††thanks: This work is dedicated in the memory of Prof. Roman Jackiw and Prof.
Trilochan Pradhan.

# A Gauge-Invariant Massive 2-form Model and its Quantization

Kumar Abhinav<EMAIL_ADDRESS>Centre for Theoretical Physics and
Natural Philosophy, Mahidol University, Nakhonsawan Campus Phayuha Khiri,
Nakhonsawan 60130, Thailand https://na.mahidol.ac.th/nas2020/kumar-abhinav-
page/

###### Abstract

An Abelian 2-form theory that maintains gauge-invariance despite having a mass
is considered in 3+1 dimensions. Though this mass owes to a non-local term the
corresponding classical equations of motion are completely local and subjected
to proper gauge fixing, yielding 3 massive degrees of freedom. Subsequently,
consistent Dirac brackets could be constructed for this system that agreed
with the subsequently obtained quantization under the non-covariant gauge
condition. The covariant gauge quantization reproduced the same spectrum
following the dissociation of a spurious massless mode from the physical
vector space. Moreover, a path-integral treatment of the theory saw ghosts
decoupling and proper BRST transformations were obtained. This 2-form field
mediates a screened interaction of Yukawa-type that mimics the Meisner effect
when topologically coupled to a fermion current.

††preprint: APS/123-QED

##  I Introduction

The interplay between gauge invariance and mass is very fundamental to
physics. In most cases, the absence of mass is synonymous with gauge
invariance [1] although it has long been established that mass and gauge
invariance can coexist when the gauge field interacts with conserved currents
[2, 3]. The observation by Proca [4] that a direct introduction of mass
reduces gauge invariance to a dynamical constraint led to the Stückelberg
model [5] wherein the presence of mass is compensated by extension of the
gauge invariance over multiple fields.

#####

As for the generation of mass, spontaneous symmetry breaking [6, 7] of an
interacting scalar field has been the most established gauge-invariant
mechanism [8, 9, 10, 11] in the as per the standard model following the
discovery of the Higgs particle [12]. Alternatively, the gauge mass can also
be generated through self-interaction of the gauge field $A_{\mu}$ in 2+1
dimensions [13, 14, 15] due to the topological Chern-Simons term [16].
Therein, the self-current term $\epsilon^{\mu\nu\rho}F_{\nu\rho}$ is conserved
by construction as the field strength
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is anti-symmetric,
which is due to the gauge invariance itself. Such topological mass generation
finds widespread application in various planar condensed matter systems
including the Hall effect and topological insulators [17]. Subsequent
extension of this topological mechanism of mass generation to the 3+1
dimensions [18, 19, 20, 21] requires the 1-form $A_{\mu}$ to interact with a
2-form $B_{\mu\nu}$ through the $B\wedge F$ term [22, 23] instead of self-
interaction. The $B\wedge F$ theory is both unitary and renormalizable and
generates gauge mass without spontaneous symmetry breaking [21].

#####

The question remains whether it is possible to construct a pure gauge theory,
i. e. one containing a single type field, in 3+1 dimensions which is massive.
Such a model was initially proposed by considering massive dispersion of the
dual 1-form field strength
$\leftindex^{*}F_{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}F_{\mu\nu}$
and was subsequently quantized [24]. In Lorentz gauge $\partial_{\mu}A^{\mu}$
this model reduces to the Proca theory. At a more fundamental level, a gauge-
invariant mass term was directly introduced to the free 1-form Lagrangian as
[25],

${\cal
L}_{A}=-\frac{1}{4}F_{\mu\nu}\left(1+\frac{m^{2}}{\partial^{2}}\right)F^{\mu\nu}.$
(1)

The apparent non-locality due to the inverse d’Alambertian $\partial^{-2}$,
responsible for a mass term $\frac{m^{2}}{2}A_{\mu}A^{\mu}$, does not hamper
locality and causality of the dynamics. Upon quantization, this theory
possesses a unitary vector space with a non-negative norm following suitable
gauge fixing [26].

#####

Whether such a non-local mass insertion will work beyond the 1-form is a
question worth asking considering the general possibility of massive gauge-
invariant dynamics without external interactions. In this work, the answer is
found to be affirmative in the case of 2-form fields as a corresponding non-
local Lagrangian of the form,

$\displaystyle{\cal
L}_{B}=\frac{1}{12}H_{\mu\nu\rho}\left(1+\frac{M^{2}}{\partial^{2}}\right)H^{\mu\nu\rho},$
$\displaystyle
H_{\mu\nu\rho}=\partial_{\mu}B_{\nu\rho}+\partial_{\nu}B_{\rho\mu}+\partial_{\rho}B_{\mu\nu},$
(2)

serves this purpose. The corresponding field strength $H_{\mu\nu\rho}$ is
invariant under the gauge transformation $B_{\mu\nu}\to
B_{\mu\nu}+\partial_{\mu}\Lambda_{\nu}-\partial_{\nu}\Lambda_{\mu}$, wherein
the gauge parameter further embodies a redundancy of its own:
$\Lambda_{\mu}\to\Lambda_{\mu}+\partial_{\mu}\sigma$. The 2-form enjoys
widespread utility across multiple areas of physics. Introduced by Kalb and
Ramond for explaining classical string dynamics [22], this tensor field
explains topological defects in superstrings [27], cosmic string interactions
[28], Abelian superfluid vortices [29, 30, 31] pertaining to type-II
superconductivity [32] and axionic charge allocation to black holes [33]. Few
recent developments in cosmology also employ 2-form models that explain
anisotropic inflation [34, 35], primordial gravitational waves with dynamical
dark energy [36] and inflatory physics [37]. A gauge-invariant mass term,
therefore, can have a considerably widespread application in physics by
providing a mass scale to the corresponding interaction mediated by the
2-form.

#####

Despite the non-locality, in this work, a consistent Dirac bracket
construction is obtained for this massive 2-form model. This construction
further complements the canonical quantization of the theory, obtained under
both covariant and non-covariant gauges. The physical mode invariably
corresponds to three massive degrees of freedom, two transverse and one
longitudinal, which gains further support from a path-integral treatment. As
expected of a massive gauge field [1] the 2-form is found to mediate a Yukawa-
type interaction when suitably coupled to charged fermions. Thus the present
theory may also model resistance-less transport, as was observed in the case
of the $B\wedge F$ model [38].

#####

The present work is organized as follows. Section II reviews the 2-form
formulation and the non-local 1-form model of Ref. [26], with possible origins
of such dynamics. The classical treatment of the non-local 2-form model is
carried out in Section III. The correct degrees of freedom are obtained from
multiple approaches, consistent with a subsequent Dirac bracket analysis that
identifies them despite the non-locality. Section IV contains covariant
canonical quantization of the system in both covariant and non-covariant
gauges, followed by a path-integral treatment. Conclusions and discussions
regarding possible further aspects make up Section V.

##  II Massive gauge dynamics in 3+1 dimensions

The free Kalb-Ramond 2-form field is described by the Lagrangian,

${\cal L}^{\rm free}_{\rm B}=\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho},$ (3)

which is invariant under the gauge transformation of $B_{\mu\nu}$. This
totally anti-symmetric field strength satisfies the Bianchi identity,

$\partial_{\mu}H_{\alpha\beta\gamma}-\partial_{\alpha}H_{\beta\gamma\mu}+\partial_{\beta}H_{\gamma\mu\alpha}-\partial_{\gamma}H_{\mu\alpha\beta}=0,$
(4)

by construction. The equation of motion for this system reads as,

$\partial_{\mu}H^{\mu\nu\rho}=0.$ (5)

Owing to the gauge redundancy, the free 2-form field contains only one
dynamical degree of freedom [22, 23]. The simplest way to see this is by re-
expressing the equation of motion in terms of the dual field-strength
$\leftindex^{*}H^{\mu}=\frac{1}{6}\epsilon^{\mu\nu\rho\sigma}H_{\nu\rho\sigma}$
as,

$\epsilon^{\mu\nu\rho\sigma}\partial_{\rho}\leftindex^{*}H_{\sigma}=0.$ (6)

Then the obvious identification $\leftindex^{*}H_{\mu}=\partial_{\mu}\varphi$
reduces ${\cal L}^{\rm free}_{\rm B}$ to the free Klein-Gordon action for
$\varphi$.

#####

Clearly, introducing a mass term of the usual form $B_{\mu\nu}B^{\mu\nu}$ to
the free 2-form Lagrangian would destroy the gauge invariance of the theory.
Instead, by virtue of interaction with a 1-form, a gauge-invariant 2-form mass
appeared in the topological $B\wedge F$ model [22, 23],

${\cal L}_{\rm
AB}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}+\frac{g}{2}\epsilon^{\mu\nu\rho\sigma}B_{\mu\nu}F_{\rho\sigma}.$
(7)

The interaction term above is an exterior product $\frac{g}{2}B\wedge F$. It
is topological in the sense that it does not depend on the metric and
therefore does not contribute to the stress-energy tensor. This term is
independently gauge-invariant for both 1- and 2-forms modulo total
derivatives, leaving the action invariant. The respective dynamics follow from
the equations of motion,

$\partial_{\mu}F^{\mu\nu}=-2g\leftindex^{*}{H}^{\nu}\quad{\rm
and}\quad\partial_{\alpha}H^{\alpha\mu\nu}=2g\leftindex^{*}{F}^{\mu\nu}.$ (8)

These equations can be decoupled into two copies of the massive dispersion,

$\left(\partial^{2}+4g^{2}\right){\cal F}=0,$ (9)

where ${\cal F}$ stands for both the field strengths $F_{\mu\nu}$ and
$H_{\mu\nu\rho}$. It was shown in Ref. [21] that this theory has three massive
degrees of freedom and thus the $B\wedge F$ model represents a massive spin-1
field. Also, this mass generation is distinct from the Higgs mechanism as
there is no symmetry breaking associated, and the theory is renormalizable.

#####

Both Higgs and $B\wedge F$ models, in addition to the well-known Stückelberg
mechanism, relied on interactions for generating gauge-invariant mass. The
Higgs mechanism already enjoys physical validation following the discovery of
the Higgs particle. As for the $B\wedge F$ term, it was shown to appear via
radiative induction from the interaction of the gauge fields with massive
fermions [39] and long-range spin currents [38]. In that sense, the
topologically massive gauge theory in 3+1 dimensions serves as an effective
description, similar to the radiatively induced Chern-Simons mass in 2+1
dimensions [40, 41, 42]. Therefore it makes sense to expect the massive 1- and
2-form models of Eq.s 1 and 2 can be obtained as effective descriptions.
Indeed, the non-local 1-form model is obtained from the Stückelberg theory,

$\displaystyle{\cal L}_{\rm ASt}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{m^{2}}{2}A_{\mu}A^{\mu}$
(10) $\displaystyle\qquad+A\cdot\partial D+\frac{1}{2m^{2}}\partial
D\cdot\partial D.$

by integrating out the Stückelberg scalar $\frac{1}{m}D$. It is to be noted
that the Stückelberg theory forms a part of the Higgs Lagrangian after the
spontaneous symmetry breaking is implemented [43]. Integrating out the 2-form
field in the $B\wedge F$ theory also leads to the same result. In 2+1
dimensions, interaction with a confined pseudo-vector fermion current in a
graphitic system can yield the same non-local 1-form mass [44]. In 1+1
dimensions, when the fermion field is integrated out in the Schwinger model
[45] and in the Ployakov’s model of gravity [46], the same effective
description emerges.

#####

On the other hand, if the 1-form is integrated out instead of the 2-form, the
topological $B\wedge F$ Lagrangian reduces to the 2-form Lagrangian ${\cal
L}_{\rm B}$. Independently, the same non-local 2-form model appears as the
effective description of a tensorial Stückelberg generalization,

$\displaystyle{\cal L}_{\rm BSt}$ $\displaystyle=$
$\displaystyle\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}-g^{2}B_{\mu\nu}B^{\mu\nu}$
(11)
$\displaystyle~{}+\frac{1}{2}B_{\mu\nu}\partial^{[\mu}D^{\nu]}-\frac{1}{16g^{2}}\partial_{[\mu}D_{\nu]}\partial^{[\mu}D^{\nu]}.$

The non-locality (isolated from the 2-form mass) is now compensated by the
‘Stückelberg 1-form’ $\frac{1}{2g}D_{\mu}$. This model serves as a non-
topological origin of the 2-form system ${\cal L}_{\rm B}$. Thus, both 1- and
2-form non-local models effectively describe multiple known models each,
representing the long-wavelength behavior of the parent theory. Following
Schwinger’s general result [2, 3], this makes sense that a gauge-invariant
mass should come through interactions [2, 3]. Therefore, a pure massive gauge
theory should be an effective description [47]. Effective models have wide-
spread use in multiple areas of physics [48] and especially in condensed
matter [49] including superconductivity [50].

