Generalized Maxwell and Weyl Equations for Massless Particles\thanks{Presented at the Fourth Mexican School on Gravitation and Mathematical Physics, December 3-8, 2000, Huatulco, Oaxaca, Mexico.}
{Valeri V. Dvoeglazov}
\baselineskip10pt I discuss generalized Maxwell and Weyl equations. They may lead to dynamics which are different from those accepted at the present time. For instance, the photon may have non-transverse components and the neutrino may be {not} in the chiral states.
math-ph0102001.tex

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\baselineskip13pt

\begin{document}

\preprint{EFUAZ FT-00-90}

\title{Generalized Maxwell and Weyl Equations for Massless
Particles\thanks{Presented at the Fourth Mexican School on Gravitation
and Mathematical Physics, December 3-8, 2000, Huatulco, Oaxaca, Mexico.}}

\author{{\bf Valeri V. Dvoeglazov}}

\address{Escuela de F\'{\i}sica, Universidad Aut\'onoma de Zacatecas\\
Apartado Postal C-580, Zacatecas 98068 Zac., M\'exico\\
E-mail: valeri@ahobon.reduaz.mx\\
URL: http://ahobon.reduaz.mx/\~~valeri/valeri.htm
}

\date{December 2000}

\maketitle

%\medskip

\begin{abstract}
\baselineskip10pt
I discuss generalized Maxwell and Weyl equations.
They may lead to dynamics which are different from those accepted
at the present time. For instance, the photon may have
non-transverse components and the neutrino may be {\it not}
in the chiral states.
\end{abstract}

%\newpage

\vskip16mm
\baselineskip13pt


The content of this talk is the following:

\begin{itemize}

\item
the van der Waerden-Sakurai derivation of the Dirac equation;

\item
the Gersten derivation of Maxwell equations and equations for
arbitrary spin;

\item
the correction of the Gersten derivation (if one considers
all solutions of the d'Alembert equation, scalar fields appear);

\item
massless particles with massive parameters;

\item
physical origins and physical conclusions.

\end{itemize}

The contribution is based on recent papers~\cite{Ger,dvo1,Ger2,dvon}.
Gersten~[1a] writes: ``We have shown how all Maxwell equations can be
derived simultaneously from first principles, similar to those which have
been used to derive the Dirac relativistic electron equation" and
concludes: ``\ldots Maxwell equations should be used as a guideline for
proper interpretation of quantum theories".

In fact, he used a method presented by van der Waerden and
Sakurai~\cite{Sak}. Let us start with the Klein-Gordon equation:
\begin{equation}
( E^2 -c^2 {\bf p}^2 -m^2 c^4 )  \Psi_{(2)} =0\,,\label{kg}
\end{equation}
hence (for $\Psi$ with {\it two} components and
$c=\hbar=1$)
\begin{equation}
(E I^{(2)}-{\bbox \sigma}\cdot {\bf p}) (E I^{(2)}+ {\bbox\sigma}\cdot
{\bf p} ) \Psi_{(2)} = m^2 \Psi_{(2)} \,.
\end{equation}
Let us denote
$\Psi_{(2)} =\eta$, then we have
\begin{mathletters} \begin{eqnarray} {(E
I^{(2)}+{\bbox \sigma}\cdot {\bf p})\over m} \eta &=&\chi\,,\\
(E I^{(2)} -
{\bbox\sigma}\cdot {\bf p} ) \chi &=& m \eta \,.  \end{eqnarray}
\end{mathletters}
In the 4-component form one has
\begin{eqnarray}
\pmatrix{-m I^{(2)}& E I^{(2)}+{\bbox \sigma}\cdot {\bf p}\cr
E I^{(2)}-
{\bbox\sigma}\cdot {\bf p}& -mI^{(2)}\cr} \pmatrix{\chi\cr\eta\cr} =0
\end{eqnarray} With the quantum-operator substitutions $E\rightarrow
i\hbar {\partial \over \partial t}$ and ${\bf p} \rightarrow -i\hbar
{\bbox\nabla}$ we recover the Dirac equation!\footnote{One can also
decompose (\ref{kg}) into the 4-component form from the beginning $$(E
I^{(4)}+{\bbox \alpha}\cdot {\bf p} +m\beta) (E I^{(4)}-{\bbox\alpha}\cdot
{\bf p} -m\beta ) \Psi_{(4)} =0$$ and then look for ${\bbox\alpha}$ and
$\beta$.}

