi$ of our St$\ddot{u}$ckelberg-modified massive Abelian
1-form theory, respectively.

At this juncture, following the sacrosanct principle of Noether’s theorem, we
define the conserved charge $Q_{(A)}$ (i.e. $Q_{(A)}=\int
d^{D-1}\,x\,J^{0}_{(A)}$) as follows:

$\displaystyle Q_{(A)}$ $\displaystyle\equiv$ $\displaystyle\int
d^{D-1}\,x\,\Big{[}-\,F^{0\nu}\,\partial_{\nu}\,\chi\pm
m\,(\partial^{0}\,\phi\mp m\ A^{0})\,\chi\Big{]}.$ (14)

Since our theory is endowed with the constraints, we have to be careful in
expanding the r.h.s. of the above equation. Taking into account the primary
constraint ($\Pi^{0}_{(A)}=-\,F^{00}\approx 0$) and the secondary constraint
($\partial_{i}\,\Pi^{i}_{(A)}\mp m\,\Pi_{\phi}\approx 0$), we obtain the
following:

$\displaystyle Q_{(A)}$ $\displaystyle=$ $\displaystyle\int
d^{D-1}\,x\,\Big{[}-\,F^{00}\,\partial_{0}\,\chi-\,F^{0i}\,\partial_{i}\,\chi\pm
m\,\Pi_{(\phi)}\,\chi\Big{]}$ (15) $\displaystyle\equiv$ $\displaystyle\int
d^{D-1}\,x\,\Big{[}\Pi^{0}_{(A)}\,(\partial_{0}\,\chi)+\Pi^{i}_{(A)}\,(\partial_{i}\,\chi)\pm
m\,\Pi_{(\phi)}\,\chi\Big{]}.$

Using Gauss’s divergence theorem, the above equation can be recast into the
following

$\displaystyle Q_{(A)}=\int
d^{D-1}\,x\,\Big{[}\Pi^{0}_{(A)}\,(\partial_{0}\,\chi)-\big{(}\partial_{i}\,\Pi^{i}_{(A)}\mp
m\,\Pi_{(\phi)}\big{)}\,\chi\Big{]},$ (16)

where the primary constraint ($\Pi^{0}_{(A)}\approx 0$) and the secondary
constraint ($\partial_{i}\,\Pi^{i}_{(A)}\mp m\,\Pi_{\phi}\approx 0$) appear
explicitly. A close look at the above equation demonstrates that we have
already derived the generator $G$ [cf. Eq. (8)] of our previous subsection as
$Q_{(A)}$ which contains the primary and secondary constraints of our modified
massive Abelian 1-form theory.

We end this subsection with the following remarks. First of all, as pointed
out earlier, it is the presence of the St$\ddot{u}$ckelberg-compensating pure
scalar field $(\phi)$ that is responsible for the conversion of the second-
class constraints of the original Proca theory into the first-class
constraints. It is obvious that, for the massless Abelian 1-form gauge theory
(where $m=0,\,\phi=0$), the constraints: $\Pi^{0}_{(A)}\approx 0$ and
$\partial_{i}\Pi^{i}_{(A)}\approx 0$ are the first-class in nature, too.
Second, the standard form of the generators for the local, continuous and
infinitesimal classical gauge symmetry transformations (see, e.g. [42, 43, 26,
8]) in the cases of the modified massive and massless theories are, in fact,
deeply connected with the Noether conserved charges which contain the first-
class constraints in a precise manner. Third, we would like to lay emphasis
that, because of the presence of the constraints, the expression for the
conserved charge (derived from the conserved Noether current) should be
expressed in a careful manner so that the constraints appear at the precise
places in it. In other words, the first-class constraints should not be
strongly set equal to zero whenever they appear in the expression for the
conserved charge. Finally, we note that the $Q_{(A)}$ in (16) reduces (with
$m=0,\,\phi=0$) to the Noether conserved charge for the Maxwell theory (i.e.
$Q_{(A)}\longrightarrow Q_{(A)}^{(M)}=\int
d^{D-1}x\,[(\partial_{0}\chi)\,\Pi^{0}_{(A)}-\chi\,(\partial_{i}\Pi^{i}_{(A)})]$)
which is equal to the generator $G^{(M)}$ in (11) where the superscript $(M)$
stands for the Maxwell theory. The latter contains the first-class constraints
(i.e. $\Pi^{0}_{(A)}\approx 0$ and $\partial_{i}\Pi^{i}_{(A)}\approx 0$) in an
explicit manner.

###  2.3 (Anti-)BRST Charges: First-Class Constraints

The central objective of our present subsection is to establish the existence
of the first-class constraints (i.e. $\Pi^{0}_{(A)}\approx
0,\;\partial_{i}\,\Pi^{i}_{(A)}\mp m\,\Pi_{\phi}\approx 0$) at the quantum
level within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism where
we demonstrate that the physicality criteria ($Q_{(a)b}|phys>=0$) on the
physical states (i.e. $|phys>$) w.r.t. the conserved and off-shell nilpotent
(anti-)BRST charges $[Q_{(a)b}]$ of our theory lead to the annihilation of the
physical states (in the total Hilbert space of states) by the operator forms
of the first-class constraints. Towards this goal in mind, we note that the
following off-shell nilpotent [$s_{(a)b}^{2}=0$] and absolutely anticommuting
$(s_{b}\,s_{ab}+s_{ab}\,s_{b}=0)$ (anti-)BRST symmetry transformations
[$s_{(a)b}$]

$\displaystyle s_{ab}\,A_{\mu}=\partial_{\mu}\,\bar{C},\quad
s_{ab}\,\bar{C}=0,\quad s_{ab}\,C=-\,i\,B,\quad s_{ab}\,B=0,\quad
s_{ab}\,\phi=\pm\,m\,\bar{C},$ $\displaystyle
s_{b}\,A_{\mu}=\partial_{\mu}\,C,\qquad s_{b}\,C=0,\quad
s_{b}\,\bar{C}=i\,B,\qquad s_{b}\,B=0,\qquad s_{b}\,\phi=\pm\,m\,C,$ (17)

leave the following (anti-)BRST invariant Lagrangian density

$\displaystyle{\cal L}_{B}$ $\displaystyle=$ $\displaystyle{\cal
L}_{(S)}+B\,(\partial\cdot A\pm
m\,\phi)+\frac{B^{2}}{2}-\,i\,\partial_{\mu}\,\bar{C}\,\partial^{\mu}\,C+i\,m^{2}\,\bar{C}\,C$
(18) $\displaystyle\equiv$
$\displaystyle-\,\frac{1}{4}\,F^{\mu\nu}\,F_{\mu\nu}+\frac{m^{2}}{2}\,A_{\mu}\,A^{\mu}\mp
m\,A^{\mu}\partial_{\mu}\phi+\frac{1}{2}\,\partial_{\mu}\phi\,\partial^{\mu}\phi+B\,(\partial\cdot
A\pm m\,\phi)$ $\displaystyle+$
$\displaystyle\frac{B^{2}}{2}-i\,\partial_{\mu}\bar{C}\,\partial^{\mu}C+i\,m^{2}\,\bar{C}\,C,$

quasi-invariant in the sense that we observe the following:

$\displaystyle s_{ab}\,{\cal
L}_{B}=\partial_{\mu}\,[B\,\partial^{\mu}\,\bar{C}],\qquad\qquad s_{b}\,{\cal
L}_{B}=\partial_{\mu}\,[B\,\partial^{\mu}\,C].$ (19)

In other words, the Lagrangian density (18) transforms to the total spacetime
derivatives under the (anti-)BRST symmetry transformations (17) thereby
rendering the action integral $S=\int d^{D}x\,{\cal L}_{B}$ invariant
$[s_{(a)b}\,S=0]$ under the (anti-)BRST symmetry transformations. Hence, the
off-shell nilpotent, continuous and infinitesimal (anti-)BRST transformations
are the symmetry transformations for the action integral of our quantum
theory.

At this crucial juncture, a few key and crucial comments are in order. First
of all, the quantum (anti-)BRST symmetry transformations (17) are the
generalizations of the classical gauge symmetry transformations (6) within the
framework of BRST formalism. Second, the local and infinitesimal classical
gauge symmetry transformation parameter $\chi\,(\vec{x},t)$ has been traded
with the fermionic $(C^{2}=\bar{C}^{2}=0,\,C\,\bar{C}+\bar{C}\,C=0)$
(anti-)ghost $(\bar{C})\,C$ fields which are invoked to maintain the unitarity
of the quantum gauge (i.e. BRST and anti-BRST) symmetry invariant BRST-
quantized theory. Third, the St$\ddot{u}$ckelberg-modified Lagrangian density
${\cal L}_{(S)}$ [cf. Eq. (2)] has been generalized to the (anti-)BRST
invariant Lagrangian density ${\cal L}_{B}$ at the quantum level that
incorporates the gauge-fixing and Faddeev-Popov ghost terms. Fourth, the
auxiliary field $B$ is the Nakanishi-Lautrup field which is invoked to
linearize the gauge-fixing term. Finally, the covariant canonical quantization
(i.e. BRST-quantization) can be performed for the Lagrangian density ${\cal
L}_{B}$ because all the dynamical fields in the theory have their counterpart
canonical conjugate momenta.

As pointed out earlier, the (anti-)BRST symmetry transformations (17) are the
infinitesimal, continuous, off-shell nilpotent and absolutely anticommuting
symmetry transformations of the action integral. As a consequence, they lead
to the derivation of the Noether currents ($J^{\mu}_{(r)},\,r=ab,\,b$) due to
the celebrated Noether theorem, namely;

$\displaystyle
J^{\mu}_{(ab)}=-\,F^{\mu\nu}\,\partial_{\nu}\,\bar{C}+B\,\partial^{\mu}\,\bar{C}\pm
m\,\bar{C}\,\partial^{\mu}\,\phi-m^{2}\,\bar{C}\,A^{\mu},$ $\displaystyle
J^{\mu}_{(b)}=-\,F^{\mu\nu}\,\partial_{\nu}\,C+B\,\partial^{\mu}\,C\pm
m\,C\,\partial^{\mu}\,\phi-m^{2}\,C\,A^{\mu},$ (20)

which are found to be conserved ($\partial_{\mu}\,J^{\mu}_{(r)}=0,\,r=ab,\,b$)
due to the following EL-EoMs that emerged out from the ${\cal L}_{B}$, namely;

$\displaystyle\partial_{\mu}\,F^{\mu\nu}=\partial^{\nu}\,B\pm
m\,\partial^{\nu}\,\phi-\,m^{2}\,A^{\nu},\qquad\Box\,\phi\mp m\,(\partial\cdot
A)=\pm m\,B,$
$\displaystyle(\Box+m^{2})\,C=0,\qquad\qquad(\Box+m^{2})\,\bar{C}=0.$ (21)

According to the basic idea behind the Noether theorem, the above conserved
currents in (20) lead to the definitions of the conserved charges
$Q_{(r)}=\int d^{(D-1)}x\,J^{0}_{(r)}$ (with $r=b,ab$). In other words, we
have the following explicit expressions for the charges

$\displaystyle Q_{ab}=\int
d^{D-1}\,x\,\Big{[}-F^{0\,i}\,(\partial_{i}\,\bar{C})+B\,\dot{\bar{C}}\pm
m\,\bar{C}\,\dot{\phi}-m^{2}\,A_{0}\,\bar{C}\big{]}\equiv\int
d^{D-1}\,x\,\big{[}B\,\dot{\bar{C}}-\dot{B}\,\bar{C}\big{]},$ $\displaystyle
Q_{b}=\int d^{D-1}\,x\,\Big{[}-F^{0\,i}\,(\partial_{i}\,C)+B\,\dot{C}\pm
m\,C\,\dot{\phi}-m^{2}\,A_{0}\,C\big{]}\equiv\int
d^{D-1}\,x\,\big{[}B\,\dot{C}-\dot{B}\,C\big{]},$ (22)

where we have used the first entry of the EL-EoMs that have been derived in
(21) from the (anti-)BRST invariant Lagrangian density ${\cal L}_{B}$. In
other words, we have used the EL-EoM: $\dot{B}=\partial_{i}\,F^{i0}\mp
m\,\partial_{0}\,\phi+m^{2}\,A_{0}$ which emerges out from the first entry of
(21).

At this stage, we define the physical states of our BRST-quantized theory
where we demand that the physical states ($|phys>$), in the total Hilbert
space of states, are those that are annihilated by the conserved and off-shell
nilpotent (anti-)BRST charges. In this context, it is pertinent to point out
that the (anti-)BRST charges are conserved [i.e.
$\dot{Q}_{(a)b}\equiv\partial_{0}Q_{(a)b}=0$]. To corroborate this claim, we
note that the applications of the EL-EoMs:
$(\Box+m^{2})\,C=0,\;(\Box+m^{2})\,\bar{C}=0$ and $(\Box+m^{2})\,B=0$, and
(ii) the Gauss divergence theorem lead to the expressions for the (anti-)BRST
charges to be zero (i.e. $\dot{Q}_{(a)b}=0$) when we apply a direct time
derivative on $Q_{(a)b}$. Furthermore, we note that the off-shell nilpotency
[$Q_{(a)b}^{2}=0$] of the (anti-)BRST charges [$Q_{(a)b}$] is also true†††We
shall see that the standard Noether (anti-)BRST charges are not off-shell
nilpotent in the cases of the modified massive and massless Abelian 2-form
theories because of the existence of the non-trivial CF-type restrictions. In
the case of our D-dimensional Abelian 1-form theory, the CF-type restriction
is trivial. This is why, we have a single (anti-)BRST invariant Lagrangian
density ${\cal L}_{B}$ [cf. Eq. (18)] and the Noether theorem leads to the
derivations of the conserved and off-shell nilpotent (anti-)BRST charges.:
$s_{ab}\,Q_{ab}=-\,i\,\\{Q_{ab},Q_{ab}\\}=0\;\Longrightarrow\;Q^{2}_{ab}=0,\;s_{b}\,Q_{b}=-\,i\,\\{Q_{b},Q_{b}\\}=0\;\Longrightarrow\;Q^{2}_{b}=0$
where we have used (i) the direct applications of the (anti-)BRST symmetry
transformation (17) on the concise forms of the (anti-)BRST charges [cf. Eq.
(22)], and (ii) the basic relationship between the continuous symmetry
transformations and their generators (as the conserved Noether charges). We
now impose the physicality criteria on the physical states (i.e. $|\,phys>$)
w.r.t. the conserved and off-shell nilpotent (anti-)BRST charges, namely;

$\displaystyle Q_{(a)b}\,|\,phys>\,=0\quad\Longrightarrow\quad
B\,|\,phys>\,=0,\qquad\quad\dot{B}\,|\,phys>\,=0,$ (23)

where we have taken onto account the non-physical nature of the (anti-)ghost
fields. At this juncture, we note that the canonical conjugate momenta w.r.t.
the basic fields $A_{\mu}$ and $\phi$ from the Lagrangian density ${\cal
L}_{B}$ are:

$\displaystyle\Pi^{\mu}_{(A)}=\frac{\partial\,{\cal
L}_{(S)}}{\partial\,(\partial_{0}\,A_{\mu})}=-\,F^{0\mu}+\eta^{\mu
0}\,B\;\Longrightarrow\;\Pi^{0}_{(A)}=B,$
$\displaystyle\Pi_{(\phi)}=\frac{\partial\,{\cal
L}_{(S)}}{\partial\,(\partial_{0}\,\phi)}=\mp\,m\,A^{0}+\partial^{0}\,\phi\equiv\partial_{0}\phi\mp\,m\,A_{0}.$
(24)

On the other hand, from the appropriate equations of motion (21), we obtain
the following

$\displaystyle\dot{B}$ $\displaystyle=$
$\displaystyle\partial_{i}E_{i}+m^{2}\,A_{0}\mp
m\,\phi\;\equiv\;\partial_{i}\Pi^{i}_{(A)}\mp m\,\Pi_{\phi},$ (25)

which is nothing but the secondary constraint of the original
St$\ddot{u}$ckelberg-modified Lagrangian density [${\cal L}_{(S)}$]. It is
interesting to point out that the Nakanishi-Lautrup field $B$ is nothing but
the expression for $\Pi^{0}_{(A)}$ (which is the P. C. emerging out from
${\cal L}_{(S)})$. Thus, it is perfectly obvious, from the physicality
criteria (23), that we have:

$\displaystyle
B\,|\,phys>\,=0\quad\Longrightarrow\quad\;\Pi^{0}_{(A)}\,|\,phys>\,=0,$
$\displaystyle\dot{B}\,|\,phys>\,=0\quad\Longrightarrow\quad\;(\partial_{i}\Pi^{i}_{(A)}\mp
m\,\Pi_{\phi})\,|\,phys>\,=0.$ (26)

The above observations show that the operator forms of the first-class
constraints of the original St$\ddot{u}$ckelberg-modified massive theory
[described by ${\cal L}_{(S)}$] annihilate the physical states. Thus, the
physicality criteria [cf. Eq. (23)] w.r.t. the conserved and off-shell
nilpotent (anti-)BRST charges $[Q_{(a)b}]$ are consistent with the Dirac
quantization conditions where the operator forms of the constraints are
required to annihilate the physical states (of a system endowed with any kind
of constraints). It is interesting to point out that the concise forms of the
(anti-)BRST charges [cf. Eq. (22)] remain the same for the modified massive
and massless Abelian 1-form theories. However, in the case of the latter (with
$m=0,\phi=0$), the second entry of (26) becomes:
$\partial_{i}\Pi^{i}_{(A)}|\,phys>\,=0$ due to the fact that
$\dot{B}=\partial_{i}E_{i}\equiv\partial_{i}\Pi^{i}_{(A)}$ is the secondary
constraint on the massless theory.

