\documentstyle[preprint,aps]{revtex}
%\documentstyle[12pt,aps]{revtex}
\begin{document}
%\hsize = 7.0in
%\widetext
\draft
\tighten
\topmargin-48pt
\evensidemargin5mm
\oddsidemargin5mm
%\preprint{EFUAZ FT-95-16-REV}
\title{About the Claimed Longitudinal Nature of the Antisymmetric
Tensor Field After Quantization}
\author{{\bf Valeri V. Dvoeglazov}\thanks{On leave of absence from
{\it Dept. Theor. \& Nucl. Phys., Saratov State University,
Astrakhanskaya ul., 83, Saratov\, RUSSIA.} Internet
address: dvoeglazov@main1.jinr.dubna.su}}
\address{
Escuela de F\'{\i}sica, Universidad Aut\'onoma de Zacatecas \\
Antonio Doval\'{\i} Jaime\, s/n, Zacatecas 98068, ZAC., M\'exico\\
Internet address: VALERI@CANTERA.REDUAZ.MX\\
URL: http://cantera.reduaz.mx/\~~valeri/valeri.htm}
%\date{First version: July, 1995. First revision:
%January 20, 1996. Second revision: December 14, 1996}
\maketitle
\begin{abstract}
It has long been claimed that the antisymmetric tensor field of the
second
rank is longitudinal after quantization. In my opinion, such a situation
produces speculations about the violation of the Correspondence
Principle.
On the basis of the Lagrangian formalism I calculate the Pauli-Lubanski
vector of relativistic spin for this field. Even at the classical level
it can be equal to zero after applications of well-known constraints.
The correct quantization procedure permits us to propose solution of
this
puzzle in the modern field theory. Obtained results develop the
previous
consideration [{\it Physica A}214 (1995) 605-618].
\end{abstract}
\pacs{PACS numbers: 03.50.De, 11.10.-z, 11.10.Ef}
\newpage
\baselineskip13pt
Quantum electrodynamics (QED) is a construct which found overwhelming
experimental confirmations (for recent reviews see, {\it e.g.},
refs.~\cite{BS1,BS2}). Nevertheless, a number of theoretical aspects
of this theory deserves more attention. First of all, they are:
the problem of ``fictious photons of helicity other than $\pm j$, as
well
as the indefinite metric that must accompany them"; the renormalization
idea, which ``would be sensible only if it was applied with finite
renormalization factors, not infinite ones (one is not allowed to
neglect
[and to subtract] infinitely large quantities)"; contradictions with the
Weinberg theorem ``that no symmetric tensor field of rank $j$ can be
constructed from the creation and annihilation operators of massless
particles of spin $j$",\, {\it etc.} They were shown at by
Dirac~\cite{Dirac1,Dirac2} and by Weinberg~\cite{Weinberg}. Moreover,
it appears now that we do not yet understand many specific features
of classical electromagnetism, first of all, the problems of
longitudinal
modes, of the gauge and of the Coulomb action-at-a-distance,
refs.~\cite{Evans,Evans1,Staru,DVO1,DVO2,DVO3,DVO4,Chubykalo}.
Secondly,
the standard model, which has been constructed on the basis of ideas,
which are similar to QED, appears to be no able to explain many
puzzles in neutrino physics.
In my opinion, all these shortcomings can be the consequences
of ignoring several important questions.
``In the classical electrodynamics of charged particles, a
knowledge of $F^{\mu\nu}$ completely determines the properties of the
system. A knowledge of $A^\mu$ is redundant there, because it is
determined only up to gauge transformations, which do not affect
$F^{\mu\nu}$\ldots Such is not the case in quantum
theory\ldots"~\cite{Huang}. We learnt, indeed, about this fact from
the Aharonov-Bohm~\cite{A1} and the Aharonov-Casher effects~\cite{A2}.
However, recently several attempts have been undertaken to explain the
Aharonov-Bohm effect classically~\cite{A3}.