#####

It makes sense to expect that these non-local gauge models will display
unitarity and local dynamics as their underlying theories do [51, 8, 9, 10,
11, 33]. The massive 1-form sector has already been shown to possess these
properties [26]. The present work analyzes the dynamics of and subsequently
quantizes the non-local 2-form sector. The quantized version of the massless
2-form has long been known [52], so is that of the $B\wedge F$ model [53]. In
the latter case, the mass-endowing topological $B\wedge F$ term allows the 1-
and 2-form sectors to retain the respective two transverse and single
longitudinal degrees of freedom from the free scenario despite their equations
of motion being coupled, yielding a massive spin-1 scenario as a whole.
Alternatively, the effective non-local 1- or 2-form descriptions are also
expected to carry three massive degrees of freedom each. This is because the
gauge-dependent part of the 1- and 2-form free propagators,

$\displaystyle\Delta_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{\partial^{2}}\left[\eta_{\mu\nu}-(1-\xi)\frac{\partial_{\mu}\partial_{\nu}}{\partial^{2}}\right]\quad{\rm
and}$ $\displaystyle\Delta_{\mu\nu,\alpha\beta}$ $\displaystyle=$
$\displaystyle-\frac{1}{\partial^{2}}\left[\eta_{\mu[\alpha}\eta_{\beta]\nu}-(1-\zeta)\frac{\eta_{[\mu[\alpha}\partial_{\beta]}\partial_{\nu]}}{\partial^{2}}\right],$
(12)

do not contribute to the effective description owing to the absolute anti-
symmetry of the field tensors and thus no gauge-fixing is required in the
process of constructing the effective theories.

#####

The expected spin-1 massive dynamics of the non-local 2-form model should be
relatively straightforward to verify with suitable quantum brackets since
there are no interactions with external fields. We shall see that the non-
local mass term does not obstruct the physical outcomes of this theory. It
serves to represent the nontrivial dynamics entirely, which is otherwise
attributed to interactions. This is directly demonstrated by shifting the
gauge fields as either,

$\displaystyle\quad A_{\mu}\rightarrow
A_{\mu}-2\frac{g}{\partial^{2}}\leftindex^{*}{H}_{\mu},\quad
B_{\mu\nu}\rightarrow B_{\mu\nu},$ $\displaystyle{\rm or}$ $\displaystyle\quad
A_{\mu}\rightarrow A_{\mu},\quad B_{\mu\nu}\rightarrow
B_{\mu\nu}+2\frac{g}{\partial^{2}}\leftindex^{*}{F}_{\mu\nu}.$ (13)

Such shifts respectively transform the $B\wedge F$ Lagrangian to either,

$\displaystyle{\cal L}_{\rm
AB}\rightarrow-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{12}H_{\mu\nu\rho}\left(1+\frac{4g^{2}}{\partial^{2}}\right)H^{\mu\nu\rho}$
$\displaystyle{\rm or}$ $\displaystyle{\cal L}_{\rm
AB}\rightarrow-\frac{1}{4}F_{\mu\nu}\left(1+\frac{4g^{2}}{\partial^{2}}\right)F^{\mu\nu}+\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}.$
(14)

The above decoupling of 1- and 2-form sectors directly shows that the non-
local term entirely represents the topological interaction. This is analogous
to the mixed Chern-Simons case in 2+1 dimensions,

${\cal
L}_{aA}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}f_{\mu\nu}f^{\mu\nu}+\kappa\epsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}a_{\rho}.$
(15)

with upper and lower case tensors representing field strengths of respective
one-forms. Similar re-definitions of fields decouple the two 1-form sectors
as,

${\cal L}_{\rm
aA}\rightarrow-\frac{1}{4}F_{1}\left(1-\frac{\kappa^{2}}{\partial^{2}}\right)F_{1}-\frac{1}{4}F_{2}F_{2},$
(16)

with $F_{1,2}$ standing for either of the field strengths. However, the gauge-
invariant mass term here has the wrong sign which is unique to the 2+1
dimensions. In the 2-form case in 3+1 dimensions, the equations of motion
display local dynamics and it is possible to construct both Dirac and quantum
brackets consistently. The path-integral treatment, however, requires the non-
locality to be compensated by an auxiliary 1-form as in Eq. 11.

###  II.1 Quantized non-local 1-form theory

Before taking up the non-local 2-form model, let us summarize the canonical
quantization of the non-local massive 1-form model obtained previously [26].
The non-local 1-form model is governed by the effective Lagrangian,

${\cal
L}_{A}=-\frac{1}{4}\left(1+\frac{m^{2}}{\partial^{2}}\right)F_{\mu\nu}F^{\mu\nu}+B\partial_{\mu}A^{\mu},$
(17)

with the mass term identified as $m=2g$. The Nakanishi-Lautrop auxiliary field
$B$ was introduced to impose the covariant gauge condition $\partial\cdot A=0$
leading to the 1-form equation of motion,

$\left(1+\frac{m^{2}}{\partial^{2}}\right)\partial_{\mu}F^{\mu\nu}=\partial^{\nu}B.$
(18)

As the above dynamics imply massless scalar dispersion of the auxiliary field
$\partial^{2}B=0$, in turn, the 1-form satisfies the following equation of
motion,

$\left(\partial^{2}+m^{2}\right)\partial^{2}A_{\mu}=0.$ (19)

Apart from the imposed gauge condition, there are no more constraints coming
out of the above equations owing to the mass term. This leaves out 3 massive
degrees of freedom. The massive mode $V_{\mu}=\partial^{2}A_{\mu}$ has a
propagator,

$K_{\mu\nu}=\frac{1}{\partial^{2}+m^{2}}\left[\eta_{\mu\nu}-\left(1-\alpha\right)\frac{\partial_{\mu}\partial_{\nu}}{\partial^{2}}\right],$
(20)

with a transverse gauge-independent part:
$\partial^{\mu}K_{\mu\nu}\left(\alpha=0\right)=0$. This is despite the
presence of mass, judiciously appearing as the real momentum-space pole of the
propagator, just like in the pure Chern-Simons propagator in 2+1 dimensions.
Since the respective pole of the mixed Chern-Simons theory of Eq. 15 is
imaginary, the non-local 1-form model (and so the $B\wedge F$ model) is
analogous only to the pure C-S case of 2+1 dimensions. Subsequently, as
expected of a massive mediating field, interaction with a conserved current
$J^{\mu}$ made of localized charge distribution
$J_{0}(\bm{x})=q_{1}\delta^{3}(\bm{x}-\bm{x}_{1})+q_{2}\delta^{3}(\bm{x}-\bm{x}_{2})$
constitute a Yukawa-type potential:

$\displaystyle V(\bm{x}_{1},\bm{x}_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\,d^{3}x\,J_{0}(\bm{x})\frac{1}{-\nabla^{2}+m^{2}}J_{0}(\bm{x})$
(21) $\displaystyle=$
$\displaystyle\frac{q_{1}q_{2}}{4\pi}\frac{e^{-m|\bm{x}_{1}-\bm{x}_{2}|}}{|\bm{x}_{1}-\bm{x}_{2}|}.$

In general, such a screened interaction may support localized structures.

#####

There is also a massless mode
$S_{\mu}=\left(\partial^{2}+4g^{2}\right)A_{\mu}$ in this model, characterized
by the Maxwell-like propagator,

$K^{\prime}_{\mu\nu}=\frac{1}{\partial^{2}}\left[\eta_{\mu\nu}-\left(1-\alpha^{\prime}\right)\frac{\partial_{\mu}\partial_{\nu}}{\partial^{2}}\right].$
(22)

However, this mode conveniently decouples from the theory having an unphysical
norm and only the massive mode $V_{\mu}$ defines the physical sub-space [26].
The physical subspace corresponds to the non-equal-time quantum bracket,

$[A_{\mu}(x),A_{\nu}(y)]=-i\eta_{\mu\nu}\Delta(x-y)+i\partial_{\mu}^{x}\partial_{\nu}^{x}\Delta^{\prime}(x-y).$
(23)

This construction follows the method of covariant canonical quantization [54]
implemented through the invariant commutator function obeying,

$\displaystyle\partial_{x,y}^{2}\Delta^{\prime}(x-y)=\Delta(x-y),$
$\displaystyle\Delta^{\prime}(x)=-\frac{1}{m^{2}}\Delta(x)+\frac{1}{m^{2}}\Delta(x,m=0),\quad{\rm
with},$ $\displaystyle\Delta(x)=-\frac{i}{(2\pi)^{3}}\int d^{4}p~{}{\rm
sgn}(p_{0})\delta(p^{2}-m^{2}){\rm e}^{-ip.x}.$ (24)

Both the equations of motion and the gauge-fixing condition conform to the
bracket and subsequently, the massive and mass-less modes get decoupled,

$\left[V_{\mu},~{}S_{\mu}\right]=0.$ (25)

Further, $S_{\mu}$ corresponds to either zero or negative norm states whereas
the massive mode $V_{\mu}$ represents semi-positive-definite states. Subjected
to the single gauge condition the theory describes a spin-1 massive particle
with three degrees of freedom.

#####

In the non-covariant temporal gauge: $A_{0}=0$, without introducing any
auxiliary fields, the $\nu=0$ component of the EOM Eq. 18 implies,

$\left(1+\frac{m^{2}}{\partial^{2}}\right)\bm{\nabla}\cdot\bm{A}=0.$ (26)

Subsequently, the remaining components of the equation of motion describe
three massive degrees of freedom:

$\left(\partial^{2}+m^{2}\right)\bm{A}=0.$ (27)

#####

As had already been mentioned, introducing a dynamical auxiliary field to
remove the 1-form non-local term leads to the Stückelberg system of Eq. 10.
Therein the original gauge redundancy now gets extended to the well-known
shared one,

$A_{\mu}\to A_{\mu}+\partial_{\mu}\Lambda,\quad D\to D-m^{2}\Lambda,$ (28)

which is consistent with the corresponding equations of motion:

$\displaystyle\left(\partial^{2}+m^{2}\right)A_{\mu}-\partial_{\mu}\partial\cdot
A=-\partial_{\mu}D,$ $\displaystyle m^{2}\partial\cdot A=-\partial^{2}D.$ (29)

As the second equation can be obtained from the first, eliminating $D$ yields
a massive vector
$Z_{\mu}=A_{\mu}+\frac{1}{m^{2}}\partial_{\mu}D=A_{\mu}-\frac{1}{\partial^{2}}\partial_{\mu}\partial\cdot
A$ with vanishing four-divergence by construction. As for gauge fixing, the
choice $\partial\cdot A-D=0$ provides massive dynamics for both $A_{\mu}$ and
$D$ having non-negative energy. The required choice of gauge parameter
$\Lambda$ invariably leads to $\partial\cdot A=0$ and $D=0$ independently,
subjected to the dynamical equations. Therefore it is equivalent to a direct
gauge choice of $\partial\cdot A=0$ that also leads to $D=0$. In any case, the
number of degrees of freedom is 3 which are all massive.

#####

The quantum versions of both Stückelberg [51] and non-local 1-form model [26]
lead to the same physical states. Thus the non-local term does not hamper the
physics subjected to consistent quantization schemes, an aspect that can be
quite useful in other non-local gauge models. Following the same logic, the
non-local 2-form theory of Eq. 2 is also expected to display similar well-
behaved dynamics despite the non-local term. In the following this indeed is
shown to be the case as physical modes are identified in a canonical set-up,
followed by consistent quantization schemes.

##  III Canonical Structure of Non-local 2-form model

A straightforward demonstration of the massive dynamics in the non-local
2-form case of Eq. 2 is to introduce a couple of totally anti-symmetric
auxiliary fields $J_{\mu\alpha\beta}$ and $\lambda_{\mu\alpha\beta}$ to the
Lagrangian as,

$\displaystyle{\cal L}_{B}$ $\displaystyle=$
$\displaystyle\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}+g^{2}\partial_{\mu}B_{\alpha\beta}J^{\mu\alpha\beta}$
(30)
$\displaystyle~{}+\lambda_{\mu\alpha\beta}\left(H^{\mu\alpha\beta}-\partial^{2}J^{\mu\alpha\beta}\right),$

with the identification $M=2g$ understood. Elimination of the auxiliary fields
by using respective equations of motion leads to the 2-form dynamical
equation,

$\left(1+\frac{4g^{2}}{\partial^{2}}\right)\partial_{\mu}H^{\mu\nu\rho}=0.$
(31)

When expressed in terms of the dual field $\leftindex^{*}H_{\mu}$, it yields 3
massive modes: $\left(\partial^{2}+4g^{2}\right)\leftindex^{*}H_{\mu}=0$
subjected to the built-in transverse-ness $\partial\cdot\leftindex^{*}H=0$,
with the additional condition,

$\left(1+\frac{4g^{2}}{\partial^{2}}\right)\epsilon^{\mu\nu\alpha\beta}\partial_{\alpha}\leftindex^{*}{H}_{\beta}=0.$
(32)

Unlike in Eq. 6, the non-local mass term prevents identification with a scalar
field. Though one can still construct a massless scalar mode as
$\left(1+\frac{4g^{2}}{\partial^{2}}\right)\leftindex^{*}H_{\mu}=\partial_{\mu}\varphi$,
it will correspond to the aforementioned spurious one that dissociates from
the physical space upon proper quantization. The $\partial^{-2}$ operator can
always be attributed to an interaction with a massless boson field and thus is
carried forward in the analysis. We will get to see that this will not
obstruct local dynamics. The 2-form generalization to the Stückelberg
mechanism in Eq. 11 justifies this aspect wherein the ‘Stückelberg 1-form’
$\frac{1}{2g}D^{\mu}$ is physical in the sense that it compensates for the
non-local term and contributes to the dynamics as well. This extended but
local system is invariant under a shared gauge transformation,