We end our Sec. 2 with the final remark that the first-class constraints are
the characteristic and decisive features of a gauge theory at the classical as
well as at the quantum level. In the latter case, within the framework of BRST
formalism, they appear as operators in the Dirac quantization conditions
through physicality criteria w.r.t. the conserved and off-shell nilpotent
versions of the (anti-)BRST charges. The nilpotency property of the
(anti-)BRST charges is very crucial when we take into account the physicality
criteria w.r.t the conserved (anti-)BRST charges within the purview of BRST
formalism.

##  3 First-Class Constraints and Generators: Modified Massive and Massless
Abelian 2-Form Theories

We begin with the following Lagrangian density for the massive Abelian 2-form
theory (with the rest mass equal to $m$) in any arbitrary D-dimension of
spacetime (see, e.g. [44])

$\displaystyle{\cal
L}_{0}^{(B)}=\frac{1}{12}\,H^{\mu\nu\lambda}\,H_{\mu\nu\lambda}-\frac{m^{2}}{4}\,B_{\mu\nu}\,B^{\mu\nu},$
(27)

where the 3-form
$H^{(3)}=d\,B^{(2)}\equiv\frac{1}{3!}\,H_{\mu\nu\lambda}\;(d\,x^{\mu}\wedge
d\,x^{\nu}\wedge d\,x^{\lambda})$ defines the field-strength tensor
$H_{\mu\nu\lambda}=\partial_{\mu}\,B_{\nu\lambda}+\partial_{\nu}\,B_{\lambda\mu}+\partial_{\lambda}\,B_{\mu\nu}$
for the Abelian 2-form $B^{(2)}=\frac{1}{2!}\,B_{\mu\nu}\;(d\,x^{\mu}\wedge
d\,x^{\nu})$ antisymmetric $(B_{\mu\nu}=-\,B_{\nu\mu})$ tensor field
$B_{\mu\nu}$. Here the operator $d$ (with $d^{2}=0)$ is the exterior
derivative of differential geometry [45-49]. The above Lagrangian density does
not respect any gauge symmetry transformation because it is endowed with the
second-class constraints in the terminology of Dirac’s prescription for the
classification scheme of constraints [5-8]. To restore the gauge symmetry
transformations (of the massless Abelian 2-form theory), we exploit the beauty
and theoretical potential of the St$\ddot{u}$ckelberg-technique of the
compensating field (e.g. a vector field $\phi_{\mu}$) in the following
replacement (see, e.g. [44])

$\displaystyle B_{\mu\nu}\longrightarrow
B_{\mu\nu}\mp\frac{1}{m}\,(\partial_{\mu}\,\phi_{\nu}-\partial_{\nu}\,\phi_{\mu})\quad\equiv\quad
B_{\mu\nu}\mp\frac{1}{m}\,\Phi_{\mu\nu},$ (28)

where the 2-form $\Phi^{(2)}=\frac{1}{2!}\,\Phi_{\mu\nu}\;(d\,x^{\mu}\wedge
d\,x^{\nu})$ defines the field-strength antisymmetric
($\Phi_{\mu\nu}=-\,\Phi_{\nu\mu}$) tensor field
$\Phi_{\mu\nu}=\partial_{\mu}\,\phi_{\nu}-\partial_{\nu}\,\phi_{\mu}$ for the
1-form $\Phi^{(1)}=\phi_{\mu}\;d\,x^{\mu}$ Lorentz vector
St$\ddot{u}$ckelberg-compensating field $(\phi_{\mu})$. The substitution (28)
into ${\cal L}_{0}^{(B)}$ leads to the following St$\ddot{u}$ckelberg-modified
Lagrangian density ${\cal L}_{(S)}^{(B)}$ for the antisymmetric tensor field,
namely;

$\displaystyle{\cal L}_{0}^{(B)}\longrightarrow{\cal
L}_{(S)}^{(B)}=\frac{1}{12}\,H^{\mu\nu\lambda}\,H_{\mu\nu\lambda}-\frac{m^{2}}{4}\,B_{\mu\nu}\,B^{\mu\nu}\pm\frac{m}{2}\,B_{\mu\nu}\,\Phi^{\mu\nu}-\frac{1}{4}\,\Phi_{\mu\nu}\,\Phi^{\mu\nu},$
(29)

which respects [i.e $\delta_{g}\,{\cal L}_{(S)}^{(B)}=0$] the following gauge
symmetry transformations

$\displaystyle\delta_{g}\,B_{\mu\nu}=(\partial_{\mu}\,\Lambda_{\nu}-\partial_{\nu}\,\Lambda_{\mu}),\qquad\delta_{g}\,\phi_{\mu}=\pm\,m\,\Lambda_{\mu}-\partial_{\mu}\Sigma,$
$\displaystyle\delta_{g}\,H_{\mu\nu\lambda}=0,\quad\qquad\qquad\qquad\;\;\delta_{g}\,\Phi_{\mu\nu}=\pm\,m\,(\partial_{\mu}\,\Lambda_{\nu}-\partial_{\nu}\,\Lambda_{\mu}),$
(30)

where $\Lambda_{\mu}$ and $\Sigma$ are the Lorentz vector and scalar gauge
symmetry transformation parameters. It is an undeniable truth that the first-
class constraints on a theory generate the infinitesimal local gauge symmetry
transformations. In our present case, there are primary and secondary
constraints on our theory. The present theory is classically described by the
Lagrangian density ${\cal L}^{(B)}_{(S)}$, for which, the canonical conjugate
momenta w.r.t. the basic fields $B_{\mu\nu}$ and $\phi_{\mu}$ are:

$\displaystyle\Pi_{(B)}^{\mu\nu}=\frac{\partial{\cal
L}_{(S)}}{\partial(\partial_{0}B_{\mu\nu})}=\frac{1}{2}\,H^{0\mu\nu}\quad\Longrightarrow\quad\Pi^{0i}_{(B)}=\frac{1}{2}\,H^{00i}\approx
0,$ $\displaystyle\Pi_{(\phi)}^{\mu}=\frac{\partial{\cal
L}_{(S)}}{\partial(\partial_{0}\phi_{\mu})}=\pm\,m\,B^{0\mu}-\Phi^{0\mu}\quad\Longrightarrow\quad\Pi^{0}_{(\phi)}=\pm\,m\,B^{00}\
-\Phi^{00}\approx 0,$
$\displaystyle\Pi^{ij}_{(B)}=\frac{1}{2}\,H^{0ij},\qquad\qquad\Pi_{(\phi)}^{i}=\pm\,m\,B^{0i}-\Phi^{0i}.$
(31)

From the above equations, it is clear that $\Pi^{0i}_{(B)}\approx 0$ and
$\Pi_{(\phi)}^{0}\approx 0$ are the primary constraints (PCs) on our theory
where we have used the Dirac notation for the idea of weakly zero. Thus, we
are allowed to take a first-order time derivative on the above PCs to obtain
the secondary constraints (SCs) on our theory. Towards this goal in mind, we
note that the following EL-EoMs, w.r.t. the basic fields $B_{\mu\nu}$ and
$\phi_{\mu}$ from ${\cal L}_{(S)}^{(B)}$, are:

$\displaystyle\partial_{\mu}H^{\mu\nu\lambda}+m^{2}\,B^{\nu\lambda}\mp\,m\,\Phi^{\nu\lambda}=0,$
$\displaystyle\partial_{\mu}[\,\pm
m\,B^{\mu\nu}-\Phi^{\mu\nu}]\equiv\pm\,m\,\partial_{\mu}B^{\mu\nu}-\partial_{\mu}\Phi^{\mu\nu}=0.$
(32)

For our simple system, the Lagrangian formulation is good enough to deal with
the derivations of the SCs. In this context, we note that the very specific
choices: $\nu=0,\;\lambda=j$ in (32) lead to the following explicit forms of
the EoMs, namely;

$\displaystyle\partial_{0}H^{00j}+\partial_{i}H^{i0j}+m^{2}\,B^{0j}\mp\,m\,\Phi^{0j}=0,$
$\displaystyle(\pm m\,\partial_{0}\,B^{00}-\,\partial_{0}\,\Phi^{00})+(\pm
m\,\partial_{i}\,B^{i0}-\,\partial_{i}\,\Phi^{i0})=0.$ (33)

Taking the inputs from (31), it is clear that we have the following time-
evolution invariance [24] of the PCs of our theory, namely;

$\displaystyle\frac{\partial\,\Pi_{(B)}^{0j}}{\partial\,t}$ $\displaystyle=$
$\displaystyle\partial_{i}\,\Pi^{ij}_{(B)}\mp\frac{m}{2}\,\Pi^{j}_{(\phi)}\approx
0,$ $\displaystyle\frac{\partial\,\Pi_{(\phi)}^{0}}{\partial\,t}$
$\displaystyle=$ $\displaystyle\partial_{i}\,\Pi^{i}_{(\phi)}\approx 0.$ (34)

The above equations of motion lead to the derivations of SCs on our theory as:
$\partial_{i}\Pi^{ij}_{(B)}\mp\frac{m}{2}\,\Pi^{j}_{(\phi)}\approx
0,\;\partial_{i}\Pi^{i}_{\phi}\approx 0$. Since all the PCs and SCs are
expressed in terms of the components of the conjugate momenta, it is clear
that they commute among themselves. Hence, these constraints are of the first-
class variety in the terminology of Dirac’s prescription for the
classification scheme of constraints [5-8]. There are no further constraints
on the theory. Using the standard formula (see, e.g. [42, 43] for details) for
the expression of the generator in terms of the first-class constraints, the
generator $G_{(B)}$ for our St$\ddot{u}$ckelberg-modified massive Abelian
2-form theory is

$\displaystyle G_{(B)}$ $\displaystyle=$ $\displaystyle\int
d^{D-1}x\,\Big{[}(\partial_{0}\,\Lambda_{i})\,\Pi^{0i}_{(B)}+(\partial_{i}\,\Lambda_{0})\,\Pi^{i0}_{(B)}-\Lambda_{j}\,\big{(}\partial_{i}\,\Pi^{ij}_{(B)}\mp\frac{m}{2}\,\Pi^{j}_{(\phi)}\big{)}$
(35) $\displaystyle-$
$\displaystyle\Lambda_{i}\,\big{(}\partial_{j}\,\Pi^{ji}_{(B)}\mp\frac{m}{2}\,\Pi^{i}_{(\phi)}\big{)}-\,\big{(}\partial_{0}\,\Sigma\mp
m\,\Lambda_{0}\big{)}\,\Pi^{0}_{(\phi)}+\Sigma\,\partial_{i}\,\Pi^{i}_{(\phi)}\Big{]},$

where $(\Lambda_{0},\Lambda_{i})$ are the components of the Lorentz vector
gauge transformation parameters $\Lambda_{\mu}$ and $\Sigma$ is the Lorentz
scalar transformation parameter which are present in the gauge symmetry
transformation of our theory in (30). The above generator can be re-expressed,
in a more transparent and useful form, as:

$\displaystyle G_{(B)}$ $\displaystyle=$ $\displaystyle\int
d^{D-1}x\,\Big{[}\big{(}\partial_{0}\,\Lambda_{i}-\partial_{i}\,\Lambda_{0}\big{)}\,\Pi^{0i}_{(B)}+\big{(}\partial_{i}\,\Lambda_{j}-\partial_{j}\,\Lambda_{i}\big{)}\,\Pi^{ij}_{(B)}$
(36) $\displaystyle-$ $\displaystyle\big{(}\partial_{0}\,\Sigma\mp
m\,\Lambda_{0}\big{)}\,\Pi^{0}_{(\phi)}-\big{(}\partial_{i}\,\Sigma\mp
m\,\Lambda_{i}\big{)}\,\Pi^{i}_{(\phi)}\Big{]}.$

The above form of the generator leads to the derivation of the gauge
transformations $(\delta_{g})$ for the basic fields (i.e. the antisymmetric
$B_{\mu\nu}$ field and vector field $\phi_{\mu}$) of our theory as

$\displaystyle\delta_{g}\,\phi_{0}\,(\vec{x},t)=-\,i\,[\phi_{0}\,(\vec{x},t),\,G_{(B)}]=-\,\partial_{0}\Sigma\pm
m\,\Lambda_{0},$
$\displaystyle\delta_{g}\,\phi_{i}\,(\vec{x},t)=-\,i\,[\phi_{i}\,(\vec{x},t),\,G_{(B)}]=-\,\partial_{i}\,\Sigma\pm
m\,\Lambda_{i},$
$\displaystyle\delta_{g}\,B_{0i}\,(\vec{x},t)=-\,i\,[B_{0i}\,(\vec{x},t),\,G_{(B)}]=\partial_{0}\,\Lambda_{i}-\partial_{i}\,\Lambda_{0},$
$\displaystyle\delta_{g}\,B_{ij}\,(\vec{x},t)=-\,i\,[B_{ij}\,(\vec{x},t),\,G_{(B)}]=\partial_{i}\,\Lambda_{j}-\partial_{j}\,\Lambda_{i},$
(37)

provided we take into account the following non-trivial equal-time canonical
commutators

$\displaystyle[\phi_{0}\,(\vec{x},t),\,\Pi^{0}_{(\phi)}\,(\vec{y},t)]\;=\;i\,\delta^{(D-1)}\,(\vec{x}-\vec{y}),$
$\displaystyle[\phi_{i}\,(\vec{x},t),\,\Pi^{j}_{(\phi)}\,(\vec{y},t)]\;=\;i\,\delta_{i}^{j}\,\delta^{(D-1)}\,(\vec{x}-\vec{y}),$
$\displaystyle[B_{0i}(\vec{x},t),\,\Pi^{0j}_{(B)}\,(\vec{y},t)]\;=\;i\,\delta_{i}^{j}\,\delta^{(D-1)}\,(\vec{x}-\vec{y}),$
$\displaystyle[B_{ij}(\vec{x},t),\,\Pi^{kl}_{(B)}\,(\vec{y},t)]\;=\;\frac{i}{2!}\,(\delta_{i}^{k}\,\delta_{j}^{l}-\delta_{i}^{l}\,\delta_{j}^{k})\,\delta^{(D-1)}\,(\vec{x}-\vec{y}),$
(38)

and the rest of the brackets for our theory are trivially zero. We have taken
the natural units ($\hbar=c=1$) in the above definition of the brackets.
Ultimately, it is clear that we have derived the local gauge symmetry
transformations in their covariant form as given in (30). Under the above
local and infinitesimal gauge symmetry transformations, it is straightforward
to check that we have the gauge invariance (i.e. $\delta_{g}\,{\cal
L}_{(S)}^{(B)}=0$).

We wrap-up this subsection with the following remarks. First, we have taken
the notations: $\Pi_{(\phi)}$ and $\Pi_{(\phi)}^{i}$ for the conjugate momenta
w.r.t. the St$\ddot{u}$ckelberg-compensating fields $\phi$ and $\phi_{\mu}$ in
the context of modified Proca theory and massive Abelian 2-form theory,
respectively, where the subscripts are the same. However, the contravariant
indices on these conjugate momenta take care of the relevance of these
definitions in their appropriate contexts. Second, it is obvious that the
generator $G_{(B)}^{(m=0)}$ for the massless Abelian 2-form gauge theory can
be obtained from (36) by taking into account the conditions: $m=0$ and
$\phi_{\mu}=0$. In other words, for the massless Abelian 2-form theory, we
have the following

$\displaystyle G_{(B)}^{(m=0)}$ $\displaystyle=$ $\displaystyle\int
d^{D-1}x\,\Big{[}(\partial_{0}\,\Lambda_{i})\,\Pi^{0i}_{(B)}+(\partial_{i}\,\Lambda_{0})\,\Pi^{i0}_{(B)}-\Lambda_{i}\,(\partial_{j}\,\Pi^{ji}_{(B)})-\Lambda_{j}\,(\partial_{i}\,\Pi^{ij}_{(B)})\Big{]},$
(39) $\displaystyle\equiv$ $\displaystyle\int
d^{D-1}x\,\Big{[}\big{(}\partial_{0}\,\Lambda_{i}-\partial_{i}\,\Lambda_{0}\big{)}\,\Pi^{0i}_{(B)}+\big{(}\partial_{i}\,\Lambda_{j}-\partial_{j}\,\Lambda_{i}\big{)}\,\Pi^{ij}_{(B)}\Big{]},$

which is derived from $G_{(B)}$ [cf. Eqs. (35), (36)] with the substitutions:
$m=0,\;\Pi^{i}_{(\phi)}=\Pi^{0}_{(\phi)}=0$ (due to $\phi_{\mu}=0$). Finally,
using the equal-time commutators appropriately, it is straightforward to note
that we obtain the gauge transformation:
$\delta_{g}\,B_{\mu\nu}=(\partial_{\mu}\,\Lambda_{\nu}-\partial_{\nu}\,\Lambda_{\mu})$
in its covariant form from (39) when we exploit and use the theoretical
strength of the appropriate commutators from (37) with the replacement:
$G_{(B)}\longrightarrow G_{(B)}^{(m=0)}$. Under this transformation (i) the
limiting case (i.e. $m=0)$ of the Lagrangian density (27) and/or (ii) the
limiting case (i.e. $m=0,\;\phi_{\mu}=0)$ of the Lagrangian density (29),
remains invariant because we observe that the field-strength tensor is gauge
invariant (i.e. $\delta_{g}\,H_{\mu\nu\lambda}=0$).