These attempts have, in my opinion, logical basis. In the mean time,
quantizing the antisymmetric tensor field led us to a new puzzle,
which until now was not drawn much attention to. It was claimed that
the
antisymmetric tensor field of the second rank is longitudinal after
quantization~\cite{Ogievet,Hayashi,Love,AVD,Sorella}. We know that the
antisymmetric tensor field (electric and magnetic fields, indeed) is
transversal in the Maxwellian classical electrodynamics. It is
doubtfully that physically longitudinal
components can be transformed into the
physically transverse ones in the $\hbar \rightarrow 0$
limit.\footnote{It is interesting to compare this
question with the group-theoretical consideration in ref.~\cite{Kim}
which
deals with the reduction of rotational degrees of freedom to
gauge degrees of freedom in infinite-momentum/zero-mass limit. The only
mentions of the transversality of the quantized antisymmetric tensor
field
see in refs.~\cite{Takahashi,Boyarkin}.} How should we manage with the
Correspondence Principle in this case? It is often concluded: one is
not
allowed to use the antisymmetric tensor field to represent the quantized
electromagnetic field in relativistic quantum mechanics. Nevertheless,
we
are convinced that a reliable theory should be constructed on the basis
of
a minimal number of ingredients (``Occam's Razor") and should have
well-defined classical limit. Therefore, in this paper I undertake a
detailed analysis of rotational properties of the antisymmetric tensor
field, I calculate the Pauli-Lubanski operator of relativistic spin
(which must define whether the quantum is in the left- or right-
polarized states or in the unpolarized state) and then conclude,
if it is possible to obtain the conventional electromagnetic theory
with photon helicities $h=\pm 1$ provided that strengths ({\it not}
potentials) are chosen to be physical variables. The particular case
also
exists when the Pauli-Lubanski vector for the antisymmetric tensor field
of the second rank is equal to zero, what corresponds to the claimed
`{\it
longitudity}' (helicity $h=0$ ?) of this field.
Researches in this area from a viewpoint of the Weinberg's $2(2j+1)$
component theory have been started in
refs.~\cite{DVA00,DVA0,DVO00,DVO01,DVO02,DVO1,DVO2,DVO3,DVO4}.
I would also like to point out that the problem at hand is directly
connected with our understanding of the nature of neutral particles,
including neutrinos~\cite{Majorana,MLC,Ziino,DVA1,DVA2,DVO5,DVO6,DVO7}.
>From a mathematical viewpoint theoretical content provided by the
space-time structure and corresponding symmetries should not depend,
what
representation space, which field operators transform on, is chosen.
I begin with the antisymmetric tensor field operator (in general,
complex-valued):
\begin{equation}
F^{\mu\nu} (x)
\,=\, \sum_\eta \int \frac{d^3 {\bf p}}{(2\pi)^3} \, {1\over 2E_p}\,
\left
[ F^{\mu\nu}_{\eta\,(+)} ({\bf p})\, a_\eta ({\bf p})\, e^{-ip\cdot x} +
F^{\mu\nu}_{\eta\, (-)} ({\bf p}) \,b_\eta^\dagger ({\bf p})\,
e^{+ip\cdot
x} \right ]\label{fop}
\end{equation}
and with the Lagrangian, including, in general, mass term:\footnote{The
massless limit ($m\rightarrow 0$) of the
Lagrangian is connected with the Lagrangians used in the conformal field
theory and in the conformal supergravity by adding the total derivative:
\begin{equation}
{\cal L}_{CFT} = {\cal L} + {1\over 2}\partial_\mu \left
( F_{\nu\alpha} \partial^\nu F^{\mu\alpha} - F^{\mu\alpha} \partial^\nu
F_{\nu\alpha} \right )\quad.
\end{equation}
The gauge-invariant
form ($F_{\mu\nu} \rightarrow F_{\mu\nu} +\partial_\nu \Lambda_\mu -
\partial_\mu \Lambda_\nu$), ref.~\cite{Ogievet},
is obtained only if one uses the Fermi procedure
{\it mutatis mutandis} by removing the additional ``phase" field
$\lambda (\partial_\mu F^{\mu\nu})^2$, with the
appropriate coefficient $\lambda$, from the Lagrangian. This has
certain analogy with the QED, where the question, whether the
Lagrangian is gauge-invariant or not, is solved depending on the
presence
of the term $\lambda (\partial_\mu A^\mu)^2$. For details
see ref.~\cite{Hayashi} and what is below.}
\begin{equation}\label{Lagran}
{\cal L} = {1\over 4} (\partial_\mu F_{\nu\alpha})(\partial^\mu
F^{\nu\alpha}) - {1\over 2} (\partial_\mu F^{\mu\alpha})(\partial^\nu
F_{\nu\alpha}) - {1\over 2} (\partial_\mu F_{\nu\alpha})(\partial^\nu
F^{\mu\alpha}) + {1\over 4} m^2 F_{\mu\nu} F^{\mu\nu} \quad.
\end{equation}
The Lagrangian leads to the equation of motion in the
following form (provided that the appropriate antisymmetrization
procedure has been taken into account):
\begin{equation} {1\over 2} ({\,\lower0.9pt\vbox{\hrule
\hbox{\vrule height 0.2 cm \hskip 0.2 cm \vrule height
0.2cm}\hrule}\,}+m^2) F_{\mu\nu} +
(\partial_{\mu}F_{\alpha\nu}^{\quad,\alpha} -
\partial_{\nu}F_{\alpha\mu}^{\quad,\alpha}) = 0 \quad,\label{PE}
\end{equation}
where ${\,\lower0.9pt\vbox{\hrule \hbox{\vrule height 0.2 cm
\hskip 0.2 cm
\vrule height 0.2 cm}\hrule}\,}
=- \partial_{\alpha}\partial^{\alpha}$.
