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{\Large \bf Translational groups as   generators of gauge transformations}
\vskip 0.5cm

{\bf Tomy Scaria}
\vskip 1.0 true cm

S. N. Bose National Centre for Basic Sciences \\
JD Block, Sector III, Salt Lake City, Calcutta -700 098, India.\\
E-mail: {\rm tomy@bose.res.in}
\end{center}
\bigskip

\centerline{\large \bf Abstract}
We examine the gauge generating nature of the translational subgroup of
Wigner's little group for the case of massless
 tensor
gauge theories and show that the gauge transformations generated by the translational group is only 
a subset of the complete set of gauge transformations.
We  also show that, just like the case of topologically massive gauge theories,
 translational groups act as generators of gauge transformations in gauge theories obtained by extending  massive gauge noninvariant theories by a
 Stuckelberg mechanism. The  representations of the translational groups
that generate gauge transformations in such
 Stuckelberg extended theories can
be obtained by the method of dimensional descent. We illustrate these with the examples of 
Stuckelberg extended first class versions of Proca, Einstein-Pauli-Fierz and 
massive Kalb-Ramond theories  in 3+1 dimensions. A detailed analysis of the 
partial gauge generation in massive and massless 2nd rank symmetric gauge 
theories is provided. The gauge transformations generated by translational
group in 2-form gauge theories are shown to explicitly manifest the 
reducibility of gauge transformations in these theories.
\vskip 0.5cm
\begin{flushleft}
{\large PACS No.(s)} 11.15.-q,  11.10.Kk,  04.20.Cv \\
{\large Keywords:} translational group, gauge transformation
\end{flushleft}
\newpage
\section{Introduction}

Wigner's little group is quite familiar to physicists mainly
because of  its role in the classification of elementary particles. 
Wigner introduced the concept
of little group in a seminal paper \cite{w} published in 1939 and showed
how particles can be classified on the basis of their spin/ helicity quantum 
numbers using the little group. Several decades later Wigner along with Kim
showed that the little group relates the internal symmetries of massive
 and massless particles \cite{kim}.   A comparatively lesser 
known facet of the little group, namely  its role as a generator of gauge 
transformations in various Abelian gauge theories, was also unraveled in the mean time. This aspect of little group was  first noticed in the contexts 
of free Maxwell theory  \cite{we,hk,hk1} and linearized Einstein gravity \cite{ng}. 
Recently, it was shown that the little group for massless particles 
acts as generators of gauge 
transformations  in the case of other gauge theories as well \cite{b}. For example,  
in addition to the Maxwell and linearized gravity theories, the
defining representation of the little group for massless particle is shown to
generate gauge transformations in 
the 3+1 dimensional Kalb-Ramond(KR) theory which is a massless 2-form gauge theory 
\cite{bc1}.  To be precise, it is the translational subgroup $T(2)$ of the Wigner's
little group for massless particles   that generate gauge transformations in 
these theories.
On the other hand, in 3+1 dimensional $B \wedge F$ theory which is a topologically 
massive gauge theory, one need to go beyond 
this translational subgroup and it is a particular representation of the 
translational group $T(3)$ that generate relevant gauge transformations in this
theory. However, as shown in \cite{bc2}, one can easily see that $T(3)$ is a
subgroup of Wigner's little group  which preserve the 
energy-momentum vector of a massless particle in 4+1 dimensions. It is further
shown in \cite{bc2} that one can systematically derive the representation of 
$T(3)$ that generate gauge transformations
 in $B \wedge F$ theory by considering the polarization vector/tensor and 
momentum 5-vector of free Maxwell and Kalb-Ramond theories in 4+1
dimensions using a method called `dimensional descent'.
Similarly, the method of dimensional descent can be employed to obtain the 
gauge generating representations of $T(1)$ for topologically massive Maxwell-Chern-Simons(MCS)
and linearized Einstein-Chern-Simons(ECS) gauge theories \cite{djt} in 2+1 dimensions by
starting respectively from Maxwell and linearized gravity theories in 3+1
dimensions \cite{bc2,sc}. Thus, dimensional descent is considered to be a 
method by which one can obtain the gauge generating representation of the 
appropriate translational group for topologically massive theories. 

In this context one may note that, apart from the usual massless gauge theories
like Maxwell theory and topologically massive gauge 
theories like MCS theory,  there are other types of gauge theories which are
obtained by converting second class 
constrained systems(in the  language of Dirac's theory of constraint dynamics) to first 
class(gauge) systems using the generalized
prescription of Batalin, Fradkin and Tyutin \cite{bft}. By such a prescription, one can obtain 
from the massive gauge noninvariant Proca theory (in 3+1 dimensions), the Stuckelberg model for vector fields  which is massive
as well as gauge invariant \cite{bb}. 
A host of questions  naturally arise at this point in connection with 
gauge transformations in these theories and their relation to translational
group. One may wonder if  translational groups 
act as generators of gauge transformation in such massive gauge theories as well. If so,
what would be the representation of these groups that generate such gauge 
transformations?  One may also ask,  what would be the role of dimensional
descent in connection with these theories. In the present study, we delve in to 
these questions and  show that the same representation of the translational 
group $T(3)$
that generates gauge transformation in  the topologically massive $B\wedge F$ theory also generates gauge
transformation in the Stuckelberg extended Proca model. This is equally  true for the Stuckelberg extended 
Einstein-Pauli-Fierz theory and for the gauge theory obtained by a Stuckelberg 
extension of massive KR theory in 3+1 dimensions. We  will also show that the 
method of dimensional descent can be used
to obtain the polarization vectors(or tensor) and momentum vectors of the Stuckelberg models and the  
representation of translational
group that generate gauge transformations in such models, by starting 
from appropriate theories in 
one higher space-time dimension.   

In this paper, we make a closer analysis of the gauge   
generation by Wigner's little group in massless tensor gauge theories like 
linearized gravity and  Kalb-Ramond(KR) theories and  unravel certain
subtle points which went unnoticed in earlier studies. (A study  of 
Wigner's little group with regard to dual gauge transformations in 
KR and other theories along with some other aspects can be found in \cite{rpm}.) 
We point out that although the translational subgroup of  Wigner's little 
group generate the whole spectrum of gauge transformations in free Maxwell
theory, the same cannot be said for the case of massless tensor gauge theories. We show explicitly that in these theories, 
the little group generates only a subset of the full range of gauge transformations.
It is well known that in massless KR theory which is constructed from a  2-form gauge field,
the generators of gauge transformations are not independent and there exists a 
{\it gauge invariance of gauge transformations} \cite{gomis,bn}. Such theories are 
known as `reducible gauge systems'. Our analysis shows that the reducibility of the
gauge transformations of  KR theory is manifested explicitly in the partial gauge generation in the theory 
 by the translational group $T(2)$. 


The organization of the paper is as follows. A review of essential results on Wigner's little group and its
connection with gauge transformations in various theories is given in section 2. In section 3 we show that, 
for Kalb-Ramond and linearized gravity theories, only a subset of gauge transformations is 
covered under the action of the translational subgroup of little group. 
The relationship between the partial gauge generation and the reducibility 
of gauge transformations in the case of  KR theory  is also  discussed in section 3.
In section 4, we discuss how
gauge transformations are generated by translational group $T(3)$  in Stuckelberg type 
of models.  Stuckelberg extended Proca theory is dealt with in section 4.1 and 
Einstein-Pauli-Fierz theory extended by the same mechanism is described in section 4.2.
A study of the gauge transformation generated by $T(3)$ in Stuckelberg extended 
massive KR theory is given in section 4.3.
In section 5, it is described how the gauge generating representation of $T(3)$ for these theories  can be obtained by dimensional
descent starting form Maxwell theory and linearized gravity in 4+1 dimensions. 
The conclusions of the work
are detailed in section 6.

Our notation is as follows. We use  $\mu, \nu $ etc. 
for denoting  indices in 3+1 dimensional space-time. The letters  
$i,j$ etc.are used for 4+1 dimensional space-time except in section 4 
where they represent spatial components of 3+1 dimensional vectors/tensors. Metric used is mostly negative.
We denote polarization vectors by $\varepsilon_{\mu}$ whereas polarization tensors
of 2-form theories are denoted by $\varepsilon_{\mu \nu}$. Polarization tensors of symmetric 2nd
rank tensor fields  are denoted by $\chi_{\mu\nu}$.   Matrix forms 
of polarization tensors denoted by $\{\chi_{\mu\nu}\}$ etc.

\section{Wigner's little group as a generator of gauge transformations in 
various theories} 
 
Here we briefly review the role of the translational subgroup of Wigner's 
little group in generating gauge transformations in various theories. 
Historically, the gauge generating property of the little group was first
 studied in  the context of free Maxwell theory \cite{we,hk,hk1} where 
it was shown that the  action of  little group on the polarization vector 
of Maxwell photons amounts to gauge transformations in momentum space.
For an explicit demonstration of this aspect of little group, it is necessary 
to obtain the maximally reduced form of the
polarization vector which displays only the physical degrees of freedom
of the theory. As is well known, free Maxwell theory is described by the
Lagrangian
\begin{equation}
{\cal{L}} = -\frac{1}{4}F_{\mu \nu }F^{\mu \nu }, ~~~~~F_{\mu \nu } =
\partial_\mu A_\nu - \partial_\nu A_\mu
\label{7}
\end{equation}
which is invariant under the gauge transformation
\begin{equation}
A_{\mu}(x) \rightarrow
A^{\prime}_{\mu}(x) =  A_{\mu}(x) + \partial_{\mu}\tilde{f}(x)
\label{7+1}
\end{equation}
where $\tilde{f}(x)$ is an arbitrary scalar function.
The Lagrangian (\ref{7}) leads to following  equation of motion:
\begin{equation}
\partial_\mu F^{\mu \nu} = 0.
\label{8}
\end{equation}
Denoting the polarization vector of a photon by $\varepsilon^{\mu}(k)$, the  gauge field $A^\mu (x)$ which is a solution of (\ref{8}), can be written as
\begin{equation}
A^{\mu}(x) = \varepsilon^{\mu}(k) e^{ik \cdot x}
\label{9}
\end{equation}
where only a single mode is considered and  the positive frequency part is suppressed  for simplicity.
In terms of the
polarization vector, the gauge transformation (\ref{7+1}) is expressed  as
\begin{equation}
\varepsilon_{\mu}(k) \rightarrow \varepsilon^{\prime}_{\mu}(k) =  \varepsilon_{\mu}(k) + if(k)k_{\mu}
\label{10}
\end{equation}
where $\tilde{f}(x)$ has been written as $\tilde{f}(x) = f(k)e^{ik \cdot x}$.
The equation of motion, in terms of the polarization vector,  will now be given
by
\begin{equation}
k^2 \varepsilon^{\mu} - k^{\mu} k_{\nu} \varepsilon^{\nu} = 0
\label{11}.
\end{equation}
The massive excitations corresponding to $k^2 \neq 0$ leads to a solution
$\varepsilon^{\mu} \propto k^{\mu}$ which can therefore be gauged away by a
suitable choice of $f(k)$ in (\ref{10}). For
massless excitations  ($k^2 = 0$), the Lorentz condition $
k_\mu \varepsilon^{\mu} =0$
 follows immediately
 from (\ref{11}).  For a photon of energy $\omega$ propagating in the $z$-direction, the 4-momentum can be written as $k^{\mu} = (\omega, 0, 0, \omega)^T$.
It then follows from ({\ref{11}) that the corresponding polarization tensor   $\varepsilon^{\mu}(k)$ takes
the form $(\varepsilon^{0}, \varepsilon^1, \varepsilon^2, \varepsilon^{0})$ which can be reduced to the
maximally reduced form
\begin{equation}
\varepsilon^{\mu}(k) =
(0, \varepsilon^1, \varepsilon^2, 0)^T
\label{13}
\end{equation}
by a suitable  gauge transformation\footnote{This procedure of obtaining the 
maximally reduced polarization vectors/tensors of
various theories by choosing a plane wave solution for the respective equation motion will henceforth be referred
 to as the ``plane wave method".} (\ref{10}) with $f(k) = \frac{i\varepsilon^0}{\omega}$. Note that the maximally reduced
form (\ref{13})
of $\varepsilon^{\mu}$ displays just the two physical degrees of freedom
$\varepsilon^1$ and $\varepsilon^2$.