$B_{\mu\nu}\to
B_{\mu\nu}+\partial_{\mu}\Lambda_{\nu}-\partial_{\nu}\Lambda_{\mu},\quad
D_{\mu}\to D_{\mu}+4g^{2}\Lambda_{\mu},$ (33)

as well as under an independent gauge transformation of the $D^{\mu}$ sector,

$D_{\mu}\to D_{\mu}+\partial_{\mu}\lambda.$ (34)

Both these transformations are further consistent with the dynamical
equations:

$\displaystyle\left(\partial^{2}+4g^{2}\right)B_{\mu\nu}-\partial_{[\mu}\left(\partial\cdot
B\right)_{\nu]}=\partial_{[\mu}D_{\nu]},$
$\displaystyle\partial^{2}D_{\mu}-\partial_{\mu}\partial\cdot
D=4g^{2}\left(\partial\cdot B\right)_{\mu},$ (35) $\displaystyle{\rm
where}\quad\left(\partial\cdot B\right)_{\mu}=\partial^{\nu}B_{\nu\mu}.$

Just like the 1-form case, the last equation above is an outcome of the first.
On eliminating the auxiliary field the dynamics becomes that of a massive
gauge-invariant yet divergence-less combined 2-form
$Z_{\mu\nu}=B_{\mu\nu}-\frac{1}{4g^{2}}\partial_{[\mu}D_{\nu]}=B_{\mu\nu}-\frac{1}{\partial^{2}}\partial_{[\mu}\left(\partial\cdot
B\right)_{\nu]}$. The natural choice for gauge in the shared gauge sector
$\left(\partial\cdot B\right)_{\mu}+D_{\mu}=0$ yields massive dispersion for
both $B_{\mu\nu}$ and $D_{\mu}$ with semi-positive definite energy, along-with
$\partial\cdot D=0$. The corresponding class of vector gauge parameter
$\Lambda_{\mu}$ is necessarily transverse $\partial\cdot\Lambda=0$.
Subsequently, the gauge parameter $\lambda$ of the independent 1-form sector
is free to implement $\partial\cdot D=0$. Therefore it is equivalent to
choosing $\left(\partial\cdot B\right)_{\mu}=0$ and $\partial\cdot D=0$
independently, which implies a massive $B_{\mu\nu}$ and on-shell vanishing
$D_{\mu}$ subjected to suitable boundary data. In this case, the equivalent
class of $\Lambda_{\mu}$ is also massive whereas that of $\lambda$ is
massless. Now considering the constraint $\partial\cdot\left(\partial\cdot
B\right)=0$ on the 2-form gauge condition itself, there are 3 massive degrees
of freedom left.

#####

As claimed before, there is no non-locality in the dynamical equations of the
massive 2-form model. Thus it is analytically efficient to stick with the non-
local version of the theory. In the following, no auxiliary fields will be
introduced to obtain the canonical quantization consistently. Instead, the
non-locality will be removed once a judicious gauge condition is imposed.
Though spurious massless modes appear as a result in some cases, they decouple
conveniently from the physical Hilbert space. This may serve as a template to
handle non-local terms in quantizing a gauge model.

###  III.1 Dynamical modes

To carry out a canonical analysis, it is useful to manifest the gauge theory
in a non-covariant form consistent with the Hamiltonian formalism [1].
Accordingly, the massive 2-form model of Eq. 2 can be re-cast as,

$\displaystyle{\cal L}_{\rm
B}=\frac{1}{2}\left(\Pi_{k}\bigodot\Pi_{k}-\Phi\bigodot\Phi\right),$
$\displaystyle\bigodot=1+\frac{4g^{2}}{\partial^{2}},$
$\displaystyle\bm{\Pi}=\partial_{0}\widetilde{\bm{B}}-\bm{\nabla}\times\overline{\bm{B}},\quad\Phi=\bm{\nabla}\cdot\widetilde{\bm{B}},$
$\displaystyle\widetilde{B}_{i}=\frac{1}{2}\epsilon_{ijk}B_{jk},\quad\overline{B}_{i}=B_{0i},$
$\displaystyle(i,j,k,\cdots)=1,2,3.$ (36)

The equations of motion of this system can be resolved in terms of the
3-vectors $\widetilde{\bm{B}}$ and $\overline{\bm{B}}$ as,

$\bigodot\partial_{\mu}H^{\mu\nu\rho}=0\Rightarrow\begin{cases}\bigodot\left(\partial_{0}\bm{\Pi}-\bm{\nabla}\Phi\right)=0,\\\
\bigodot\bm{\nabla}\times\bm{\Pi}=0.\end{cases}$ (37)

The Bianchi identity of Eq. 4, that owes only to the inherent anti-symmetry of
$H_{\mu\nu\rho}$, now takes the form,

$\partial_{0}\Phi-\bm{\nabla}\cdot\bm{\Pi}=0.$ (38)

.A massive dispersion can now be obtained simply by substituting one field in
terms of the other:

$\partial^{2}\bigodot\mathfrak{G}=\left(\partial^{2}+4g^{2}\right)\mathfrak{G}=0\qquad\mathfrak{G}=\left(\bm{\Pi},\,\Phi\right),$
(39)

similar to the massive 1-form case before. The massive fields
$\left(\bm{\Pi},\,\Phi\right)$ represent three degrees of freedom following
the scalar constraint of Eq. 38. At this level, there are no spurious massless
modes since $\left(\bm{\Pi},\,\Phi\right)$ are gauge-invariant quantities. A
hint of the spurious massless modes appears if $\bigodot\mathfrak{G}$ are
taken as field variables. Following Bianchi’s identity, this mode also
contains three degrees of freedom, and from the second of Eq. 37 it is also
irrotational.

#####

The presence of an antisymmetric source field $J_{\mu\nu}=-J_{\nu\mu}$,

${\cal
L}_{B}=\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}+\frac{1}{2}B_{\mu\nu}J^{\mu\nu},$
(40)

modifies the equations of motion to,

$\displaystyle\bigodot\partial_{\mu}H^{\mu\nu\rho}=J^{\nu\rho}$
$\displaystyle\Rightarrow\begin{cases}\bigodot\left(\partial_{0}\bm{\Pi}-\bm{\nabla}\Phi\right)=\widetilde{\bm{J}},\\\
\bigodot\bm{\nabla}\times\bm{\Pi}=\overline{\bm{J}},\end{cases}$ (41)
$\displaystyle\widetilde{\rm J}_{i}=\frac{1}{2}\epsilon_{ijk}{\rm
J}_{jk},\quad\overline{\rm J}_{i}={\rm J}_{0i},$

with the Bianchi identity prevailing. The current conservation yields,

$\partial_{\mu}J^{\mu\nu}=0\Rightarrow\begin{cases}\partial_{0}\overline{\bm{J}}-\bm{\nabla}\times\widetilde{\bm{J}}=0,\\\
\bm{\nabla}\cdot\overline{\bm{J}}=0,\end{cases}$ (42)

wherein the second equation is actually a subsidiary of the first111We would
get $\partial_{0}\bm{\nabla}\cdot\overline{\bm{J}}=0$ which leads to
$\bm{\nabla}\cdot\overline{\bm{J}}=f(\bm{x})$. The inhomogeneous part
$f(\bm{x})$ can safely be set equal to zero as its form is not going to affect
the dynamics.. So there are three independent current components that resolve
the massive dynamics,

$\left(\partial^{2}+4g^{2}\right)\bm{\Pi}=\partial_{0}\widetilde{\bm{J}}+\bm{\nabla}\times\overline{\bm{J}},\quad\left(\partial^{2}+4g^{2}\right)\Phi=\bm{\nabla}\cdot\widetilde{\bm{J}},$
(43)

having 3 degrees of freedom.

###  III.2 The gauge redundancy

The gauge dependence comes into effect in terms of the dynamical fields
$\widetilde{\bm{B}}$ and $\overline{\bm{B}}$, the latter being redundant (a
Lagrange multiplyer) as its conjugate momentum is absent in the Hamiltonian.
The equations of motions 37 then take the forms,

$\displaystyle\partial_{0}^{2}\bigodot\widetilde{\bm{B}}=\bm{\nabla}\left(\bm{\nabla}\cdot\bigodot\widetilde{\bm{B}}\right)+\partial_{0}\bm{\nabla}\times\bigodot\overline{\bm{B}}$
$\displaystyle{\rm and}~{}$
$\displaystyle\partial_{0}\bm{\nabla}\times\bigodot\widetilde{\bm{B}}=\bm{\nabla}\times\bm{\nabla}\times\bigodot\overline{\bm{B}},$
(44)

whereas the Bianchi identity vanishes identically as usual. The gauge
transformations of the 2-form are now expressed as,

$\widetilde{\bm{B}}\rightarrow\widetilde{\bm{B}}+\bm{\nabla}\times\bm{\Gamma},\qquad\overline{\bm{B}}\rightarrow\overline{\bm{B}}+\partial_{0}\bm{\Gamma}+\bm{\nabla}\Gamma.$
(45)

with gauge parameter $\Gamma_{\mu}=\left(\Gamma,\,-\bm{\Gamma}\right)$.
Particular gauge fixing is now required to identify the degrees of freedom. On
adopting the standard covariant gauge,

$\partial_{\mu}B^{\mu\nu}=0\Rightarrow\begin{cases}\partial_{0}\overline{\bm{B}}+\bm{\nabla}\times\widetilde{\bm{B}}=0,\\\
\bm{\nabla}\cdot\overline{\bm{B}}=0,\end{cases}$ (46)

one obtains the massive dispersion,

$\left(\partial^{2}+4g^{2}\right)\mathfrak{B}=0,\quad\mathfrak{B}=\widetilde{\bm{B}},\,\overline{\bm{B}}.$
(47)

Among the four gauge conditions of Eq. 46, the last one is a subsidiary of the
other three owing to the inherent anti-symmetry of $B^{\mu\nu}$, and it makes
$\overline{\bm{B}}$ transverse. Given non-zero mass, this transverse field
vanishes in the rest frame, leaving out 3 massive degrees of freedom
represented by $\widetilde{\bm{B}}$. In contrast, the massless case has the
transverse components of $\widetilde{\bm{B}}$ equated to the redundant field
$\overline{\bm{B}}$, leaving out a single longitudinal degree of freedom.
Starting from arbitrary fields, the explicit gauge choices that lead to the
covariant gauge,

$\bm{\Gamma}=-\frac{1}{\partial^{2}}\left(\partial_{0}\overline{\bm{B}}+\bm{\nabla}\times\widetilde{\bm{B}}\right),\qquad\Gamma=\frac{1}{\partial^{2}}\bm{\nabla}\cdot\overline{\bm{B}}.$
(48)

are themselves invariably constrained as $\partial\cdot\Gamma=0$ and thus only
3 independent constraint conditions can be imposed. Since these gauge
parameters are further redundant as
$\Gamma_{\mu}\rightarrow\Gamma_{\mu}+\partial_{\mu}\gamma$, one subsequently
gets $\partial^{2}\gamma=0$. Non-trivial choices for $\gamma$ are consistent
with the chosen covariant gauge.

#####

Sacrificing manifest covariance makes it simpler to resolve the dynamical
modes [1] as redundant fields can directly be eliminated as a gauge choice:
$\overline{\bm{B}}=0$222It is equivalent to the temporal gauge of one-form
gauge theory.. This is achieved by choosing three components of $\Gamma_{\mu}$
left out after fixing $\gamma$. Then the second of the equations of motion in
44 eludes to,

$\bigodot\bm{\nabla}\times\widetilde{\bm{B}}=f(\bm{x}),$ (49)

Since $f(\bm{x})$ is arbitrary and independent of time, a choice of
$f(\bm{x})=0$ does not hamper the system dynamics. As a consequence, the other
remaining equation of motion reads as,

$\left(\partial^{2}+4g^{2}\right)\widetilde{\bm{B}}=0,$ (50)

with 3 free physical components. Crucial to this result, due to the presence
of mass, the condition $\bigodot\bm{\nabla}\times\widetilde{\bm{B}}=0$ does
not constrain the components of $\widetilde{\bm{B}}$. This ‘temporal’ gauge is
equivalent to the covariant gauge in the rest frame with
$p^{\mu}=\left(2g,\,\bm{0}\right)$, a correspondence that is absent in the
massless case. In the latter scenario, instead, the condition
$\bm{\nabla}\times\widetilde{\bm{B}}=0$ negates two more degrees of freedom.