It is this equation for antisymmetric-tensor-field components
that follows from the Proca-Bargmann-Wigner consideration,
which is characterized by the equations:\footnote{In the textbooks
the equations with the ``renormalized" potentials $A^\mu \rightarrow 2m
A^\mu$ are usually used. This ``renormalization" can change the
asymptotic $m\rightarrow 0$ behaviour of classical potentials.
Therefore,
until the complete investigation of this question one should use
the form (\ref{Proca1},\ref{Proca2}) which follows from the Dirac
equations satisfied by the symmetric spinor of the second rank.}
\begin{mathletters} \begin{eqnarray} &&\partial_\alpha
F^{\alpha\mu} + {m\over 2} A^\mu = 0 \quad, \label{Proca1}\\
&&2 m
F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu \quad,
\label{Proca2}
\end{eqnarray}
\end{mathletters}
provided that $m\neq 0$ and in the final expression one takes into
account
the Klein-Gordon equation $({\,\lower0.9pt\vbox{\hrule \hbox{\vrule
height
0.2 cm \hskip 0.2 cm \vrule height 0.2 cm}\hrule}\,} - m^2) F_{\mu\nu}=
0$. The latter expresses relativistic dispersion relations $E^2 -{\bf
p}^2
=m^2$ and it follows from the coordinate Lorentz transformation
laws~\cite[\S 2.3]{Ryder}.
Following the variation procedure given, {\it e.g.}, in
refs.~\cite{Corson,Barut,Bogoliubov} one can obtain that
for rotations $x^{\mu^\prime} = x^\mu + \omega^{\mu\nu} x_\nu$
the corresponding variation of the wave function is found
from the formula:
\begin{equation}
\delta F^{\alpha\beta} = {1\over 2} \omega^{\kappa\tau}
{\cal T}_{\kappa\tau}^{\alpha\beta,\mu\nu} F_{\mu\nu}\quad.
\end{equation}
The generators of infinitesimal transformations are then defined as
\begin{eqnarray}
\lefteqn{{\cal T}_{\kappa\tau}^{\alpha\beta,\mu\nu} \,=\,
{1\over 2} g^{\alpha\mu} (\delta_\kappa^\beta \,\delta_\tau^\nu \,-\,
\delta_\tau^\beta\,\delta_\kappa^\nu) \,+\,{1\over 2} g^{\beta\mu}
(\delta_\kappa^\nu\delta_\tau^\alpha \,-\,
\delta_\tau^\nu\, \delta_\kappa^\alpha) +\nonumber}\\
&+&\,
{1\over 2} g^{\alpha\nu} (\delta_\kappa^\mu \, \delta_\tau^\beta \,-\,
\delta_\tau^\mu \,\delta_\kappa^\beta) \,+\, {1\over 2}
g^{\beta\nu} (\delta_\kappa^\alpha \,\delta_\tau^\mu \,-\,
\delta_\tau^\alpha \, \delta_\kappa^\mu)\quad.
\end{eqnarray}
It is ${\cal T}_{\kappa\tau}^{\alpha\beta,\mu\nu}$, the generators of
infinitesimal transformations,
that enter in the formula for the relativistic spin tensor:
\begin{equation}
J_{\kappa\tau} = \int d^3 {\bf x} \left [ \frac{\partial {\cal
L}}{\partial ( \partial F^{\alpha\beta}/\partial t )} {\cal
T}^{\alpha\beta,\mu\nu}_{\kappa\tau} F_{\mu\nu} \right ]\quad.
\label{inv}
\end{equation}
As a result one obtains:
\begin{eqnarray}
J_{\kappa\tau} &=& \int d^3 {\bf x} \left [ (\partial_\mu F^{\mu\nu})
(g_{0\kappa} F_{\nu\tau} - g_{0\tau} F_{\nu\kappa}) - (\partial_\mu
F^\mu_{\,\,\,\,\kappa}) F_{0\tau} + (\partial_\mu F^\mu_{\,\,\,\,\tau})
F_{0\kappa} + \right. \nonumber\\
&+& \left. F^\mu_{\,\,\,\,\kappa} ( \partial_0 F_{\tau\mu} +
\partial_\mu F_{0\tau} +\partial_\tau F_{\mu 0}) -
F^\mu_{\,\,\,\,\tau}
( \partial_0 F_{\kappa\mu} +\partial_\mu F_{0\kappa} +\partial_\kappa
F_{\mu 0}) \right ]\quad. \label{gene}
\end{eqnarray}
If agree that the
orbital part of the angular momentum \begin{equation} L_{\kappa\tau} =
x_\kappa \Theta_{0\,\tau} - x_\tau \Theta_{0\,\kappa} \quad,
\end{equation}
with $\Theta_{\tau\lambda}$ being the energy-momentum tensor, does not
contribute to the Pauli-Lubanski operator when acting on the
one-particle free states (as in the Dirac $j=1/2$ case), then
the Pauli-Lubanski 4-vector is constructed as
follows~\cite[Eq.(2-21)]{Itzykson}
\begin{equation}
W_\mu = -{1\over 2} \epsilon_{\mu\kappa\tau\nu} J^{\kappa\tau} P^\nu
\quad,
\end{equation}
with $J^{\kappa\tau}$ defined by Eqs.