Before going to the role little group in generating gauge transformations,
we digress briefly to recapitulate the essential aspects of Wigner's little
group.
The Wigner's little group is defined as the subgroup of homogeneous
 Lorentz group that preserves the
 energy-momentum vector of a particle:
\begin{equation}
{W^\mu}_\nu k^\nu = k^\mu
\label{1}
\end{equation}
It is obvious that, in 3+1 dimensions,  the little group for a massive particle
is the three dimensional rotation group $SO(3)$. On the other hand, for a
massless  particle, the little group is isomorphic to the Euclidean group $E(2)$ which is a
semi-direct product of $SO(2)$ and $T(2)$ - the group of translations in the
2-dimensional plane \cite{hk1}. The  explicit representation of Wigner's
little group which preserves the 4-momentum $k^\mu = (\omega, 0, 0, 0)^T$ of a photon
of energy $ \omega $ moving in the $z$-direction is given by \cite{we}
\begin{equation}
W_4(p, q;  \phi) =
\left( \begin{array}{cccc}
1+ \frac{p^2 + q^2 }{2} & p\cos \phi - q \sin \phi & p\sin \phi + q \cos \phi  & -\frac{p^2 + q^2 }{2} \\
p & \cos \phi & \sin \phi  & -p \\
q & -\sin \phi & \cos \phi  & -q \\
\frac{p^2 + q^2 }{2} &  p\cos \phi -q \sin \phi & p\sin \phi + q \cos \phi  & 1
-\frac{p^2 + q^2 }{2}
\end{array}\right).
\label{2}
\end{equation}
Here $p,q$ are   real parameters.
This little group element can be written as
\begin{equation}
W_4(p, q;  \phi) = W(p, q) R(\phi)
\label{3}
\end{equation}
where
\begin{equation}
W(p, q) \equiv W_4(p, q;  0) = \left( \begin{array}{cccc}
1+ \frac{p^2 + q^2 }{2} & p & q  & -\frac{p^2 + q^2 }{2} \\
p & 1 & 0 & -p \\
q & 0 & 1 & -q \\
\frac{p^2 + q^2 }{2} &  p & q  & 1 -\frac{p^2 + q^2 }{2}
\end{array}\right)
\label{4}
\end{equation}
is  a particular representation of the  translational subgroup  $T(2)$ of
the little group and $ R(\phi)$ represents a $SO(2)$ rotation about the $z$-axis.
Note that the representation $W(p, q)$ satisfies the  relation $ W(p, q)W(\bar{p},\bar{q}) = W(p+\bar{p}, q+\bar{q})$. 

Under the action  of the translational group $T(2)$ in (\ref{4}),  the 
maximally reduced polarization
vector (\ref{13}) of Maxwell theory transforms  as follows:
\begin{equation}
\varepsilon^{\mu} \rightarrow \varepsilon^{\prime \mu} = {W^{\mu}}_{\nu}(p, q) \varepsilon^{\nu} = \varepsilon^{\mu} +
 \left( \frac{p\varepsilon^1 + q\varepsilon^2}{\omega}\right)k^{\mu}~.
\label{14}
\end{equation}
Clearly, this can be identified as a gauge transformation of the form of 
(\ref{10}) by choosing $f(k) = \frac{p\varepsilon^1 + q\varepsilon^2}{i\omega}$ 
 thus displaying the gauge generating property
of Wigner's little group for massless particles in free Maxwell theory. Conversely, any general gauge transformation 
(in momentum space) in Maxwell theory can be viewed as resulting from the action of translational
group $W(p, q)$ on the polarization vector of the theory.
 
The same translational group $T(2)$ in the representation (\ref{4}) generates gauge 
transformations in 3+1 dimensional linearized gravity \cite{sc} and Kalb-Ramond 
theory \cite{bc1}. However, as we will show in the next section, the transformations generated by  
 $T(2)$ does not include the whole spectrum  of gauge 
transformations in these 
theories.
On the other hand, in the case of $B\wedge F$ theory which is
 obtained by
topologically coupling Maxwell field to a Kalb-Ramond theory, the translational
 group (\ref{4}) fails to act as gauge generator. 
It is to be noted that  $B\wedge F$ theory is a topologically massive gauge 
theory where gauge invariance co-exists with mass \cite{al}. The generator of gauge 
transformation in $B\wedge F$ theory is shown in \cite{bc1} 
to be the translational group $T(3)$ in the representation
\begin{equation}
D(p, q, r) = \left( \begin{array}{cccc}
1 & p & q & r \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array} \right)
\label{53-1}
\end{equation}
(where $p, q, r$ are real parameters). 
The representation $D(p, q, r)$ of $T(3)$ which
generates gauge transformation in a topologically massive theory is
isomorphic to the translational subgroup of Wigner's little group for
massless particles which
generates gauge transformation in 4+1 dimensional Maxwell theory.
The close relationship between these different representations of $T(3)$ is
analyzed in detail in \cite{bc2}. Moreover a method, called dimensional descent, is
introduced in \cite{bc2} by which one can obtain the polarization tensor/vector and the
gauge generating representation of translational group for a topologically 
massive gauge theory, by starting from the gauge transformation 
properties of a suitable massless gauge theory living 
in a space-time of one higher dimension.
This relies on the mapping of the energy-momentum 5-vector of a massless 
particle moving along the extra space dimension
 to the energy-momentum 4-vector of a massive particle at rest in 3+1 
dimensions.
For example, the polarization tensor and vector of $B\wedge F$ theory
in 3+1 dimensions and the representation (\ref{53-1}) of the translational group $T(3)$
which generate gauge transformation can be systematically derived 
 by starting from
4+1 dimensional Kalb-Ramond and Maxwell theories. One of the aims of the present
 study is to show that by a similar dimensional descent respectively  from 4+1 dimensional Maxwell theory, linearized  gravity and massless KR theory
 one can arrive at the relevant results for Stuckelberg extended Proca,
Einstein-Pauli-Fierz and massive KR theories in 3+1 dimensions.
The concept of dimensional descent will be further elaborated in section 6.

\section{Partial gauge generation by $T(2)$ in massless tensor gauge theories}

It was argued respectively in \cite{bc1} and \cite{ng,sc} that the translational 
group $T(2)$  in the representation (\ref{4}) generates gauge transformations 
in 3+1 dimensional 
Kalb-Ramond theory and linearized gravity.  As pointed out in \cite{ng} 
the gauge generation by little group in linearized gravity is subject to 
certain
restrictions and therefore forms only a part of the full range of gauge 
transformations available in the theory. 
Here  we make a closer examination of this partial gauge generation by $T(2)$ 
in linearized gravity  revealing some aspects which went unnoticed before. We 
also show that $T(2)$ generates only a restricted set of gauge transformations
in massless KR theory, another point that was missed earlier. 
Furthermore, we compare and contrast the 
gauge transformations generated by $W(p,q)$ in these two cases and show that in the
case of KR theory, the reducible nature of its gauge transformations 
 is reflected in the 
partial
gauge generation by the translational group.

 We first consider linearized gravity\footnote{In linearized
 gravity, the metric $g_{\mu\nu}$ is
assumed to be close to the  flat background part $\eta_{\mu\nu}$ and one 
writes $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$ with deviation $ |h_{\mu\nu}|<< 1$. 
Raising and lowering of indices are done by $\eta^{\mu\nu}$ and $\eta_{\mu\nu}$ respectively.} which is governed by the 
Lagrangian
\begin{equation}
{\cal L}_L^E = \frac{1}{2}h_{\mu\nu} \left[ R^{\mu\nu}_L - \frac{1}{2} \eta^{\mu\nu} R_L\right].
\label{44}
\end{equation}
where $ R^{\mu\nu}_L$ is the linearized Ricci tensor given by
\begin{equation}
 R^{\mu\nu}_L = \frac{1}{2}(- \Box h^{\mu\nu} + \partial^\mu \partial_\alpha  h^{\alpha \nu} + \partial^\nu\partial_\alpha  h^{\alpha \mu} - \partial^\mu \partial^\nu h)
\label{45}
\end{equation}
with $h = h^\alpha_\alpha$ and $R_L = {R^\alpha_L}_\alpha$. 
The equation of motion following from this Lagrangian is 
\begin{equation}
- \Box h^{\mu\nu} + \partial^\mu \partial_\alpha  h^{\alpha \nu} + \partial^\nu\partial_\alpha  h^{\alpha \mu}
 - \partial^\mu \partial^\nu h + \eta^{\mu\nu}(\Box h -\partial_\alpha \partial_\beta h^{\alpha \beta}) = 0.
\label{46}
\end{equation}
which can be seen to be invariant under the gauge transformation 
\begin{equation}
h^{\mu\nu}(x) \rightarrow h'^{\mu\nu}(x) = h^{\mu\nu}(x) + \partial^\mu \tilde{\zeta}^\nu (x) 
+ \partial^\nu  \tilde{\zeta}^\mu (x).
\label{47}
\end{equation} 
Adopting the ansatz (analogous to (\ref{9}))
\begin{equation}
h^{\mu\nu} = \chi^{\mu\nu} (k) e^{ik.x} + c.c.
\label{48}
\end{equation}
where $\chi^{\mu\nu}$ is the symmetric  polarization tensor for linearized gravity,
the gauge transformation (\ref{47}) can be written in the momentum space as
\begin{equation}
 \chi^{\mu\nu} (k)  \rightarrow {\chi}'^{\mu\nu} (k) = \chi^{\mu\nu} (k) + k^\mu \zeta^\nu (k) +  k^\nu \zeta^\mu (k).
\label{50}
\end{equation}
(Here we have written the arbitrary vector function $\tilde{\zeta}^\mu (x) = \zeta^\mu (k) e^{ik.x}$.) 
Now, following the plane wave method as described in \cite{sc}, one can obtain  the 
maximally reduced form of polarization tensor corresponding to a particle with the 
4-momentum $k^\mu = (\omega , 0,0, \omega )^T$ and is given by 
\begin{equation}
 \{\chi^{\mu\nu}\} = \left(
\begin{array}{cccc}
0 & 0 & 0 & 0\\
0 & a & b & 0 \\
0 & b & -a & 0 \\
0 & 0 & 0 & 0
\end{array}
\right).
\label{53}
\end{equation}
(For another derivation see \cite{weinberg1}.)
Here $a$ and $b$ are are free parameters  representing the two physical 
degrees of freedom in 4-dimensional linearized gravity\footnote{Linearized gravity in $d$ dimensions has $\frac{d(d-3)}{2}$ degrees of freedom \cite{weinberg1}.}.
Notice that the maximally reduced form (\ref{53}) of the polarization tensor satisfies the 
momentum space harmonic gauge condition,
\begin{equation}
k_\mu \chi^{\mu}_\nu = \frac{1}{2}k_\nu \chi^{\mu}_\mu .
\label{533}
\end{equation}