#####

Physical degrees of freedom equivalently get resolved in the light-cone gauge.
The equations of motion in the momentum space,

$\displaystyle\bigodot{}_{p}\left(p^{2}B^{\mu\nu}-p^{\mu}B^{\nu}+p^{\nu}B^{\mu}\right)=0,$
(51) $\displaystyle{\rm where}~{}\bigodot{}_{p}=1-\frac{4g^{2}}{p^{2}},\quad
B^{\mu}=p_{\alpha}B^{\alpha\mu},$

are now subjected to the light-cone coordinates
$p^{\mu}=\left(p^{+},p^{-},p^{2},p^{3}\right)$ with $p^{+}\neq 0$ [57]. The
secondary gauge parameter $\gamma$ is chosen to fix the primary parameter
component $\Gamma^{+}=0$. Then the remaining components $\Gamma^{-,l}$, with
$l=2,3$, can be chosen to fix the light-cone gauge $B^{+\mu}=0$, which
essentially is equivalent to the ‘temporal gauge’ above. As a result one
obtains,

$B^{+}=0,\quad B^{-}=p^{l}B^{-l},\quad B^{l}=p^{+}B^{-l}-p^{m}B^{ml}.$ (52)

Different components of the equation of motion then culminate to,

$\displaystyle\left(\mu=+,~{}\nu=-\right):\quad\bigodot{}_{p}p^{l}B^{-l}=0,$
(53)
$\displaystyle\left(\mu=+,~{}\nu=l\right):\quad\bigodot{}_{p}B^{-l}=\frac{p^{m}}{p^{+}}\bigodot{}_{p}B^{ml},$
(54) $\displaystyle\left(\mu=-,~{}\nu=l\right):\quad(p^{2}-4g^{2})B^{-l}=0.$
(55)

Two massive degrees of freedom can now be identified for $l=2,3$ and a third
one $B^{ml}\equiv B^{23}$ is a result of substituting from Eq. 54333In the
massless case $B^{-l}\propto B^{ml}$ in Eq. 54 leaving out a single massless
degree of freedom..

###  III.3 The Hamiltonian analysis

The present theory’s canonical treatment requires adequately handling the
constraints to identify physical variables so that correct canonical brackets
can be constructed. In terms of the dynamical 3-vectors, the non-local 2-form
Lagrangian is expressed as,

$\displaystyle{\cal
L}_{B}=\frac{1}{2}\Bigg{[}\left(\partial_{0}\widetilde{\bm{B}}-\bm{\nabla}\times\overline{\bm{B}}\right)_{i}\bigodot\left(\partial_{0}\widetilde{\bm{B}}-\bm{\nabla}\times\overline{\bm{B}}\right)_{i}$
$\displaystyle\qquad\qquad\quad-\bm{\nabla}\cdot\widetilde{\bm{B}}\bigodot\bm{\nabla}\cdot\widetilde{\bm{B}}\Bigg{]}.$
(56)

The corresponding conjugate momentum components are,

$\displaystyle\widetilde{\Pi}_{i}=\frac{\partial{\cal
L}_{B}}{\partial\partial_{0}\widetilde{B}_{i}}=\bigodot\left(\partial_{0}\widetilde{\bm{B}}-\bm{\nabla}\times\overline{\bm{B}}\right)_{i}$
$\displaystyle{\rm and}~{}$
$\displaystyle\overline{\Pi}_{i}=\frac{\partial{\cal
L}_{B}}{\partial\partial_{0}\overline{B}_{i}}=0,$ (57)

wherein the second equation depicts constraints. The non-local operator
$\bigodot$ is maintained in the definition of momentum as it gets negated by
gauge-fixing without hampering the physical outcome. As there are constraints
involved, it is customary to employ the method of Dirac brackets [45] to
construct the canonical brackets of this system. For this purpose, the
canonical Hamiltonian is obtained through the Legendre transformation as,

$\displaystyle{\cal H}_{\rm Can}$ $\displaystyle=$
$\displaystyle\widetilde{\Pi}_{i}\,\partial_{0}\widetilde{B}_{i}-{\cal L}_{B}$
(58) $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\widetilde{\Pi}_{i}\frac{1}{\bigodot}\widetilde{\Pi}_{i}+\Phi\bigodot\Phi\right)+\overline{\bm{B}}\cdot\left(\bm{\nabla}\times\widetilde{\bm{\Pi}}\right).$

The primary constraints of the system are labeled as,

$\overline{\Theta}_{i}=\overline{\Pi}_{i}=0,$ (59)

which are needed to be included in the primary Hamiltonian,

$\displaystyle{\cal H}_{\rm P}$ $\displaystyle=$ $\displaystyle{\cal H}_{\rm
Can}+\lambda_{i}\overline{\Theta}_{i}$ (60) $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\widetilde{\Pi}_{i}\frac{1}{\bigodot}\widetilde{\Pi}_{i}+\Phi\bigodot\Phi\right)+\overline{\bm{B}}\cdot\left(\bm{\nabla}\times\widetilde{\bm{\Pi}}\right)$
$\displaystyle\quad+\lambda_{i}\overline{\Theta}_{i},$

through Lagrangian multipliers $\lambda_{i}$. Then the equal-time Poisson
brackets pertaining to this system are defined as,

$\displaystyle\left\\{\mathfrak{A}(\bm{x},t),\,\mathfrak{B}(\bm{y},t)\right\\}_{\rm
PB}$
$\displaystyle\qquad=\int\,d^{3}z\,\Bigg{[}\frac{\partial\mathfrak{A}(\bm{x},t)}{\partial\widetilde{B}_{i}(\bm{z},t)}\frac{\partial\mathfrak{B}(\bm{y},t)}{\partial\widetilde{\Pi}_{i}(\bm{z},t)}-\frac{\partial\mathfrak{A}(\bm{x},t)}{\partial\widetilde{\Pi}_{i}(\bm{z},t)}\frac{\partial\mathfrak{B}(\bm{y},t)}{\partial\widetilde{B}_{i}(\bm{z},t)}+\frac{\partial\mathfrak{A}(\bm{x},t)}{\partial\overline{B}_{i}(\bm{z},t)}\frac{\partial\mathfrak{B}(\bm{y},t)}{\partial\overline{\Pi}_{i}(\bm{z},t)}-\frac{\partial\mathfrak{A}(\bm{x},t)}{\partial\overline{\Pi}_{i}(\bm{z},t)}\frac{\partial\mathfrak{B}(\bm{y},t)}{\partial\overline{B}_{i}(\bm{z},t)}\Bigg{]},$
(61)

over the complete phase-space
$\left\\{\left(\widetilde{B}_{i},\,\overline{B}_{i},\,\widetilde{\Pi}_{i},\,\overline{\Pi}_{i}\right)\right\\}$.
The non-trivial fundamental Poisson brackets in the complete phase-space are,

$\displaystyle\left\\{\widetilde{B}_{i}(\bm{x},t),\,\widetilde{\Pi}_{j}(\bm{y},t)\right\\}_{\rm
PB}=\delta_{ij}\delta^{3}\left(\bm{x}-\bm{y}\right),$
$\displaystyle\left\\{\overline{B}_{i}(\bm{x},t),\,\overline{\Pi}_{j}(\bm{y},t)\right\\}_{\rm
PB}=\delta_{ij}\delta^{3}\left(\bm{x}-\bm{y}\right),$ (62)

with $\overline{\Pi}_{i}=0$ depicting the constraint hyper-surface. Any
variable $\mathfrak{C}(\bm{x},t)$ that vanishes on this hyper-surface is
termed as weakly vanishing: $\mathfrak{C}(\bm{x},t)\approx 0$.

#####

The constraints evolve with time according to the primary Hamiltonian as,

$\displaystyle\partial_{0}\overline{\Theta}_{i}$ $\displaystyle=$
$\displaystyle\left\\{\overline{\Theta}_{i}(\bm{x},t),\,\int\,d^{3}y\,{\cal
H}_{\rm P}(\bm{y},t)\right\\}_{\rm PB}$ (63) $\displaystyle\equiv$
$\displaystyle\int\,d^{3}y\,\left\\{\overline{\Pi}_{i}(\bm{x},t),\,\overline{\bm{B}}(\bm{y},t)\cdot\left(\bm{\nabla}_{\rm
y}\times\widetilde{\bm{\Pi}}(\bm{y},t)\right)\right\\}_{\rm PB}$
$\displaystyle=$ $\displaystyle-\left(\bm{\nabla}_{\rm
x}\times\widetilde{\bm{\Pi}}(\bm{x},t)\right)_{i}.$

Since the constraint itself vanishes weakly it is weakly stationary too,
yielding the secondary constraints,

$\Theta_{i}(\bm{x},t):=-\left(\bm{\nabla}\times\widetilde{\bm{\Pi}}(\bm{x},t)\right)_{i}\approx
0.$ (64)

These secondary constraints are found to be stationary over the whole phase
space,

$\displaystyle\partial_{0}\Theta_{i}(\bm{x},t)$
$\displaystyle\quad\equiv\int\,d^{3}y\,\left\\{\left(\bm{\nabla}_{\rm
x}\times\widetilde{\bm{\Pi}}(\bm{x},t)\right)_{i},\,\frac{1}{2}\Phi(\bm{y},t)\bigodot\Phi(\bm{y},t)\right\\}_{\rm
PB}$ $\displaystyle\quad=0,$ (65)

and thus the chain terminates. Now we are left with two sets of first-class
constraints $\overline{\Theta}_{i}$ and $\Theta_{i}$ since they commute among
themselves following the fundamental Poisson brackets. As the presence of
first-class constraints marks gauge redundancy [45] an equal number of gauge
fixing conditions are needed. By the ‘temporal’ gauge before we choose these
conditions to be,

$\overline{\Xi}_{i}=\overline{B}_{i}\approx 0,\quad{\rm
and}\quad\Xi_{i}=\bigodot\left(\bm{\nabla}\times\widetilde{\bm{B}}\right)_{i}\approx
0.$ (66)

The non-vanishing equal-time Poisson brackets of the constraints and gauge
fixing conditions are,

$\displaystyle\left\\{\overline{\Theta}_{i}(\bm{x},t),\,\overline{\Xi}_{j}(\bm{y},t)\right\\}_{\rm
PB}=\delta_{ij}\delta^{3}\left(\bm{x}-\bm{y}\right),$
$\displaystyle\left\\{\Theta_{i}(\bm{x},t),\,\Xi_{j}(\bm{y},t)\right\\}_{\rm
PB}=\bigodot{\cal P}_{ij}\delta^{3}(\bm{x}-\bm{y}),$ $\displaystyle{\cal
P}_{ij}=\delta_{ij}\bm{\nabla}^{2}-\partial_{i}\partial_{j},$ (67)

designating the constraints to be second class. Thus the Lagrange multipliers
can be obtained for $\overline{\bm{\Xi}},\,\bm{\Xi}$ being stationary which
leads to $\lambda_{i}=0$. On defining a 4-component variable
$\Psi_{A}=\left(\overline{\bm{\Theta}},\,\bm{\Theta},\,\overline{\bm{\Xi}},\,\bm{\Xi}\right)$
with $A=1,2,3,4$ the matrix of the Poisson brackets among the components of
$\Psi_{A}$ is obtained as,

$\mathfrak{C}=\begin{pmatrix}0&0&-\mathbb{I}&0\\\ 0&0&0&-\bigodot{\cal P}\\\
\mathbb{I}&0&0&0\\\ 0&\bigodot{\cal
P}&0&0\end{pmatrix}\,\delta^{3}\left(\bm{x}-\bm{y}\right).$ (68)

In this matrix, each element is a $3\times 3$ matrix. It is the inverse of the
matrix $\mathfrak{C}$:

$\mathfrak{C}^{-1}=\begin{pmatrix}0&0&\mathbb{I}&0\\\ 0&0&0&{\cal
P}^{-1}\bigodot^{-1}\\\ -\mathbb{I}&0&0&0\\\ 0&-{\cal
P}^{-1}\bigodot^{-1}&0&0\end{pmatrix}\,\delta^{3}\left(\bm{x}-\bm{y}\right),$
(69)

that appears in the Dirac brackets. Since the operator $\bigodot^{-1}$ is
local, the Dirac brackets will not suffer from the concerned ambiguity, and
the initial decision to adopt a non-local canonical momentum is thus
validated. Another subtlety here is that ${\cal P}$ is transverse and thus
${\cal P}^{-1}$ does not exist, a consequence of the secondary gauge
invariance $\Gamma_{\mu}\rightarrow\Gamma_{\mu}+\partial_{\mu}\gamma$.
However, on adopting the regularization (gauge-fixing): ${\cal
P}^{-1}_{ij}=\bm{\nabla}^{-2}\left[\delta_{ij}-(1-\xi)\bm{\nabla}^{-2}\partial_{i}\partial_{j}\right]$,
analogous to the 1-form case, it is easily seen that the final results stay
independent of the parameter $\xi$444Actually, the only non-trivial bracket
turns to be longitudinal.. On defining the Dirac bracket as usual:

$\displaystyle\left\\{\mathfrak{A}(\bm{x},t),\,\mathfrak{B}(\bm{y},t)\right\\}_{\rm
D}$ $\displaystyle=$
$\displaystyle\left\\{\mathfrak{A}(\bm{x},t),\,\mathfrak{B}(\bm{y},t)\right\\}_{\rm
PB}$ (70)
$\displaystyle\quad-\int\,d^{3}z\,d^{3}w\,\left\\{\mathfrak{A}(\bm{x},t),\,\Psi_{A}(\bm{z},t)\right\\}_{\rm
PB}\mathfrak{C}^{-1}_{AB}(\bm{z},\bm{w},t)\left\\{\Psi_{A}(\bm{w},t),\,\mathfrak{B}(\bm{y},t)\right\\}_{\rm
PB},$

the dynamical variables now satisfy,

$\displaystyle\left\\{\widetilde{B}_{i}(\bm{x},t),\,\overline{\Pi}_{j}(\bm{y},t)\right\\}_{\rm
D}=0=\left\\{\overline{B}_{i}(\bm{x},t),\,\widetilde{\Pi}_{j}(\bm{y},t)\right\\}_{\rm
D},$
$\displaystyle\left\\{\overline{B}_{i}(\bm{x},t),\,\overline{\Pi}_{j}(\bm{y},t)\right\\}_{\rm
D}=0,\quad{\rm and}$
$\displaystyle\left\\{\widetilde{B}_{i}(\bm{x},t),\,\widetilde{\Pi}_{j}(\bm{y},t)\right\\}_{\rm
D}=\frac{\partial_{i}\partial_{j}}{\bm{\nabla}^{2}}\delta^{3}(\bm{x}-\bm{y}).$
(71)

Therefore, the dynamics is rightly confined to the field $\widetilde{\bm{B}}$
which is subjected to the ‘non-restrictive condition’
$\bm{\Xi}=\bigodot\left(\bm{\nabla}\times\widetilde{\bm{B}}\right)\approx 0$.
The primary Hamiltonian now simplifies to,

${\cal H}_{\rm P}={\cal H}_{\rm
Can}+\lambda_{i}\overline{\Theta}_{i}=\frac{1}{2}\left(\widetilde{\Pi}_{i}\frac{1}{\bigodot}\widetilde{\Pi}_{i}+\Phi\bigodot\Phi\right),$
(72)

subjected to the constraint $\overline{\bm{\Xi}}\approx 0$ and the result
$\lambda_{i}=0$. In the framework of the Dirac bracket, the constraint
$\bm{\Xi}$ does not make individual components of $\widetilde{\bm{B}}$
interdependent. In fact, on imposing $\bm{\Xi}=0$ along with
$\overline{\bm{\Xi}}=0$ essentially transforms the Hamiltonian to,

${\cal H}_{\rm
P}=\frac{1}{2}\left[\left(\partial_{0}\widetilde{\bm{B}}\right)^{2}+\left(\partial_{i}\widetilde{\bm{B}}\right)^{2}+4g^{2}\left(\widetilde{\bm{B}}\right)^{2}\right],$
(73)

along-with a local momentum
$\widetilde{\bm{\Pi}}=\partial_{0}\widetilde{\bm{B}}$. This is a massive
3-vector, consistent with the only non-trivial Dirac bracket of Eq. 71 that
manifests a longitudinal projection. Noticeably, the gauge choice removes the
non-local operator. Quantization can follow as usual by upgrading the Dirac
brackets to quantum ones, subject to the constraints which will then be
operator-valued.

##  IV Quantization of the 2-Form model

Responsible for the gauge-invariant 2-form mass, the non-local term persists
in the canonical momenta at the classical level, although we have identified
the correct physical modes. Choosing a particular gauge cures this problem,
but that is not explicit in the construction of Dirac brackets as the
constraints are implemented only weakly. More importantly, we want to maintain
manifest Lorentz covariance as much as possible while quantizing, leading to
the physical spectrum expected to be gauge-invariant.

#####

These goals are most efficiently met by introducing covariant, non-equal time
fundamental quantum brackets [24] that demonstrate clear equivalence between
covariant and non-covariant gauge choices. The covariant Lorentz gauge removes
the non-locality at the expense of spurious massless modes that dissociate
from the physical space. Only the physical mode survives on implementing the
temporal gauge, though its agreement with covariance needs validation.

#####

On the other hand, since the explicit non-locality is retained unless the
Lagrangian (or the Hamiltonian) is directly gauge-fixed, a path-integral
quantization scheme will work only when auxiliary fields are introduced to
compensate for the same, more so as there are no equations of motion involved.
This manifestly covariant approach necessarily utilizes the tensorial
Stückelberg extension of Eq. 11 to arrive at correct degrees of freedom with
the gauge redundancy compensated by the duly decoupled ghost sector. In the
following, each of these treatments is carried out in detail.

###  IV.1 Quantization in the $R_{\xi}$ gauge

The covariant gauge condition $\left(\partial\cdot B\right)_{\mu}=0$ is
implemented in the massive 2-form Lagrangian through the $R_{\xi}$
prescription,

${\cal L}_{\rm
B}=\frac{1}{12}H_{\mu\nu\rho}\left(1+\frac{4g^{2}}{\partial^{2}}\right)H^{\mu\nu\rho}-\frac{\xi}{2}\left(\partial_{\mu}B^{\mu\nu}\right)^{2}.$
(74)

Doing so invariably removes manifest gauge invariance and thus necessitates a
ghost sector that conveniently decouples in the Abelian case. However, gauge-
invariance can still be tracked through dependence on the gauge parameter
$\xi$. The corresponding equations of motion have the form,

$\left(1+\frac{4g^{2}}{\partial^{2}}\right)\partial_{\mu}H^{\mu\alpha\beta}=\xi\partial^{[\alpha}\left(\partial\cdot
B\right)^{\beta]},$ (75)

which can be resolved in two parts,

$\displaystyle\partial^{2}\left(\partial^{2}+4g^{2}\right)B_{\mu\nu}=0,$
$\displaystyle\partial^{2}\left(\partial\cdot B\right)_{\mu}=0.$ (76)

The second equation above imposes the gauge condition $\left(\partial\cdot
B\right)_{\mu}=0$ given the boundary values are chosen judiciously. As a
direct consequence, Eq. 75 reduces to a free massive dispersion:

$\left(\partial^{2}+4g^{2}\right)B_{\mu\nu}=0,$ (77)

and thus the additional massless mode in the first of Eq.s 76 is removed.

#####

The gauge condition cannot be implemented directly at the quantum level as the
fields become operators [1]. Before going into that, the gauge invariance of
the outcomes of this theory can be checked at the Classical level. The
$R_{\xi}$ gauge propagator has the form,

$\displaystyle\langle\left[B_{\mu\nu}(x),B_{\alpha\beta}(y)\right]\rangle=K_{\mu\nu,\alpha\beta}\delta(x-y),$
$\displaystyle
K_{\mu\nu,\alpha\beta}=\frac{1}{\partial^{2}+4g^{2}}\left[\eta_{\mu[\alpha}\eta_{\beta]\nu}-(1+\xi)\frac{\eta_{[\mu[\alpha}\partial_{\beta]}\partial_{\nu]}}{\partial^{2}}\right].$
(78)

Naturally, it respects the anti-symmetry of the constituent fields but does
not conform to the covariant gauge condition owing to the $\xi$-dependent
part. Still, the physical processes do respect the gauge condition since for a
conserved external current $J_{\alpha\beta}$,

$J^{\alpha\beta}(y)\partial_{x}^{\mu}K_{\mu\nu,\alpha\beta}\delta(x-y)=0.$
(79)

This is because the gauge-dependent contribution to the term
$J^{\alpha\beta}(y)K_{\mu\nu,\alpha\beta}\delta(x-y)$ is a total derivative
for $J_{\alpha\beta}$ being conserved. Given the current is further anti-
symmetric and time-independent the massive mode leads to the potential energy,

$\displaystyle V(\bm{x}_{1},\bm{x}_{2})$
$\displaystyle=\frac{1}{2}\int\,d^{3}x\,J^{\mu\nu}(\bm{x})K_{\mu\nu,\alpha\beta}J^{\alpha\beta}$
$\displaystyle=2\int\,d^{3}x\left(\widetilde{J}_{i}\frac{1}{-\nabla^{2}+4g^{2}}\widetilde{J}_{i}-\overline{J}_{i}\frac{1}{-\nabla^{2}+4g^{2}}\overline{J}_{i}\right).$
(80)

In 3+1-dimension the requirement for the current tensor can be fulfilled as
$J_{\mu\nu}=\epsilon_{\mu\nu\alpha\beta}\partial^{\alpha}j^{\beta}$ where
$j_{\mu}$ can represent charged fermions [38, 60]. This current is
topologically conserved since its 4-divergence vanishes irrespective of the
system dynamics [38, 61]. Therefore the interaction potential,

$\displaystyle V(\bm{x}_{1},\bm{x}_{2})$
$\displaystyle=2\int\,d^{3}x\left(j_{0}\frac{1}{-\nabla^{2}+4g^{2}}j_{0}-j_{i}\frac{1}{-\nabla^{2}+4g^{2}}j_{i}\right),$
(81)

can lead to the Yukawa-like interaction of Eq. 21 between localized charges
and currents in a gauge-invariant manner, akin to the Meisner-like effect
inferred to in the case of the $B\wedge F$ theory [38].

#####

The manifestly covariant quantization of the present model is achieved through
the procedure of Klein-Gordon Divisor [24] through postulating the covariant
fundamental quantum bracket555The non-locality here is conveniently confined
to the gauge-dependent term. Further, the overall sign is important for
correctly obtaining the physical modes.,

$\displaystyle\left[B_{\mu\nu}(x),B_{\alpha\beta}(y)\right]=i\eta_{\mu[\alpha}\eta_{\beta]\nu}\Delta(x-y)-\frac{i}{\xi}\left(1+\frac{4g^{2}}{\partial_{x}^{2}}+\xi\right)\eta_{[\mu[\alpha}\partial^{x}_{\beta]}\partial^{x}_{\nu]}\Delta^{\prime}(x-y),$
(82)

at unequal times, following the same definitions in Eq.s 24 with the
substitution $m\to 2g$. This bracket is consistent with the dynamics of both
Eq.s 75 and 76 as well as with the exact gauge condition $\left(\partial\cdot
B\right)_{\mu}=0$. However the same is not true for individual massive and
massless dynamics. Thus the fundamental bracket overlooks the system’s
constraints in the sense that the physical mode is not entirely isolated,
though the gauge condition is respected. The latter owes to the $R_{\xi}$
procedure of incorporating the gauge condition directly into the action. For
an externally imposed gauge condition, as is the case for the non-covariant
(‘temporal’) gauge or even the covariant gauge chosen explicitly, the bracket
need not be consistent with the gauge condition even if it is consistent with
the equations of motion and thus again over-counts the degrees of freedom [1].
In any case, only an additional condition like the Gupta-Bleuler theorem on
the physical subspace:

$\partial_{\mu}B^{\mu\nu\,(+)}|{\rm Phys.}\rangle=0,$ (83)

can isolate the physical mode(s), since $\left(\partial\cdot B\right)_{\mu}$
behaves like a massless field.

#####

One can also introduce a Nakanishi-Lautrup field $C_{\mu}$ to modify the
gauge-fixing term as,

$-\frac{\xi}{2}\left(\partial_{\mu}B^{\mu\nu}\right)^{2}\rightarrow-
C_{\nu}\partial_{\mu}B^{\mu\nu}+\frac{1}{2\xi}C_{\mu}C^{\mu}.$ (84)

It has the operational advantage that the gauge parameter does not appear in
the 2-form sector and a manifestly gauge-independent treatment can be
performed without having to ‘fix’ $\xi$. The $C_{\mu}$ sector now imbibes the
gauge condition: $C_{\mu}=\xi\partial^{\alpha}B_{\alpha\mu}$ whereas the
2-form equation reduces to,

$\left(1+\frac{4g^{2}}{\partial^{2}}\right)\partial_{\mu}H^{\mu\alpha\beta}=\partial^{[\alpha}C^{\beta]}.$
(85)

Subsequently, the fields separate as,

$\displaystyle\partial^{2}\left(\partial^{2}+4g^{2}\right)B^{\alpha\beta}=0,\quad{\rm
with},$ $\displaystyle\partial^{2}C_{\mu}=0,$ (86)

since $\partial\cdot C=0$. The massless mode persists and owes to the
covariant gauge condition, implemented by the second equation above. This is
apparent from the dual 1-form ($\leftindex^{*}H_{\mu}$) theory that hosts the
massive mode only given its explicit gauge independence666For a massive case
$\left(\partial^{2}+m^{2}\right)\phi=0$ one can always construct a massless
mode $\left(1+\frac{m^{2}}{\partial^{2}}\right)\phi$. But the non-locality
here cannot be removed, unlike the present case.. This massless mode will be
shown to represent states with non-positive norms that lie outside the
physical Fock space. Following the fundamental bracket, the auxiliary field
$C_{\mu}$ commutes both with the 2-form field and itself:

$\left[C_{\mu}(x),B_{\alpha\beta}(y)\right]=0=\left[C_{\mu}(x),C_{\nu}(y)\right].$
(87)

As a result, it dissociates from the dynamical sector as expected and
corresponds to zero-norm states.