(\ref{inv},\ref{gene}). The 4-momentum operator $P^\nu$ can be replaced
by
its eigenvalue when acting on the plane-wave eigenstates.
One should choose space-like normalized vector $n^\mu n_\mu = -1$, for
example $n_0 =0$,\, ${\bf n} = \widehat {\bf p} = {\bf p} /\vert {\bf
p}\vert$.\footnote{Let me remind that the helicity operator is
connected with the Pauli-Lubanski vector in the following manner $({\bf
J}
\cdot \widehat {\bf p}) = ({\bf W} \cdot \widehat {\bf p})/ E_p$, see
ref.~\cite{Shirok}. The choice of ref.~\cite[p.147]{Itzykson}, $n^\mu =
\left ( t^\mu - p^\mu {p\cdot t \over m^2} \right ) {m\over \mid {\bf p}
\mid}$, with $t \equiv (1,0,0,0)$ being a time-like vector, is also
possible but it leads to some oscurities in the procedure of taking the
massless limit. These oscurities will be clarified in a separate paper.}
After lengthy calculations in a spirit
of~\cite[p.58,147]{Itzykson} one can find the explicit form of the
relativistic spin:
\begin{mathletters} \begin{eqnarray} && (W_\mu \cdot n^\mu) = - ({\bf
W}\cdot {\bf n}) = -{1\over 2} \epsilon^{ijk} n^k J^{ij}
p^0\quad,\label{PL1}\\ && {\bf J}^k = {1\over 2} \epsilon^{ijk} J^{ij} =
\epsilon^{ijk} \int d^3 {\bf x} \left [ F^{0i} (\partial_\mu F^{\mu j})
+
F_\mu^{\,\,\,\,j} (\partial^0 F^{\mu i} +\partial^\mu F^{i0} +\partial^i
F^{0\mu} ) \right ]\quad.\label{PL2} \end{eqnarray} \end{mathletters}
Now
it becomes obvious that the application of the generalized Lorentz
conditions (which are quantum versions of free-space dual Maxwell's
equations) leads in such a formulation to the absence of
electromagnetism
in a conventional sense. The resulting Kalb-Ramond field is
longitudinal
(helicity $h=0$). All the components of the angular momentum tensor for
this case are identically equated to zero. The discussion of this fact
can
also be found in ref.~\cite{Hayashi,DVO2}. This situation can
occur in the particular choice of the normalization of the operators
$J_{\mu\nu}$ and $g_\mu \equiv J_{\mu\nu} P^\nu$ only.
One of the possible ways to obtain helicities $h=\pm 1$
is a modification of the electromagnetic field tensor like ref.~[30q],
{\it i.e.}, introducing the non-Abelian electrodynamics~\cite{Evans1}:
\begin{equation}
F_{\mu\nu}\quad \Rightarrow \quad
{\bf G}_{\mu\nu}^{a} = \partial_\mu A_\nu^{(a)\, \ast}
- \partial_\nu A_\mu^{(a)\, \ast} -i{e\over \hbar}[ A_\mu^{(b)},
A_\nu^{(c)}] \quad,
\end{equation}
where $(a),\,(b),\,(c)$ are the vector components in the
$(1),\,(2),\,(3)$
circular basis~\cite{Evans,Evans1}. In the other words, one can add some
ghost field (the ${\bf B}^{(3)}$ field) to the antisymmetric tensor
$F_{\mu\nu}$. As a matter of fact this induces hypotheses on a massive
photon and/or an additional displacement current. I can agree with the
{\it possibility} of the ${\bf B}^{(3)}$ field concept
(while it is required {\it rigorous} elaboration in the terminology
of the modern quantum field theory), but, at the moment, I
prefer to avoid any auxiliary constructions (even they are valuable in
intuitive explanations and generalizations). If these non-Abelian
constructions exist they should be deduced from a more general theory on
the basis of some fundamental postulates, {\it e.g.}, in a spirit of
refs.~\cite{DVA0,DVA2,DVO95}. In the procedure of the quantization one
can reveal the important case, when the transversality (in the meaning
of existence of $h=\pm 1$) of
the antisymmetric tensor field is preserved. This conclusion is related
with existence of the dual tensor $\widetilde F^{\mu\nu}$, with
possibility of the Bargmann-Wightman-Wigner-type quantum field theory
revealed in ref.~\cite{DVA0}\footnote{The remarkable feature of the
Ahluwalia {\it et al.} consideration is: boson and its antiboson can
possess opposite relative parities.} and with normalization questions.