It is now easy to show that the action of the translational group 
$W(p, q) $ (\ref{4}) on the polarization tensor (\ref{53})  is equivalent 
to a gauge transformation. 
$$\{ \chi^{\mu \nu}\} \rightarrow \{\chi'^{\mu\nu}\}= W(p, q) \{\chi^{\mu\nu}\} W^T(p, q)$$
\begin{equation}
 = \{\chi^{\mu\nu}\} +\left(
\begin{array}{cccc}
((p^2 - q^2)a + 2pqb)  & (pa + qb) & (pb -qa) & ((p^2 - q^2)a + 2pqb) \\
(pa + qb) &0 & 0 & (pa + qb) \\
(pb -qa)  & 0 & 0 & (pb -qa) \\
((p^2 - q^2)a + 2pqb) &( pa + qb) &( pb -qa) & ((p^2 - q^2)a + 2pqb)
\end{array}\right).
\label{544}
\end{equation}
The above transformation can be cast in the form of a gauge transformation 
(\ref{50}) with the
following choice for  the arbitrary functions $\zeta^\mu(k)$ \cite{sc}:
\begin{equation}
\zeta^1 = \frac{pa + qb}{\omega},~~~~\zeta^2 = \frac{pb -qa}{\omega},~~~~
\zeta^0 = \zeta^3 = \frac{(p^2 - q^2)a + 2pqb}{2\omega}.
\label{555}
\end{equation}
However, since $k^\mu = (\omega , 0,0, \omega )^T$,  a general gauge 
transformation for this polarization tensor (\ref{53}) has the form
$$\{\chi^{\mu\nu} \} \rightarrow \{\chi'^{\mu\nu}\}= \{\chi^{\mu\nu} \} + \{ k^\mu \zeta^\nu \} +  
\{k^\nu \zeta^\mu\} $$
\begin{equation}
 = \{\chi^{\mu\nu}\} +\omega\left(
\begin{array}{cccc}
2 \zeta^0   & \zeta^1 & \zeta^2  & (\zeta^0 + \zeta^3) \\
\zeta^1 &0 & 0 & \zeta^1 \\
\zeta^2  & 0 & 0 & \zeta^2 \\
(\zeta^0 + \zeta^3) &\zeta^1 &\zeta^2 & 2 \zeta^3
\end{array}\right).
\label{577}
\end{equation}
Upon comparing the above form of general gauge transformation with the one
generated by $W(p,q)$ given in (\ref{544}), it becomes clear that the latter is
only a special case of the former as the relations in
(\ref{555})  restricts the number of independent components of the arbitrary 
vector  $ \zeta^\mu$. Therefore, the translational subgroup $T(2)$ of 
Wigner's little  group for massless particles generates only a subset of the 
full set of gauge transformations in linearized gravity. The reason for
this partial gauge transformation is as follows. We must notice that our 
starting point of gauge generation by $W(p,q)$ in linearized gravity is the 
maximally reduced polarization tensor (\ref{53}) which contains just the 
physical sector of the theory (in the reference frame where $k^\mu = 
(\omega , 0,0, \omega )^T$) and is devoid of any arbitrary variables to 
begin with.
Hence in the gauge generation by $W(p,q)$, we must rely entirely on the two
parameters $p$ and $q$ of the translational group to manufacture the 
gauge equivalence class of the state corresponding the polarization tensor
(\ref{53}). However, the gauge freedom in linearized gravity is represented by
the arbitrary vector variable $ \zeta^\mu$ having four components. Naturally,
in the gauge generation by $W(p,q)$ in linearized gravity, only two of the four
components of $ \zeta^\mu$  remain independent (as is evident from 
(\ref{577}))  when expressed in terms of the two parameters $(p,q)$ of the 
translational group and therefore the gauge generation is only partial. 
It was noted in \cite{ng} that the gauge generation by the little group in 
linearized gravity is subject to the `Lorentz condition' $k_\mu \zeta^\mu (k) = 0$. This also can be seen from the third relation $\zeta^0 = \zeta^3$ in 
(\ref{555}). Thus, our present analysis  have unravelled all the constraints 
behind the partial gauge generation by Wigner's little group in linearized 
gravity. In contrast, the gauge freedom in free Maxwell theory is represented
by a single arbitrary scalar variable $f(k)$ (\ref{10}) which can be expressed 
(without any restrictions) in
terms of the two parameters of  $W(p,q)$ in the gauge generation by little
group as is evident from (\ref{14}). 
Hence translational subgroup of Wigner's little group  generates the full set of gauge 
transformations in Maxwell theory.

We now consider the gauge transformations generated by the translational group $W(p,q)$ 
in Kalb-Ramond (KR) theory which has a second rank antisymmetric tensor 
as its basic field. The KR theory is described by the Lagrangian,
\begin{equation}
{\cal L} = \frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda}; ~~~~H_{\mu\nu\lambda} = 
\partial_\mu B_{\nu\lambda} + \partial_\nu B_{\lambda\mu} + \partial_\lambda B_{\mu\nu} 
\label{199}
\end{equation}
where $B_{\mu\nu}$ is the 2nd rank antisymmetric gauge field;
\begin{equation}
B_{\mu\nu} = - B_{\nu\mu}.
\label{1991}
\end{equation}
The equation of motion is 
\begin{equation}
\partial_\mu H^{\mu\nu\lambda} = 0.
\label{1992}
\end{equation}
The KR theory is invariant under the gauge transformation
\begin{equation}
B_{\mu\nu}(x) \rightarrow   B'_{\mu\nu}(x) = B_{\mu\nu}(x) + \partial_\mu F_\nu (x) - \partial_\nu F_\mu (x)
\label{1993}
\end{equation}
where $F_\mu (x)$ are arbitrary functions.
However, these gauge transformations are not all independent. 
One can see that under the transformation 
\begin{equation}
F_\mu (x)
 \rightarrow F'_\mu (x) = F_\mu (x) + \partial_\mu \beta (x)
\label{1993+1}
\end{equation} 
 (where $\beta (x)$ is
an arbitrary scalar function) the gauge transformation (\ref{1993}) remains invariant.
In particular, if $F_\mu = \partial_\mu \Lambda$,  the gauge transformation vanish 
trivially. 
This is known as  the `gauge invariance of gauge transformations' and is  a typical
property of reducible gauge theories\footnote{Notice a crucial difference in the
 case of
linearized gravity which has the  symmetric tensor $h_{\mu\nu}$ as its underlying gauge field. 
Under a transformation of the type (\ref{1993+1}),  
 the gauge transformation (\ref{47}) changes. 
This shows that, unlike KR theory, there is no `gauge invariance of gauge transformation' in 
linearized gravity which is not a reducible gauge system.} where the generators of gauge transformation
are not all independent \cite{gomis}. Since the components of the arbitrary
field $F_\mu (x)$ (which represents the gauge freedom in the KR theory) 
are not all independent and there exists some superfluity in the 
 gauge transformation (\ref{1993}).
The maximally reduced form of the antisymmetric polarization tensor $\varepsilon^{\mu \nu}$
associated with the 2-form potential $B^{\mu \nu} (= \varepsilon^{\mu \nu} e^{ik.x}$) of
KR theory is obtained in \cite{bc1} using plane wave method as
\begin{equation}
\{\varepsilon^{\mu\nu}\} = \varepsilon^{12}
  \left( \begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0
 \end{array} \right).
\label{19}
\end{equation}
Notice that, similar to  Maxwell and linearized gravity theories,  this form of the polarization tensor satisfies a `Lorentz condition' 
\begin{equation}
k_\mu \varepsilon^{\mu \nu} = 0
\label{18+1-1}
\end{equation}
corresponding to $\partial_\mu B^{\mu \nu} = 0$. 
In the case of KR theory, the counterpart of the momentum space gauge 
transformation (\ref{50}) is given by 
\begin{equation}
\varepsilon^{\mu \nu}(k) \rightarrow \varepsilon'^{\mu \nu}(k) = \varepsilon^{\mu \nu}(k) + i(k^{\mu}f^{\nu}(k) - k^{\nu}f^{\mu}(k))
\label{18+1}
\end{equation}
where $f_{\mu}(k)$ are arbitrary and independent functions of $k$ (with $F_\mu (x) = f_\mu (k) e^{ik.x}$). 
In terms of the polarization tensor
$ \varepsilon^{\mu \nu} $ the equation of motion of KR theory can be written as 
\begin{equation}
k_\mu [k^\mu \varepsilon^{\nu \lambda} + k^\nu \varepsilon^{ \lambda\mu} + k^\lambda \varepsilon^{\mu \nu} ]= 0.
\label{18+1-11}
\end{equation}
The transformation of $ \{\varepsilon_{\mu\nu}\}$ 
under the translational subgroup $W(p, q)$
(\ref{4}) of Wigner's little group, can be written as
\begin{equation}
\{\varepsilon ^{\mu \nu} \}\rightarrow \{\varepsilon'^{\mu \nu}\}  = W(p,q) \{\varepsilon^{\mu \nu}\}  W^T (p,q) =
 \{\varepsilon^{\mu \nu} \}+ \varepsilon^{12}\left( \begin{array}{cccc}
0 & -q & p & 0 \\
q & 0& 0 & q \\
-p & 0 & 0 &  -p \\
0 & -q & p & 0
\end{array} \right)
\label{20}
\end{equation}
This can be cast in the
form of  (\ref{18+1}) with
\begin{equation}
f^1 = \frac{-q\varepsilon^{12}}{i\omega},~~~
f^2 =  \frac{p\varepsilon^{12}}{i\omega}, ~~~
f^3 = f^0.
\label{2001}
\end{equation}
 As in the case of linearized gravity, on account of the 
requirement $f^3 = f^0$, the gauge transformations generated by the 
translational group fails to include the entire set of gauge transformations
in KR theory too. Analogous to (\ref{577}), the general form of gauge 
transformation (\ref{18+1}) in the matrix form is 
\begin{equation}
\{\varepsilon^{\mu \nu} \} \rightarrow \{\varepsilon'^{\mu \nu} \} = \{\varepsilon^{\mu \nu}\}  + \omega \left( \begin{array}{cccc}
0 & f^1 & f^2 & f^0 - f^3 \\
-f^1 & 0& 0 &  -f^1\\
-f^2 & 0 & 0 &  - f^2 \\
 f^3  -  f^0 & f^1 &  f^2 & 0
\end{array} \right)
\label{200}
\end{equation}
which makes it quite explicit that the transformation (\ref{20}) 
 does not exhaust (\ref{200}), but is only a special case  (where $f^0 = f^3$) of it.
Here again, for the case of gauge transformation (\ref{20}) generated by the translational
group $W(p,q)$, the arbitrary vector function $f^\mu(k)$ which correspond
to the gauge freedom of KR theory satisfy the `Lorentz condition' 
$k_\mu f^\mu(k) = 0$ since $k^\mu = (\omega , 0,0, \omega )^T $ correspond to a KR quantum propagating in the $z$-direction. 

Similar to the gauge generation  (\ref{544}) in linearized gravity, the 
transformation (\ref{20}) is an attempt to generate the gauge equivalence class
of the completely physical (maximally reduced) polarization tensor (\ref{19})
of KR theory using only the two parameters of the translational group $W(p,q)$
while the full gauge freedom of the theory is represented by the arbitrary
4-vector variable $f^\mu$. Here again, the components $f^1$ and $f^2$ of
$f^\mu$ can be expressed in terms of the parameters $p, q$ of the 
 translational group $W(p,q)$ and they remain independent of each other as  
can be seen from (\ref{2001}). However, unlike in the case of linearized 
gravity, the other two components ($f^0, f^3$)
 are
independent of the parameters (and of the maximally reduced polarization
 tensor) and are left completely undetermined subject only
to the  constraint $f^0 = f^3$.
Thus, in the gauge generation by $W(p,q)$ in KR theory, corresponding to any given pair  
($f^1, f^2$) there exists a continuum of allowed choices for $f^0 (= f^3$)
representative of the invariance of gauge transformations (\ref{1993}) 
under (\ref{1993+1}). Therefore, the partial gauge generation by $W(p,q)$
in massless KR theory clearly exhibits the reducibility of its gauge transformations.
The  reducibility of the gauge transformation (\ref{1993}) is manifested
in the special choice
(\ref{2001}) which makes the transformation (\ref{20}) of 
 the maximally reduced polarization tensor $\varepsilon^{\mu\nu}$ effected by
$W(p,q)$, a gauge transformation of the KR theory. This may be compared to
the gauge generation (\ref{544}) in linearized gravity by $W(p,q)$ where all the components of the arbitrary vector variable $\zeta^\mu$  are expressed in terms of the
parameters $(p,q)$ (see (\ref{555})) hence indicating the absence of any reducibility
in the gauge transformation of the theory.