##  4 Noether Conserved Charges and Constraints: Modified Massive and Massless
Abelian 2-Form Theories

According to the basic tenets of the Noether theorem, the existence of the
infinitesimal, continuous and local gauge symmetry transformations (30) leads
to the following Noether’s current for the modified massive Abelian 2-form
theory [described by (29)], namely;

$\displaystyle J^{\mu}_{(B)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,H^{\mu\nu\lambda}\,(\partial_{\nu}\,\Lambda_{\lambda}-\partial_{\lambda}\,\Lambda_{\nu})+(\pm
m\,B^{\mu\nu}-\Phi^{\mu\nu})\ (\pm\,m\,\Lambda_{\nu}-\partial_{\nu}\,\Sigma)$
$\displaystyle\equiv$ $\displaystyle
H^{\mu\nu\lambda}\,(\partial_{\nu}\,\Lambda_{\lambda})+(\pm
m\,B^{\mu\nu}-\Phi^{\mu\nu})\ (\pm\,m\,\Lambda_{\nu}-\partial_{\nu}\,\Sigma),$
(40)

where the totally antisymmetric property of $H_{\mu\nu\lambda}$ has been used.
The conservation law ($\partial_{\mu}\,J^{\mu}_{(B)}=0$) can be proven in a
straightforward fashion by using the EL-EoMs w.r.t the basic fields
$B_{\mu\nu}$ and $\phi_{\mu}$ of our theory [cf. Eq. (32)]. In addition, we
have to keep in mind that the sum of the antisymmetric and symmetric indices
always turns out to be zero. The conserved Noether charge [$Q_{(B)}=\int
d^{D-1}x\,J^{0}_{(B)}$], derived from the above Noether conserved current
[$J^{\mu}_{(B)}$], is as follows:

$\displaystyle Q_{(B)}\equiv\int
d^{D-1}x\,\Big{[}\frac{1}{2}\,H^{0\nu\lambda}\,(\partial_{\nu}\,\Lambda_{\lambda}-\partial_{\lambda}\,\Lambda_{\nu})+(\pm
m\,B^{0\nu}-\Phi^{0\nu})\
(\pm\,m\,\Lambda_{\nu}-\partial_{\nu}\,\Sigma)\Big{]}.$ (41)

Since our theory is endowed with constraints [cf. Eqs. (31, (34)] we have to
be careful in expanding the r.h.s. of (41) so that the presence of
constraints, at appropriate places, is taken care of. In other words, we have
the following explicit expression for the Noether conserved charge for our
modified massive Abelian 2-form theory, namely;

$\displaystyle Q_{(B)}$ $\displaystyle=$ $\displaystyle\int
d^{D-1}x\,\Big{[}(\partial_{0}\Lambda_{i}-\partial_{i}\Lambda_{0})\,\Pi^{0i}_{(B)}+(\partial_{i}\Lambda_{j}-\partial_{j}\Lambda_{i})\,\Pi^{ij}_{(B)}$
(42) $\displaystyle+$
$\displaystyle(\,\pm\,m\,\Lambda_{0}-\partial_{0}\Sigma)\,\Pi^{0}_{(\phi)}+(\,\pm\,m\,\Lambda_{i}-\partial_{i}\Sigma)\,\Pi^{i}_{(\phi)}\Big{]}.$

Using (i) the theoretical strength of the Gauss divergence theorem, and (ii) a
simple algebraic trick: $\pm
m\,\Lambda_{i}\,\Pi^{i}_{(\phi)}\equiv\pm\,({m}/{2})\,[\Lambda_{i}\,\Pi^{i}_{(\phi)}+\Lambda_{j}\,\Pi^{j}_{(\phi)}],$
we can re-express the above conserved Noether charge [i.e. $Q_{(B)}$] as
follows

$\displaystyle Q_{(B)}$ $\displaystyle=$ $\displaystyle\int
d^{D-1}x\,\Big{[}(\partial_{0}\Lambda_{i}-\partial_{i}\Lambda_{0})\,\Pi^{0i}_{(B)}-\Lambda_{i}\,\big{(}\partial_{j}\Pi^{ji}_{(B)}\mp\frac{m}{2}\,\Pi^{i}_{(\phi)}\big{)}$
(43) $\displaystyle-$
$\displaystyle\Lambda_{j}\,\big{(}\partial_{i}\Pi^{ij}_{(B)}\mp\frac{m}{2}\,\Pi^{j}_{(\phi)}\big{)}+(\,\pm\,m\,\Lambda_{0}-\partial_{0}\Sigma)\,\Pi^{0}_{(\phi)}-\Sigma\,(\partial_{i}\Pi^{i}_{(\phi)})\Big{]},$

where the primary constraints $(\Pi^{0i}_{(B)}\approx
0,\,\Pi^{0}_{(\phi)}\approx 0$) and the secondary constraints of our theory
$(\partial_{i}\Pi^{ij}_{(B)}\mp({m}/{2})\,\Pi^{j}_{(\phi)}\approx
0,\,\partial_{i}\Pi^{i}_{(\phi)}\approx 0)$ appear in an explicit form. A
close look at (43) demonstrates that, in reality, we have obtained the
generators $G_{(B)}$ in terms of the constraints [cf. Eq. (35)] that has been
derived in our previous section (cf. Sec. 3).

We end this section with the following interesting remarks. First, we have
established a deep relationship between (i) the Noether conserved charge
(derived from the infinitesimal, local and continuous gauge symmetry
transformations), and (ii) the standard formula for the generator of the
infinitesimal gauge symmetry transformations (expressed in terms of the first-
class constraints of the original St$\ddot{u}$ckelberg-modified massive
classical gauge theory). Second, it is straightforward to note that in the
limit: $m=0,\,\phi_{\mu}=0,$ we obtain the Noether conserved charge
[$Q_{(B)}^{(m=0)}$] from the charges (42) and/or (43) as follows

$\displaystyle Q_{(B)}^{(m=0)}$ $\displaystyle=$ $\displaystyle\int
d^{D-1}x\,\Big{[}(\partial_{0}\Lambda_{i}-\partial_{i}\Lambda_{0})\,\Pi^{0i}_{(B)}-(\partial_{i}\Pi^{ij}_{(B)})\,\Lambda_{j}-(\partial_{j}\Pi^{ji}_{(B)})\,\Lambda_{i})\Big{]}$
(44) $\displaystyle\equiv$ $\displaystyle\int
d^{D-1}x\,\Big{[}(\partial_{0}\Lambda_{i}-\partial_{i}\Lambda_{0})\,\Pi^{0i}_{(B)}+(\partial_{i}\Lambda_{j}-\partial_{j}\Lambda_{i})\,\Pi^{ij}_{(B)}\Big{]},$

where the first-class constraints $(\Pi^{0i}_{(B)}\approx
0,\,\partial_{i}\,\Pi^{ij}_{(B)}\approx 0$) of the massless Abelian 2-form
theory appear explicitly. Finally, due to the presence of constraints on our
modified massive and massless Abelian 2-form gauge theories, one has to be
careful in the derivations of the Noether conserved charges where the
constraints can not to be set strongly equal to zero.

##  5 First-Class Constraints and Their Appearance at the Quantum Level: BRST
Approach

This section is divided into two parts. In subsection 5.1, we consider the
massless Abelian 2-form theory within the framework of BRST formalism and show
that the physicality criteria with the conserved and off-shell nilpotent
(anti-)BRST charges lead to the annihilation of the physical states (present
in the total quantum Hilbert space of states) by the operator forms of the
first-class constraints of the original classical Abelian 2-form gauge theory
(with zero rest mass). This exercise is repeated for the St$\ddot{u}$ckelberg-
modified massive Abelian 2-form theory in our subsection 5.2 to establish the
existence of the first-class constraints, within the purview of BRST
formalism, at the quantum level.

###  5.1 Massless Abelian 2-Form Theory: (Anti-)BRST Charges

We begin with the following off-shell nilpotent [$s_{(a)b}^{2}=0$] (anti-)BRST
symmetries $s_{(a)b}$ [which are the generalizations of the classical
infinitesimal and local gauge symmetry transformations‡‡‡We have taken
purposely a minus sign in the gauge transformation because it is consistent
with the (anti-)BRST symmetry transformations:
$s_{ab}\,B_{\mu\nu}=-\,(\partial_{\mu}\,\bar{C}_{\nu}-\partial_{\nu}\,\bar{C}_{\mu})$
and $s_{b}\,B_{\mu\nu}=-\,(\partial_{\mu}\,C_{\nu}-\partial_{\nu}\,C_{\mu})$
which have been precisely derived from the superfield approach to BRST
formalism in the context of Abelian 2-form gauge theory where the
horizontality condition has played a decisive role (see, e.g. [29]).:
$\delta_{g}\,B_{\mu\nu}=-\,(\partial_{\mu}\,\Lambda_{\nu}-\partial_{\nu}\,\Lambda_{\mu})$
under which the classical Lagrangian density ${\cal
L}_{0}=({1}/{12})\,(H^{\mu\nu\lambda}\,H_{\mu\nu\lambda})$, for our massless
theory, remains invariant], namely;

$\displaystyle
s_{ab}B_{\mu\nu}=-(\partial_{\mu}{\bar{C}}_{\nu}-\partial_{\nu}{\bar{C}}_{\mu}),\quad
s_{ab}{\bar{C}}_{\mu}=-\partial_{\mu}{\bar{\beta}},\qquad
s_{ab}C_{\mu}={\bar{B}}_{\mu},$ $\displaystyle s_{ab}\phi=\rho,\quad
s_{ab}\beta=-\lambda,\;\;s_{ab}B_{\mu}=\partial_{\mu}\rho,\quad
s_{ab}\bigl{[}\rho,\lambda,{\bar{\beta}},{\bar{B}}_{\mu},H_{\mu\nu\kappa}\bigr{]}=0,$
$\displaystyle
s_{b}B_{\mu\nu}=-(\partial_{\mu}C_{\nu}-\partial_{\nu}C_{\mu}),\qquad
s_{b}C_{\mu}=-\partial_{\mu}\beta,\qquad s_{b}\bar{C}_{\mu}=-B_{\mu},$
$\displaystyle s_{b}\phi=\lambda,\quad s_{b}\bar{\beta}=-\rho,\quad
s_{b}\bar{B}_{\mu}=-\,\partial_{\mu}\lambda,\quad
s_{b}\bigl{[}\rho,\lambda,\beta,B_{\mu},H_{\mu\nu\kappa}\bigr{]}=0.$ (45)

The above (anti-)BRST transformations are the symmetries of the following
coupled (but equivalent) Lagrangian densities (see, e.g. [40] for details)

$\displaystyle{\cal L}_{B}$ $\displaystyle=$
$\displaystyle\frac{1}{12}\,H^{\mu\nu\kappa}\,H_{\mu\nu\kappa}+B^{\mu}\,\Bigl{(}\partial^{\nu}B_{\nu\mu}-\partial_{\mu}\phi\Bigr{)}+B\cdot
B+\partial_{\mu}{\bar{\beta}}\,\partial^{\mu}\beta$ (46) $\displaystyle+$
$\displaystyle(\partial_{\mu}{\bar{C}}_{\nu}-\partial_{\nu}{\bar{C}}_{\mu})\,(\partial^{\mu}C^{\nu})+(\partial\cdot
C-\lambda)\,\rho+(\partial\cdot{\bar{C}}+\rho)\,\lambda,$ $\displaystyle{\cal
L}_{\bar{B}}$ $\displaystyle=$
$\displaystyle\frac{1}{12}\,H^{\mu\nu\kappa}\,H_{\mu\nu\kappa}+{\bar{B}}^{\mu}\,\Bigl{(}\partial^{\nu}\,B_{\nu\mu}+\partial_{\mu}\phi\Bigr{)}+{\bar{B}}\cdot{\bar{B}}+\partial_{\mu}\,{\bar{\beta}}\,\partial^{\mu}\beta$
(47) $\displaystyle+$
$\displaystyle(\partial_{\mu}{\bar{C}}_{\nu}-\partial_{\nu}{\bar{C}}_{\mu})\,(\partial^{\mu}C^{\nu})+(\partial\cdot
C-\lambda)\,\rho+(\partial\cdot{\bar{C}}+\rho)\,\lambda,$

where the Nakanishi-Lautrup auxiliary fields ($B_{\mu},\,\bar{B}_{\mu}$) are
related to each-other by the CF-type restriction:
$B_{\mu}-\bar{B}_{\mu}-\partial_{\mu}\phi=0$. Here $\phi$ is a pure scalar
field that is present in the (anti-)BRST invariant Lagrangian densities [cf.
Eqs. (46), (47)] of our theory. The fermionic (anti-)ghost fields
$(\bar{C}_{\mu})C_{\mu}$ are the generalization of the Lorentz vector gauge
transformation parameter ($\Lambda_{\mu}$) of the classical gauge symmetry
transformation:
$\delta_{g}\,B_{\mu\nu}=-\,(\partial_{\mu}\,\Lambda_{\nu}-\partial_{\nu}\,\Lambda_{\mu})$
and they carry the ghost numbers $(-1)+1$, respectively. The bosonic
(anti-)ghost fields $(\bar{\beta})\beta$ are endowed with the ghost numbers
$(-2)+2$, respectively, and the fermionic
($\rho^{2}=\lambda^{2}=0,\,\rho\,\lambda+\lambda\,\rho=0$) auxiliary
(anti-)ghost fields $(\rho)\,\lambda$ carry the ghost numbers $(-1)+1$,
respectively, because $\rho=-\,({1}/{2})\,(\partial\cdot\bar{C})$ and
$\lambda=({1}/{2})\,(\partial\cdot C)$. The conserved (anti-)BRST Noether
currents, with the superscripts $(ab)$ and $(b)$, namely;

$\displaystyle
J^{\mu}_{(ab)}=\rho\,{\bar{B}}^{\mu}-(\partial^{\mu}{\bar{C}}^{\nu}-\partial^{\nu}{\bar{C}}^{\mu})\,{\bar{B}}_{\nu}-(\partial^{\mu}C^{\nu}-\partial^{\nu}C^{\mu})\,\partial_{\nu}{\bar{\beta}}-\lambda\;\partial^{\mu}{\bar{\beta}}-H^{\mu\nu\kappa}\;(\partial_{\nu}\bar{C}_{\kappa}),$
$\displaystyle
J^{\mu}_{(b)}=(\partial^{\mu}{\bar{C}}^{\nu}-\partial^{\nu}{\bar{C}}^{\mu})\;\partial_{\nu}\beta-(\partial^{\mu}C^{\nu}-\partial^{\nu}C^{\mu})\,B_{\nu}-\lambda\;B^{\mu}-\rho\;\partial^{\mu}\beta-H^{\mu\nu\kappa}\;(\partial_{\nu}C_{\kappa}),$
(48)

lead to the definitions of the conserved (anti-)BRST charges:

$\displaystyle
Q_{ab}={\displaystyle\int}d^{D-1}x\,\Bigr{[}\rho\,{\bar{B}}^{0}-(\partial^{0}{\bar{C}}^{i}-\partial^{i}{\bar{C}}^{0})\,{\bar{B}}_{i}-(\partial^{0}C^{i}-\partial^{i}C^{0})\,\partial_{i}{\bar{\beta}}-\lambda\;\partial^{0}{\bar{\beta}}-H^{0ij}\;(\partial_{i}\bar{C}_{j})\Bigl{]},$
$\displaystyle
Q_{b}={\displaystyle\int}d^{D-1}x\,\Bigr{[}(\partial^{0}{\bar{C}}^{i}-\partial^{i}{\bar{C}}^{0})\,\partial_{i}\beta-(\partial^{0}C^{i}-\partial^{i}C^{0})B_{i}-\lambda\;B^{0}-\rho\;\partial^{0}\beta-H^{0ij}\;(\partial_{i}C_{j})\Bigl{]}.$
(49)

It turns out, however, that the above charges are not nilpotent
[$Q_{(a)b}^{2}\neq 0$] of order two. We would like to point out that (unlike
the cases of the modified massive and massless Abelian 1-form theories where
the CF-type restriction is trivial) for our present massless Abelian 2-form
theory, there exits a non-trivial (anti-)BRST invariant (i.e.
$s_{(a)b}\,\\{B_{\mu}-\bar{B}_{\mu}-\partial_{\mu}\phi\\}=0$) CF-type
restriction where the Nakanishi-Lautrup auxiliary fields
$(B_{\mu},\,\bar{B}_{\mu})$ and the scalar field $(\phi)$ are restricted to
obey: $B_{\mu}-\bar{B}_{\mu}-\partial_{\mu}\phi=0$. This is primarily the
reason behind the observation that the standard conserved Noether (anti-)BRST
charges $Q_{(a)b}$ are non-nilpotent (i.e. $Q_{(a)b}^{2}\neq 0$). This is not
true for the modified massive and massless Abelian 1-form theories within the
framework of BRST formalism (cf. Subsec. 2.3).