I choose the field operator, Eq. (\ref{fop}), such that:
\begin{mathletters}
\begin{eqnarray}
F^{i0}_{(+)} ({\bf p})
&=& E^i ({\bf p})\quad,\quad F^{jk}_{(+)} ({\bf p}) = - \epsilon^{jkl}
B^l
({\bf p})\quad;\\
F^{i0}_{(-)} ({\bf p}) &=& \tilde F^{i0} ({\bf p}) = B^i ({\bf
p})\quad,\quad F^{jk}_{(-)} ({\bf p}) = \tilde F^{jk} ({\bf p}) =
\epsilon^{jkl} E^l ({\bf p})\quad,
\end{eqnarray} \end{mathletters}
where
$\tilde F^{\mu\nu} = {1\over 2} \epsilon^{\mu\nu\rho\sigma}
F_{\rho\sigma}$ is the tensor dual to $F^{\mu\nu}$; and
$\epsilon^{\mu\nu\rho\sigma} = - \epsilon_{\mu\nu\rho\sigma}$\, , \,
$\epsilon^{0123} = 1$ is the totally antisymmetric Levi-Civita tensor.
After lengthy but standard calculations one achieves:\footnote{Of
course,
the question of the behaviour of vectors ${\bf E}_\eta$ and ${\bf
B}_\eta$
and/or of creation and ahhihilation operators with respect to the parity
operation in this particular case deserves detailed elaboration.}
\begin{eqnarray}
{\bf J}^k &=& \sum_{\eta\eta^\prime}\int \frac{d^3 {\bf p}}{(2\pi)^3
2E_p}
\left \{ \frac{i\epsilon^{ijk}{\bf E}^i_\eta ({ \bf p})
{\bf B}^j_{\eta^\prime} ({\bf p})}{2}
\left [ a_\eta ({\bf p}) b^\dagger_{\eta^\prime} ({\bf p}) +
a_{\eta^\prime} ({\bf p}) b^\dagger_\eta ({\bf p}) +
b^\dagger_{\eta^\prime} ({\bf p}) a_\eta ({\bf p}) +
b^\dagger_\eta ({\bf p}) a_{\eta^\prime} ({\bf p}) \right ] -
\right.\nonumber\\
&-&\left. \frac{i{\bf p}^k ({\bf E}_\eta ({\bf p}) \cdot {\bf
E}_{\eta^\prime} ({\bf p}) + {\bf B}_\eta ({\bf p}) \cdot {\bf
B}_{\eta^\prime})
- i {\bf E}^k_{\eta^\prime} ({\bf p}) ({\bf p}\cdot {\bf E}_{\eta} ({\bf
p}))
- i {\bf B}^k_{\eta^\prime} ({\bf p}) ({\bf p}\cdot {\bf B}_{\eta}
({\bf p}))}{2E_p}\times\right.\nonumber\\
&\times&\left. \left [ a_\eta
({\bf p}) b^\dagger_{\eta^\prime} ({\bf p}) + b^\dagger_\eta ({\bf p})
a_{\eta^\prime} ({\bf p})\right ] \right \}
\end{eqnarray}
One should choose normalization conditions. For
instance, one can use the analogy with the (dual) classical
electrodynamics:\footnote{Different choices of the normalization could
still lead to equating the spin operator to zero or even to the other
values of helicity, which differ from $\pm 1$. The question is: what
cases are realized in the Nature and what processes do correspond to
every case?}
\begin{mathletters} \begin{eqnarray} &&({\bf E}_\eta ({\bf
p}) \cdot {\bf E}_{\eta^\prime} ({\bf p}) + {\bf B}_\eta ({\bf p}) \cdot
{\bf B}_{\eta^\prime}) = 2E_p \delta_{\eta\eta^\prime}\quad,\\ &&{\bf
E}_\eta \times {\bf B}_{\eta^\prime} = {\bf p}\delta_{\eta\eta^\prime}
-{\bf p}\delta_{\eta, -\eta^\prime}\quad. \end{eqnarray}
\end{mathletters}
These conditions still imply that ${\bf E}\perp {\bf B} \perp
{\bf p}$. Finally, one obtains
\begin{equation} {\bf J}^k = - i\sum_\eta
\int \frac{d^3 {\bf p}}{(2\pi)^3} \, \frac{{\bf p}^k}{2E_p}
\left [ a_\eta ({\bf p})
b_{-\eta}^\dagger ({\bf p}) +b_\eta^\dagger ({\bf p}) a_{-\eta}
({\bf p}) \right ]\quad.\label{spin}
\end{equation} If we want to describe
states with the definite helicity quantum number (photons) we should
assume that $b^\dagger_\eta ({\bf p})= i a^\dagger_\eta ({\bf p})$ what
is
reminiscent with the Majorana-like
theories~\cite{DVA1,DVA2,Bil}.\footnote{Of course, the imaginary unit
can
be absorbed by the corresponding re-definition of negative-frequency
solutions.} One can take into account the prescription of the normal
ordering and set up the commutation relations in the form:
\begin{equation} \left [a_\eta ({\bf p}), a_{\eta^\prime}^\dagger ({\bf
k})\right ]_{-} = (2\pi)^3 \delta ({\bf p}-{\bf k})
\delta_{\eta,-\eta^\prime}\quad. \label{cr} \end{equation} After acting
the operator (\ref{spin}) on the physical states, {\it i.e.},
$a_h^\dagger
({\bf p}) \vert 0>$ , we are convinced that antisymmetric tensor field
can
describe particles with helicities to be equal to $\pm 1$). One can see
that the origins of this conclusion are the possibilities of different
definitions of the field operator (and its normalization), the existence
of the `{\it antiparticle}' for the particle described by antisymmetric
tensor field. The latter statement is related with the Weinberg
discussion of the connection between helicity and representations of the
Lorentz group~[47a]. Next, I would like to point out that the
Proca-like
equations for antisymmetric tensor field with {\it mass}, {\it e.g.},
Eq.