Notice that the transformation (\ref{1993+1}) is of same form as the 
gauge transformation (\ref{7+1}) of Maxwell theory where the generator of 
gauge transformations is $W(p,q)$. Hence, one may consider that the `gauge
transformation (\ref{1993+1}) of gauge transformations' in KR theory
as being generated by a translational subgroup $W(p,q)$ of little group for
massless particles. Therefore, in KR theory which is a 2-form gauge theory,
  two independent elements of the translational group $W(p,q)$ are involved 
in generating gauge transformations, one for the underlying 2-form field
$B_{\mu\nu}$ and the other for the field $F_\mu$ which correspond to the gauge 
freedom of the theory.
In the gauge generation for massless theories by the translational group
$W(p,q)$, we therefore perceive an appealing 
hierarchical structure starting from the  Maxwell (1-form) 
and KR (2-form) theories; namely  in a $n$-form theory, $n$  elements of the
translational group $W(p,q)$ being involved in gauge 
generation. It is expected that this hierarchical structure continues for higher
form gauge theories as well where 3 independent little group elements
generating `gauge transformations' in 3-form theory and so on.

It is now clear that, in the above second rank tensor gauge theories, the set of gauge 
transformations generated by the translational subgroup of Wigner's  
little group of massless particles is not exhaustive, but form only  special cases of the full range of gauge 
transformations of these theories. In the case of KR theory which is a  2-form gauge theory, the partial gauge generation
by translational subgroup of Wigner's little group is related to the reducible nature of gauge transformations
of the theory.  Similar reducibility of gauge transformations are exhibited by higher form Abelian
gauge theories \cite{bn}. The connection between little group and gauge transformations in such theories 
is under investigation and will be reported later.

\section{Massive gauge theories}
In this section we study the relationship between the translational groups and gauge transformation
in gauge theories which are obtained from massive theories through  Stuckelberg mechanism.
Only 3+1 dimensional
theories are considered in this section.
\subsection{Massive vector gauge theory}
One can render the 4-dimensional Proca theory (which does not possess any  gauge symmetry) gauge invariant
by  Stuckelberg mechanism with the introduction of a new scalar field  $\alpha
(x)$
as follows;
\begin{equation}
{\cal{L}} = -\frac{1}{4}F_{\mu \nu }F^{\mu \nu } + \frac{m^2}{2} (A_\mu + \partial_\mu \alpha)
(A^\mu + \partial^\mu \alpha)
\label{554}
\end{equation}
The Lagrangian remains invariant under the transformations
\begin{equation}
A_{\mu}(x) \rightarrow
A^{\prime}_{\mu}(x) =  A_{\mu}(x) + \partial_{\mu} \Lambda (x), ~~~~~~\alpha (x)  \rightarrow  \alpha '(x) = \alpha (x) -\Lambda (x)
\label{54}
\end{equation}
where $\Lambda (x)$ is an arbitrary scalar function.
The equations of motion for the theory are
\begin{equation}
-\partial^\nu F_{\mu \nu } + m^2 (A_\mu + \partial_\mu \alpha)  = 0,~~~~~ \partial^\mu (A_\mu + \partial_\mu \alpha) = 0.
\label{55}
\end{equation}
One must notice that  by
operating $\partial_\mu $ on the first equation in (\ref{55}) one yields the second. Hence the
latter is consequence of the former. This implies that the gauge transformation
 of the $\alpha $-field can be deduced by knowing that of the $A^\mu$-field. 
Similar to (\ref{9}), here we adopt  the ansatz $A_{\mu}(x) = \varepsilon_\mu \exp (ik.x)$ and $\alpha (x)
= \tilde{\alpha}(k) \exp (ik.x)$. As before, $\varepsilon_\mu (k)$ is the polarization vector of the field $A_\mu(x)$ and $\tilde{\alpha}(k)$ is a particular
Fourier component of $\alpha (x)$ . In terms
of the polarization vector $\varepsilon_\mu (k)$, the equations of motion become respectively,
\begin{equation}
k^\nu(k_\mu \varepsilon_\nu - k_\nu \varepsilon_\mu ) + m^2(\varepsilon_\mu + ik_\mu \tilde{\alpha}) = 0, ~~~
ik^\nu(\varepsilon_\nu + ik_\nu \tilde{\alpha}) = 0.
\label{566}
\end{equation}
Analogous to (\ref{55}), the second equation in (\ref{566}) is a consequence of the first one.
For massless excitations $k^2 = 0$, the second equation of the above pair of
equations gives the Lorentz condition  $ k_\nu \varepsilon^\nu = 0$ which
when substituted in the first gives,
\begin{equation}
\varepsilon_\mu  =- i k_\mu \tilde{\alpha}.
\label{57}
\end{equation}
Since this is a solution proportional to the 4-momentum $k_\mu$, it can be gauged away by an appropriate
choice of the gauge. Thus massless excitations are gauge artefacts. For $k^2 = M^2 $ (massive excitations),
the equations of motion (\ref{566}) becomes,
\begin{equation}
(m^2 - M^2) \varepsilon^\mu + k^\mu  k_\nu \varepsilon^\nu + im^2  k^\mu \tilde{\alpha}  = 0, ~~~ \tilde{\alpha} = \frac{ik_\nu \varepsilon^\nu}{M^2}.
\label{58}
\end{equation}
Substituting  the second equation in (\ref{58}) in  the first yields,
\begin{equation}
(m^2 - M^2) \varepsilon^\mu + k_\nu \varepsilon^\nu k^\mu (1 - \frac{m^2}{M^2})
=0.
\label{59}
\end{equation}
Now, (\ref{59}) can be satisfied only if $M= m$. Therefore, one can conclude that the mass of the excitation is given by $m$ itself and  the rest frame momentum 4-vector of the
theory can be written as $ k^\nu = (m, 0, 0, 0)$. In the rest frame, the second
equation in (\ref{566}) gives
\begin{equation}
\varepsilon_0 = -i m \tilde{\alpha}.
\label{60}
\end{equation}
Therefore, the polarization vector of $A^\mu (x)$ field in (\ref{554}) can be written as
\begin{equation}
\varepsilon^{\mu} = (-i m \tilde{\alpha}, \varepsilon^1, \varepsilon^2, \varepsilon^3)^T
\label{61}
\end{equation}
The maximally reduced form of the polarization vector can be obtained from
(\ref{61}), by a gauge transformation with the choice $\Lambda (x) = \alpha (x)$ and is given by
\begin{equation}
\varepsilon^{\mu} = (0, \varepsilon^1, \varepsilon^2, \varepsilon^3)^T
\label{62}
\end{equation}
with the free components $\varepsilon^1, \varepsilon^2, \varepsilon^3 $
representing the three physical degrees of freedom of the theory.
One must note  that (\ref{62}) is of the same form as that of the $B\wedge F$ theory polarization vector \cite{bc1}.  Therefore, just as in the case of 
$B\wedge F$ theory, the action of representation $D(p,q,r)$ (\ref{53-1}) of 
$T(3)$ on the polarization vector  (\ref{62}) amounts to a gauge transformation
 in Stuckelberg extended Proca theory:
\begin{equation}
\varepsilon^{\mu} \rightarrow \varepsilon^{\prime \mu} = {D^{\mu}}_{\nu}(p,q,r)\varepsilon^{\nu} = \varepsilon^{\mu} + \frac{i}{m}(p\varepsilon^1 +
q \varepsilon^2 + r  \varepsilon^3)k^{\mu}
\label{622}
\end{equation}
The above transformation can be cast in the form of the momentum space 
gauge transformation 
\begin{equation}
\varepsilon^{\mu}\rightarrow \varepsilon^{\mu} + 
ik^\mu \lambda (k)
\label{6221}
\end{equation}
 (where $\Lambda(x)= \lambda (k) e^{ik.x}$) corresponding 
to the field $A(x)$, by choosing the field $ \Lambda(x)$ such that 
\begin{equation}
\lambda (k)= \frac{(p\varepsilon^1 +
q \varepsilon^2 + r  \varepsilon^3)}{m}. 
\label{6222}
\end{equation}

As mentioned before, it is possible to obtain the gauge transformation property of
$\alpha$ field from that of the $A^\mu$ field for which we now proceed as follows.
Consider the second relation in (\ref{58}); 
\begin{equation}
\tilde{\alpha} = \frac{ik_\mu \varepsilon^\mu}{m^2}.
\label{6223}
\end{equation}
and let $\varepsilon^\mu$ undergo the gauge transformation (\ref{6221}) which has the effect
of making a corresponding transformation in $\alpha$ field as  
\begin{equation}
\tilde{\alpha} \rightarrow \tilde{\alpha}'  = \frac{ik_\mu (\varepsilon^\mu + ik^\mu \lambda )}{m^2} =\frac{ik_\mu \varepsilon^\mu}{m^2} - \lambda = \tilde{\alpha} - \lambda  .
\label{62233}
\end{equation}
Here $\lambda$ is given by (\ref{6222}) corresponding to the gauge transformation generated by the
translational group $T(3)$ in the $A^\mu(x)$ field. Notice that the above equation (\ref{62233})
correspond to the second equation in (\ref{54}). We have thus obtained the gauge
transformation generated in the $\alpha$ field by $T(3)$ from that in the $A_\mu (x)$-field. It follows therefore that $\alpha$ field can be gauged away
completely by a suitable gauge fixing condition (unitary gauge) and it does not
appear in the physical spectrum of the theory.  

Hence it is obvious that the representation $D(p,q,r)$  of $T(3)$ generates 
gauge transformation in the massive vector gauge theory governed  by (\ref{554}).