We have developed [40] a systematic theoretical procedure to obtain the off-
shell nilpotent (anti-)BRST charges from the non-nilpotent standard Noether
(anti-)BRST charges where we have exploited (i) the appropriate EL-EoMs that
emerge out from the specific Lagrangian density, and (ii) the Gauss divergence
theorem. The off-shell nilpotent versions of the (anti-)BRST charges
[$Q_{(a)b}^{(1)}$] for our massless Abelian 2-form gauge theory are [40]

$\displaystyle Q_{ab}\longrightarrow Q_{ab}^{(1)}$ $\displaystyle=$
$\displaystyle{\displaystyle\int}\;d^{D-1}x\Bigr{[}(\partial^{0}\bar{B}^{i}-\partial^{i}\bar{B}^{0})\,\bar{C}_{i}+(\partial^{0}{C}^{i}-\partial^{i}{C}^{0})\,\partial_{i}\bar{\beta}$
$\displaystyle-$
$\displaystyle(\partial^{0}\bar{C}^{i}-\partial^{i}\bar{C}^{0})\,\bar{B}_{i}+\rho\,\bar{B}^{0}-2\,\bar{\beta}\,\dot{\lambda}-\lambda\,\dot{\bar{\beta}}\Bigl{]},$
$\displaystyle Q_{b}\longrightarrow Q_{b}^{(1)}$ $\displaystyle=$
$\displaystyle{\displaystyle\int}\;d^{D-1}x\;\Bigr{[}(\partial^{0}B^{i}-\partial^{i}B^{0})\,C_{i}-(\partial^{0}{\bar{C}}^{i}-\partial^{i}{\bar{C}}^{0})\;\partial_{i}\beta$
(50) $\displaystyle+$ $\displaystyle
2\,\beta\,\dot{\rho}-(\partial^{0}C^{i}-\partial^{i}C^{0})B_{i}-\lambda
B^{0}-\rho\,\dot{\beta}\Bigl{]},$

where it can be explicitly checked that:

$\displaystyle s_{b}\,Q_{b}^{(1)}$ $\displaystyle=$
$\displaystyle-\,i\,\\{Q_{b}^{(1)},\;Q_{b}^{(1)}\\}=0\quad\Longrightarrow\quad[Q_{b}^{(1)}]^{2}=0,$
$\displaystyle s_{ab}\,Q_{ab}^{(1)}$ $\displaystyle=$
$\displaystyle-\,i\,\\{Q_{ab}^{(1)},\;Q_{ab}^{(1)}\\}=0\quad\Longrightarrow\quad[Q_{ab}^{(1)}]^{2}=0.$
(51)

We are now in the position to impose the physicality criteria:
$Q_{(a)b}^{(1)}\,|phys>\,=0$ on the physical states that are present in the
total Hilbert space of states. Since the (anti-)ghost fields are not the
physical fields, we note that the above criteria lead to the following:

$\displaystyle Q_{b}^{(1)}\,|\,phys>\,=0\quad\Longrightarrow\quad
B_{i}\,|\,phys>\,=0,\;\;\;\quad(\partial_{0}B_{i}-\partial_{i}B_{0})\,|\,phys>\,=0,$
$\displaystyle
Q_{ab}^{(1)}\,|\,phys>\,=0\quad\Longrightarrow\quad\bar{B}_{i}\,|\,phys>\,=0,\;\;\;\quad(\partial_{0}\bar{B}_{i}-\partial_{i}\bar{B}_{0})\,|\,phys>\,=0.$
(52)

To explicitly see the appearance of the first-class constraints, we focus on
the Lagrangian density ${\cal L}_{B}$ and ${\cal L}_{\bar{B}}$ and define the
conjugate momenta w.r.t. the gauge field $(B_{\mu\nu})$:

$\displaystyle\Pi^{\mu\nu}_{(B)}=\frac{\partial{\cal
L}_{B}}{\partial(\partial_{0}B_{\mu\nu})}=\frac{1}{2}\,H^{0\mu\nu}\,+\frac{1}{2}\,(\eta^{0\mu}\,B^{\nu}-\eta^{0\nu}\,B^{\mu}),$
$\displaystyle\Pi^{\mu\nu}_{(\bar{B})}=\frac{\partial{\cal
L}_{\bar{B}}}{\partial(\partial_{0}B_{\mu\nu})}=\frac{1}{2}\,H^{0\mu\nu}\,+\frac{1}{2}\,(\eta^{0\mu}\,\bar{B}^{\nu}-\eta^{0\nu}\,\bar{B}^{\mu}).$
(53)

The above conjugate momenta have the following components:

$\displaystyle\Pi^{0i}_{(B)}=\frac{1}{2}\,B^{i}\equiv-\,\frac{1}{2}\,B_{i},\qquad\qquad\qquad\Pi^{ij}_{(B)}=\frac{1}{2}\,H^{0ij},$
$\displaystyle\Pi^{0i}_{(\bar{B})}=\frac{1}{2}\,\bar{B}^{i}\equiv-\,\frac{1}{2}\,\bar{B}_{i},\qquad\qquad\qquad\Pi^{ij}_{(\bar{B})}=\frac{1}{2}\,H^{0ij}.$
(54)

Thus, we note that the primary constraints $\Pi_{(B)}^{0i}\approx 0$ and/or
$\Pi_{(\bar{B})}^{0i}\approx 0$ are connected with the Nakanishi-Lautrup
auxiliary fields $(B_{i}$ and/or $\bar{B}_{i})$ as far as the Lagrangian
densities ${\cal L}_{B}$ and ${\cal L}_{\bar{B}}$ are concerned. Furthermore,
from the following EL-EoMs that are derived from ${\cal L}_{B}$ and ${\cal
L}_{\bar{B}}$, respectively, namely;

$\displaystyle\partial_{\mu}H^{\mu\nu\lambda}+(\partial^{\nu}B^{\lambda}-\partial^{\lambda}B^{\nu})=0,\quad\qquad\partial_{\mu}H^{\mu\nu\lambda}+(\partial^{\nu}\bar{B}^{\lambda}-\partial^{\lambda}\bar{B}^{\nu})=0,$
(55)

we find that, for the choices: $\nu=0,\,\lambda=j$, we have the following:

$\displaystyle\partial_{i}H^{0ij}=(\partial^{0}B^{j}-\partial^{j}B^{0}),\qquad\qquad\partial_{i}H^{0ij}=(\partial^{0}\bar{B}^{j}-\partial^{j}\bar{B}^{0}).$
(56)

Using the definitions of the components of the conjugate momenta from (54), we
note that we have obtained the following relationship between the first-class
constraints and the specific combination (e.g.
$\partial_{0}B_{i}-\partial_{0}B_{0},\;\partial_{0}\bar{B}_{i}-\partial_{0}\bar{B}_{0}$)
of the derivatives on the Nakanishi-Lautrup axillary fields
$(B_{i},\bar{B}_{i})$ of the Lagrangian densities ${\cal L}_{B}$ and ${\cal
L}_{\bar{B}}$:

$\displaystyle
2\,\partial_{i}\Pi^{ij}_{(B)}=-\,(\partial_{0}B_{j}-\partial_{j}B_{0}),\quad\qquad
2\,\partial_{i}\Pi^{ij}_{(\bar{B})}=-\,(\partial_{0}\bar{B}_{j}-\partial_{j}\bar{B}_{0}).$
(57)

A close look at the equations (54) and (57) demonstrate that we have obtained
the same kinds of quantization conditions (i.e. restrictions on the physical
states) from the conserved and nilpotent versions of the (anti-)BRST charges
(with $B_{i}\longleftrightarrow\bar{B}_{i}$). Ultimately, we claim, however,
that we have obtained the conditions where the operator forms of the first-
class constraints annihilate the physical states of our theory. In other
words, mathematically, we have obtained the following conditions on the
physical states (from both the charges):

$\displaystyle B_{i}\,|phys>\,=0\qquad$ $\displaystyle\Longrightarrow$
$\displaystyle\qquad\Pi^{0i}_{(B)}\,|phys>\,=0,$
$\displaystyle(\partial_{0}B_{i}-\partial_{i}B_{0})\,|phys>\,=0\qquad$
$\displaystyle\Longrightarrow$
$\displaystyle\qquad\partial_{i}\Pi^{ij}_{(B)}\,|phys>\,=0.$ (58)

We conclude that, within the framework of BRST formalism, the operator forms
of the first-class constraints (i.e. $\Pi^{0i}_{(B)}\approx 0$ and
$\partial_{i}\Pi^{ij}_{(B)}\approx 0)$ of the massless Abelian 2-form theory
annihilate the physical states of the quantum theory which is consistent with
the requirements of the Dirac-quantization conditions (for the physical
systems that are endowed with the constraints of any kind/variety [5-8] in
Dirac’s terminology).

We end this subsection with a few remarks. First, the Noether theorem does not
lead to the off-shell nilpotent (anti-)BRST charges whenever the non-trivial
CF-type restriction(s) exist on a BRST-quantized theory (see, e.g. [40]).
Second, to obtain the conserved and off-shell nilpotent (anti-)BRST charges
from the standard Noether non-nilpotent (anti-)BRST charges, one has to
exploit $(i)$ the appropriate EL-EoMs from the appropriate Lagrangian density,
and $(ii)$ the beauty and strength of the Gauss divergence theorem. Finally,
it is the physicality criteria with the conserved and off-shell nilpotent
versions of the (anti-)BRST charges that trace the trajectory of the elevation
of the first-class constraints from the classical level to the quantum level
within the framework of of BRST formalism.

###  5.2 Modified Massive Abelian 2-Form Theory: BRST Formalism

We begin with the following (anti-)BRST invariant coupled (but equivalent)
Lagrangian densities (${\cal L}_{b}$ and ${\cal L}_{\bar{b}}$) for the
St$\ddot{u}$ckelberg-modified massive Abelian 2-form theory§§§We have taken a
specific sign with a specific term of the Lagrangian densities because these
signs satisfy all the requirements that are essential for an (anti-)BRST
invariant theory. The uniqueness of these Lagrangian densities has been proven
in our very recent work (see, e.g. [50] for details). in any arbitrary
D-dimension of spacetime (see, e.g. [40] for details)

$\displaystyle{\cal L}_{b}$ $\displaystyle=$
$\displaystyle\frac{1}{12}\,H_{\mu\nu\eta}H^{\mu\nu\eta}-\frac{1}{4}\,m^{2}\,B_{\mu\nu}B^{\mu\nu}-\frac{1}{4}\,\Phi_{\mu\nu}\Phi^{\mu\nu}+\frac{1}{2}\,m\,B_{\mu\nu}\Phi^{\mu\nu}-B^{2}$
(59) $\displaystyle-$ $\displaystyle
B\left(\partial\cdot\phi+m\,\varphi\right)+B^{\mu}B_{\mu}+B^{\mu}\left(\partial^{\nu}B_{\nu\mu}-\partial_{\mu}\varphi+m\,\phi_{\mu}\right)-m^{2}\,\bar{\beta}\beta$
$\displaystyle+$
$\displaystyle\left(\partial_{\mu}\bar{C}_{\nu}-\partial_{\nu}\bar{C}_{\mu}\right)\left(\partial^{\mu}C^{\nu}\right)-\left(\partial_{\mu}\bar{C}-m\,\bar{C}_{\mu}\right)\left(\partial^{\mu}C-m\,C^{\mu}\right)+\partial_{\mu}\bar{\beta}\,\partial^{\mu}\beta$
$\displaystyle+$
$\displaystyle\left(\partial\cdot\bar{C}+\rho+m\,\bar{C}\right)\lambda+\left(\partial\cdot
C-\lambda+m\,C\right)\rho,$ $\displaystyle{\cal L}_{\bar{b}}$ $\displaystyle=$
$\displaystyle\frac{1}{12}\,H_{\mu\nu\eta}H^{\mu\nu\eta}-\frac{1}{4}\,m^{2}\,B_{\mu\nu}B^{\mu\nu}-\frac{1}{4}\,\Phi_{\mu\nu}\Phi^{\mu\nu}+\frac{1}{2}\,m\,B_{\mu\nu}\Phi^{\mu\nu}-{\bar{B}}^{2}$
(60) $\displaystyle+$
$\displaystyle\bar{B}\left(\partial\cdot\phi-m\,\varphi\right)+\bar{B}_{\mu}\bar{B}^{\mu}+\bar{B}^{\mu}\left(\partial^{\nu}B_{\nu\mu}+\partial_{\mu}\varphi+m\,\phi_{\mu}\right)-m^{2}\,\bar{\beta}\beta$
$\displaystyle+$
$\displaystyle\left(\partial_{\mu}\bar{C}_{\nu}-\partial_{\nu}\bar{C}_{\mu}\right)(\partial^{\mu}C^{\nu})-\left(\partial_{\mu}\bar{C}-m\,\bar{C}_{\mu}\right)\left(\partial^{\mu}C-m\,C^{\mu}\right)+\partial_{\mu}\bar{\beta}\,\partial^{\mu}\beta$
$\displaystyle+$
$\displaystyle\left(\partial\cdot\bar{C}+\rho+m\,\bar{C}\right)\lambda+\left(\partial\cdot
C-\lambda+m\,C\right)\rho,$

where the fermionic (anti-)ghost fields $(\bar{C}_{\mu})C_{\mu}$ and
$(\bar{C})C$ carry the ghost numbers $(-\,1)+1$, respectively, and the bosonic
(anti-)ghost fields $(\bar{\beta})\beta$ are endowed with the ghost numbers
$(-\,2)+2$, respectively. In addition, there are fermionic auxiliary fields
$(\rho)\lambda$ with the ghost numbers (-1)+1, respectively. All the other
symbols are same as in the subsection 5.1 but for the additional Nakanishi-
Lautrup type auxiliary fields $(\bar{B})B$ which are invoked to linearize the
gauge-fixing terms for the St$\ddot{u}$ckelberg-compensating vector field
$\phi_{\mu}$. We have the rest mass for all the basics fields, with well-
defined gauge-fixing terms, in our modified massive Abelian 2-form theory
which is denoted by the symbol $m$. The above coupled (but equivalent)
Lagrangian densities respect the following (anti-)BRST symmetry
transformations (see, e.g. [40, 50] for details):

$\displaystyle
s_{ab}B_{\mu\nu}=-\,(\partial_{\mu}\bar{C}_{\nu}-\partial_{\nu}\bar{C}_{\mu}),\qquad
s_{ab}\bar{C}_{\mu}=-\,\partial_{\mu}\bar{\beta},\qquad
s_{ab}\phi_{\mu}=\partial_{\mu}\bar{C}-m\,\bar{C}_{\mu},$ $\displaystyle
s_{ab}C_{\mu}=\bar{B}_{\mu},\quad s_{ab}\beta=-\,\lambda,\quad
s_{ab}\bar{C}=-\,m\,\bar{\beta},\quad s_{ab}C=\bar{B},\quad
s_{ab}B=-\,m\,\rho,$ $\displaystyle s_{ab}\varphi=\rho,\qquad
s_{ab}B_{\mu}=\partial_{\mu}\rho,\qquad
s_{ab}[\bar{B},\rho,\lambda,\bar{\beta},\bar{B}_{\mu},H_{\mu\nu\kappa}]=0,$
$\displaystyle
s_{b}B_{\mu\nu}=-\,(\partial_{\mu}C_{\nu}-\partial_{\nu}C_{\mu}),\qquad
s_{b}C_{\mu}=-\,\partial_{\mu}\beta,\qquad
s_{b}\phi_{\mu}=\partial_{\mu}C-\,m\,C_{\mu},$ $\displaystyle
s_{b}\bar{C}_{\mu}=-\,B_{\mu},\quad s_{b}\bar{\beta}=-\,\rho,\quad
s_{b}C=-\,m\,\beta,\quad s_{b}\bar{C}=B,\quad s_{b}\bar{B}=-\,m\,\lambda,$
$\displaystyle s_{b}\varphi=\lambda,\qquad
s_{b}\bar{B}_{\mu}=-\,\partial_{\mu}\lambda,\qquad
s_{b}[B,\rho,\lambda,\beta,B_{\mu},H_{\mu\nu\kappa}]=0.$ (61)

It is interesting to point out that the above symmetry transformations are
off-shell nilpotent of order two (i.e. $s_{(a)b}^{2}=0$) and their absolute
anticommutativity property (i.e. $\\{s_{b},s_{ab}\\}=0$) is also satisfied
except in the two specific cases which are as follows:

$\displaystyle\\{s_{b},\,s_{ab}\\}\,B_{\mu\nu}=\partial_{\mu}(B_{\nu}-\bar{B}_{\nu})-\partial_{\nu}(B_{\mu}-\bar{B}_{\mu}),$
$\displaystyle\\{s_{b},\,s_{ab}\\}\,\Phi_{\mu}=\partial_{\mu}(B+\bar{B})+m\,(B_{\mu}-\bar{B}_{\mu}).$
(62)

However, we have the following (anti-)BRST invariant CF-type restrictions

$\displaystyle B_{\mu}-\bar{B}_{\mu}-\partial_{\mu}\phi=0,\qquad
B+\bar{B}+m\,\phi=0,$ (63)

on our modified massive Abelian 2-form theory within the framework of BRST
formalism. The above restrictions have been derived from the superfield
approach to BRST formalism (see, e.g. [29]). It can be checked that we have:
$\\{s_{b},s_{ab}\\}\,B_{\mu\nu}=0$ and $\\{s_{b},s_{ab}\\}\,\phi_{\mu}=0$ in
equation (62) due to (63). Thus, the (anti-)BRST symmetry transformations,
listed in (61), are off-shell nilpotent and absolutely anticommuting in
nature. Taking the precise forms of the off-shell nilpotent (anti-)BRST
symmetry transformations for the appropriate fields, it is straightforward to
check that we have the following:

$\displaystyle s_{(a)b}\,[B_{\mu}-\bar{B}_{\mu}-\partial_{\mu}\phi]=0,\qquad
s_{(a)b}\,[B+\bar{B}+m\,\phi]=0.$ (64)

Thus, the CF-type restrictions in (63), on our St$\ddot{u}$ckelberg-modified
massive Abelian 2-form theory, are physical in the sense that both of them are
(anti-)BRST invariant.