(\ref{PE}) can possess tachyonic solutions, see for the discussion in
ref.~\cite{DVO1}. Therefore, in a massive case the states can be
``partly" tachyonic states mathematically. We then deal always with the
problem of the choice of normalization conditions which could permit us
to
describe both transversal and longitudinal {\it physical} modes of the
$j=1$ field.
In conclusion, I calculated the Pauli-Lubanski vector of relativistic
spin on the basis of the N\"otherian symmetry
method~\cite{Corson,Barut,Bogoliubov}. Let me remind that it is a part
of
the angular momentum vector, which is conserved as a consequence of
the rotational invariance. After explicit~\cite{Hayashi} (or
implicit~\cite{AVD}) applications of the constraints (the generalized
Lorentz condition) in the Minkowski space, the antisymmetric tensor
field
becomes `{\it longitudinal}' in the meaning that the angular momentum
operator is equated to zero (the sense which was attached by the authors
of the works~\cite{Ogievet,Hayashi,Sorella,AVD}). I proposed one of
possible ways to resolve this contradiction with the Correspondence
Principle in refs.~\cite{DVO1,DVO2,DVO3,DVO4}. Another hypothesis has
been proposed by Evans~\cite{Evans,EVANS,Evans1}, in which the
component of the Pauli-Lubanski vector generalized
to the isovector space $(1), (2), (3)$ has been identified
with the new ${\bf B}^{(3)}$ field of electromagnetism.\footnote{See
also
the paper of Chubykalo and Smirnov-Rueda~\cite{Chubykalo}. The paper on
connections between the Chubykalo and Smirnov-Rueda `{\it
action-at-a-distance}' construct and the $B(3)$ theory was submitted
(private communication from A. Chubykalo).} The present article
continues
these researches. The conclusion achieved is: the antisymmetric tensor
field can describe both the Maxwellian $j=1$ field and the Kalb-Ramond
$j=0$ field. Nevertheless, I still think that the physical nature of
the
$E=0$ solution re-discovered in refs.~\cite{Gian,DVA00}, its connections
with the Evans-Vigier ${\bf B}^{(3)}$ field, ref.~\cite{Evans,Evans1},
with Avdeev-Chizhov $\delta^\prime$- type transversal solutions~[7b],
which cannot be interpreted as relativistic particles, as well as with
my
concept of $\chi$ boundary functions, ref.~\cite{DVO4} are not
completely
explained until now. Finally, while I do not have any intention to
doubt
theoretical results of the ordinary quantum electrodynamics I am sure
that
the questions put forth in this note (as well as in previous papers of
both mine and other groups) should be explained properly.
\medskip
{\it Acknowledgements.}
I am thankful to Profs. D. V. Ahluwalia, A. E. Chubykalo,
A.~F.~Pashkov and S. Roy for stimulating discussions. After
the writing of the preliminary version of the manuscript I received the
paper of Prof. M. W. Evans ``The Photomagneton and Photon
Helicity"~\cite{EVANS}, devoted to a consideration of the similar
topics, but from very different standpoints. This was a motivation for
revising the preliminary version of the present manuscript.
I am delighted by the referee
reports on the papers~\cite{DVO1,DVO2,DVO3,DVO4} from ``Journal
of Physics A". In fact, they helped me to learn many useful things.
I am grateful to Zacatecas University for a professorship.
This work has been supported in part by el Mexican Sistema
Nacional de Investigadores, el Programa de Apoyo a la Carrera Docente
and by the CONACyT, M\'exico under the research project 0270P-E.
\begin{references}
\footnotesize{
\baselineskip13pt
\bibitem{A1} Y. Aharonov and D. Bohm, Phys. Rev. {\bf 115} (1959) 485
\bibitem{A2} Y. Aharonov and A. Casher, Phys. Rev. Lett. {\bf 53} (1984)
319
\bibitem{DVA00} D. V. Ahluwalia and D. J. Ernst, Mod. Phys. Lett.