\subsection{Massive symmetric tensor gauge theory}
Consider the massive and gauge noninvariant Einstein-Pauli-Fierz (EPF) theory
in 3+1 dimension as given by the Lagrangian,
\begin{equation}
{\cal L}_L^{EPF} = \frac{1}{2}h_{\mu\nu} \left[ R^{\mu\nu}_L - 
\frac{1}{2} \eta^{\mu\nu} R_L\right] - \frac{\mu^2}{2}\left((h_{\mu\nu})^2
 - h^2 \right).
\label{63}
\end{equation}
Just as the Proca theory (section 4.1) can be made gauge invariant by Stuckelberg
mechanism, the linearized EPF theory also can be made  gauge invariant by
introducing the an additional vector field $A^\mu$ as follows:
\begin{equation}
{\cal L}_L^{EPF} = \frac{1}{2}h_{\mu\nu} \left[ R^{\mu\nu}_L -
\frac{1}{2} \eta^{\mu\nu} R_L\right] - \frac{\mu^2}{2}\left(\left(h_{\mu\nu} +
\partial_\mu A_\nu + \partial_\nu A_\mu          \right)^2 - \left(h +
2 \partial \cdot A\right)^2\right).
\label{64}
\end{equation}
The  transformations
\begin{equation}
h_{\mu\nu} \rightarrow h'_{\mu\nu} = h_{\mu\nu} + \partial_\mu \Lambda_\nu
+ \partial_\nu \Lambda_\mu
\label{65}
\end{equation}
\begin{equation}
A_\mu (x)\rightarrow A'_\mu (x) = A_\mu (x) - \Lambda_\mu (x)
\label{66}
\end{equation}
represent the gauge symmetry of the theory described by (\ref{64}).
The equation of motion for $h_{\mu\nu}$ is
$$- \Box h^{\mu\nu} + \partial^\mu \partial_\alpha  h^{\alpha \nu} + \partial^\nu\partial_\alpha  h^{\alpha \mu}
 - \partial^\mu \partial^\nu h + \eta^{\mu\nu}(\Box h -\partial_\alpha
\partial_\beta h^{\alpha \beta})$$
\begin{equation}
 - \mu^2 \left[ (h^{\mu\nu} +
\partial^\mu A^\nu + \partial^\nu A^\mu) - \eta^{\mu\nu}(h + 2 \partial \cdot A)\right] = 0
\label{67}
\end{equation}
and that for $A_\mu$ is \begin{equation}
\Box A^\mu + \partial_\nu h^{\nu\mu} -  \partial^\mu h -\partial^\mu (\partial \cdot A)  = 0.
\label{68}
\end{equation}
Analogous to the case of massive vector gauge theory discussed before, the equation of motion
(\ref{68}) for $A^\mu$ can be obtained from (\ref{67}) by applying the operator $\partial_\nu$.
Therefore, gauge transformation of $A^\mu$  is obtainable by knowing the gauge transformation of 
the $h^{\mu\nu}$ field. 
As in the previous cases we employ the plane wave method to obtain the 
maximally reduced polarization
tensor $\chi_{\mu \nu}$  and vector $\varepsilon_\mu$  involved in $h_{\mu \nu}$ and $A_\mu$ respectively. Considering only the negative frequency part of a single  mode in the corresponding mode expansions, we write,
\begin{equation}
h_{\mu \nu}(x) = \chi_{\mu \nu}(k) e^{ik.x}
\label{69}
\end{equation}
\begin{equation}
A_\mu(x) = \varepsilon_\mu (k) e^{ik.x}.
\label{70}
\end{equation}
In terms of the polarization tensor $\chi_{\mu \nu}$ and vector $\varepsilon_\mu$  the gauge transformations (\ref{65}) and (\ref{66})  respectively can be written as
\begin{equation}
\chi_{\mu\nu} \rightarrow \chi_{\mu\nu}' = \chi_{\mu\nu} + ik_\mu \zeta_\nu
+ ik_\nu \zeta_\mu
\label{70+1}
\end{equation}
\begin{equation}
\varepsilon_\mu \rightarrow \varepsilon_\mu' = \varepsilon_\mu - \zeta_\mu
\label{70+2}
\end{equation}
where $\Lambda_\mu (x) = \zeta_\mu (k) \exp (ik.x)$.
Substituting (\ref{69}) and (\ref{70}) in (\ref{67}), one gets
$$k^2\chi^{\mu\nu} - k^\mu k_\alpha \chi^{\alpha \nu} -
k^\nu k_\alpha \chi^{\alpha \mu }
+  k^\mu k^\nu \chi
+\eta^{\mu\nu}(-k^2 \chi +k_\alpha k_\beta \chi^{\alpha \beta})$$
\begin{equation}
-\mu^2 \left[ \chi^{\mu\nu} + ik^\mu\varepsilon^\nu + ik^\nu\varepsilon^\mu
-\eta^{\mu\nu}(\chi + 2ik_\alpha\varepsilon^\alpha) \right] = 0.
\label{71}
\end{equation}
Contracting with $\eta_{\mu\nu}$ and considering only massless ($k^2 = 0$)
excitations the above equation reduces to
\begin{equation}
2k_{\mu\nu}\chi^{\mu\nu} + \mu^2\left[3(\chi + 2ik_\mu\varepsilon^\mu)\right] =
0.
\label{72}
\end{equation}
The solution of the above equation is
\begin{equation}
\chi^{\mu\nu} = -i(k^\mu\varepsilon^\nu + k^\nu\varepsilon^\mu).
\label{73}
\end{equation}
Hence it is also the solution of (\ref{71}) with $k^2 = 0$.
It is obvious that this solution is a gauge artefact since one can choose the
arbitrary vector field $\Lambda_\mu = A_\mu$ so as to make this solution vanish.

Next we consider the massive case ($k^2 = M^2, M \neq 0$) and consider the $(00)$
component of the equation of motion (\ref{71})  which, by a straightforward algebra,
can be reduced to
\begin{equation}
{\chi^1}_1 + {\chi^2}_2 + {\chi^3}_3 = 0
\label{74}
\end{equation}
Similarly the ($0i$) component of (\ref{71}) gives
\begin{equation}
{\chi_{0i}} = -iM \varepsilon_i
\label{75}
\end{equation}
Now, the ($ij$) component of (\ref{71}) is given by
\begin{equation}
k^2\chi_{ij} - \eta_{ij}k^2(\chi - \chi^{00}) - \mu^2[\chi_{ij} - \eta_{ij}
(\chi + 2iM\varepsilon^0)] =0
\label{76}
\end{equation}
Using (\ref{74}), the above equation can be reduced to
\begin{equation}
k^2\chi_{ij} - \mu^2[\chi_{ij} - \eta_{ij}
(\chi + 2iM\varepsilon^0)] =0
\label{77}
\end{equation}
For $i= j = 1, 2, 3$ respectively in (\ref{77}), we have  the following set of equations;
$$k^2\chi_{11} - \mu^2[\chi_{00} - \chi_{22} - \chi_{33}]
- 2iM\varepsilon_0 =0, $$
$$k^2\chi_{22} - \mu^2[\chi_{00} - \chi_{11} - \chi_{33}]
- 2iM\varepsilon_0 =0, $$
$$k^2\chi_{33} - \mu^2[\chi_{00} - \chi_{11} -\chi_{22}]
- 2iM\varepsilon_0 =0. $$
Adding the above three equations together and subsequently using (\ref{74}), we
arrive at
\begin{equation}
\chi_{00} = -2 iM\varepsilon_0
\label{78}
\end{equation}
When $i\neq j$, the equation (\ref{77}) reduces to
\begin{equation}
( \mu^2 - M^2) \chi_{ij} = 0.
\label{79}
\end{equation}
At this juncture notice that only two of the three components $\chi_{ii}, i= 1,
2, 3 $ are
independent on account of the equation (\ref{74}). Also, the $\chi_{00}$ and
$\chi_{0i}$ components can be set equal to zero by choosing the arbitrary 
field
$\Lambda_\mu$ to be $A_\mu$. Therefore, if $\chi_{ij} =0$
(for $ i\neq j$) in the above equation (\ref{79}), the number of independent components of $\chi_{\mu \nu}$ will  be only two. Since this is not the case, we can satisfy the equation (\ref{79})
only if $\mu^2 = M^2 $. Thus we see that the parameter $\mu$  represents the mass of the physical excitations of the field $h_{\mu \nu}$ and that its polarization
tensor is
\begin{equation}
\{ \chi_{\mu \nu} \} = \left( \begin{array} {cccc}
-2 i\mu\varepsilon_0 & -i\mu\varepsilon_1 & -i\mu\varepsilon_2 & -i\mu\varepsilon_3 \\
-i\mu\varepsilon_1   & \chi_{11}        &   \chi_{12}      &  \chi_{13}  \\
-i\mu\varepsilon_2   & \chi_{12}        &   \chi_{22}      &  \chi_{23}  \\
-i\mu\varepsilon_3   & \chi_{13}        &  \chi_{23}       &  \chi_{33} \end{array} \right)
\label{80}
\end{equation}
where $\chi_{11} + \chi_{22} + \chi_{33} = 0$ (see (\ref{74})).
As mentioned before, by choosing the  field $\Lambda_\mu$ to be $A_\mu$
and making a gauge transformation, the above form of the polarization tensor can be converted
 to its maximally reduced form given by
\begin{equation}
\{ \chi_{\mu \nu} \} = \left( \begin{array} {cccc}
0 & 0 & 0 & 0 \\
0 &  \chi_{11}        &   \chi_{12}      &  \chi_{13}  \\
0 &  \chi_{12}        &   \chi_{22}      &  \chi_{23}  \\
0 &  \chi_{13}        &  \chi_{23}       &  \chi_{33} \end{array} \right);~~~ \chi_{11} + \chi_{22} + \chi_{33} = 0.
\label{81}
\end{equation}
Our next task is to show explicitly that it is possible to obtain the gauge transformation of $A^\mu$
from that of $h^{\mu\nu}$. For this purpose we 
consider now the equation of motion (\ref{68}) corresponding to the vector field $A_\mu$ and
the associated polarization tensor $\varepsilon_\mu$. Substituting (\ref{69}) and (\ref{70}) in
 (\ref{68})(or by contracting (\ref{71}) with $k_\nu$ ) we get,  
\begin{equation}
k^2\varepsilon^\mu -  k^\mu k_\nu\varepsilon^\nu - ik_\nu \chi^{\nu \mu} + ik^\mu \chi = 0.
\label{82}
\end{equation}
On making a gauge transformation (\ref{70+1}) in the polarization tensor $\chi^{\mu\nu}$, the polarization
vector $\varepsilon^\mu$ in (\ref{82}) automatically undergoes a gauge transformation. 
\begin{equation}
k^2\varepsilon'^\mu -  k^\mu k_\nu\varepsilon'^\nu =  ik_\nu (\chi^{\nu \mu} + ik^\mu \zeta^\nu
+ ik^\nu \zeta^\mu)
 - ik^\mu (\chi + 2ik^\nu \zeta_\nu).
\label{82+1}
\end{equation}
This implies 
\begin{equation}
k^2\varepsilon'^\mu -  k^\mu k_\nu\varepsilon'^\nu =  [ik_\nu \chi^{\nu \mu} -ik^\mu \chi ] -k^2 \zeta^\mu + k^\mu (k.\zeta)
\label{82+2}
\end{equation} 
From  equation (\ref{82}), substitute for the expression inside the square bracket in (\ref{82+2}) to obtain
\begin{equation}
[k^2\varepsilon'^\mu -  k^\mu k_\nu\varepsilon'^\nu ] -[k^2\varepsilon^\mu -  k^\mu k_\nu\varepsilon^\nu ] = -k^2 \zeta^\mu + k^\mu (k.\zeta)
\label{82+3}
\end{equation}
It is now easy to see that this relation can be satisfied only if $\varepsilon'^\mu -\varepsilon ^\mu = - \zeta^\mu $ in agreement with
the previous relation (\ref{70+2}).
Therefore a knowledge of the gauge transformation of $h_{\mu\nu}$ is enough to deduce the gauge transformation 
property of $A_\mu$-field. Like the $\alpha$-field in Stuckelberg extended
Proca theory, this $A_\mu$-field too disappears from the physical spectrum.
 