According to Noether’s theorem, the infinitesimal, continuous, off-shell
nilpotent and absolutely anticommuting (anti-)BRST symmetry transformations
lead to the derivations of the following (anti-)BRST charges (see, e.g. [40]
for details)

$\displaystyle Q_{ab}$ $\displaystyle=$ $\displaystyle\int
d^{D-1}x\Big{[}-H^{0ij}\,\big{(}\partial_{i}\bar{C}_{j})-\big{(}\partial^{0}\bar{C}^{i}-\partial^{i}\bar{C}^{0}\big{)}\,\bar{B}_{i}+\big{(}\partial^{0}\bar{C}-m\,\bar{C}^{0}\big{)}\,\bar{B}$
$\displaystyle-$
$\displaystyle\big{(}\partial_{i}\bar{C}-m\,\bar{C}_{i}\big{)}\,\big{(}\Phi^{0i}-m\,B^{0i}\big{)}+m\,\big{(}\partial^{0}C-m\,C^{0}\big{)}\,\bar{\beta}$
$\displaystyle-$
$\displaystyle\big{(}\partial^{0}C^{i}-\partial^{i}C^{0}\big{)}(\partial_{i}\bar{\beta})-\lambda\,\partial^{0}\bar{\beta}+\rho\,\bar{B}^{0}\Big{]},$
$\displaystyle Q_{b}$ $\displaystyle=$ $\displaystyle\int
d^{D-1}x\Big{[}-H^{0ij}\,\big{(}\partial_{i}C_{j})-\big{(}\partial^{0}C^{i}-\partial^{i}C^{0}\big{)}\,B_{i}-\big{(}\partial^{0}C-m\,C^{0}\big{)}\
B$ (65) $\displaystyle-$
$\displaystyle\big{(}\Phi^{0i}-m\,B^{0i}\big{)}\,\big{(}\partial_{i}C-m\,C_{i}\big{)}-m\,\big{(}\partial^{0}\bar{C}-m\,\bar{C}^{0}\big{)}\,\beta$
$\displaystyle+$
$\displaystyle\big{(}\partial^{0}\bar{C}^{i}-\partial^{i}\bar{C}^{0}\big{)}(\partial_{i}\beta)-\rho\,\partial^{0}\beta-\lambda\,B^{0}\Big{]},$

which are the generators of the (anti-)BRST symmetry transformations. However,
these conserved charges are found to be non-nilpotent [i.e. $Q_{(a)b}^{2}\neq
0$]. A systematic theoretical methodology has been developed in our earlier
work (see, e.g. [40]) where we have exploited the theoretical strength of (i)
the Gauss divergence theorem, and (ii) appropriate EL-EoMs from the suitable
Lagrangian density, to obtain the conserved and nilpotent (anti-)BRST charges
$[Q_{(a)b}^{(2)}]$ from the standard but non-nilpotent versions of the
conserved (anti-)BRST charges $[Q_{(a)b}]$ that have been derived from the
basic principle behind the Noether theorem. The conserved and off-shell
nilpotent versions of the (anti-)BRST charges are [40]:

$\displaystyle Q_{b}$ $\displaystyle\longrightarrow$ $\displaystyle
Q_{b}^{(2)}=\int
d^{D-1}x\,\Big{[}(\partial^{0}B^{i}-\partial^{i}B^{0})\,C_{i}+(\dot{B}+m\,B^{0})\,C+m\,(\dot{\bar{C}}-m\,\bar{C}^{0})\,\beta$
$\displaystyle-$
$\displaystyle\,(\partial^{0}\bar{C}^{i}-\partial^{i}\bar{C}^{0})\,\partial_{i}\beta\,+\,2\,\dot{\rho}\,\beta-\rho\,\dot{\beta}-\lambda\,B^{0}-(\partial^{0}C^{i}-\partial^{i}C^{0})\,B_{i}-(\dot{C}-m\,C^{0})\,B\Big{]},$
$\displaystyle Q_{ab}$ $\displaystyle\longrightarrow$ $\displaystyle
Q_{ab}^{(2)}=\int
d^{D-1}x\,\Big{[}(\partial^{0}\bar{B}^{i}-\partial^{i}\bar{B}^{0})\,\bar{C}_{i}-(\bar{B}^{0}-m\,\bar{B}^{0})\,\bar{C}+(\partial^{0}C^{i}-\partial^{i}C^{0})\,\partial_{i}\bar{\beta}$
(66) $\displaystyle-$ $\displaystyle
m\,(\dot{C}-mC^{0})\,\bar{\beta}+2\,\dot{\lambda}\bar{\beta}-\lambda\,\dot{\bar{\beta}}+\rho\,\bar{B}^{0}+(\dot{\bar{C}}-m\bar{C}^{0})\,\bar{B}-(\partial^{0}\bar{C}^{i}-\partial^{i}\bar{C}^{0})\,\bar{B}_{i}\Big{]},$

where the superscript $(2)$ has been taken to distinguish these charges from
our subsection 5.1 where the conserved and nilpotent charges [cf. Eq. (50)]
have been denoted by the superscript $(1)$. It is an elementary exercise to
check that we have

$\displaystyle s_{b}\,Q_{b}^{(2)}$ $\displaystyle=$
$\displaystyle-\,i\,\\{Q_{b}^{(2)},\;Q_{b}^{(2)}\\}=0\quad\Longrightarrow\quad[Q_{b}^{(2)}]^{2}=0,$
$\displaystyle s_{ab}\,Q_{ab}^{(2)}$ $\displaystyle=$
$\displaystyle-\,i\,\\{Q_{ab}^{(2)},\;Q_{ab}^{(2)}\\}=0\quad\Longrightarrow\quad[Q_{ab}^{(2)}]^{2}=0,$
(67)

where the l.h.s. can be readily computed by using the (anti-)BRST symmetry
transformations (61) and the explicit expressions for the nilpotent
(anti-)BRST charges from (66).

The stage is set now to exploit the beauty and strength of the physicality
criteria with the help of conserved and off-shell nilpotent (anti-)BRST
charges $Q_{(a)b}^{(2)}$, namely;

$\displaystyle Q_{b}^{(2)}\,|phys>\,=0\qquad$ $\displaystyle\Longrightarrow$
$\displaystyle\qquad B_{i}\,|phys>\,=0,$ $\displaystyle\Longrightarrow$
$\displaystyle\qquad B\,|phys>\,=0,$ $\displaystyle\Longrightarrow$
$\displaystyle\qquad(\partial^{0}B^{i}-\partial^{i}B^{0})\,|phys>\,=0,$
$\displaystyle\Longrightarrow$
$\displaystyle\qquad(\dot{B}+m\,B^{0})\,|phys>\,=0.$ (68)

In the above, we have taken into consideration only the BRST charge
$[Q_{b}^{(2)}]$ which is off-shell nilpotent and conserved because we shall
get the same conditions from the anti-BRST charge $[Q_{ab}^{(2)}]$ except that
the Nakanishi-Lautrup auxiliary fields $(B,\,B_{i})$ will be replaced by
$(\bar{B},\,\bar{B}_{i})$. However, these notational changes will not alter
any physical consequences as far as the constraint analysis, at the quantum
level, is concerned. To observe the physical consequences from the conditions
(68), it is essential that we focus on the following conjugate momenta
(derived from the perfectly BRST invariant Lagrangian density ${\cal L}_{b}$),
namely:

$\displaystyle\Pi^{\mu\nu}_{(B)}=\frac{\partial{\cal
L}_{b}}{\partial(\partial_{0}B_{\mu\nu})}=\frac{1}{2}\,H^{0\mu\nu}\,+\frac{1}{2}\,(\eta^{0\mu}\,B^{\nu}-\eta^{0\nu}\,B^{\mu}),$
$\displaystyle\Pi^{\mu}_{(\phi)}=\frac{\partial{\cal
L}_{b}}{\partial(\partial_{0}\phi_{\mu})}=+m\,B^{0\mu}-\Phi^{0\mu}-\eta^{0\mu}\,B,$
(69)

which lead to the derivations of the following components of the conjugate
momenta:

$\displaystyle\Pi^{0i}_{(B)}=-\,\frac{1}{2}\,B_{i},\qquad\qquad\Pi^{ij}_{(B)}=-\,\frac{1}{2}\,H^{0ij},$
$\displaystyle\Pi^{0}_{(\phi)}=-\,B_{i},\qquad\qquad\Pi^{i}_{(\phi)}=+m\,B^{0i}-\Phi^{0i}.$
(70)

The above expressions demonstrate that, within the framework of BRST
formalism, all the components of momenta are non-zero and, hence, the
covariant canonical quantization (i.e. BRST-quantization) can be performed in
a straightforward manner. In addition to (70), we require the following EL-
EoMs, w.r.t the basic fields $B_{\mu\nu}$ and $\phi_{\mu}$, namely;

$\displaystyle\partial_{\mu}H^{\mu\nu\lambda}+m^{2}\,B^{\nu\lambda}-m\,\Phi^{\nu\lambda}+(\partial^{\nu}B^{\lambda}-\partial^{\lambda}B^{\nu})=0,$
$\displaystyle\partial_{\mu}\Phi^{\mu\nu}-m\,\partial_{\mu}\,\Phi^{\mu\nu}+\partial^{\nu}\,B+m\,B^{\nu}=0,$
(71)

that are derived from the perfectly BRST-invariant Lagrangian density ${\cal
L}_{b}$ [cf. Eq. (59)]. Taking the choices: $\nu=0,\lambda=j$ in (70), we
obtain the following

$\displaystyle(\partial^{0}\,B^{i}-\partial^{i}\,B^{0})=2\,\big{[}\partial_{j}\,\Pi^{ji}_{(B)}-\frac{m}{2}\,\Pi^{i}_{(\phi)}\big{]},$
$\displaystyle(\dot{B}+m\,B^{0})=-\,\big{[}\partial_{i}\,\Pi^{i}_{(\phi)}\big{]},$
(72)

where we have used the notations from (70) for the components of the conjugate
momenta. It is crystal clear now that we have the following consequences from
(68) in terms of explicit forms of the first-class constraints:

$\displaystyle B\,|\,phys>\,=0\qquad$ $\displaystyle\Longrightarrow$
$\displaystyle\qquad\Pi^{0}_{(\phi)}\,|\,phys>\,=0,$ $\displaystyle
B_{i}\,|\,phys>\,=0\qquad$ $\displaystyle\Longrightarrow$
$\displaystyle\qquad\Pi^{0i}_{(B)}\,|\,phys>\,=0,$
$\displaystyle(\dot{B}+m\,B^{0})\,|\,phys>\,=0\qquad$
$\displaystyle\Longrightarrow$
$\displaystyle\qquad\partial_{i}\,\Pi^{i}_{(\phi)}\,|\,phys>\,=0,$
$\displaystyle(\partial^{0}\,B^{i}-\partial^{i}\,B^{0})\,|\,phys>\,=0\qquad$
$\displaystyle\Longrightarrow$
$\displaystyle\qquad\big{[}\partial_{j}\,\Pi^{ji}_{(B)}-\frac{m}{2}\,\Pi^{i}_{(\phi)}\big{]}\,|\,phys>\,=0.$
(73)

Ultimately, we conclude that the operator forms of the first-class constraints
[$\Pi^{0i}_{(B)}\approx 0,\;\Pi^{0}_{(\phi)}\approx
0,\;(\partial_{j}\,\Pi^{ji}_{(B)}-\frac{m}{2}\,\Pi^{i}_{(\phi)})\approx
0,\;\partial_{i}\,\Pi^{i}_{(\phi)}\approx 0$] of the St$\ddot{u}$ckelberg-
modified massive Abelian 2-form theory [classically described by the
Lagrangian density (29)] annihilate the physical states (in the total quantum
Hilbert of states) within the framework of BRST formalism through the
celebrated physicality criteria w.r.t. the off-shell nilpotent and conserved
(anti-)BRST charges for our modified massive Abelian 2-form theory. A
noteworthy point, at this stage, is the observation that we have not taken
into account any condition on the physical state $(|phys>)$ from the term:
$-\,\lambda\,B^{0}$ from the BRST charge $Q_{b}^{(2)}$. This due to the fact
that (i) the auxiliary ghost field $\lambda$ is not the basic [i.e.
$\lambda=(\frac{1}{2})\,(\partial\cdot C+m\,C)$] field in the theory, and (ii)
the canonical conjugate momentum $[\Pi_{(\phi)}]$, w.r.t. the $\phi$ field for
the Lagrangian density (59), is: $\Pi_{(\phi)}=-\,B^{0}$ which is not a
constraint. In other words, the Nakanishi-Lautrup auxiliary field $B^{0}$ is
not connected with any kinds of constraints (that are present in our modified
massive Abelian 2-form theory at the classical level). To be precise, the
scalar field $\phi$ appears only at the quantum level in our theory.

##  6 Modifications in St$\ddot{u}$ckelberg-Technique: 2D Proca and 4D Massive
Abelian 2-Form Theories

In this section, we discuss the modifications in the St$\ddot{u}$ckelberg-
technique of the compensating fields in the context of (i) the 2D massive
Abelian 1-form (i.e. Proca) theory, and (ii) the 4D massive Abelian 2-form
theory. Thus, our present section is divided into two subsections to capture
the above appropriate modifications separately. For this section to be self-
contained, we have taken some theoretical materials from our earlier works
[40, 50].

###  6.1 2D Massive Abelian 1-Form Theory: Special Features

We begin with the following Lagrangian density for the two (1 + 1)-dimensional
(2D) Abelian 1-form (i.e. Proca) theory (see, e.g. [41, 38, 39])

$\displaystyle{\cal
L}_{(P)}^{(2D)}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{m^{2}}{2}A_{\mu}\,A^{\mu}\equiv+\frac{1}{2}E^{2}+\frac{m^{2}}{2}A_{\mu}\,A^{\mu},$
(74)

where, in the specific two (1 + 1)-dimensions of spacetime, the field-strength
tensor $F_{\mu\nu}=\partial_{\mu}\,A_{\nu}-\partial_{\nu}\,A_{\mu}$ has only
one independent and existing component (i.e.
$F_{01}=\partial_{0}\,A_{1}-\partial_{1}\,A_{0}=E$) which is nothing but a
pseudo-scalar (electric) field that changes sign under the parity operation.
In any arbitrary dimension of spacetime, the usual St$\ddot{u}$ckelberg-
technique, can be written in the following form in the language of the
differential 1-forms, namely;

$\displaystyle A^{(1)}\longrightarrow A^{(1)}\mp\frac{1}{m}\,d\,\Phi^{(0)},$
(75)

where the 0-form $\Phi^{(0)}=\phi$ is a pure scalar field without any
tensorial index and the operator $d$ (with $d^{2}=0$) is the exterior
derivative of the differential geometry [45-49]. Taking into account
$d=\partial_{\mu}\;d\,x^{\mu}$ and $A^{(1)}=A_{\mu}\;d\,x^{\mu}$, we find that
the above relationship (75) is nothing but the usual St$\ddot{u}$ckelberg-
technique for the replacement: $A_{\mu}\longrightarrow
A_{\mu}\mp\frac{1}{m}\,\partial_{\mu}\phi$ which leads to the derivation of
the St$\ddot{u}$ckelberg-modified Lagrangian density $[{\cal L}_{(S)}^{(2D)}]$
from ${\cal L}_{(P)}^{(2D)}$ as

$\displaystyle{\cal L}_{(P)}^{(2D)}\longrightarrow{\cal
L}_{(S)}^{(2D)}=\frac{1}{2}\,E^{2}+\frac{m^{2}}{2}A_{\mu}\,A^{\mu}\mp
m\,A_{\mu}\,\partial^{\mu}\,\phi+\frac{1}{2}\,\partial_{\mu}\,\phi\,\partial^{\mu}\,\phi,$
(76)

which respects the following gauge symmetry transformations for 2D theory,
namely;

$\displaystyle\delta_{g}\,A_{\mu}=\partial_{\mu}\,\Sigma,\qquad\delta_{g}\,\phi=\pm\,m\,\Sigma,\qquad\delta_{g}\,E=0,$
(77)

because of the presence of the first-class constraints on the
St$\ddot{u}$ckelberg-modified Lagrangian density for the 2D Proca theory in
(76) where the infinitesimal and local gauge symmetry transformation parameter
is $\Sigma(\vec{x},t)$. It is pertinent to point out that the second-class
constraints of ${\cal L}_{(P)}^{(2D)}$ have been converted into the first-
class constraints¶¶¶For our 2D theory, the constraints are:
$\Pi^{0}_{(A)}\approx 0$ and $\partial_{i}\,\Pi^{(i=1)}_{(A)}\mp
m\,\Pi_{(\phi)}\equiv\partial_{1}\,E\mp m\,\Pi_{(\phi)}\approx 0$ where
$\Pi^{\mu}_{(A)}=-\,\varepsilon^{0\mu}\,E$ defines $\Pi^{0}_{(A)}\approx 0$
and $\Pi^{(i=1)}_{(A)}=E$ and we also have $\Pi_{(\phi)}=\dot{\phi}\mp
m\,A_{0}$ as the conjugate momentum w.r.t. the pure scalar field ($\phi$).
These first-class constraints consist of the primary constraint
($\Pi^{0}_{(A)}\approx 0$) and the secondary constraint ($\partial_{1}\,E\mp
m\,\Pi_{(\phi)}\approx 0$) which commute with each-other.  due to the presence
of the St$\ddot{u}$ckelberg-compensating field $(\phi)$ which happens to be a
pure scalar field.