A{\bf 7} (1992) 1967
\bibitem{DVA0} D. V. Ahluwalia, M. B. Johnson and T. Goldman,
Phys. Lett. B{\bf 316} (1993) 102; D. V. Ahluwalia and T. Goldman, Mod.
Phys. Lett. A{\bf 8} (1993) 2623
\bibitem{DVA1} D. V. Ahluwalia, M. B. Johnson and T. Goldman,
Mod. Phys. Lett. A{\bf 9} (1994) 439; Acta Phys. Polon. B{\bf 25}
(1994) 1267
\bibitem{DVA2} D. V. Ahluwalia, Int. J. Mod. Phys. A{\bf 11} (1996) 1855
\bibitem{AVD} L. V. Avdeev and M. V. Chizhov, Phys. Lett. B{\bf
321} (1994) 212; {\it A Queer Reduction of Degrees of Freedom.}
Preprint JINR E2-94-263 (hep-th/9407067), Dubna, July 1994
\bibitem{Barut} A. O. Barut, {\it Electrodynamics and Classical
Theory of Fields and Particles.} (Dover Pub., Inc., New York, 1980)
\bibitem{Bil} S. M. Bilen'ky, J. Phys. Suppl. G{\bf 17} (1991) S251
\bibitem{Bogoliubov} N. N. Bogoliubov and D. V. Shirkov, {\it
Introduction to the Theory of Quantized Fields.} (John Wiley \& Sons
Ltd., 1980)
\bibitem{Boyarkin} O. M. Boyarkin, Izvest. VUZ:fiz. No. 11 (1981) 29
[English translation: Sov. Phys. J. {\bf 24} (1981) 1003]
\bibitem{Chubykalo} A. E. Chubykalo and R. Smirnov-Rueda, Phys. Rev.
E{\bf
53} (1996) 5373
\bibitem{Love} T. E. Clark and S. T. Love, Nucl. Phys. B{\bf 223}
(1983)
135; T. E. Clark, C. H. Lee and S. T. Love, ibid B{\bf 308} (1988)
379
\bibitem{Corson} E. M. Corson, {\it Introduction to Tensors, Spinors,
And
Relativistic Wave-Equations.} (Hafner, New York)
\bibitem{Dirac1} P. A. M. Dirac, in {\it Mathematical Foundations of
Quantum Theory.} Ed. by A. R. Marlow. (Academic Press, 1978),~p.~1
\bibitem{Dirac2} P. A. M. Dirac, in {\it Directions in Physics.} Ed. by
H. Hora and J. R. Shepanski. (John Wiley \& Sons, New York, 1978), p.
32
\bibitem{BS1} V. V. Dvoeglazov, R. N. Faustov and Yu. N. Tyukhtyaev,
Mod. Phys. Lett. A{\bf 8} (1993) 3263
\bibitem{BS2} V. V. Dvoeglazov, Yu. N. Tyukhtyaev and R. N. Faustov,
Phys. Part. Nucl. {\bf 25}~(1994)~58
\bibitem{DVO00} V. V. Dvoeglazov, Hadronic J. {\bf 16} (1993) 423;
ibid 459
\bibitem{DVO01} V. V. Dvoeglazov Yu. N. Tyukhtyaev and S. V. Khudyakov,
Izvest. VUZ:fiz. No. 9 (1994) 110 [English translation: Russ. Phys. J.
{\bf 37} (1994) 898]
\bibitem{DVO02} V. V. Dvoeglazov, Rev. Mex. Fis. {\it (Proc. of the XVII
Symp. on Nucl. Phys. Oaxtepec, M\'exico. Jan. 4-7,
1994)} {\bf 40}, Suppl. 1 (1994) 352
\bibitem{DVO95} V. V. Dvoeglazov, in {\it Proc. of the IV Wigner Symp.
Guadalajara, M\'exico, Aug. 7-11, 1995.} (World Scientific, Singapore,
1996), p. 231; Nuovo Cimento B{\bf 111} (1996) 483
\bibitem{DVO1} V. V. Dvoeglazov, Helv. Phys. Acta {\bf 70} (1997) 677
\bibitem{DVO2} V. V. Dvoeglazov, Helv. Phys. Acta {\bf 70} (1997) 686
\bibitem{DVO3} V. V. Dvoeglazov, Helv. Phys. Acta, accepted
\bibitem{DVO4} V. V. Dvoeglazov, {\it Can the Weinberg-Tucker-Hammer
Equations Describe the Electromagnetic Field?} Preprint EFUAZ
FT-94-09-REV
(hep-th/9410174), Zacatecas, Oct. 1994
\bibitem{DVO5} V. V. Dvoeglazov, Rev.