#####

For the convenience of notations, the massive and massless modes are now
respectively represented as,

$\displaystyle V_{\mu\nu}(x)=\partial^{2}B_{\mu\nu}(x)\quad{\rm and}$
$\displaystyle S_{\mu\nu}(x)=\left(\partial^{2}+4g^{2}\right)B_{\mu\nu}(x).$
(88)

These fields individually satisfy the covariant gauge-fixing conditions
$\partial_{\mu}V^{\mu\nu}=0=\partial_{\mu}S^{\mu\nu}$ and their corresponding
quantum brackets are,

$\displaystyle\left[V_{\mu\nu}(x),V_{\alpha\beta}(y)\right]$ $\displaystyle=$
$\displaystyle
i4g^{2}\left(4g^{2}\eta_{\mu[\alpha}\eta_{\beta]\nu}+\eta_{[\mu[\alpha}\partial^{x}_{\beta]}\partial^{x}_{\nu]}\right)\Delta(x-y),$
$\displaystyle\left[S_{\mu\nu}(x),S_{\alpha\beta}(y)\right]$ $\displaystyle=$
$\displaystyle-\frac{i}{\xi}4g^{2}\left(1+\frac{4g^{2}}{\partial_{x}^{2}}+\xi\right)\eta_{[\mu[\alpha}\partial^{x}_{\beta]}\partial^{x}_{\nu]}\Delta(x-y,\,g=0),$
$\displaystyle\left[V_{\mu\nu}(x),S_{\alpha\beta}(y)\right]$ $\displaystyle=$
$\displaystyle 0.$ (89)

The massive and massless modes conveniently dissociate. The spuriousness of
the latter is strongly suggested by the exclusive presence of $\xi$ in its
defining commutator777A simpler calculation corresponds to the Landau gauge:
$\xi\to\infty$, where the commutators are no longer non-local and the massless
sector becomes exactly zero-normed.. It will be seen that they represent
states with non-positive norms. Further assurance comes from the commutator of
the dual 1-form,

$\left[{}^{*}H_{\mu}(x),^{*}H_{\nu}(y)\right]=\frac{i}{2}\left(4g^{2}\eta_{\mu\nu}-\partial_{\mu}\partial_{\nu}\right)\Delta(x-y),$
(90)

which is consistent with Eq. 32 and the default transverse-ness
$\partial\cdot\leftindex^{*}H=0$. This bracket is relatively more complicated
than what naïvely follows from the relevant equation of motion, as there is no
gauge redundancy in the dual sector. The corresponding physical states have
positive norms as required.

#####

Taking the equal-time limit of the commutators disrupts manifest covariance by
separating spatial and temporal components, but it leads to the physical modes
more clearly. On going to the momentum space,

$\displaystyle b_{\mu\nu}({\bf q})=\frac{i}{\sqrt{2(2\pi)^{2}E_{q}}}\int
d^{3}x~{}e^{iq.x}\overleftrightarrow{\partial_{0}}B_{\mu\nu}(x),$
$\displaystyle E_{q}^{2}={\bf q}^{2}+4g^{2},$ (91)

the fundamental bracket corresponds to,

$\left[b_{\mu\nu}({\bf q}),b^{\dagger}_{\alpha\beta}({\bf
q}^{\prime})\right]=\eta_{\mu[\alpha}\eta_{\beta]\nu}\delta^{3}({\bf q}-{\bf
q}^{\prime}),$ (92)

among positive and negative frequency components. On resolving the vector
components,

$\displaystyle\left[\widetilde{b}_{i}({\bf
q}),\widetilde{b}^{\dagger}_{j}({\bf
q}^{\prime})\right]=\delta_{ij}\delta^{3}\left({\bf q}-{\bf
q}^{\prime}\right)=-\left[\overline{b}_{i}({\bf
q}),\overline{b}^{\dagger}_{j}({\bf q}^{\prime})\right],$
$\displaystyle\left[\widetilde{b}_{i}({\bf q}),\overline{b}^{\dagger}_{j}({\bf
q}^{\prime})\right]=0.$ (93)

the previously identified mode $\widetilde{b}_{i}({\bf q})$
$\left(\widetilde{B}_{i}(x)\right)$ describes physical states with positive
norm,

$\widetilde{\langle i,\,{\bf q}}|\widetilde{i,\,{\bf
q}^{\prime}\rangle}=\langle 0|\widetilde{b}_{i}({\bf
q})\widetilde{b}^{\dagger}_{i}({\bf q}^{\prime})|0\rangle=\delta^{3}\left({\bf
q}-{\bf q}^{\prime}\right).$ (94)

The states corresponding to $\overline{b}_{i}({\bf q})$
$\left(\overline{B}_{i}(x)\right)$ will have negative norms, which dissociate
from the physical ones. To specify the physical degrees of freedom, the
covariant gauge condition in Eq. 83 implies that,

$\displaystyle\overline{\bm{b}}_{\rm T}({\bf q})|{\rm
Phys.}\rangle=\frac{1}{E_{q}}\bm{q}\times\widetilde{\bm{b}}_{\rm T}({\bf
q})|{\rm Phys.}\rangle,$ $\displaystyle\overline{b}_{\rm L}({\bf q})|{\rm
Phys.}\rangle=0.$ (95)

Here, the suffixes ${\rm L,T}$ stand for longitudinal and transverse
components respectively. The ‘unphysical’ $\overline{b}_{i}({\bf q})$
components get eliminated, leaving behind 3 massive components of
$\widetilde{\bm{b}}({\bf q})$888It is to be kept in mind that the
$R_{\xi}$-gauge quantum brackets are not consistent with individual massive
and massless dynamics.. To arrive at the physical spectrum one can introduce a
polarization basis,

$b_{\mu\nu}\left(\bm{q}\right)=\sum_{\lambda,\lambda^{\prime}=0}^{3}\,\beta_{\lambda\lambda^{\prime}}\left(\bm{q}\right)\varepsilon_{\mu\nu}\left(\bm{q};\,\lambda,\lambda^{\prime}\right).$
(96)

Given the anti-symmetry of $b_{\mu\nu}$ the polarization tensors can be chosen
as
$\varepsilon_{\mu\nu}\left(\bm{q};\,\lambda,\lambda^{\prime}\right)=\varepsilon_{[\mu}\left(\bm{q};\,\lambda\right)\varepsilon_{\nu]}\left(\bm{q};\,\lambda^{\prime}\right)$
making the operators $b_{\lambda,\lambda^{\prime}}\left(\bm{q}\right)$ anti-
symmetric as well. A natural choice for the set of basis vectors
$\varepsilon_{\mu}\left(\bm{q};\,\lambda\right)$ is that for a Proca field
[66],

$\displaystyle\varepsilon_{\mu}\left(\bm{q};\,0\right)=\frac{q_{\mu}}{2g},\quad\varepsilon_{\mu}\left(\bm{q};\,n\right)=\left(0,-\bm{\varepsilon}\left(\bm{q};\,n\right)\right),$
$\displaystyle\varepsilon_{\mu}\left(\bm{q};\,3\right)=\left(\frac{|\bm{q}|}{2g},-\frac{q_{0}}{2g}\hat{\bm{q}}\right),\quad
n=1,2,$ (97)

which complete an orthonormal basis about the momentum vector as999These
relations further ensure that the operators
$b_{\lambda,\lambda^{\prime}}\left(\bm{q}\right)$ also satisfy the commutation
relations prescribed for the operators $b_{\mu\nu}\left(\bm{q}\right)$.,

$\displaystyle\varepsilon\left(\bm{q};\,\lambda\right)\cdot\varepsilon\left(\bm{q};\,\lambda^{\prime}\right)=\eta_{\lambda,\lambda^{\prime}},$
$\displaystyle\sum_{\lambda=0}^{3}\eta_{\lambda\lambda}\varepsilon_{\mu}\left(\bm{q};\,\lambda\right)\varepsilon_{\nu}\left(\bm{q};\,\lambda\right)=\eta_{\mu\nu},$
$\displaystyle
q\cdot\varepsilon\left(\bm{q};\,\lambda\right)=2g\eta_{0\lambda}.$ (98)

The longitudinal $\left(\varepsilon_{\mu}\left(\bm{q};\,3\right)\right)$ and
temporal $\left(\varepsilon_{\mu}\left(\bm{q};\,0\right)\right)$ components
exist due to the non-zero mass and unlike the Proca theory, the absence of any
built-in constraints allows all four basis vectors. Thus these basis vectors
are equally applicable for the dual 1-form theory, thereby reaffirming 3
massive degrees of freedom subjected to a gauge condition. Presently, the
covariant gauge condition translates to,

$\displaystyle\sum_{\lambda=1}^{3}\bar{\beta}_{\lambda}\left(\bm{q}\right)\varepsilon^{\mu}_{\lambda}\left(\bm{q};\,\lambda^{\prime}\right)|{\rm
Phys.}\rangle=0,$
$\displaystyle\bar{\beta}_{\lambda}=\beta_{0\lambda},~{}\lambda=1,2,3,$ (99)

which implies $\bar{\beta}_{\lambda}\left(\bm{q}\right)|{\rm Phys.}\rangle=0$
for $\lambda=1,2,3$ that follows from the explicit basis vectors. Therefore
the covariant gauge Hamiltonian,

$H^{C}=\frac{1}{2}\int
d^{3}x\left[\dot{\widetilde{\bm{B}}}\cdot\dot{\widetilde{\bm{B}}}+\partial_{i}\widetilde{\bm{B}}\cdot\partial_{i}\widetilde{\bm{B}}+4g^{2}\widetilde{\bm{B}}\cdot\widetilde{\bm{B}}-\dot{\overline{\bm{B}}}\cdot\dot{\overline{\bm{B}}}-\partial_{i}\overline{\bm{B}}\cdot\partial_{i}\overline{\bm{B}}-4g^{2}\overline{\bm{B}}\cdot\overline{\bm{B}}\right],$
(100)

ends up having the normal-ordered expectation value in the physical subspace
as,

$\displaystyle\langle{\rm Phys.}|H^{C}|{\rm Phys.}\rangle=\langle{\rm
Phys.}|\int
d^{3}q\,E_{q}\sum_{\lambda=1}^{3}\widetilde{\beta}^{\dagger}_{\lambda}\widetilde{\beta}_{\lambda}|{\rm
Phys.}\rangle,$
$\displaystyle\quad\widetilde{\beta}_{\lambda}=\frac{1}{2}\epsilon_{\lambda\alpha\beta}\beta_{\alpha\beta},~{}\lambda,\alpha,\beta=1,2,3,$
(101)

with 3 massive components. As is well-known for the massless limit, vectors
$\varepsilon_{\mu}\left(\bm{q};\,0/3\right)$ are undefined as the reference
vector $q_{\mu}$ cannot be normalized to 1. Subjected to the covariant gauge,
the longitudinal component of $\overline{\bm{\beta}}$ vanishes whereas its
transverse components exactly cancel out with those of
$\widetilde{\bm{\beta}}$, leaving behind only the massless longitudinal
component of $\widetilde{\bm{\beta}}$.

#####

An equivalent picture emerges on considering the momentum-space commutators of
massive and massless modes,

$\displaystyle\left[v_{\mu\nu}({\bf q}),v^{\dagger}_{\alpha\beta}({\bf
q}^{\prime})\right]=4g^{2}\left(4g^{2}\eta_{\mu[\alpha}\eta_{\beta]\nu}-\eta_{[\mu[\alpha}q_{\beta]}q_{\nu]}\right)\delta^{3}({\bf
q}-{\bf q}^{\prime}),$ $\displaystyle\left[s_{\mu\nu}({\bf
q}),s^{\dagger}_{\alpha\beta}({\bf
q}^{\prime})\right]=4g^{2}\eta_{[\mu[\alpha}q_{\beta]}q_{\nu]}\delta^{3}({\bf
q}-{\bf q}^{\prime}),\quad\left[v_{\mu\nu}({\bf
q}),s^{\dagger}_{\alpha\beta}({\bf q}^{\prime})\right]=0.$ (102)

Clearly, these two sectors dissociate. On resolving to non-covariant
components $\left(\widetilde{\bm{v}}({\bf q}),\overline{\bm{v}}({\bf
q})\right)$ and $\left(\widetilde{\bm{s}}({\bf q}),\overline{\bm{s}}({\bf
q})\right)$, they also dissociate among themselves. Both
$\widetilde{\bm{s}}({\bf q})$ and $\overline{\bm{s}}({\bf q})$ turn out to
have negative norm states:

$\displaystyle\left[\widetilde{s}_{i}({\bf
q}),\widetilde{s}^{\dagger}_{j}({\bf q}^{\prime})\right]$ $\displaystyle=$
$\displaystyle 4g^{2}\left(q_{i}q_{j}-{\bf
q}^{2}\delta_{ij}\right)\delta^{3}\left({\bf q}-{\bf q}^{\prime}\right)$ (103)
$\displaystyle=$ $\displaystyle\left[\overline{s}_{i}({\bf
q}),\overline{s}^{\dagger}_{j}({\bf q}^{\prime})\right],$

reconfirming the spuriousness of this massless mode. That the physical
subspace of Eq. 83 is free of these negative norm states is evident as Eq. 75
leads to,

$S^{\mu\nu}=\left(1+\frac{4g^{2}}{\partial^{2}}+\xi\right)\partial^{[\mu}\left(\partial\cdot
B\right)^{\nu]},$ (104)

wherein the non-local factor disappears in the Landau gauge.