Now we study the gauge transformation properties of the field $h_{\mu \nu}$ 
under the action of the little group. It is easy to see that, similar to the case of $B\wedge F$ theory,
the representation $D(p, q, r)$ (\ref{53-1}) of $T(3)$ generate gauge 
transformation of the massive field. The action of $D(p, q, r)$ on the polarization tensor
$\{\chi_{\mu\nu}\}$ (\ref{81}) is given by,
$$\{\chi_{\mu\nu}\} \rightarrow  \{\chi_{\mu\nu}\}' =  D(p, q, r) \{\chi_{\mu\nu}\} D^T(p, q, r) = $$
\begin{equation}
\{\chi_{\mu\nu}\} + \left( \begin{array}{cccc}
\left(\begin{array}{c}p(p\chi_{11} + q\chi_{12} + r\chi_{13}) \\
+ q(p\chi_{12} + q\chi_{22} + r\chi_{23}) \\
+ r( p\chi_{13} + q\chi_{23} + r\chi_{33})  \end{array}\right)
&\left(\begin{array}{c} p\chi_{11} + q\chi_{12} \\+ r\chi_{13} \end{array}\right) &
\left(\begin{array}{c}  p\chi_{12} + q\chi_{22} \\ + r\chi_{23} \end{array} \right)&
\left(\begin{array}{c}  p\chi_{13} + q\chi_{23} \\
+ r\chi_{33}  \end{array}\right)  \\
p\chi_{11} + q\chi_{12} + r\chi_{13} & 0 & 0 & 0 \\
p\chi_{12} + q\chi_{22} + r\chi_{23} & 0 & 0 & 0 \\
 p\chi_{13} + q\chi_{23} + r\chi_{33} & 0 & 0 & 0 \\
\end{array} \right)
\label{87+1}
\end{equation}
By choosing
$$ \zeta_0 = \frac{1}{2} (p\zeta_1 + q\zeta_2 + r\zeta_3) $$
$$ \zeta_1 = \frac{1}{\mu} (p\chi_{11} + q\chi_{12} + r\chi_{13}) $$
$$ \zeta_2 = \frac{1}{\mu} (p\chi_{12} + q\chi_{22} + r\chi_{23}) $$
$$ \zeta_3 = \frac{1}{\mu}(p\chi_{13} + q\chi_{23} + r \chi_{33});~~~~\chi_{11} + \chi_{22} + \chi_{33} =0 $$
it is straightforward to see that (\ref{87+1}) is of the form (\ref{70+1}) 
which is the gauge transformation of $\chi_{\mu\nu}$. Notice that when one 
makes the choices for  the components $\zeta_1, \zeta_2, \zeta_3$ in terms of 
the parameters $p,q,r$ of the translational group $T(3)$, the component 
$\zeta_0$ gets automatically fixed. Therefore, in  the gauge transformation 
(\ref{87+1}) generated by the representation $D(p,q,r)$ of $T(3)$ only the
three space components of the field $\zeta_\mu$ remain arbitrary. 
 However, in the complete set of gauge transformations (\ref{70+1}) all the 
four components of $\zeta_\mu$ should be chosen independent of one another.  
Hence, the above gauge transformations (\ref{87+1})  generated by the 
translational group $D(p,q,r)$ does not exhaust the complete set of gauge 
transformations
available to the massive symmetric tensor gauge theory. This  is because 
of the fact that in order to generate the entire  gauge equivalence 
class of the maximally reduced polarization tensor (\ref{81}) we require
four independent variables (corresponding to the four components of the 
abitrary vector function $\zeta_\mu (k)$ which represents the gauge freedom) whereas the translational group $T(3)$
provides only three independent parameters.

Therefore,  in the present case (\ref{64}) of massive tensor gauge theory,
a partial set of  gauge transformations are generated  the representation 
$D(p,q, r)$ (\ref{53-1}) of the translational group $T(3)$. The gauge 
transformation
of the $A_\mu$-field can be obtained from that of the $h_{\mu\nu}$-field, though
 the former one does not appear in the physical spectrum of the theory.

\subsection{Massive antisymmetric tensor gauge theory}

Here we show that the translational group $T(3)$ generates the
full range of gauge transformations in the Stuckelberg extended 
massive Kalb-Ramond theory. Similar to the massless KR theory discussed in
section3, the gauge transformation in Stuckelberg extended massive KR theory
is also reducible.  Though the 
analysis in this case closely resembles that of Stuckelberg extended EPF theory
detailed before, here the reducibility  of the  
gauge transformation  is manifested in the  gauge 
generation by $T(3)$. 

The Lagrangian of the Stuckelberg extended
massive KR theory is 
\begin{equation}
{\cal L} = \frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda} - \frac{m^2}{4} (B_{\mu\nu} + \partial_\mu A_\nu - 
\partial_\nu A_\mu ) (B^{\mu\nu} + \partial^\mu A^\nu -
\partial^\nu A^\mu )
\label{n1}
\end{equation}
with $B_{\mu\nu} = -B_{\nu\mu}$ and $~H_{\mu\nu\lambda} =
\partial_\mu B_{\nu\lambda} + \partial_\nu B_{\lambda\mu} +
\partial_\lambda B_{\mu\nu}$. It can be easily verified that this theory
is invariant under the joint gauge transformations 
\begin{equation}
B_{\mu\nu}(x) \rightarrow B_{\mu\nu}(x) + \partial_\mu F_\nu
(x) - \partial_\nu F_\mu (x)
\label{n2}
\end{equation}
and
\begin{equation}
A_\mu (x)\rightarrow A_\mu (x) - \Lambda_\mu (x).
\label{n3}
\end{equation}
Here we must notice that the transformation (\ref{n2}) is  reducible exactly
as  in the case of massless KR theory; i.e.,
the transformation (\ref{n2}) remains invariant if we make the change 
$F_\mu (x) \rightarrow F_\mu (x) + \partial_\mu \beta (x)$. Therefore, there
exist a `gauge invariance of gauge transformation' in the theory described by
(\ref{n1}) also.
The equation of motion corresponding to $ B^{\nu\lambda}(x)$ is given by
\begin{equation}
\partial_\mu H^{\mu\nu\lambda} + m^2 (B^{\nu\lambda} + \partial^\nu A^\lambda
- \partial^\lambda  A^\nu ) = 0 
\label{n4}
\end{equation}
and that corresponding to $A^\nu$ is 
\begin{equation}
\partial_\mu (B^{\mu\nu} + \partial^\mu A^\nu - \partial^\nu A^\mu ) = 0.
\label{n5}
\end{equation}
As in the case of the Stuckelberg extended massive theories considered 
previously in sections 4.1 and 4.2, the equation of motion (\ref{n5}) for
$A^\nu$ can be obtained from the equation (\ref{n4})  by the 
 application of the operator $\partial_\lambda$ upon 
the latter equation. Hence,  one can  obtain the gauge 
transformation property (\ref{n3}) of the $A^\nu$ field from that of the 
$ B^{\nu\lambda}$ field. (This can be easily achieved by a straightforward
procedure  similar to the ones adopted before in the cases
 of Stuckelberg extended Proca and EPF theories for the same purpose 
and hence is not elaborated here again.) 

Our next task is to obtain the maximally reduced polarization tensor 
$\varepsilon^{\mu\nu}(k)$ corresponding to the antisymmetric field 
$B^{\mu\nu}(x)$
so that the role of $T(3)$ as a generator of gauge transformations in 
(\ref{n1}) can be studied. For this purpose, as usual we use the ansatz 
\begin{equation} 
B^{\mu\nu}(x) = \varepsilon^{\mu\nu}(k) e^{ik\cdot x},~~~~A^{\mu}(x) = \varepsilon^{\mu}(k)  e^{ik\cdot x}
\label{n6}
\end{equation}
and employ the plane wave method. 
The momentum space gauge transformation corresponding to (\ref{n2}) 
now has the same form as (\ref{18+1});
\begin{equation}
\varepsilon^{\mu \nu}(k) \rightarrow  \varepsilon^{\mu \nu}(k) + (k^{\mu}f^{\nu}(k) - k^{\nu}f^{\mu}(k))
\label{nn}
\end{equation}
Then the equation of motion
 (\ref{n4}) can be written (in the momentum space) as 
\begin{equation}
-k^2 \varepsilon^{\nu\lambda} - k^\nu k_\mu \varepsilon^{\lambda\mu} - k^\lambda k_\mu \varepsilon^{\mu \nu} + m^2 (\varepsilon^{\nu\lambda} + ik^\nu \varepsilon^{\lambda} - ik^\lambda \varepsilon^{\nu}) = 0.
\label{n7}
\end{equation}
If $k^2 = 0$ (massless excitations), the above equations reduces to 
\begin{equation}
- k^\nu k_\mu \varepsilon^{\lambda\mu} - k^\lambda k_\mu \varepsilon^{\mu \nu} + m^2 (\varepsilon^{\nu\lambda} + ik^\nu \varepsilon^{\lambda} - ik^\lambda \varepsilon^{\nu}) = 0
\label{n8}
\end{equation}
the most general solution for which is 
\begin{equation}
\varepsilon^{ \nu\lambda}(k) = C(ik^\nu \varepsilon^{\lambda} - ik^\lambda 
\varepsilon^{\nu}) + D (\epsilon^{\nu\lambda\tau\sigma} 
k_\tau\varepsilon_\sigma )
\label{n9}
\end{equation}
where $C$ and $D$ are constants to be fixed.
Substituting this solution (\ref{n9}) in (\ref{n8}), we can easily see that
$C = -1$ and $D = 0$.
Therefore, the solution to (\ref{n7}) corresponding to  massless excitations
is 
\begin{equation}
\varepsilon^{ \nu\lambda}(k) = -ik^\nu \varepsilon^{\lambda} + ik^\lambda
\varepsilon^{\nu}.
\label{n10}
\end{equation} 
However, such solutions can be gauged away by choosing the arbitrary field $\Lambda^\mu (x)  = A^\mu (x)$ which shows that massless excitations are gauge artefacts. 