Mex. Fis. {\it (Proc. of the XVIII Oaxtepec Symp. on Nucl.
Phys., Oaxtepec, M\'exico, January 4-7, 1995)} {\bf 41}, Suppl. 1
(1995) 159
\bibitem{DVO6} V. V. Dvoeglazov, Int. J. Theor. Phys. {\bf 34} (1995)
2467
\bibitem{DVO7} V. V. Dvoeglazov, Nuovo Cimento A{\bf 108} (1995) 1467
\bibitem{Evans} M. W. Evans, Mod. Phys. Lett. B{\bf 7} (1993) 1247;
Physica B{\bf 182} (1992) 227, 237; ibid {\bf 183} (1993) 103; ibid
{\bf 190} (1993) 310; Found. Phys. {\bf 24} (1994) 1519, 1671; ibid
{\bf
25} (1995) 383; Found. Phys. Lett. {\bf 7} (1994) 67, 209, 379, 437,
577, 591; ibid {\bf 8} (1995) 83, 187, 253, 279; F. Farahi, Y. Atkas and
M. W. Evans, J. Mol. Structure, {\bf 285} (1993) 47; M. W. Evans and
F.
Farahi, ibid {\bf 300} (1993) 435
\bibitem{EVANS} M. W. Evans, Physica A{\bf 214} (1995) 605
\bibitem{Evans1} M. W. Evans and J.-P. Vigier, {\it Enigmatic Photon.}
Vol. 1 \& 2 (Kluwer Academic Pub., Dordrecht, 1994-95)
\bibitem{Gian} E. Gianetto, Lett. Nuovo Cim. {\bf 44} (1985) 140
\bibitem{Kim} D. Han, Y. S. Kim and D. Son, Phys. Lett. {\bf 131}B
(1983)
327; Y. S. Kim, in {\it Proc. of the IV Wigner Symposium, Guadalajara,
M\'exico, Aug. 7-11, 1995.} (World Scientific, 1996), p. 1
\bibitem{Hayashi} K. Hayashi, Phys. Lett. B{\bf 44} (1973) 497;
M. Kalb and P. Ramond, Phys. Rev. D{\bf 9} (1974) 2273
\bibitem{Huang} K. Huang, {\it Quarks, Leptons and Gauge Fields.}
(World Scientific, Singapore, 1992), p. 57
\bibitem{Itzykson} C. Itzykson and J.-B. Zuber, {\it Quantum Field
Theory.}
(McGraw-Hill Book Co., New York, 1980)
\bibitem{Sorella} V. Lemes, R. Renan and S. P. Sorella, {\it
Algebraic Renormalization of Antisymmetric Tensor Matter Field.}
Preprint
HEP-TH/9408067, Aug. 1994
\bibitem{Majorana} E. Majorana, Nuovo Cim. {\bf 14} (1937)
171 [English translation: D. A. Sinclair, Tech. Trans.
TT-542, National Research Council of Canada]
\bibitem{MLC} J. A. McLennan, Phys. Rev. {\bf 106} (1957) 821;
K. M. Case, Phys. Rev. {\bf 107} (1957) 307
\bibitem{Ogievet} V. I. Ogievetski\u{\i} and I. V. Polubarinov,
Yadern. Fiz. {\bf 4} (1966) 216 [English translation:
Sov. J. Nucl. Phys. {\bf 4} (1967) 156]
\bibitem{A3} H. Rubio, J. M. Getino and O. Rojo, Nuovo Cim. {\bf 106}B
(1991) 407
\bibitem{Ryder} L. H. Ryder, {\it Quantum Field Theory.} (Cambridge
Univ. Press, 1985)
\bibitem{Shirok} Yu. M. Shirokov, ZhETF {\bf 33} (1957) 861, 1196
[English
translation: Sov. Phys. JETP {\bf 6} (1958) 664, 919]; Chou Kuang-chao
and M. I. Shirokov, ZhETF {\bf 34} (1958) 1230 [English translation:
Sov. Phys. JETP {\bf 7} (1958) 851
\bibitem{Staru} A. Staruszkiewicz, Acta Phys. Polon. B{\bf 13} (1982)
617; ibid {\bf 14} (1983) 63, 67, 903; ibid {\bf 15} (1984) 225;
ibid {\bf 23} (1992) 591
\bibitem{Takahashi} Y. Takahashi and R. Palmer, Phys. Rev. D{\bf 1}
(1970)
2974
\bibitem{Weinberg} S. Weinberg, Phys. Rev. B{\bf 134} (1964) 882; ibid
B{\bf 138} (1965) 988
\bibitem{Ziino} G. Ziino, Ann. Fond. L. de Broglie {\bf 14} (1989)
427; ibid {\bf 16} (1991) 343; A. O. Barut and G. Ziino, Mod. Phys.
Lett.
A{\bf 8} (1993) 1011
}
\end{references}
\end{document}