#####

As for the massive mode, however, both $\widetilde{\bm{v}}({\bf q})$ and
$\overline{\bm{v}}({\bf q})$ correspond to semi-positive norms:

$\displaystyle\left[\widetilde{v}_{i}({\bf
q}),\widetilde{v}^{\dagger}_{j}({\bf
q}^{\prime})\right]=4g^{2}\left[\left({\bf
q}^{2}+4g^{2}\right)\delta_{ij}-q_{i}q_{j}\right]\delta^{3}\left({\bf q}-{\bf
q}^{\prime}\right)$ $\displaystyle\left[\overline{v}_{i}({\bf
q}),\overline{v}^{\dagger}_{j}({\bf q}^{\prime})\right]=4g^{2}\left[{\bf
q}^{2}\delta_{ij}-q_{i}q_{j}\right]\delta^{3}\left({\bf q}-{\bf
q}^{\prime}\right).$ (105)

The longitudinal component of $\overline{\bm{v}}({\bf q})$ has a vanishing
norm whereas its transverse components can be expressed in terms of those of
$\widetilde{\bm{v}}({\bf q})$ as per Eq. 95 in the covariant gauge. In
general, all the components of $\widetilde{\ast}({\bf q})$ can never be
replaced by those of $\overline{\ast}({\bf q})$. Unlike $s_{\mu\nu}({\bf q})$,
the states corresponding to $v_{\mu\nu}({\bf q})$ are a part of the physical
Fock space and it serves our purpose that they have positive norms.

###  IV.2 Quantization in non-covariant gauge

Since the field $\overline{\bm{B}}$ acts as a Lagrange multiplier, the non-
covariant gauge choice $\overline{\bm{B}}=0$ resolves the physical mode with
relative ease, but at the expense of manifest covariance. As we have seen from
Eq.s 44 this constrains the remaining field as
$\bm{\nabla}\times\bigodot\widetilde{\bm{B}}=0$ imbibing it with massive
dynamics. In this way, the residual gauge redundancy $B_{ij}(x)\to
B_{ij}(x)+\partial_{i}\Lambda_{j}\left(\bm{x}\right)-\partial_{j}\Lambda_{i}\left(\bm{x}\right)$
after setting $\overline{\bm{B}}=0$ has also been taken care off.
Subsequently, the Lagrangian simplifies to,

$\displaystyle{\cal L}_{\rm B}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\partial_{0}\widetilde{\bm{B}}\cdot\bigodot\partial_{0}\widetilde{\bm{B}}-\bm{\nabla}\cdot\widetilde{\bm{B}}\bigodot\bm{\nabla}\cdot\widetilde{\bm{B}}\right)$
(106) $\displaystyle\to$
$\displaystyle-\frac{1}{2}\widetilde{\bm{B}}\cdot\bigodot\left[\partial^{2}_{0}\widetilde{\bm{B}}-\bm{\nabla}\left(\bm{\nabla}\cdot\widetilde{\bm{B}}\right)\right]$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\widetilde{\bm{B}}\cdot\left(\partial^{2}+4g^{2}\right)\widetilde{\bm{B}},$

wherein in the middle step we removed total derivatives. There are no spurious
massless modes in this gauge. As for the quantization, the physical sub-space
is now defined as $\overline{\bm{B}}^{(+)}|{\rm Phys.}\rangle=0$ that houses
the fundamental commutator,

$\left[\widetilde{B}_{i}(x),\,\widetilde{B}_{j}(y)\right]=i\delta_{ij}\Delta(x-y).$
(107)

It is consistent with the massive vector dynamics and reduces to the equal-
time commutator,

$\left[\widetilde{B}_{i}({\bf x},t),\,\dot{\widetilde{B}}_{j}({\bf
y},t)\right]=i\delta_{ij}\delta^{3}({\bf x}-{\bf y}).$ (108)

Although trivially consistent with the gauge condition, however, this bracket
is not consistent with the residual gauge condition
$\bigodot\bm{\nabla}\times\widetilde{\bm{B}}=0$ since,

$\displaystyle\left[\left(\bigodot\bm{\nabla}\times\widetilde{\bm{B}}\right)_{i}({\bf
x},t),\,\dot{\widetilde{B}}_{j}({\bf y},t)\right]$
$\displaystyle\qquad=-i\left(1-\frac{4g^{2}}{\nabla^{2}_{x}}\right)\epsilon_{ijk}\partial_{k}^{x}\delta^{3}({\bf
x}-{\bf y})\neq 0.$ (109)

The complete consistency is achieved by postulating a ‘longitudinal’ quantum
bracket instead,

$\left[\widetilde{B}_{i}({\bf x},t),\,\dot{\widetilde{B}}_{j}({\bf
y},t)\right]=i\frac{\partial^{x}_{i}\partial^{x}_{j}}{\nabla_{x}^{2}}\delta^{3}({\bf
x}-{\bf y}).$ (110)

The inherent validation of this construction comes from the Dirac bracket
analysis we had carried out, particularly from the last one of Eq.s 71, as in
the temporal gauge the canonical conjugate of $\widetilde{B}_{i}$ is
$\widetilde{\Pi}_{i}=\dot{\widetilde{B}}_{i}$. Again, the gauge fixing cures
the non-locality imposed by the operator $\bigodot$.

#####

In the momentum space the fundamental bracket takes the form,

$\left[\widetilde{b}_{i}({\bf q}),\,\widetilde{b}^{\dagger}_{j}({\bf
q}^{\prime})\right]=\frac{q_{i}q_{j}}{\bm{q}^{2}}\delta^{3}\left({\bf q}-{\bf
q}^{\prime}\right),$ (111)

leading to non-negative norms as required. However, owing to the on-shell
restriction, the residual gauge condition becomes an identity in the momentum
space. Therefore one can still work with the fundamental commutator identical
to the covariant case in Eq. 93. This suggests that the respective
quantization processes are equivalent; in the sense that the commutator of the
gauge-independent fields,

$\left[\Pi_{i}({\bf x},t),\,\Phi({\bf
y},t)\right]=i\partial^{x}_{i}\delta^{3}({\bf x}-{\bf y}),$ (112)

is unaffected by the choice of gauge101010The non-equal-time commutators
satisfy the micro-causality condition as $\Delta(x-y)$ (and thus
$\Delta^{\prime}(x-y)$) vanishes for space-like separations. The equal-time
condition serves as a special case of that and thus represents a causally
definite condition for comparing quantization in the two gauges.. Formally,
the inherent covariance of the non-covariant gauge quantization is assured
through the Dirac-Schwinger covariance condition [1],

$\displaystyle\left[T^{T}_{00}({\bf x},t),\,T^{T}_{00}({\bf y},t)\right]$
$\displaystyle\qquad=i\left(T^{T}_{0i}({\bf x},t)+T^{T}_{0i}({\bf
y},t)\right)\partial^{x}_{i}\delta^{3}({\bf x}-{\bf y}),$ (113)

linking Hamiltonian and momentum densities that follow from the energy-
momentum tensor,

$T^{T}_{\mu\nu}=\partial_{\mu}\widetilde{\bm{B}}\cdot\partial_{\nu}\widetilde{\bm{B}}-\frac{1}{2}\eta_{\mu\nu}\left(\partial_{\rho}\widetilde{\bm{B}}\cdot\partial^{\rho}\widetilde{\bm{B}}-4g^{2}\widetilde{\bm{B}}\cdot\widetilde{\bm{B}}\right).$
(114)

in this particular gauge. As for the energy spectrum, the situation is much
simpler now. Since Eq. 100 is free from $\overline{\bm{B}}$ the same mode
distribution of energy in Eq. 101 is simply reproduced. Expectantly, the
energy of the system is gauge-independent.

###  IV.3 Path integral Quantization

Following the canonical quantization of the current model that led to
physically well-behaved modes, the question persists as to whether such an
explicit non-local model will behave well when quantum corrections are
included. It finds assurance since the $B\wedge F$ model is renormalizable
[21] and we can expect that to carry over to an effective description. A
somewhat equivalent approach is to quantize the theory in the path integral
procedure [45]. Since it is implemented at the level of generating function,
the off-shell path integral quantization is valid up to all orders [69]. The
present Abelian fields, upon gauge fixing, are supposed to decouple from the
ghost sector having negative norm states. Subsequently, the gauge redundancy
is replaced by the BRST transformations [70, 69, 71].

#####

Given the path integral approach is off-shell, the removal of the non-local
term in the dynamical equation through gauge-fixing is not possible. In fact,
given the standard gauge conditions, the ghost sector would be insensitive to
the non-locality. Thus it becomes necessary to introduce auxiliary fields to
compensate for the non-locality so that the actual degrees of freedom can be
identified. The physics remains unchanged in doing so since the integration of
the auxiliary fields is included in the generating function. In the following,
the extended Lagrangian of Eq. 11 is used for the path integral treatment,
preceded by that of the 1-form counterpart (Eq. 10) for the sake of
completion.

####  IV.3.1 Massive 1-form case

The Stückelberg Lagrangian is invariant under the combined gauge
transformations of Eq. 28. By choosing the shared gauge parameter $\Lambda$ we
may fix a gauge condition, say, the covariant one $\partial\cdot A=D$. Such a
gauge is known to yield positive-definite energies [51]. Subsequently, the
Lagrangian is extended by the gauge fixing and the corresponding ghost term
as,

$\displaystyle{\cal L}_{A}\to{\cal L}_{A}+{\cal L}_{\rm GF}+{\cal L}_{\rm
Ghost},~{}{\rm where},$ $\displaystyle{\cal L}_{\rm
GF}=-\frac{\chi}{2}\left(\partial\cdot A-D\right)^{2},$ $\displaystyle{\cal
L}_{\rm Ghost}=\partial_{\mu}\overline{c}\partial_{\mu}c-m^{2}\overline{c}c.$
(115)

The complete partition function now looks like this,

${\cal Z}\left[J_{\mu}\right]={\cal N}_{0}\int{\cal
D}[A_{\mu},D,\,\overline{c},\,c]\exp\left(i\int d^{4}x~{}{\cal L}_{A}+{\cal
L}_{\rm GF}+{\cal L}_{\rm Ghost}+A_{\mu}J^{\mu}+DJ\right).$ (116)

Herein, the external sources $\left(J^{\mu},\,J\right)$ are introduced for
generating $n$-point functions and ${\cal N}_{0}$ is the overall normalization
factor. $\left(\overline{c},\,c\right)$ are the anti-commuting (anti-)ghost
fields that carry a negative degree of freedom each. They decouple from the
rest of the action in this Abelian case, leaving the physical dynamics to the
$A_{\mu}$ sector. The gauge invariance of the original Stückelberg action is
now traded for invariance under the nilpotent BRST transformations [51],

$\displaystyle\delta A_{\mu}=\partial_{\mu}c,\quad\delta D=-m^{2}c,$
$\displaystyle\delta\overline{c}=-\chi\left(\partial\cdot
A-D\right),\quad\delta c=0.$ (117)

#####

The scalar field $D$ is physical since it represents the non-locality already
present in the original Lagrangian, unlike the Nakanishi-Lautrup fields
introduced to incorporate a gauge. The total number of degrees of freedom then
counts as $4+1-2\times 1=3$ which are already identified as massive. As
discussed before, a gauge choice of $\partial\cdot A=0$ instead would have
worked equally well, only the ghosts would have been massless. Since a single
scalar gauge parameter $\Lambda$ exists, the treatments corresponding to
different gauge choices are almost identical. It will not be the case for the
2-form analog which has two distinct gauge sectors.

####  IV.3.2 Massive 2-form case

As has been explained before, the path integral quantization of the massive
2-form model starts with the extended Lagrangian in Eq, 11 wherein the
nonlinearity is compensated by an auxiliary 1-form. This model is invariant
under two sets of gauge transformations. The one in Eq. 33 is shared by 2- and
1-forms whereas the other given in Eq. 34 is confined to the 1-form sector.
The covariant gauge $f_{\mu}:=\left(\partial\cdot B\right)_{\mu}=0$ choice in
the shared gauge sector endows the 2-form with massive. According to the
Faddeev-Popov prescription [45] the gauge-fixing affects a ghost action of the
form,

${\cal L}^{B}_{\rm
Ghost}=\frac{1}{2}\partial_{[\mu}\overline{c}_{\nu]}\partial^{[\mu}c^{\nu]},$
(118)

in addition to the gauge-fixing term,

${\cal L}^{B}_{\rm
GF}=-\frac{\xi}{2}\left(\partial_{\mu}B^{\mu\nu}\right)^{2}.$ (119)

The ghost action itself is redundant as the vector ghosts themselves behave as
1-form,

$\overline{c}_{\mu}\to\overline{c}_{\mu}+\partial_{\mu}\overline{\kappa},\quad
c_{\mu}\to c_{\mu}+\partial_{\mu}\kappa,$ (120)

which mandates an additional ghost-of-ghost Lagrangian,

${\cal L}^{\overline{c},c}_{\rm
Ghost}=\partial_{\mu}\overline{d}\partial^{\mu}d+\partial_{\mu}\overline{e}\partial^{\mu}e.$
(121)

Although $\left(\overline{d},d,\overline{e},e\right)$ are Grassman fields they
obey statistics opposite to that of
$\left(\overline{c}_{\mu},\,c_{\mu}\right)$, and thus represent positive
degrees of freedom [43]. The ghost sector further needs to be gauge-fixed
through the term,

${\cal L}^{\overline{c},c}_{\rm
GF}=-\frac{\eta}{2}\left(\partial\cdot\overline{c}\right)\left(\partial\cdot
c\right).$ (122)