The two (1 + 1)-dimensional (2D) spacetime is very special for the massive
Abelian 1-form theory because the usual St$\ddot{u}$ckelberg-technique (with a
pure scalar field $\phi$) gets modified and it can incorporate a pseudo-scalar
field ($\tilde{\phi}$), too. In other words, we find that (75) changes to the
following form (in the special 2D case), namely;

$\displaystyle A^{(1)}\longrightarrow
A^{(1)}\mp\frac{1}{m}\,d\,\Phi^{(0)}\pm\frac{1}{m}\,*\,d\,\tilde{\Phi}^{(0)},$
(78)

where the 0-form $\tilde{\Phi}^{(0)}=\tilde{\phi}$ is a pseudo-scalar field
(which changes sign under the parity operation) and $*$ is the Hodge duality
operation so that the parity symmetry is maintained in our theory. In the
language of the vector field $(A_{\mu})$, we have the following replacement:

$\displaystyle A_{\mu}\longrightarrow
A_{\mu}\mp\frac{1}{m}\,\partial_{\mu}\phi\mp\frac{1}{m}\,\varepsilon_{\mu\nu}\,\partial^{\nu}\,\tilde{\phi}.$
(79)

Here $\varepsilon_{\mu\nu}$ is the 2D Levi-Civita tensor with the choices:
$\varepsilon_{01}=+1=\varepsilon^{10},\,\,\varepsilon_{10}=\varepsilon^{01}=-1$
and
$\varepsilon_{\mu\nu}\,\varepsilon^{\mu\nu}=-\,2!,\,\varepsilon_{\mu\nu}\,\varepsilon^{\mu\lambda}=-\,\delta^{\lambda}_{\nu},$
etc. As a side remark, we point out that the 2D field-strength tensor
$F_{\mu\nu}$ (with only $F_{01}=E$ as the independent and existing component)
can also be expressed in terms of the Levi-Civita tensor
$(\varepsilon_{\mu\nu})$ as:
$F_{01}=E\equiv-\,\varepsilon^{\mu\nu}\,\partial_{\mu}A_{\nu}.$ The modified
St$\ddot{u}$ckelberg-technique (with a pure scalar field $\phi$ and a pseudo-
scalar field $\tilde{\phi}$) remains invariant under the following discrete
duality symmetry transformations:

$\displaystyle
A_{\mu}\to\mp\,i\,\varepsilon_{\mu\nu}A^{\nu},\qquad\qquad\;\phi\to\mp\,i\,\tilde{\phi},\qquad\qquad\tilde{\phi}\to\mp\,i\,\phi.$
(80)

We call the above transformations as the “duality” symmetry transformations
because the origin of the basic symmetry transformation for the gauge field
(i.e. $A_{\mu}\longrightarrow\mp\,i\,\varepsilon_{\mu\nu}A^{\nu}$) owes its
origin to the self-duality condition on the 1-form field $A^{(1)}$ in 2D
spacetime. In other words, we have the following explicit form:

$\displaystyle*\,A^{(1)}\;\equiv\;A_{\mu}\;\bigl{[}*\,(dx^{\mu})\bigr{]}\equiv\;\tilde{A}_{\mu}\;dx^{\mu}\quad\;\Longrightarrow\;\quad\tilde{A}_{\mu}=-\,\varepsilon_{\mu\nu}A^{\nu}.$
(81)

Thus, it is clear that the self-duality condition on the basic 1-form
$(A^{(1)}=dx^{\mu}A_{\mu})$ field $A_{\mu}$ is the root-cause behind the
transformations: $A_{\mu}\longrightarrow\mp\,i\,\varepsilon_{\mu\nu}\,A^{\nu}$
(modulo some signs and $`i$’ factors). Under the replacement (79), we have the
following explicit replacements:

$\displaystyle A_{0}$ $\displaystyle\longrightarrow$ $\displaystyle
A_{0}\mp\frac{1}{m}\,(\partial_{0}\phi-\partial_{1}\tilde{\phi}),$
$\displaystyle A_{1}$ $\displaystyle\longrightarrow$ $\displaystyle
A_{1}\mp\frac{1}{m}\,(\partial_{1}\phi-\partial_{0}\tilde{\phi}),$
$\displaystyle F_{01}$ $\displaystyle\longrightarrow$ $\displaystyle
F_{01}\pm\frac{1}{m}\,\Box\,\tilde{\phi}.$ (82)

The last entry implies that: $E\longrightarrow
E\pm\frac{1}{m}\,\Box\,\tilde{\phi}$. However, we note that, in the
St$\ddot{u}$ckelberg-modified Lagrangian density (76), we have the kinetic
term for the gauge field as: $(1/2)\,E^{2}$ in the 2D spacetime. Thus, for the
2D modified massive Proca theory, we shall have higher derivative terms in the
following change [due to (79)], namely;

$\displaystyle\frac{1}{2}\,E^{2}\longrightarrow\frac{1}{2}\,(E\pm\frac{1}{m}\,\Box\,\tilde{\phi})^{2},$
(83)

which will be problematic for a renormalizable and consistent theory in the
two (1 + 1)-dimensions of spacetime. We can get rid of this problem, if we
take the help of the Klein-Gorden equation of motion:
$(\Box+m^{2})\,\tilde{\phi}=0\Longrightarrow\Box\,\tilde{\phi}=-\,m^{2}\,\tilde{\phi}$
in an ad-hoc and arbitrary manner at this stage. However, from the
appropriately defined Lagrangian density for the modified 2D Proca theory,
ultimately, we should obtain the EL-EoM: $(\Box+m^{2})\,\tilde{\phi}=0$ so
that the sanctity of this substitution can be explicitly corroborated. Taking
the input of the on-shell condition:
$\Box\,\tilde{\phi}=-\,m^{2}\,\tilde{\phi}$, we observe that we have the
following transformation for the kinetic term of the 2D St$\ddot{u}$ckelberg-
modified Proca theory (where $\phi$ and $\tilde{\phi}$ are present as the
compensating fields), namely;

$\displaystyle\frac{1}{2}\,E^{2}\longrightarrow\frac{1}{2}\,(E\mp\,m\,\tilde{\phi})^{2},$
(84)

which is a renormalizable term in the 2D spacetime. Finally, we have the
following gauge-fixed Lagrangian density for the modified 2D Proca theory
[with the modified form of the St$\ddot{u}$ckelberg-technique in (79)],
namely;

$\displaystyle{\cal L}_{(2D)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,{(E\mp m\,\tilde{\phi})}^{2}\pm
m\,E\,\tilde{\phi}-\frac{1}{2}\,\partial_{\mu}\,\tilde{\phi}\,\,\partial^{\mu}\,\tilde{\phi}+\frac{m^{2}}{2}A_{\mu}\,A^{\mu}$
(85) $\displaystyle\mp$ $\displaystyle
m\,A_{\mu}\,\partial^{\mu}\,\phi+\frac{1}{2}\,\partial_{\mu}\,\phi\,\partial^{\mu}\,\phi-\frac{1}{2}\,(\partial\,\cdot
A\pm m\,\phi)^{2},$

where the last term is nothing but the gauge-fixing term in the ‘t Hooft gauge
(see, e.g. [41]). At this stage, we lay emphasis on three crucial points
connected with (85). First, it can be checked that the gauge-fixed Lagrangian
density respects the discrete duality symmetry transformations:
$A_{\mu}\to\mp\,i\,\varepsilon_{\mu\nu}A^{\nu},\;\phi\to\mp\,i\,\tilde{\phi},\;\tilde{\phi}\to\mp\,i\,\phi$
along with $(\partial\,\cdot A)\to\pm i\,E,\;E\to\pm i\,(\partial\,\cdot A)$.
The latter two transformations are due to:
$A_{\mu}\to\mp\,i\,\varepsilon_{\mu\nu}A^{\nu}$. Second, we can check
explicitly that the EL-EoMs∥∥∥This shows that, in going from (83) to (84),
whatever we have chosen in an ad-hoc and arbitrary manner is the correct way
to get rid of the higher order derivative terms for our 2D theory. This is due
to the fact that the Lagrangian density (85) produces
$(\Box+m^{2})\,\tilde{\phi}=0$ as the EL-EoM.: $(\Box+m^{2})\,\phi=0$ and
$(\Box+m^{2})\,\tilde{\phi}=0$ are satisfied from the gauge-fixed Lagrangian
density. Finally, we note that the kinetic term for the pseudo-scalar field
$\tilde{\phi}$ carries a negative sign in comparison to the scalar field
$\phi$. Such fields with negative kinetic terms (but with well-defined rest
mass) are exotic fields which are relevant in the context of the cyclic,
bouncing and self-accelerated cosmological models of the Universe [51-53] and
they are also a set of possible candidates for the dark matter [54, 55] in the
modern literature on this challenging topic of research activity.

Within the framework of BRST formalism, the above gauge-fixed Lagrangian
density can be generalized to incorporate the Faddeev-Popov (FP) ghost terms
as follows

$\displaystyle{\cal L}_{(B)}^{(2D)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,{(E\mp m\,\tilde{\phi})}^{2}\pm
m\,E\,\tilde{\phi}-\frac{1}{2}\,\partial_{\mu}\,\tilde{\phi}\,\,\partial^{\mu}\,\tilde{\phi}+\frac{m^{2}}{2}A_{\mu}\,A^{\mu}\mp
m\,A_{\mu}\,\partial^{\mu}\,\phi$ (86) $\displaystyle+$
$\displaystyle\frac{1}{2}\,\partial_{\mu}\,\phi\,\partial^{\mu}\,\phi-\frac{1}{2}\,[(\partial\,\cdot
A)\pm
m\,\phi]^{2}-\,i\,\partial_{\mu}\,\bar{C}\,\partial^{\mu}\,C+i\,m^{2}\,\bar{C}\,C,$

where the (anti-)ghost fields $(\bar{C})C$ are fermionic (i.e.
$C^{2}=\bar{C}^{2}=0,\;\bar{C}\,C+C\,\bar{C}=0$) in nature and they are
invoked to maintain the unitarity in the theory. The subscript $(B)$ on the
Lagrangian density stands for its (anti-)BRST and (anti-)co-BRST invariant
form. We can linearize the quadratic kinetic and gauge-fixing terms by
invoking a couple of Nakanishi-Lautrup type auxiliary fields (${\cal B},B$) as

$\displaystyle{\cal L}_{(B,\,{\cal B})}^{(2D)}$ $\displaystyle=$
$\displaystyle{\cal B}\,{(E\mp m\,\tilde{\phi})}-\frac{1}{2}\,{\cal B}^{2}\pm
m\,E\,\tilde{\phi}-\frac{1}{2}\,\partial_{\mu}\,\tilde{\phi}\,\,\partial^{\mu}\,\tilde{\phi}+\frac{m^{2}}{2}A_{\mu}\,A^{\mu}+\frac{1}{2}\,\partial_{\mu}\,\phi\,\partial^{\mu}\,\phi$
(87) $\displaystyle\mp$ $\displaystyle
m\,A_{\mu}\,\partial^{\mu}\,\phi+B\,(\partial\cdot A\pm
m\,\phi)+\frac{1}{2}\,{B}^{2}-\,i\,\partial_{\mu}\,\bar{C}\,\partial^{\mu}\,C+i\,m^{2}\,\bar{C}\,C,$

which respects the generalized version of the classical discrete duality
symmetry transformations (80) in the following form at the quantum level:

$\displaystyle\phi\to\mp\,i\,\tilde{\phi},\qquad\tilde{\phi}\to\mp\,i\,\phi,\qquad
A_{\mu}\to\mp\,i\,\varepsilon_{\mu\nu}A^{\nu},\qquad B\to\mp i\,{\cal
B},\quad{\cal B}\to\mp i\,B,$ $\displaystyle C\to\pm
i\,\bar{C},\quad\qquad\;\;\bar{C}\to\pm i\,C,\quad\quad\;(\partial\,\cdot
A)\to\pm i\,E\qquad E\to\pm i\,(\partial\,\cdot A).$ (88)

It is very interesting to point out that the Lagrangian density (87) respects,
in addition to a couple of discrete duality symmetry transformations in (88),
a set of six continuous symmetry transformations that incorporate four
fermionic symmetry transformations and two bosonic. In precise terms, the four
fermionic (i.e. nilpotent) symmetry transformations are the BRST, anti-BRST,
co-BRST and anti-co-BRST symmetry transformations. On the other hand, there
exists a unique bosonic symmetry transformation that turns up as a suitable
anticommutator of the (anti-)BRST and (anti-)co-BRST transformations [38, 39].
Finally, there exists a ghost-scale symmetry transformation that is also
bosonic in nature. These total (i.e. discrete and continuous) symmetries
entail upon the 2D modified Proca theory to become a field-theoretic model for
the Hodge theory (see, e.g. [39] for details).

As far as our present investigation is concerned, we mention here only two
fermionic (i.e. nilpotent) BRST and co-BRST (i.e. dual BRST) symmetry
transformations which are denoted by $s_{b}$ and $s_{d}$, respectively. These
are assimilated as follows [38, 39]

$\displaystyle s_{b}\,A_{\mu}=\partial_{\mu}\,C,\qquad\;\;s_{b}\,C=0,\qquad
s_{b}\,\bar{C}=+\,i\,B,\qquad s_{b}\,\phi=\pm m\,C,$ $\displaystyle
s_{b}\,B=0,\qquad\qquad\;\;s_{b}\,E=0,\qquad\;s_{b}\,{\cal B}=0,\qquad\qquad
s_{b}\,{\tilde{\phi}}=0,$ $\displaystyle
s_{d}\,A_{\mu}=-\varepsilon_{\mu\nu}\partial^{\nu}\,\bar{C},\quad
s_{d}\,\bar{C}=0,\quad s_{d}\,C=-\,i\,{\cal B},\quad\quad s_{d}\,{\cal
B}=0,\qquad s_{d}\,\phi=0,$ $\displaystyle s_{d}\,B=0,\qquad\qquad
s_{d}\,\tilde{\phi}=\mp\,m\,\bar{C},\qquad s_{d}\,(\partial\cdot A)=0,\qquad
s_{d}\,E=\Box\,\bar{C},$ (89)

which demonstrate that (i) the electric field (i.e. $E$), owing its origin to
the exterior derivative, remains invariant under $s_{b}$, and (ii) the gauge-
fixing term (i.e. $\partial\cdot A$), tracing back its origin to the co-
exterior derivative, remains invariant under the dual-BRST (i.e. co-BRST)
symmetry transformations. Hence, the nomenclatures of the BRST and co-BRST
symmetry transformations are appropriate because of their origin (and
connection with the exterior and co-exterior derivatives). The infinitesimal,
continuous and off-shell nilpotent symmetry transformations lead, according to
the Noether theorem, to the derivations of the conserved current and
corresponding conserved charges. These latter charges (which are off-shell
nilpotent of order two) are as follows (see, e.g. [38, 39] for details):

$\displaystyle Q_{b}=\int dx\,[B\,\dot{C}-\dot{B}\,C],\qquad\qquad Q_{d}=\int
dx\,[{\cal B}\,\dot{\bar{C}}-\dot{\cal B}\,\bar{C}],$ (90)

where the pair $(B,{\cal B})$ are the Nakanishi-Lautrup auxiliary fields and
$(\bar{C})C$ are the fermionic (anti-)ghost fields of the Lagrangian density
(87). It is clear, from the latter, that we have: ${\cal B}=(E\mp
m\,\tilde{\phi})$ and $B=-\,(\partial\cdot A\pm m\,\phi)$. Further, we note
that the conjugate momenta w.r.t. the massive gauge field $(A_{\mu})$, from
(87), is as follows

$\displaystyle\Pi^{\mu}_{(A)}=\frac{\partial{\cal
L}_{(B)}}{\partial(\partial_{0}A_{\mu})}=-\,\varepsilon^{0\mu}\,{\cal
B}+\eta^{0\mu}\,B\mp m\,\varepsilon^{0\mu}\,\tilde{\phi},$ (91)

where we have taken the help of the covariant form of the electric field:
$E=-\,\varepsilon^{\mu\nu}\,\partial_{\mu}A_{\nu}$ and the subscript $(A)$ on
the conjugate momenta denotes that the field $A_{\mu}$ is invoked (in the
canonical definition of the conjugate momenta). It is clear that we have the
following components of the momenta in the 2D Minkowskian spacetime, namely;

$\displaystyle\Pi^{0}_{(A)}=B\equiv-\,(\partial\cdot A\pm
m\,\phi),\qquad\qquad\Pi^{(i=1)}_{(A)}={\cal B}\pm m\,\tilde{\phi}\equiv E.$
(92)

In addition, we have the following EL-EoM w.r.t the massive gauge field
$A_{\mu}$, namely;

$\displaystyle\varepsilon^{\mu\nu}\,\partial_{\nu}\,{\cal
B}+\partial^{\mu}\,B\pm
m\,\varepsilon^{\mu\nu}\,\partial_{\nu}\,\tilde{\phi}=m^{2}\,A^{\mu}\mp
m\,\partial^{\mu}\,\phi,$ (93)

which, for the choice: $\mu=0$, leads to

$\displaystyle\dot{B}=\partial_{1}\,({\cal B}\pm m\,\tilde{\phi})\mp
m\,(\dot{\phi}\mp m\,A_{0})\quad\equiv\quad\partial_{1}\,E\mp
m\,\Pi_{(\phi)},$ (94)

where $\Pi_{(\phi)}=\dot{\phi}\mp m\,A_{0}$ is the canonical conjugate
momentum w.r.t. the pure scalar field $\phi$ and we have also used ${\cal
B}=E\mp m\,\tilde{\phi}$ that emerges out from (87).

We are in the position now to impose the physicality criterion w.r.t. the off-
shell nilpotent and conserved BRST charge $Q_{(b)}$ [cf. Eq. (90)]. Since the
(anti-)ghost fields are not physically important, we note that we have the
following:

$\displaystyle Q_{(b)}\,|\,phys>=0\qquad\Longrightarrow\qquad
B\,|phys>=0,\;\qquad\;\;\dot{B}\,|phys>=0.$ (95)

A close and careful look at (92) and (94) shows that, in the language of the
first-class constraints of the St$\ddot{u}$ckelberg-modified massive 2D Proca
theory [classically described by the Lagrangian density (76)], we have the
following:

$\displaystyle B\,|phys>=0\qquad$ $\displaystyle\Longrightarrow$
$\displaystyle\qquad\Pi^{0}_{(A)}\,|phys>=0$ $\displaystyle\Longrightarrow$
$\displaystyle\qquad-\,(\partial\cdot A\pm m\,\phi)\,|phys>=0,$
$\displaystyle\dot{B}\,|phys>=0\qquad$ $\displaystyle\Longrightarrow$
$\displaystyle\qquad(\partial_{1}\,E\mp m\,\Pi_{(\phi)})\,|phys>=0$
$\displaystyle\Longrightarrow$
$\displaystyle\qquad-\,\partial_{0}(\partial\cdot A\pm m\,\phi)\,|phys>=0.$
(96)

Hence, the physical states (in the total quantum Hilbert space of states) are
those that are annihilated by the conserved and off-shell nilpotent BRST
charge in any arbitrary dimension of spacetime (where the special 2D spacetime
is also included). This automatically implies that the operator forms of the
first-class constraints annihilate the physical states (existing in the total
quantum Hilbert space of states) at the quantum level.