Next we consider the massive case, $k^2 = M^2,~(M\neq 0)$ where it is
possible to go to the rest frame and one has the momentum 4-vector 
$k^\mu = (M, 0,0,0)^T$.  In the rest frame,
the equation of motion (\ref{n7}) 
reduces to 
\begin{equation}
(m^2 -M^2) \varepsilon^{ \nu\lambda} -M(k^\nu \varepsilon^{\lambda 0} + k^\lambda \varepsilon^{ 0\nu }) + m^2 (ik^\nu \varepsilon^{\lambda} - ik^\lambda
\varepsilon^{\nu}) = 0.
\label{n11}
\end{equation}
Note that, since the polarization tensor $\varepsilon^{ \nu\lambda}$ is antisymmetric,
all its diagonal entries are automatically zero. Considering the components
of (\ref{n11}) for which $(\nu =0, \lambda = i)$, we have 
\begin{equation}
 \varepsilon^{ i0} = iM \varepsilon^{ i}
\label{n12}
\end{equation}
For $(\nu =i, \lambda = j)$ with $i\neq j$, the equation (\ref{n11}) gives 
\begin{equation}
(m^2 -M^2) \varepsilon^{ij} = 0.
\label{n13}
\end{equation}
This leads to two possibilities; either $\varepsilon^{ij} = 0$ or $ M^2 = m^2$.
The former possibility can be ruled out by the following reasoning.
Since (\ref{n1}) is the first class version (obtained by a Stuckelberg 
extension mechanism)
of massive KR theory possessing three physical degrees of
freedom, the former too must inherit the same number of degrees of freedom. 
However, the $\varepsilon^{ i0}$ elements can all be made to vanish by the gauge choice $\Lambda_\mu = A_\mu$. Therefore the possibility
 $\varepsilon^{ij} = 0$ leads to a null theory and hence should be discounted. 
Therefore, we have $M^2 = m^2$ which  is also consistent with the degrees of 
freedom counting. Finally, analogous to (\ref{81}), the maximally reduced 
form of the polarization tensor corresponding to (\ref{n1}) is given by,
\begin{equation}
\{\varepsilon^{\mu\nu}\} = \left( \begin{array} {cccc}
0 & 0 & 0 & 0 \\
0 &  0        &   \varepsilon^{12}      &  \varepsilon^{13}  \\
0 &     -\varepsilon^{12}     &     0    &  \varepsilon^{23}  \\
0 &   -\varepsilon^{13}      &  -\varepsilon^{23}       &  0 \end{array} \right)
\label{n14}
\end{equation}
As in the case of Stuckelberg extended EPF theory, the $A_\mu$-field  disappears from the physical spectrum in this case also.
Here it must be emphasized that the maximally reduced polarization tensor
of $B\wedge F$ theory also has the same form (\ref{n14}). This is not 
surprising since the physical sector of $B\wedge F$ theory is equivalent to
massive KR theory whose first class version is the theory (\ref{n1}) under
consideration now. 
It is now straight forward to see that the translational group $T(3)$ in the 
representation $D(p, q, r)$ (\ref{53-1}) generates the full set  of gauge 
transformations in the theory described by (\ref{n1}) also. The action of 
$D(p, q, r)$ on (\ref{n14}) is given by,
$$\{\varepsilon^{\mu\nu}\}\rightarrow \{\varepsilon^{\mu\nu}\}' = D(p, q, r) \{\varepsilon^{\mu\nu}\} D^T(p, q, r)$$
\begin{equation}
=\{\varepsilon^{\mu\nu}\} + \left( \begin{array} {cccc}
0 & -q\varepsilon^{12} - r \varepsilon^{13} & p\varepsilon^{12} - r \varepsilon^{23} & p\varepsilon^{13} + q \varepsilon^{23} \\
q\varepsilon^{12} + r \varepsilon^{13} & 0 & 0& 0 \\
-p\varepsilon^{12} + r \varepsilon^{23}& 0 & 0& 0 \\
- p\varepsilon^{13} - q \varepsilon^{23}& 0 & 0& 0 \\
\end{array} \right)
\label{n15}
\end{equation}
This can be considered to be the gauge 
transformations of the form (\ref{nn}) with the choice 
\begin{equation}
f^1 =\frac{1}{m}(q\varepsilon^{12} + r \varepsilon^{13}),~~~
f^2 =  \frac{1}{m}(-p\varepsilon^{12} + r \varepsilon^{23}), ~~~
f^3 = \frac{-1}{m}( p\varepsilon^{13} + q \varepsilon^{23})
\label{n16}
\end{equation}
Here one must clearly note that the component $f^0$ remains completely 
undetermined and does not depend at all either on the parameters $p,q,r$ of 
$T(3)$ or on the maximally reduced polarization tensor of the theory 
whereas the other components $f^1, f^2, f^3$ are determined by these
parameters and the elements of the polarization tensor.
Interestingly, it is exactly in the same fashion as in the present case (of 
Stuckelberg  extended massive KR theory) that gauge transformations 
of $B\wedge F$ theory are generated by the translational group $D(p,q,r)$ (we
refer to \cite{bc1} for details). Hence, analogous to the gauge transformation
generated by $W(p,q)$ in massless KR theory, for any given set of ($f^1, f^2, f^3$)
we have a continuum of values for $f^0$, representing the reducibility of
the gauge transformation in the underlying 2-form field  both in  Stuckelberg
extended first class version of massive KR theory and in the $B\wedge F$ theory.
Therefore, the complete independence of the time-component of $f^\mu$ on the 
maximally reduced polarization tensor or on the parameters of the group 
$D(p,q,r)$ is a consequence of the reducibility of the gauge transformations 
of these theories.

Analogous to the hierarchical structure involving the elements of $T(2)$ 
present in the gauge generation in massless $n$-form theories, there is 
a hierarchical structure in the gauge transformations generated by $T(3)$ 
in massive $n$-form theories also. In section 4.1 we have seen that an element
of $T(3)$ generates gauge transformation in massive 1-form theory (the 
Stuckelberg extended Proca theory). In the massive 2-form  (the
Stuckelberg extended massive KR) theory, two elements of $T(3)$ are involved
as generators of gauge transformation, one element for the gauge transformation
of the field $B^{\mu\nu}$ and a second element for the `gauge transformation'
$F^\mu \rightarrow F^\mu + \partial^\mu \beta$ which correspond to the 
reducibility of (\ref{n2}). (This transformation is of the same form as 
the first transformation in (\ref{54}) corresponding to massive vector theory
and hence may be considered to be generated by $T(3)$.)  


\section{Dimensional Descent}

Dimensional descent \cite{bc2} is a method by which one can obtain the
 energy-momentum
vector, polarization tensor and the gauge generating representation of the translational
subgroup of Wigner's little group etc in a  massive gauge theory living in a
certain space time
dimension from similar results for gauge theories in one higher dimension. In this sense,
dimensional descent is a unification scheme for the results presented in the previous sections
regarding various gauge theories. Similar ideas are earlier used in the context
of string
theory where a massive particle is viewed as a massless particle in one higher dimension
with the mass being considered as the momentum component along the additional dimension \cite{blt}.

We begin our discussion of dimensional descent by noting that, the translational group $T(3)$ which generates gauge transformation
in 3+1 dimensional  $B\wedge F$ theory and in the massive excitations of Stuckelberg extended Proca and EPF theories, is an invariant
 subgroup of $E(3)$.  Now, just as $E(2)$ is the generator of gauge transformation in 4-dimensional
Maxwell theory, $E(3)$ generates gauge transformation in 5-dimensional Maxwell theory. This
indicates that the generators of gauge transformations in the above mentioned 
massive gauge theories and  5-dimensional Maxwell theory 
are related. This relationship is explicitly demonstrated through the method
of dimensional descent which we will describe below in the present context. 

An element of Wigner's little group in 5 dimensions can be written as
\begin{equation}
W_5(p,q,r; \psi, \phi, \eta) =
\left( \begin{array}{ccccc}
1+ \frac{p^2 + q^2 + r^2}{2} & p & q & r & -\frac{p^2 + q^2 + r^2}{2} \\
p & & & & -p \\
q & &R(\psi, \phi, \eta) & & -q \\
r & & & & -r  \\
\frac{p^2 + q^2 + r^2}{2} &  p & q & r & 1 -\frac{p^2 + q^2 + r^2}{2}
\end{array}\right)
\label{88}
\end{equation}
where $p,q,r$ are  any real numbers, while $R(\psi, \phi, \eta) \in
SO(3)$,  with $(\psi, \phi, \eta)$ being a triplet of Euler angles.
 The above result can be derived by following the standard treatment(see, for
example \cite{we}). The
corresponding element of the translational group $T(3)$ can be trivially
obtained by setting $R(\psi, \phi, \eta)$ to be the identity matrix and
will be denoted by $W(p,q,r) = W_5(p,q,r; 0)$. \\
 
Let us now consider free Maxwell theory in 5-dimensions,
\begin{equation}
{\cal L} = -{\frac{1}{4}}{F^{ij}F_{ij}} ; \hskip 1.0cm i
,j = 0, 1, 2, 3, 4.
\label{89}
\end{equation}
For a photon of energy $\omega$ (in 5-dimensional space-time) propagating in the $i = 4$ direction,
the momentum 5-vector is given by
\begin{equation}
k^i = (\omega, 0, 0, 0, \omega)^T.
\label{90}
\end{equation}
By following the plane wave method and proceeding exactly as in section 2, one can show that the
maximally reduced form of the polarization vector of the photon is
\begin{equation}
\varepsilon^{i} = (0, \varepsilon^1,\varepsilon^2,\varepsilon^3 , 0)^T
\label{91}
\end{equation}
where $\varepsilon^1,\varepsilon^2,\varepsilon^3$ represent the three 
transverse degrees of freedom
(since the polarization vector satisfies the `Lorentz gauge'
$\varepsilon^{i}k_i = 0$).
If we now suppress the last row of the column matrices  $k^{i}$ (\ref{90}) and $\varepsilon^{i}$ (\ref{91}), we end up respectively with the energy-momentum 4-vector and the polarization vector 
 of Stuckelberg extended Proca model in 3+1 dimensions. This is equivalent to applying
the projection operator given by the matrix
\begin{equation}
{\cal P}= diag(1, 1, 1, 1, 0)
\label{92}
\end{equation}
to the momentum 5-vector (\ref{90}) and the polarization vector (\ref{91}).
Similarly, it is  possible to derive the polarization tensor of
Stuckelberg extended EPF theory (\ref{64}) from that  of linearized  Einstein gravity
 in 5-dimensions by a  procedure analogous to the one
  described above. As done in the 3+1 dimensional case, one can easily 
show that the maximally reduced form of the polarization tensor of 4+1 
dimensional linearized gravity is 
\begin{equation}
\{ \chi^{ij} \} = \left( \begin{array} {ccccc}
0 & 0 & 0 & 0 & 0 \\
0 &  \chi^{11}        &   \chi^{12}      &  \chi^{13}  & 0 \\
0 &  \chi^{12}        &   \chi^{22}      &  \chi^{23}   & 0 \\
0 &  \chi^{13}        &  \chi^{23}       &  \chi^{33}   & 0 \\
0 & 0 & 0 & 0 & 0 \end{array} \right); ~~~~ \chi^{11} + \chi^{22} + \chi^{33} =0.
\label{94}
\end{equation}
By suppressing the
last row and  column of the polarization tensor(\ref{94}), 
one obtains the polarization tensor (\ref{81})  of the  Stuckelberg extended 
EPF model in 4-dimensions. 
 
Now we study  the gauge transformation properties of polarization vector 
(\ref{91}) and polarization tensor (\ref{94})  in relation to the translational subgroup
$T(3)$ of Wigner's little group $E(3)$\footnote{In this regard, we recollect 
a comment made in \cite{sc} that translational subgroup of Wigner's little group
for massless particles generates gauge transformations only in the 3+1 
dimensional version of linearized gravity, but not in its higher dimensional 
versions. This was mistakenly ascribed to the mismatch in the number of 
degrees of freedom ($\frac{d(d-3)}{2}$) in higher dimensional linearized gravity and the
number of parameters ($d-2$) of the translational subgroup of Wigner's little group for massless particles in $d> 4$. 
However, this need to be amended as this mismatch is
of no consequence in this regard and it must be stated that the translational
subgroup generates gauge transformations for linearized gravity any dimension $d\geq 4$.    
  }. For this purpose, consider the  action of $W(p, q, r)$ on
$\varepsilon^{i}$.
\begin{equation}
\varepsilon^{i} \rightarrow {\varepsilon'^{i}} = \varepsilon^{i} + \delta \varepsilon^{i} =
{{W_5 (p, q, r)}^{i}}_{j} \varepsilon^{j} = \varepsilon^{i} +
( p\varepsilon^1 + q\varepsilon^2 + r\varepsilon^3)\frac{k^i}{\omega}
\label{95}
\end{equation}
This is indeed a gauge transformation in (4+1) dimensional Maxwell theory.
Applying the projection operator ${\cal P}$  (\ref{92}) on (\ref{95}) yields
\begin{equation}
\delta {\varepsilon}^{\mu} = {\cal P}\delta\varepsilon^{i}
= \frac{1}{\omega}( p\varepsilon^1 + q\varepsilon^2 + r\varepsilon^3)k^{\mu}
\label{96}
\end{equation}
Here ${\varepsilon}^{\mu} = (0,\varepsilon^1,\varepsilon^2,\varepsilon^3 )^T$ corresponds to the 
polarization vector in the Stuckelberg extended Proca theory and $k^\mu$ is the
 momentum vector of a particle at rest in 3+1 dimensions. (Here, the time component $\omega$  
of a 5-dimensional a massless particle, moving along the extra 5th dimension 
is identified with the mass $\omega$ of a massive particle at rest 
in 4-dimensional space-time.)
Modulo an $i$ factor, this is precisely how the polarization vectors  in the 
massive gauge theory (\ref{554}) transforms under gauge transformation (see Section 4.1).
The form of (\ref{96}), makes it obvious that 
\begin{equation}
\delta \varepsilon^{\mu} = D(p,q,r) \varepsilon^{\mu} - {\varepsilon}^\mu
\label{97}
\end{equation}
where
$D(p, q, r)$ is given by (\ref{53-1}). Thus, in this fashion we are able to 
derive  the gauge generating representation $D(p, q, r)$ of $T(3)$ in a massive vector field by a
judicious application of
the projection operator ${\cal P}$  (\ref{92}) from the gauge transformation
relation of a higher dimensional massless gauge theory.