On the other hand, the gauge-fixing condition $f_{\mu}$ itself is a
divergence-less term. This restriction needs to be implemented in terms of
another ghost term as,

${\cal L}^{f}_{\rm Ghost}=\frac{1}{2}\partial_{\mu}f\partial^{\mu}f.$ (123)

The bosonic ghost field $f$ represents a positive degree of freedom. Finally,
the individual gauge redundancy of the $D_{\mu}$ sector is handled through the
ghost action,

${\cal L}^{D}_{\rm Ghost}=\partial_{\mu}\overline{g}\partial^{\mu}g,$ (124)

with usual Grassman scalars and a gauge-fixing part,

${\cal L}^{D}_{\rm GF}=-\frac{\varepsilon}{2}\left(\partial\cdot
D\right)^{2}.$ (125)

All the individual parts now are collected to form the complete Lagrangian,

$\displaystyle{\cal L}_{B}^{PI}$ $\displaystyle=$ $\displaystyle{\cal
L}_{B}+{\cal L}^{B}_{\rm Ghost}+{\cal L}^{B}_{\rm GF}+{\cal
L}^{\overline{c},c}_{\rm Ghost}+{\cal L}^{\overline{c},c}_{\rm GF}+{\cal
L}^{f}_{\rm Ghost}+{\cal L}^{D}_{\rm Ghost}+{\cal L}^{D}_{\rm
GF}+B_{\mu\nu}J^{\mu\nu}$ (126) $\displaystyle=$
$\displaystyle\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}-g^{2}B_{\mu\nu}B^{\mu\nu}+\frac{1}{2}B_{\mu\nu}\partial^{[\mu}D^{\nu]}-\frac{1}{16g^{2}}\partial_{[\mu}D_{\nu]}\partial^{[\mu}D^{\nu]}+\frac{1}{2}\partial_{[\mu}\overline{c}_{\nu]}\partial^{[\mu}c^{\nu]}-\frac{\xi}{2}\left(\partial_{\mu}B^{\mu\nu}\right)^{2}$
$\displaystyle~{}+\partial_{\mu}\overline{d}\partial^{\mu}d+\partial_{\mu}\overline{e}\partial^{\mu}e-\frac{\eta}{2}\left(\partial\cdot\overline{c}\right)\left(\partial\cdot
c\right)+\frac{1}{2}\partial_{\mu}f\partial^{\mu}f+\partial_{\mu}\overline{g}\partial^{\mu}g-\frac{\varepsilon}{2}\left(\partial\cdot
D\right)^{2}+B_{\mu\nu}J^{\mu\nu}+D_{\mu}j^{\mu},$

with proper sources $\left(J_{\mu\nu},\,j\right)$ for the respective fields.
Correspondingly, the generating function now takes the form,

${\cal Z}\left[J_{\mu\nu},\,j_{\mu}\right]={\cal N}_{0}\int{\cal
D}[B_{\mu\nu},D_{\mu},\overline{c}_{\mu},c_{\mu},\overline{d},\,d,\,\overline{e},\,e,\,f,\,\overline{g},\,g]\exp\left(i\int
d^{4}x~{}{\cal L}_{B}^{PI}\right).$ (127)

The anti-symmetric tensor field $B_{\mu\nu}$ has 6 degrees of freedom whereas
the 1-form $D_{\mu}$ has 4. The (anti-)ghost fields
$\left(\overline{c}_{\mu},c_{\mu},f,\,\overline{g},\,g\right)$ posses
$-4-4-1-1-1=-11$ degrees of freedom in total, whereas that number for
(anti-)ghost of (anti-)ghost fields
$\left(\overline{d},\,d,\,\overline{e},\,e\right)$ is $1+1+1+1=4$. Therefore
the total degrees of freedom of the system is $6+4-11+4=3$ as required. We
identify all of them as massive since $D_{\mu}$ can dynamically be replaced by
$B_{\mu\nu}$ which has a massive dispersion in the implemented covariant
gauge. The gauge-fixing terms can further be extended by introducing suitable
Nakanishi-Lautrup fields like before and the gauge parameters
$\left(\xi,\,\eta,\,\varepsilon\right)$ can be fixed as required.

#####

The alternate gauge fixing choice $\left(\partial\cdot
B\right)_{\mu}+D_{\mu}=0$ might feel appealing since it leads to massive
dynamics for both $B_{\mu\nu}$ and $D_{\mu}$ fields with positive energies.
The respective physical subspace is suitably defined as,

$\left[\left(\partial\cdot B\right)_{\mu}+D_{\mu}\right]|{\rm Phys}\rangle=0.$
(128)

However, the path integral quantization is off-shell and subjected to the
discussion following Eq. 35 this situation is equivalent to the gauge
condition $\left(\partial\cdot B\right)_{\mu}=0$. To appreciate that, let us
consider the general gauge transformation,

$\left(\partial\cdot B\right)_{\mu}+D_{\mu}\to\left(\partial\cdot
B\right)_{\mu}+D_{\mu}+\partial^{2}\Lambda_{\mu}-\partial_{\mu}\partial\cdot\Lambda+4g^{2}\Lambda_{\mu}+\partial_{\mu}\lambda=0,$
(129)

that achieves the desired gauge condition involving both the gauge parameters.
Taking a divergence reduces the condition to,

$\partial\cdot D\to\partial\cdot
D+4g^{2}\partial\cdot\Lambda+\partial^{2}\lambda=0.$ (130)

On substituting for $\lambda$ in Eq. 129 one gets,

$\left(\partial\cdot B\right)_{\mu}+D_{\mu}\to\left(\partial\cdot
B\right)_{\mu}+D_{\mu}-\frac{1}{\partial^{2}}\partial_{\mu}\partial\cdot
D+\left(\partial^{2}+4g^{2}\right)\left(\Lambda_{\mu}-\frac{1}{\partial^{2}}\partial_{\mu}\partial\cdot\Lambda\right)=0.$
(131)

Effectively, we have a transverse gauge-fixing function
$f_{\mu}=\left(\partial\cdot
B\right)_{\mu}+D_{\mu}-\frac{1}{\partial^{2}}\partial_{\mu}\partial\cdot D$,
that requires a bosonic ghost of the form in Eq. 123 limited to the
$\Lambda_{\mu}$ sector. The gauge fixing itself will now implement the massive
yet gauge-invariant fermionic vector ghost action,

${\cal L}^{B}_{\rm
Ghost}=-\frac{1}{2}\overline{c}_{\mu}\left(1+\frac{4g^{2}}{\partial^{2}}\right)\left(\partial^{2}\eta^{\mu\nu}-\partial^{\mu}\partial^{\nu}\right)c_{\nu},$
(132)

that replaces Eq. 118. Interestingly, the non-local nature of the initial
Lagrangian appears in the ghost sector, although the auxiliary 1-form had been
introduced. The vector ghost sector still harbors the redundancy of Eq. 120
requiring two sets of ghost-of-ghost fields as before. Further, the gauge
parameter in Eq. 131 has the form,

$\Lambda_{\mu}=\frac{1}{\partial^{2}+4g^{2}}\left[\left(\partial\cdot
B\right)_{\mu}+D_{\mu}-\frac{1}{\partial^{2}}\partial_{\mu}\partial\cdot
D\right],$ (133)

which is local yet transverse: $\partial\cdot\Lambda=0$. Then from Eq. 130 the
auxiliary 1-form sector becomes dissociated as,

$\partial\cdot D\to\partial\cdot D+\partial^{2}\lambda=0.$ (134)

Therefore the gauge condition $d=\partial\cdot D=0$ can still be implemented
independently, requiring scalar (anti-)ghosts of Eq. 124. Thus the number of
(anti-)ghost components remains the same as that corresponding to the gauge
choice of $\left(\partial\cdot B\right)_{\mu}=0$, leaving out the correct
number of physical degrees of freedom.

#####

The lack of gauge symmetry in the final Lagrangian is again compensated by an
extensive nilpotent (anti-)BRST symmetry [72, 73]. We list here the set of
BRST transformations that leave the action in Eq. 126 invariant,

$\displaystyle\delta
B_{\mu\nu}=\partial_{\mu}c_{\nu}-\partial_{\nu}c_{\mu},\quad\delta
D_{\mu}=4g^{2}c_{\mu}+\partial_{\mu}g,$
$\displaystyle\delta\overline{c}_{\mu}=-\xi\left(\partial\cdot
B\right)_{\mu}+\partial_{\mu}d,\quad\delta c_{\mu}=\partial_{\mu}e,$
$\displaystyle\delta\overline{d}=-\frac{\eta}{2}\partial\cdot\overline{c},\quad\delta
d=0,$ $\displaystyle\delta\overline{e}=-\frac{\eta}{2}\partial\cdot
c,\quad\delta e=0,$ $\displaystyle\delta\overline{g}=-\varepsilon\partial\cdot
D,\quad\delta g=0,$ (135)

where $\delta^{2}=0$ due to nilpotency. A similar set of complementing anti-
BRST transformations also exists. As mentioned before, invariance under these
transformations ensures quantization up to all orders [69]. Since all the
(anti-)ghosts decouple from the dynamical fields in this Abelian case, the
physically relevant dynamics no longer contain any negative-norm states.
Subsequently, the 2-form quantum propagator can directly be obtained through
functional derivatives as,

$K^{\mu\nu,\alpha\beta}(x,y)=\frac{-i}{{\cal
Z}\left[J_{\mu\nu}\right]}\frac{\delta^{2}{\cal
Z}\left[J_{\mu\nu}\right]}{\delta J_{\mu\nu}(x)\delta
J_{\alpha\beta}(y)}\bigg{\rvert}_{J_{\mu\nu}=0},$ (136)

with all the properties intact, including the gauge conditions obeyed by the
propagator in Eq. 78.

##  V Summary and Conclusions

The present work analyzed the non-local 2-form model with gauge-invariant
massive dispersion and its quantization. The non-locality does not hamper the
dynamics of the system as the classical equations of motion turn out to be
well-behaved depicting 3 massive degrees of freedom. Given the full gauge,
invariance is intact due to the non-local mass term, the latter can be carried
through the canonical analysis as proper gauge fixing leads to local dynamics
and a sensible Dirac bracket structure. On adopting the covariant quantization
approach, though the system housed a spurious mass-less mode under the
covariant gauge, it dissociates from the physical vector space. Under the non-
covariant gauge condition, only the massive mode appeared which is in
agreement with the Dirac brackets. The gauge invariance of this scheme led to
the same physical spectrum for both gauge choices. Finally, the path-integral
quantization required the removal of the non-locality in terms of an auxiliary
1-form given the off-shell nature of the approach. Subsequently, the ghosts
dissociated properly in the covariant gauge, leaving out the correct physical
mode. A set of BRST transformations is given that replaced the gauge
redundancy.

#####

The present model is interesting as it is dual to the non-local 1-form model
which in turn leads to the Stückelberg system. This is a generalization of the
duality between free 2-form and free scalar field in 3+1 dimensions, which
serves as the massless limit of the present case. On the same footing, the
generalized Stückelberg model of Eq. 11 is dual to the standard Stückelberg
theory of Eq. 10. Usually, such dualities enhance the understanding of an
applicable system already studied under one of the dual theories. In
particular, a dual theory having a larger redundancy (since the number of
degrees of freedom is the same) may lead to interesting possibilities based on
how the redundancy is fixed. Given the Stückelberg model is well-understood
[51], these ‘extended duals’ are also highly expected to be sensible theories.
In particular, we hope to study the renormalization of the non-local 2-form
model in the near future. This model deserves further study in the presence of
fermion interactions given the $B\wedge F$ model yields the London equations
[38] by mediating the short-range interaction.

#####

The non-local 1-form model has recently been used to demonstrate early
anisotropic deceleration and late time acceleration of the Universe [74]. This
belongs to a larger class of cosmological models that are generalizations to
Proca theory [75] and beyond [76, 77]. such models provide a larger range of
parameters than that in scalar cosmology with the ghosts decoupling. The
present model is a valid candidate for such applications since it corresponds
to Proca-like dynamics. Apart from having the same 3 massive degrees of
freedom, a direct way to see this correspondence is to extend the Lagrangian
of Eq. 2 as,

${\cal
L}_{B}\to\frac{1}{2}\partial_{\mu}\varphi\left(1+\frac{M^{2}}{\partial^{2}}\right)^{-1}\partial^{\mu}\varphi\to\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi-M\varphi\partial\cdot
K+\frac{1}{2}K_{\mu}\left(\partial^{2}+M^{2}\right)K^{\mu},$ (137)

where $M^{2}=4g^{2}$. The on-shell elimination of $\varphi$ now makes
$K_{\mu}$ to obey,

$\left(\partial^{2}+M^{2}\right)K^{\mu}=0,\quad\partial\cdot K=0,$ (138)

which essentially is the Proca dynamics. This is distinct from the usual
massive Kalb-Ramond field which is a combination of axion and Proca fields
[78]. The current model may serve as a fruitful dark matter candidate, in
addition to possibly extending the already known role of the Kalb-Ramond field
during inflation [36], given the short-range gauge-invariant interaction. We
plan to study these aspects in the near future.

###### Acknowledgements.

The author greatly appreciates valuable input from Dr. Vivek M. Vyas during
the inception of this work. This work enjoys the support of Mahidol
University.

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