We end this subsection with a crucial remark that is connected with the
St$\ddot{u}$ckelberg-modified Proca theory in the two (1+1)-dimension of
spacetime which has already been proven (see, e.g. [56, 39] for details) to be
a massive model of Hodge theory where the Hodge decomposition theorem (see,
e.g. [39, 56] for details) can be performed in the quantum Hilbert space of
states. The most symmetric state, in the above decomposition, is the harmonic
state that is annihilated by both the basic conserved and nilpotent charges of
our theory (which are nothing but the conserved and off-shell nilpotent BRST
and co-BRST charges). Thus, the physical state ($|phys>$), in the total
Hilbert space of states, is the harmonic state for a field-theoretic model of
Hodge theory. Hence, the physicality criterion w.r.t. the off-shell nilpotent
and conserved co-BRST charge leads to the following conditions, namely;

$\displaystyle Q_{d}\,|phys>=0\qquad\Longrightarrow\qquad{\cal
B}\,|phys>=0,\quad\qquad\dot{\cal B}\,|phys>=0,$ (97)

which can be equivalently expressed as follows

$\displaystyle{\cal B}\,|phys>=0\qquad\Longrightarrow\qquad(E\mp
m\,\tilde{\phi})\,|phys>=0,$ $\displaystyle\dot{\cal
B}\,|phys>=0\qquad\Longrightarrow\qquad\partial_{0}\,(E\mp
m\,\tilde{\phi})\,|phys>=0.$ (98)

At this stage, first of all, we note that $Q_{b}$ and $Q_{d}$ are related to
each-other (i.e. $Q_{b}\longleftrightarrow Q_{d}$) by the discrete duality
symmetry transformations: $B\longrightarrow\mp i\,{\cal B},\;{\cal
B}\longrightarrow\mp i\,B,\;C\longrightarrow\pm
i\,\bar{C},\;\bar{C}\longrightarrow\pm i\,C$ that are present in the full set
of discrete duality symmetry transformations (88). Thus, we conclude that
$Q_{b}\,|phys>=0$ implies that the operator form of the first-class
constraints must annihilate the physical states. On the other hand, the
physicality criterion w.r.t. the conserved and off-shell nilpotent co-BRST
charge $Q_{d}$ (i.e. $Q_{d}\,|phys>=0$) implies that the dual versions of the
operator form of the first-class constraints must annihilate the physical
states of a 2D field-theoretic model of Hodge theory, too. Mathematically, we
capture these statements as follows

$\displaystyle B\,|phys>=0\quad\Longrightarrow\quad-\,(\partial\cdot A\pm
m\,\phi)\,|phys>=0\qquad\Longleftrightarrow$ $\displaystyle{\cal
B}\,|phys>=0\quad\Longrightarrow\quad(E\mp m\,\tilde{\phi})\,|phys>=0,$
$\displaystyle\dot{B}\,|phys>=0\quad\Longrightarrow\quad-\,\partial_{0}(\partial\cdot
A\pm m\,\phi)\,|phys>=0\quad\,\Longleftrightarrow$ $\displaystyle\dot{\,}{\cal
B}|phys>=0\quad\Longrightarrow\quad\partial_{0}\,(E\mp
m\,\tilde{\phi})\,|phys>=0,$ (99)

where we have used the duality symmetry transformations: $(\partial\cdot
A)\longrightarrow\pm i\,E,\;E\longrightarrow\pm i\,(\partial\cdot
A),\;B\longrightarrow\mp i\,{\cal B},\;{\cal B}\longrightarrow\mp
i\,B,\;\phi\longrightarrow\mp i\,\tilde{\phi},\;\tilde{\phi}\longrightarrow\mp
i\,\phi$ which are a part of the full set of discrete duality symmetry
transformations that have been written explicitly in (88). Lastly, we would
like to comment that for $\mu=1$, we obtain from (93), the expression for the
time derivative on the auxiliary field ${\cal B}$ as: $\dot{\cal
B}=-\,[\partial_{1}\,(\partial\cdot A)\pm m\,(\dot{\tilde{\phi}}\pm
m\,A_{1})]$ which is the dual of the secondary first-class constraint:
$\partial_{1}\,E\mp m\,\Pi_{\phi}\approx 0$ that annihilates the physical
state due to $\dot{B}\,|phys>=0$ in Eq. (96). It is the observations in (99)
that have been responsible to prove that the massless 2D Abelian 1-form gauge
theory (and the 2D non-Abelian 1-form gauge theory) provide (i) a new class of
topological field theories (TFTs) which capture a few key aspects of Witten-
type TFTs [31] and some salient features of Schwarz-type TFTs [32], and (ii)
the perfect field-theoretic examples of Hodge theory. The topological nature
of the 2D massless Abelian 1-form theory happens because both the degrees of
freedom of the 2D photon (in the Abelian case) can be gauged away due to the
presence of BRST and co-BRST symmetries (see, e.g. [51, 33] for details).

###  6.2 4D Massive Abelian 2-Form Theory: Special Observations

Against the backdrop of the Subsec. 6.1, we would like to discuss the
modification in the standard St$\ddot{u}$ckelberg-technique of the
compensating field(s) in the context of a massive Abelian 2-form theory in a
very specific four (3 + 1)-dimension of spacetime. We begin with the
Lagrangian density ${\cal L}^{(B)}_{0}$ [cf. Eq. (27)] and express the
standard St$\ddot{u}$ckelberg-technique (28), valid in any arbitrary
D-dimension of spacetime, in the differential-form language (in terms of the
2-forms) as follows

$\displaystyle B^{(2)}\longrightarrow B^{(2)}\mp\frac{1}{m}\,d\,\Phi^{(1)},$
(100)

where the 1-form $\Phi^{(1)}=\phi_{\mu}\;d\,x^{\mu}$ defines the
St$\ddot{u}$ckelberg-compensating field $(\phi_{\mu})$ which turns out to be a
Lorentz vector. In the specific four (3 + 1)-dimensional (4D) Minkowskian flat
spacetime, one can generalize (100) by incorporating an axial-vector field
$\tilde{\phi}_{\mu}$ which is defined through the pseudo 1-form
$\tilde{\Phi}^{(1)}=\tilde{\phi}_{\mu}\;d\,x^{\mu}$ and using the basic
concepts behind the exterior derivative and the Hodge duality $*$ operation on
the 4D spacetime, as follows:

$\displaystyle B^{(2)}\longrightarrow
B^{(2)}\mp\frac{1}{m}\,d\,\Phi^{(1)}\mp\frac{1}{m}\,*\,d\,\tilde{\Phi}^{(1)}.$
(101)

We point out that the above St$\ddot{u}$ckelberg-modification has been made
such that the parity symmetry is respected and maintained in our theory
(unlike the theory of weak interactions where the parity symmetry is
violated). In terms of the antisymmetric ($B_{\mu\nu}=-\,B_{\nu\mu}$) tensor
field, we can re-express (101) as follows

$\displaystyle B_{\mu\nu}$ $\displaystyle\longrightarrow$ $\displaystyle
B_{\mu\nu}\mp\frac{1}{m}\,(\partial_{\mu}\,\phi_{\nu}-\partial_{\nu}\,\phi_{\mu})\mp\frac{1}{m}\,\varepsilon_{\mu\nu\lambda\xi}\,\partial^{\lambda}\,\tilde{\phi}^{\xi},$
(102)

which respects the following set of discrete duality symmetry transformations:

$\displaystyle
B_{\mu\nu}\longrightarrow\mp\frac{i}{2!}\,\varepsilon_{\mu\nu\lambda\xi}\,B^{\lambda\xi},\qquad\phi_{\mu}\longrightarrow\pm
i\,\tilde{\phi_{\mu}},\qquad\tilde{\phi_{\mu}}\longrightarrow\mp
i\,\phi_{\mu}.$ (103)

This observation is exactly similar to our discussion in Subsec. 6.1 where we
have discussed the “self-duality” in the context of the modified Abelian
1-form (i.e. Proca) theory in 2D [cf. Eqs. (80), (81]. Using the notation from
the ordinary 2-form $\Phi^{(2)}=\frac{1}{2!}\,\Phi_{\mu\nu}\;(dx^{\mu}\wedge
dx^{\nu})$ where
$\Phi_{\mu\nu}=\partial_{\mu}\phi_{\mu}-\partial_{\nu}\phi_{\mu}$ and taking
the symbol
$\tilde{f}_{\mu\nu}=\varepsilon_{\mu\nu\lambda\xi}\,\partial^{\lambda}\,\tilde{\phi}^{\xi}$,
the modified St$\ddot{u}$ckelberg technique of the replacement (102) can be
re-expressed in two different ways as

$\displaystyle B_{\mu\nu}$ $\displaystyle\longrightarrow$ $\displaystyle
B_{\mu\nu}\mp\frac{1}{m}(\,\Phi_{\mu\nu}\mp\tilde{f}_{\mu\nu})\equiv
B_{\mu\nu}\mp\frac{1}{m}\,\Big{(}\Phi_{\mu\nu}+\frac{1}{2}\,\varepsilon_{\mu\nu\lambda\xi}\,\tilde{\Phi}^{\lambda\xi}\Big{)},$
(104)

where
$\tilde{\Phi}_{\mu\nu}=\partial_{\mu}\tilde{\phi}_{\nu}-\partial_{\nu}\tilde{\phi}_{\mu}$.
The latter is defined through the pseudo 2-form:
$\tilde{\Phi}^{(2)}=\frac{1}{2!}\,\tilde{\Phi}_{\mu\nu}\;(d\,x^{\mu}\wedge
d\,x^{\nu})$ which is nothing but an antisymmetric pseudo tensor field of
rank-2 and it is the origin of the kinetic term
[$+(1/4)\,\tilde{\Phi}^{\mu\nu}\,\tilde{\Phi}_{\mu\nu}$] for the axial-vector
field $\tilde{\phi}_{\mu}$. With the input from (104), it is clear that we
have the following transformations for the mass term and the field-strength
tensor of the starting Lagrangian density ${\cal L}^{(B)}_{0}$ [cf. Eq. (27)]

$\displaystyle-\,\frac{m^{2}}{4}\,B_{\mu\nu}\,B^{\mu\nu}\longrightarrow$
$\displaystyle-$
$\displaystyle\,\frac{m^{2}}{4}\,B_{\mu\nu}\,B^{\mu\nu}\pm\frac{m}{2}\,B_{\mu\nu}\,(\Phi^{\mu\nu}+\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}\,\tilde{\Phi}_{\rho\sigma})$
(105) $\displaystyle-$
$\displaystyle\,\frac{1}{4}\,\Phi_{\mu\nu}\,\Phi^{\mu\nu}+\frac{1}{4}\tilde{\Phi}_{\mu\nu}\,\tilde{\Phi}^{\mu\nu},$
$\displaystyle H_{\mu\nu\lambda}\longrightarrow H_{\mu\nu\lambda}$
$\displaystyle\mp$
$\displaystyle\frac{1}{m}\,(\partial_{\mu}\,\Phi_{\nu\lambda}+\partial_{\nu}\,\Phi_{\lambda\mu}+\partial_{\lambda}\,\Phi_{\mu\nu})$
(106) $\displaystyle\mp$
$\displaystyle\frac{1}{m}\,(\partial_{\mu}\,\tilde{f}_{\nu\lambda}+\partial_{\nu}\,\tilde{f}_{\lambda\mu}+\partial_{\lambda}\,\tilde{f}_{\mu\nu}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

where we note that the second term of (106) is equal to zero due to :
$\Phi_{\mu\nu}=\partial_{\mu}\phi_{\nu}-\partial_{\nu}\phi_{\mu}.$ We also lay
emphasis on the observation that there is a relative sign difference between
the kinetic terms of field $\phi_{\mu}$ and $\tilde{\phi}_{\mu}$,
respectively, in (105). We shall come back to this point later in our Sec. 7
when we shall make some concluding remarks.

With the inputs from the equations (105) and (106), we note that the standard
kinetic term of the Lagrangian density ${\cal L}_{0}^{(B)}$ [cf. Eq. (27)]
transforms as follows

$\displaystyle\frac{1}{12}\,H_{\mu\nu\lambda}\,H^{\mu\nu\lambda}\;\longrightarrow\;\frac{1}{12}\,H_{\mu\nu\lambda}\,H^{\mu\nu\lambda}\mp\frac{1}{6\,m}\,H_{\mu\nu\lambda}\,\Sigma^{\mu\nu\lambda}+\frac{1}{12\,m^{2}}\,\Sigma_{\mu\nu\lambda}\,\Sigma^{\mu\nu\lambda},$
(107)

where we have introduced a new notation:
$\Sigma_{\mu\nu\lambda}=\partial_{\mu}\,\tilde{f}_{\nu\lambda}+\partial_{\nu}\,\tilde{f}_{\lambda\mu}+\partial_{\lambda}\,\tilde{f}_{\mu\nu}$
which makes the transformation (106) as: $H_{\mu\nu\lambda}\longrightarrow
H_{\mu\nu\lambda}\mp(1/m)\,\Sigma_{\mu\nu\lambda}$. The latter has been used
in the derivation of the transformation (107). With the help of (105) and
(107), we note that the starting Lagrangian density ${\cal L}_{0}^{(B)}$ [cf.
Eq. (27)] for the massive Abelian 2-form theory transforms to the following
St$\ddot{u}$ckelberg-modified massive Abelian 2-form theory

$\displaystyle{\cal L}_{0}^{(B)}\longrightarrow{\cal
L}_{S}^{(m)}=\frac{1}{12}\,H_{\mu\nu\lambda}\,H^{\mu\nu\lambda}\mp\frac{1}{6\,m}\,H_{\mu\nu\lambda}\,\Sigma^{\mu\nu\lambda}+\frac{1}{12\,m^{2}}\,\Sigma_{\mu\nu\lambda}\,\Sigma^{\mu\nu\lambda}$
$\displaystyle-$ $\displaystyle\frac{m^{2}}{4}B_{\mu\nu}\,B^{\mu\nu}$
$\displaystyle\pm\frac{m}{2}\,B_{\mu\nu}\,[\Phi^{\mu\nu}+\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}\,\tilde{\Phi}_{\rho\sigma}]-\,\frac{1}{4}\,\Phi_{\mu\nu}\,\Phi^{\mu\nu}+\frac{1}{4}\tilde{\Phi}_{\mu\nu}\,\tilde{\Phi}^{\mu\nu},$
(108)

where there are higher derivative terms in the second and third terms of the
4D Lagrangian density ${\cal L}_{S}^{(m)}$ and the superscript $(m)$ denotes
that we have obtained (108) with the modified version of the
St$\ddot{u}$ckelberg-technique of the compensating fields (102) in 4D.