We now consider  the polarization matrix $\{\chi^{ij}\} $ (\ref{94}) of 
linearized gravity in 5-dimensions for which
the transformation under the action of $W(p,q,r)$  is given by,
$$\{\chi^{ij}\} \rightarrow \{\chi '^{ij}\} = W(p, q, r)\{\chi^{ij}\} W^T (p, q, r)
=\{\chi^{ij}\} + \delta \{\chi^{ij}\}$$
where,
\begin{equation}
\delta \{\chi^{ij}\} =  \{\delta \chi^{ij} \}
= \left(
\begin{array}{ccccc}
ap+ bq+rc & a & b & c & ap+ bq+rc \\
a & 0 & 0 & 0 & a \\
b & 0 & 0 & 0 &  b \\
c & 0 & 0 & 0 & c\\
ap+ bq+rc & a & b & c & ap+ bq+rc
\end{array}
\right)
\label{98}
\end{equation}where,
$a = p\chi^{11}+q\chi^{12}+ r\chi^{13},~ b = p\chi^{12}+q\chi^{22}+ 
r\varepsilon^{23}$ and  
$c = p\chi^{13}+q\chi^{23}+ r\chi^{33}$ with $ \chi^{11} + \chi^{22} +\chi^{33}=0$.
Again this can be easily recognized as a gauge transformation\footnote{Exactly
 as in the case of 4-dimensional case, this gauge transformation forms only a
subset of the full set of gauge transformation available in 5-dimensional linearized gravity.}  in (4+1)
dimensional linearized gravity involving massless quanta, as $\delta \chi^{ij}$ can be expressed as $ (k^{i} \zeta^{j}(k) + k^{j}\zeta^{i}(k))$
with suitable choice for $\zeta^{i}(k)$, where $k^i = (\omega , 0, 0, 0, \omega )^T$.
By applying the projection operator
${\cal P}$ on (\ref{98}), we get
the change (under gauge transformation) in the 3+1 dimensional polarization matrix
$ \{ {\chi}^{\mu \nu}\}$ (of Stuckelberg extended EPF theory), by the formula,
$\delta \{ {\chi}^{\mu \nu}\} = {\cal P}\delta \{\chi^{ij}\}{\cal P}^T$. This simply
amounts to a deletion of the last row and column of $\delta \{\chi^{ij}\} $.
The result
can be expressed more compactly as
\begin{equation}
\delta \{\chi^{\mu\nu}\} = (D\{\chi^{\mu \nu}\}D^T - \{\chi^{\mu \nu}\})
\label{99}
\end{equation}
where $D=D(p,q,r)$ has already been defined in (\ref{53-1}).
This has the precise form of gauge transformation of the
 polarization matrix
of  Stuckelberg extended EPF model, since it can be cast in the form
\begin{equation}
\delta {\chi}^{\mu \nu} = (k^{\mu} \zeta^{\nu}(k) + k^{\nu}\zeta^{\mu}(k))
\label{100}
\end{equation}
for a suitable $\zeta^{\mu}(k)$, where $k^{\mu} = (\mu, 0, 0, 0)^T$. Here we have identified $\omega$ with $\mu$.
 
Clearly the
 generators $T_1 = \frac{\partial D}{\partial p}, T_2 =
\frac{\partial D}{\partial q}, T_3 = \frac{\partial D}{\partial r} $ provide a
Lie algebra basis for the group $T(3)$. One can verify,
\begin{equation}
[T_1, T_2] = [T_2, T_3] = [T_3, T_1] = 0
\label{101}
\end{equation}
They also satisfy
\begin{equation}
{T_1}^2 = {T_2}^2 = {T_3}^2 = T_1 T_2 = T_1 T_3 = T_2 T_3 = 0
\label{102}
\end{equation}
so that
\begin{equation}
 D(p, q, r) = e^{pT_1 + qT_2 + rT_3} = 1 + pT_1 + qT_2 + rT_3
\label{103}
\end{equation}
The change in the polarization vector  $\varepsilon^\mu$ can be expressed
as the action of a Lie algebra element,
\begin{equation}
\delta \varepsilon^\mu = (pT_1 + qT_2 + rT_3) \varepsilon^\mu
\label{103+1}
\end{equation}
Besides this, $D(p,q,r)$ also preserves the 4-momentum of a particle at rest.
Thus we have shown, how this representation
(\ref{53-1}) of $T(3)$ can be connected to the Wigner's little group for
massless particle in 4+1 dimension through appropriate projection in the
intermediate steps, where the massless particles moving in  4+1 dimensions
can be associated with a massive particle at rest in 3+1 dimensions.

The method of dimensional descent as applied in the case of 3+1 dimensional
$B\wedge F$ theory
was earlier discussed in \cite{bc2} where it was shown that one can arrive at
the representation $D(p,q,r)$ of translational group $T(3)$ by considering
the gauge transformation properties of 4+1 dimensional massless KR theory.
Since the physical sectors of Stuckelberg extended massive KR theory
and $B\wedge F$ theory are equivalent (and hence posses identical 
  rest frame momentum 4-vectors and maximally reduced 
polarization tensors), it is possible to obtain the gauge generating
representation $D(p,q,r)$ of $T(3)$ for Stuckelberg extended
massive KR theory using dimensional descent proceeding exactly as 
was done in \cite{bc2} for the case of $B\wedge F$ theory.

In earlier studies, the method of dimensional descent was used to relate
the gauge transformation properties of a massless gauge theory to those
of a topologically massive gauge theory inhabiting a lower dimensional space-time. 
Here we have demonstrated that dimensional descent is equally applicable
to massive gauge theories obtained by a Stuckelberg extension of the 
configuration space of massive theories. Hence we conclude that dimensional
descent relates gauge transformation of a massless gauge theory to that of
an appropriate lower dimensional massive gauge theory.
This is true for both topologically massive gauge theories and massive theories
rendered gauge invariant by Stuckelberg mechanism.
 Exactly as described above, dimensional descent 
may also be employed to connect theories living in 3+1 dimensions to those in
2+1 dimensions \cite{bc2,sc}.



\section{Conclusion}

In the present work, apart from making certain amendments to some of the earlier
 studies on the role of  translational groups in generating gauge 
transformations in Abelian gauge theories, we have shown that translational
groups can act as gauge generators also for massive theories rendered gauge
invariant by a Stuckelberg mechanism. 
We started our discussion with a  brief review (in section 2) of the essential 
aspects of Wigner's little group and its 
translational subgroup in relation to their role in generating gauge 
transformations.  

 We pointed out in section 3, that for massless tensor gauge theories (linearized gravity and Kalb-Ramond theories), although the translational subgroup $T(2)$ of
little group for massless particles generate gauge transformations, such
gauge transformations forms only a subset of the whole range of gauge transformations available in the theories. 
 We showed that  partial gauge generation 
in Kalb-Ramond theory by the translational group $T(2)$ manifestly exhibits 
the 
reducibility of gauge transformations in the theory. In the gauge generation
by $T(2)$ in KR theory, only the two of the components of the arbitrary
vector field (corresponding to the gauge freedom of the theory) can be
expressed in terms of the parameters of $T(2)$ and elements of the maximally
reduced polarization tensor where as the other two components are left 
completely undetermined subject to a single constraint. This is shown 
to be related to the reducibility of the gauge transformations of the theory. 
In contrast,
for the case of linearized gravity  where   any such reducibility is absent,
in gauge transformations are generated by $T(2)$  all the components of the arbitrary vector field  representing  gauge freedom are dependent on the 
parameters of $T(2)$ as well as on the polarization tensor. 
Thus the presence of any reducibility of gauge transformations 
is reflected in the gauge generation by the translational subgroup of Wigner's
little group for massless particles. 
Moreover, there exists an interesting  hierarchical structure in the 
gauge generation by the translational  group with $n$ independent elements of
$T(2)$
involved in the gauge generation in an $n$-form massless gauge theory
in 3+1 dimensions. This was explicitly demonstrated in the cases of $n=1,2$
and is expected to be valid for $n>2$ also. The relationship of Wigner's little
group with gauge transformations in higher form gauge theories is under
further study and will be reported elsewhere. 

In section 4, we have shown that for massive 3+1 dimensional theories rendered
gauge invariant by Stuckelberg mechanism, the gauge transformations 
are generated by a particular representation of the translational group $T(3)$.
In earlier studies, this representation of the  translational group was shown to generate
gauge transformations  in the topologically massive $B\wedge F$ theory.
Therefore, we find that the gauge transformations are generated by $T(3)$
in all massive gauge theories irrespective of whether the concerned theory 
is a topologically
massive gauge theory or a first class constraint system  obtained
via Stuckelberg extension of a massive theory. 
As examples, we considered massive vector gauge 
(the Stuckelberg extended Proca) theory in section 4.1,  massive symmetric 2nd rank 
tensor gauge (the Stuckelberg extended Einstein-Pauli-Fierz) theory in section 
4.2 and 
 massive 2-form gauge (the Stuckelberg extended  massive Kalb-Ramond)
theory in section 4.3. 
Similar to the partial gauge generation by $T(2)$  in linearized gravity,  gauge transformations generated by $T(3)$ in Stuckelberg
extended Einstein-Pauli-Fierz
theory is only a subset of the full range gauge transformation available
to the theory.  On the other hand full set of gauge transformations are
generated by $T(3)$ in Stuckelberg extended  massive Kalb-Ramond theory whose
gauge transformations are reducible.  This reducibility is reflected in the 
gauge generation by $T(3)$ in Stuckelberg extended  massive Kalb-Ramond theory
as one of the components of the arbitrary field which represents the gauge
freedom is left independent of the parameters of $T(3)$ as well as of
the polarization tensor of the theory. 
Finally, a hierarchical structure similar to the one noticed with massless 
$n$-form gauge theories is present in the case of massive $n$-form gauge theories as well, namely $n$ independent elements of the translational group $T(3)$
being involved in the gauge generation in a massive $n$-form gauge theory.
  
 Finally in section 5, we have shown how the different gauge generating
representations of various translational groups can be connected by the method
of dimensional descent. Previously, the method of dimensional descent was
applied only to obtain the gauge generating representations of translational
groups for the case of topologically massive gauge theories. In the present work, we
demonstrated that dimensional descent can also be useful for studying the gauge 
transformation properties of  gauge theories obtained from massive gauge noninvariant theories  
through Stuckelberg mechanism. 
We have illustrated this by deriving the polarization vector (tensor) and gauge
generating representation of $T(3)$ for 3+1 dimensional Stuckelberg extended
Proca (Einstein-Pauli-Fierz) theory by starting from Maxwell (linearized gravity) theory in 4+1 dimensions. Thus it is clear that dimensional 
descent is applicable to all massive gauge theories; i.e., topologically
massive gauge theories as well as gauge theories obtained by rendering 
 gauge invariance to massive theories through  Stuckelberg mechanism.
Therefore, the method of  dimensional descent gives a unified picture of the 
relationship between gauge transformations in various Abelian gauge theories and
 translational subgroups Wigner's little groups in different space-time dimensions. One may conclude that
translational groups, in its different representations,  act as generators of 
gauge transformations in different Abelian gauge theories (massless as well as
massive), although such transformations sometimes constitute only a subset
of the whole set of gauge transformations of the concerned theory. 


{\bf Acknowledgment:} Author is thankful to Dr. R. Banerjee for  suggesting 
this  problem and for  several useful comments. 
Thanks are also due to Dr. B. Chakraborty for many 
 illuminating discussions regarding this work.

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\end{document}