Our present discussion will center around the Lagrangian density (108) where
all its terms will be well-defined, consistent and renormalizable in the
specific four (3 + 1)-dimensions of Minkowskian flat spacetime using the on-
shell condition for the basic field $\tilde{\phi}_{\mu}$. This systematic
exercise has been performed in a very recent paper by us (see e.g. [50] for
details). Thus, we skip many theoretical and mathematical steps because it
will be only an academic exercise if we repeat those steps here. Finally, we
state that we have derived the coupled (but equivalent) (anti-)BRST and
(anti-)co-BRST invariant Lagrangian densities in the four (3 + 1)-dimensions
of spacetime where the generalizations of (i) the discrete duality symmetry
transformation (103), and (ii) the continuous (dual-)gauge symmetry
transformations have been taken into account (see e.g. [50, 30] for details).
However, the numerical factors associated with the gauge-fixing terms and
Faddeev-Popove ghost terms are quite different from our present discussion of
the D-dimensional modified massive Abelian 2-form theory (cf. Subsec. 5.2 for
details). To be consistent with notations adopted in Subsec. 5.2, we
concentrate on the following (co-)BRST invariant Lagrangian density (${\cal
L}_{\cal B}$) with the appropriate generalization of (108) that incorporates
into it the proper gauge-fixing and FP-ghost terms, namely;

$\displaystyle{\cal L}_{\cal B}$ $\displaystyle=$ $\displaystyle{\cal
B}_{\mu}\,{\cal B}^{\mu}+{\cal
B}^{\mu}\,\Big{[}\frac{1}{2}\,\varepsilon_{\mu\nu\lambda\xi}\,\partial^{\nu}\,B^{\lambda\xi}+m\,\tilde{\phi}_{\mu}-\partial_{\mu}\tilde{\phi}\Big{]}-\,\frac{m^{2}}{4}\,B_{\mu\nu}\,B^{\mu\nu}$
$\displaystyle-$
$\displaystyle\frac{1}{4}\,\Phi_{\mu\nu}\,\Phi^{\mu\nu}+\frac{1}{4}\tilde{\Phi}_{\mu\nu}\,\tilde{\Phi}^{\mu\nu}+\frac{m}{2}\,B_{\mu\nu}\,\Big{[}\Phi^{\mu\nu}+\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}\,\tilde{\Phi}_{\rho\sigma}\Big{]}$
(109) $\displaystyle+$ $\displaystyle
B^{\mu}\,\Big{[}\partial^{\nu}\,B_{\nu\mu}+m\,\phi_{\mu}-\partial_{\mu}\,\phi\Big{]}\,+B^{\mu}\,B_{\mu}-B\,\Big{(}\partial\cdot\phi+m\,\phi\Big{)}$
$\displaystyle-$ $\displaystyle B^{2}+{\cal
B}\,\Big{(}\partial\cdot\tilde{\phi}+m\,\tilde{\phi}\Big{)}+{\cal
B}^{2}+\partial_{\mu}\,\bar{\beta}\,\partial^{\mu}\,\beta-\,m^{2}\,\bar{\beta}\,\beta$
$\displaystyle+$
$\displaystyle(\partial_{\mu}\,\bar{C}_{\nu}-\partial_{\nu}\,\bar{C}_{\mu})\,(\partial^{\mu}\,C^{\nu})-(\partial_{\mu}\,\bar{C}-m\,\bar{C}_{\mu})\,(\partial^{\mu}\,C-m\,C^{\mu})$
$\displaystyle+$
$\displaystyle\Big{(}\partial\cdot\bar{C}+m\,\bar{C}+\rho\Big{)}\,\lambda+\Big{(}\partial\cdot
C+m\,C-\lambda\Big{)}\,\rho,$

where bosonic (anti-)ghost fields $(\bar{\beta})\beta$ carry the ghost numbers
(-2)+2, respectively, and the fermionic (anti-)ghost fields
$(\bar{C}_{\mu})C_{\mu}$ are endowed with the ghost numbers (-1)+1,
respectively. We have also linearized the kinetic term and gauge-fixing term
for $B_{\mu\nu}$ and an axial-vector field $\tilde{\phi}_{\mu}$ by invoking
Nakanishi-Lautrup type auxiliary fields $({\cal B}_{\mu},{\cal B})$. It will
be noted that we also have a pseudo-scalar field $\tilde{\phi}$ in our
theory******A close look at (109) shows that it is the generalization of the
perfectly BRST invariant Lagrangian density (${\cal L}_{b}$) that has been
quoted in (59) of our Subsec. 5.2. For the sake of simplicity, we have not
bothered about the numerical factors that are present in various terms of the
Lagrangian density that has been taken into account in our earlier works [30,
50] where the discrete and continuous symmetries are precisely respected at
every stage. For our present Lagrangian density (109), we have considered only
the infinitesimal, continuous and off-shell nilpotent (co-)BRST symmetries
$(s_{d})s_{b}$. (see, e.g. [50, 40, 30] for details). Our (co-)BRST invariant
Lagrangian density (109) also contains additional (anti-) ghost fields
$(\bar{C})C$ which are fermionic
$(C^{2}=\bar{C}^{2}=0,C\,\bar{C}+\bar{C}\,C=0)$ in nature with the ghost
numbers (-1)+1, respectively. Finally, the FP ghost terms contain a set of
auxiliary (anti-)ghost fields $(\rho)\,\lambda$ with ghost numbers (-1)+1,
respectively. The Lagrangian density (109) respects the following (co-)BRST
symmetry transformations $(s_{d})s_{b}$

$\displaystyle
s_{d}\,B_{\mu\nu}=-\,\varepsilon_{\mu\nu\lambda\xi}\,\partial^{\lambda}\,\bar{C}^{\xi},\quad
s_{d}\,\bar{C}_{\mu}=-\,\partial_{\mu}\,\bar{\beta},\quad s_{d}\,C_{\mu}={\cal
B}_{\mu},$ $\displaystyle s_{d}\,\beta=-\,\lambda,\quad
s_{d}\,\tilde{\phi}_{\mu}=+(\partial_{\mu}\,\bar{C}-m\,\bar{C}_{\mu}),\quad
s_{d}\,C={\cal B},\quad s_{d}\,\bar{C}=-m\,\bar{\beta},$ $\displaystyle
s_{d}\,\tilde{\phi}=-\rho,\quad s_{d}\,[\partial^{\nu}\,B_{\nu\mu},\,{\cal
B}_{\mu},\,B_{\mu},\,{\cal
B},\,\phi_{\mu},\,\Phi_{\mu\nu},\,\phi,\,\bar{\beta},\,\lambda,\,\rho]=0,$
$\displaystyle
s_{b}\,B_{\mu\nu}=-\,(\partial_{\mu}\,C_{\nu}-\partial_{\nu}\,C_{\mu}),\quad
s_{b}\,C_{\mu}=-\,\partial_{\mu}\,\beta,\quad s_{b}\bar{C}_{\mu}=-\,B_{\mu},$
$\displaystyle s_{b}\,\bar{\beta}=-\,\rho,\quad
s_{b}\,\phi_{\mu}=+(\partial_{\mu}C-m\,C_{\mu}),\quad s_{b}\,\bar{C}=B,\quad
s_{b}\,\phi=+\lambda,$ $\displaystyle s_{b}\,C=-\,m\,\beta,\quad
s_{b}\,[H_{\mu\nu\lambda},\,B,\,\lambda,\,\rho,\,B_{\mu},\,{\cal
B}_{\mu},\,\beta,\,{\cal
B},\,\tilde{\phi}_{\mu},\,\tilde{\phi}_{\mu\nu},\,\tilde{\phi}]=0,$ (110)

which are characterized by the observations that (i) the field strength tensor
$(H_{\mu\nu\lambda})$, owing its origin to the exterior derivative, remains
invariant under $s_{b}$, and (ii) the gauge-fixing term
$(\partial^{\nu}B_{\nu\mu})$, tracing back its origin to the co-exterior
derivative of differential geometry [45-49], remains invariant under $s_{d}$.
The above (co-)BRST transformations are the symmetry transformations for the
action integral $S=\int d^{4}x\,{\cal L}_{\cal B}$ because we observe that
${\cal L}_{\cal B}$ transform under $(s_{d})s_{b}$ to the total spacetime
derivatives as:

$\displaystyle s_{d}\,{\cal L}_{\cal B}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\,\Big{[}-m\,\varepsilon^{\mu\nu\lambda\xi}\,\phi_{\nu}\,\partial_{\lambda}\,\bar{C}_{\xi}-\,(\partial^{\mu}\,\bar{C}^{\nu}-\partial^{\nu}\,\bar{C}^{\mu})\,{\cal
B}_{\nu}+{\cal B}^{\mu}\,\rho$ $\displaystyle+$
$\displaystyle(\partial^{\mu}\,\bar{C}-m\,\bar{C}^{\mu})\,{\cal
B}-\lambda\,(\partial^{\mu}\,\bar{\beta})\Big{]},$ $\displaystyle s_{b}\,{\cal
L}_{\cal B}$ $\displaystyle=$
$\displaystyle-\,\partial_{\mu}\,\Big{[}m\,\varepsilon^{\mu\nu\lambda\xi}\,\tilde{\phi}_{\nu}\,\partial_{\lambda}\,C_{\xi}+(\partial^{\mu}\,C^{\nu}-\partial^{\nu}\,C^{\mu})\,B_{\nu}+\lambda\,B^{\mu}$
(111) $\displaystyle+$
$\displaystyle(\partial^{\mu}\,C-m\,C^{\mu})\,B+\rho\,(\partial^{\mu}\,\beta)\Big{]}.$

According to the basic principles behind Noether’s theorem, the existence of
the continuous, infinitesimal and off-shell nilpotent
$(s_{d}^{2}=0,\,s_{b}^{2}=0)$ (co-)BRST symmetry transformations
$(s_{d})s_{b}$ lead to the derivations of the conserved (co-)BRST currents and
corresponding conserved charges. It has been shown, however, in our earlier
work (see, e.g. [40]) that the conserved Noether BRST charge is not off-shell
nilpotent. A systematic method has been developed to obtain the off-shell
nilpotent BRST charge [$Q_{b}^{(2)}$] from the non-nilpotent $(Q_{b}^{2}\neq
0)$ Noether BRST charge $(Q_{b})$. The conserved and off-shell nilpotent
version of the BRST charge has been taken into consideration [cf. Eq. (66)] in
our Subsec. 5.2 and a detailed analysis of the constraints from the
physicality criterion (w.r.t. the conserved and off-shell nilpotent BRST
charge) has been performed [cf. Eq. (73)] where we observe that the operator
forms of the first-class constraints of the original St$\ddot{u}$ckelberg-
modified massive Abelian 2-form theory (described by the Lagrangian density
${\cal L}_{S}^{(B)}$ [cf. Eq. (29)]) annihilate [cf. Eq. (68)] the physical
states (in the total Hilbert space of quantum states).

To obtain the off-shell nilpotent and conserved co-BRST charge
$(Q_{d}^{(2)})$, first of all, we derive the standard Noether conserved
current and corresponding conserved charge ($Q_{d}$). We wish to perform this
exercise in explicit fashion because the co-BRST and anti-co-BRST charges have
not been discussed in our earlier work [40] where we have developed a
systematic theoretical method to obtain the nilpotent (anti-)BRST charges from
the non-nilpotent standard Noether (anti-)BRST charges. In this context, first
of all, we note that the Noether conserved co-BRST current ($J_{(d)}^{\mu}$)
is

$\displaystyle J^{\mu}_{d}$ $\displaystyle=$
$\displaystyle(s_{d}B_{\alpha\beta})\,\frac{\partial{\cal L}_{\cal
B}}{\partial(\partial_{\mu}B_{\alpha\beta})}+(s_{d}\tilde{\phi}_{\alpha})\,\frac{\partial{\cal
L}_{\cal
B}}{\partial(\partial_{\mu}\tilde{\phi}_{\alpha})}+(s_{d}C_{\alpha})\,\frac{\partial{\cal
L}_{\cal
B}}{\partial(\partial_{\mu}C_{\alpha})}+(s_{d}\bar{C}_{\alpha})\,\frac{\partial{\cal
L}_{\cal B}}{\partial(\partial_{\mu}\bar{C}_{\alpha})}$ (112) $\displaystyle+$
$\displaystyle(s_{d}\tilde{\phi})\,\frac{\partial{\cal L}_{\cal
B}}{\partial(\partial_{\mu}\tilde{\phi})}+(s_{d}\bar{C})\,\frac{\partial{\cal
L}_{\cal B}}{\partial(\partial_{\mu}\bar{C})}+(s_{d}C)\,\frac{\partial{\cal
L}_{\cal B}}{\partial(\partial_{\mu}C)}+(s_{d}\beta)\,\frac{\partial{\cal
L}_{\cal B}}{\partial(\partial_{\mu}\beta)}$ $\displaystyle+$
$\displaystyle(\partial^{\mu}\bar{C}^{\nu}-\partial^{\nu}\bar{C}^{\mu})\,{\cal
B}_{\nu}+\lambda\,\partial^{\mu}\bar{\beta}+m\,\varepsilon^{\mu\nu\lambda\xi}\,\phi_{\nu}\,\partial_{\lambda}\bar{C}_{\xi}-\rho\,{\cal
B}^{\mu}-(\partial^{\mu}\bar{C}-m\,\bar{C}^{\mu})\,{\cal B},$

which, ultimately, turns out to be

$\displaystyle J^{\mu}_{d}$ $\displaystyle=$
$\displaystyle\varepsilon^{\mu\nu\lambda\xi}\,\Big{[}(m\,\phi_{\nu}-B_{\nu})\,(\partial_{\lambda}\bar{C}_{\xi})+\frac{m}{2}\,(\partial_{\nu}\bar{C}-m\,\bar{C}_{\nu})\,B_{\lambda\xi}\Big{]}+m\,(\partial^{\mu}C-m\,C^{\mu})\,\bar{\beta}$
(113) $\displaystyle-$
$\displaystyle(\partial^{\mu}C^{\nu}-\partial^{\nu}C^{\mu})\,(\partial_{\nu}\bar{\beta})+\tilde{\Phi}^{\mu\nu}\,(\partial_{\nu}\bar{C}-m\,\bar{C}_{\nu})+(\partial^{\mu}\bar{C}-m\,\bar{C}^{\mu})\,{\cal
B}$ $\displaystyle-$
$\displaystyle(\partial^{\mu}\bar{C}^{\nu}-\partial^{\nu}\bar{C}^{\mu})\,{\cal
B}_{\nu}+\rho\,{\cal B}^{\mu}-\lambda\,\partial^{\mu}\bar{\beta}.$

The conservation law $(\partial_{\mu}J^{\mu}_{(d)}=0)$ can be proven by using
the following EL-EoMs w.r.t. the fields:
$B_{\mu\nu},\tilde{\phi}_{\mu},C_{\mu},\bar{C}_{\mu},C,\bar{C},\beta,\tilde{\phi}$
from the Lagrangian density (109), namely;

$\displaystyle\varepsilon^{\mu\nu\lambda\xi}\,\partial_{\lambda}{\cal
B}_{\xi}-m^{2}\,B^{\mu\nu}+m\,(\Phi^{\mu\nu}+\frac{1}{2}\,\varepsilon^{\mu\nu\rho\sigma}\,\tilde{\Phi}_{\rho\sigma})-(\partial^{\mu}B^{\nu}-\partial^{\nu}B^{\mu})=0,$
$\displaystyle\partial_{\mu}\big{(}\tilde{\Phi}^{\mu\nu}-\frac{m}{2}\,\varepsilon^{\mu\nu\lambda\xi}\,B_{\lambda\xi}\big{)}=m\,{\cal
B}^{\nu}-\partial^{\nu}{\cal B},$
$\displaystyle\partial_{\mu}(\partial^{\mu}C^{\nu}-\partial^{\nu}C^{\mu})+\partial^{\nu}\lambda=m\,(\partial^{\nu}C-m\,C^{\nu}),$
$\displaystyle\partial_{\mu}(\partial^{\mu}\bar{C}^{\nu}-\partial^{\nu}\bar{C}^{\mu})-\partial^{\nu}\rho=m\,(\partial^{\nu}\bar{C}-m\,\bar{C}^{\nu}),$
$\displaystyle\partial_{\mu}\,(\partial^{\mu}\,\bar{C}-m\,\bar{C}^{\mu})=m\,\rho,\quad\partial_{\mu}\,(\partial^{\mu}\,C-m\,C^{\mu})=-\,m\,\lambda,$
$\displaystyle(\Box+m^{2})\,\bar{\beta}=0,\quad\partial\cdot{\cal
B}=-\,m\,{\cal B}.$ (114)

Some of the additional EoMs that emerge out from the above are:

$\displaystyle\Box\cdot{\cal B}-m\,(\partial\cdot{\cal
B})=0,\qquad\qquad(\Box+m^{2})\,{\cal B}=0,$
$\displaystyle\varepsilon^{\mu\nu\rho\sigma}\,\partial_{\rho}\big{(}m\,\phi_{\sigma}-B_{\sigma}\big{)}-(\partial^{\mu}{\cal
B}^{\nu}-\partial^{\nu}{\cal
B}^{\mu})-m\,(\tilde{\Phi}^{\mu\nu}+\frac{m}{2}\,\varepsilon^{\mu\nu\rho\sigma}B_{\rho\sigma})=0.$
(115)

The last entry, in the above equation, is nothing but the first entry of (114)
which is very useful in the proof of the conservation law
$(\partial_{\mu}\,J^{\mu}_{d}=0)$.

From the conserved current (113), one can compute the expression for the
conserved co-BRST charge $(Q_{d}=\int d^{3}x\ J^{0}_{d})$ as follows:

$\displaystyle Q_{d}$ $\displaystyle=$ $\displaystyle\int
d^{3}x\Big{[}m\,(\partial^{0}\,C-m\,C^{0})\,\bar{\beta}-(\partial^{0}\,C^{i}-\partial^{i}\,C^{0})\,\partial_{i}\,\bar{\beta}+\Phi^{0i}\,(\partial_{i}\,\bar{C}-m\,\bar{C}_{1})$
(116) $\displaystyle+$
$\displaystyle(\partial^{0}\,\bar{C}-m\,\bar{C}^{0})\,{\cal
B}-(\partial^{0}\,\bar{C}^{i}-\partial^{i}\,\bar{C}^{0})\,{\cal
B}_{i}+\rho\,{\cal B}^{0}-\lambda\,\partial^{0}\,\bar{\beta}$ $\displaystyle+$
$\displaystyle\varepsilon^{0ijk}\big{\\{}(m\,\phi_{i}-B_{i})\,\partial_{j}\,\bar{C}_{k}+\frac{m}{2}\,(\partial_{i}\,\bar{C}-m\,\bar{C}_{i})\,B_{jk}\big{\\}}\Big{]}.$

Using the basic principle behind the continuous symmetries and their
generators as the conserved Noether charges, it is straightforward to check
that the following is true, namely;

$\displaystyle s_{d}\,Q_{d}=-\,i\,\\{Q_{d},\,Q_{d}\\}\neq
0\quad\Longrightarrow\quad Q_{d}^{2}\neq 0,$ (117)

where the l.h.s can be computed from the direct application of the co-BRST
symmetry transformations ($s_{d}$) on the conserved charge ($Q_{d}$) which
lead to the following:

$\displaystyle s_{d}\,Q_{d}=\int d^{3}x\Big{[}m\,(\partial^{0}\,{\cal
B}-m\,{\cal B}^{0})\,\bar{\beta}-(\partial^{0}\,{\cal
B}^{i}-\partial^{i}\,{\cal B}^{0})\,\partial_{i}\,\bar{\beta}\Big{]}\neq 0.$
(118)

Hence, we conclude that the standard Noether conserved co-BRST charge
$(Q_{d})$ is not off-shell nilpotent $(Q_{d}^{2}\neq 0)$ of order two. To
obtain the off-shell nilpotent (i.e. $[Q_{d}^{(2)}]^{2}=0)$ version of the co-
BRST charge $(Q_{d}^{(2)}=0)$, we follow our proposal (see, e.g. [40] for
details) where the elegant combinations of (i) the Gauss divergence theorem,
(ii) the suitable equations of motion, and (iii) the off-shell nilpotent co-