In the $S=1$ and $m=0$ case we proceed in a similar way:
\begin{equation}
\left ( {E^2\over c^2}   - {\bf p}^{\,2} \right ) {\bbox \Psi}_{(3)}
= \left ( {E\over c} I^{(3)} - {\bf S}\cdot {\bf p} \right )
\left ( {E\over c} I^{(3)} + {\bf S}\cdot {\bf p} \right )
{\bbox \Psi}_{(3)} - \pmatrix{p_x\cr p_y \cr p_z\cr}
\left ({\bf p}\cdot {\bbox \Psi}_{(3)}\right ) =0
\,,\quad \mbox{Eq. (9) of~[1a]}
\nonumber
\end{equation}
where $({\bf S}^i)^{jk} = -i\epsilon^{ijk}$.\footnote{Please note
the difference $({\bbox\sigma}\cdot {\bf p})^2 ={\bf p}^{\,2}$,
but $({\bf S}\cdot {\bf p})^3 = ({\bf S}\cdot {\bf p})$; the spin-1
matrices are {\it singular} (sic!).}
Gersten found that solutions are defined by
\begin{mathletters}
\begin{eqnarray}
&&\left ( {E\over c} I^{(3)} +{\bf S}\cdot {\bf p} \right )
{\bbox\Psi}_{(3)} =0\quad\qquad\mbox{Eq. (10) of~[1a]}\nonumber\\
&&\left ({\bf p}\cdot {\bbox \Psi}_{(3)}\right ) =0
\qquad\qquad\mbox{Eq. (11) of~[1a]}\nonumber
\end{eqnarray}
\end{mathletters}
and their complex conjugates; the latter may be interpreted
as the ones for opposite helicity.
If one assumes ${\bbox\Psi}_{(3)} = {\bf E} - i{\bf B}$ then after quantum
operator substitutions:
\begin{mathletters}
\begin{eqnarray}
&&{i\hbar\over c} {\partial {\bbox \Psi}_{(3)} \over \partial t}
= -\hbar {\bbox\nabla}\times {\bbox\Psi}_{(3)}\,,\\
&&-i\hbar {\bbox \nabla}\cdot {\bbox\Psi}_{(3)} =0\,,
\end{eqnarray}
\end{mathletters}
or
\begin{mathletters}
\begin{eqnarray}
&&{\bbox\nabla}\times ({\bf E}-i{\bf B}) = -{i\over c} {\partial ({\bf
E}-i{\bf B}) \over \partial t}\,,\\
&& {\bbox\nabla} \cdot ({\bf E} - i{\bf B}) =0\,.
\end{eqnarray} \end{mathletters}
Please note that the Planck constant was cancelled out!
Separating into real and imaginary parts will lead to
\begin{equation}
{\bbox \nabla}\times {\bf
\vec{E}}=-\frac{1}{c}\frac{\partial {\bf \vec{B}}}{\partial t} \,,
\end{equation}

\begin{equation}
{\bbox \nabla }\times {\bf \vec{B}}=\frac{1}{c}\frac{\partial {\bf
\vec{E}}}{\partial t} \,,
\end{equation}

\begin{equation}
{\bbox \nabla }\cdot {\bf \vec{E}}= 0\,,
\end{equation}

\begin{equation}
{\bbox \nabla }\cdot {\bf \vec{B}}=0\,.
\end{equation}
The situation is the same if one starts with the complex conjugate function
${\bbox\Psi}^\ast_{(3)} = {\bf E}+i{\bf B}$.

{\it Lagrangian.} The Lagrangian for this theory has recently been
presented in~\cite{Ger2,Hott}:
\begin{equation}
{\Lambda} = -c {\bbox\Psi}^\dagger_{(3)} ({E\over c} I^{(3)}
+{\bf S}\cdot {\bf p} )\ {\bbox \Psi}_{(3)}\,.
\end{equation}
However, it was shown in~\cite{dvoo} that such a form of Lagrangians is
{\it not} a scalar (it is a zero component of a 4-vector; see
also~\cite{sud}).

{\it The relativistic spin generalization.} We have
\begin{equation}
W_\mu W^\mu = -s(s+1) p_\mu p^\mu I^{(s)}
\end{equation}
and
\begin{equation}
W^\mu p_\mu =0\,
\end{equation}
by definition. Hence,
\begin{equation}
(W_\mu -s p_\mu) (W^\mu +sp^\mu ) =-s (2s+1) p_\mu p^\mu I^{(s)}\,,
\end{equation}
which is similar to Eq. (9) of ref.~[1a] if one imposes $p_\mu p^\mu
=m^2=0$.

Therefore, we have
\begin{mathletters}
\begin{eqnarray}
&&({\bf S}\cdot {\bf p} - s p_0 ) {\bbox\Psi}_{(s)}=0\,,\\
&&({\bf S} p_0 +i {\bf S}\times {\bf p} - s {\bf p}) {\bbox \Psi}_{(s)}
=0\,.  \end{eqnarray} \end{mathletters}
It is easy to see that in the
$S=1$ case one can recover the previous consideration~[1b]. The second
equation can be considered as a subsidiary condition
\begin{eqnarray}
\pmatrix{0&0&0\cr p_x&p_y&p_z\cr 0&0&0\cr} {\bbox\Psi}_{(3)} =0\,.
\end{eqnarray}
In~\cite{dvo1} we corrected his claim. The equation (9) of ref.~[1a]
is satisfied also under the choice
\begin{mathletters}
\begin{eqnarray}
&&\left ( {E\over c} I^{(3)} +  {\bf S}\cdot {\bf p} \right )
{\bbox\Psi}_{(3)} ={\bf p} \chi\,,\\
&&\left ({\bf p}\cdot {\bbox \Psi}_{(3)}\right ) ={E\over c} \chi\, ,
\end{eqnarray}
\end{mathletters}
due to $({\bf S}\cdot {\bf p}) {\bf p} =0$.
The above set leads to
\begin{mathletters}
\begin{eqnarray}
&&{\bbox \nabla}\times {\bf
E}=-\frac{1}{c}\frac{\partial {\bf B}}{\partial t} + {\bbox
\nabla} {\it Im} \chi \,, \label{1}\\
&&{\bbox \nabla }\times {\bf B}=\frac{1}{c}\frac{\partial {\bf
E}}{\partial t}  +{\bbox \nabla} {\it Re} \chi\,,\label{2}\\
&&{\bbox \nabla}\cdot {\bf E}=-{1\over c} {\partial \over \partial
t} {\it Re}\chi \,,\label{3}\\
&&{\bbox \nabla }\cdot {\bf B}= {1\over
c} {\partial \over \partial t} {\it Im} \chi \,,  \label{4}
\end{eqnarray}
\end{mathletters}
with an additional scalar field.
It is also possible that
\begin{mathletters}
\begin{eqnarray}
&&\left ( {E\over c} I^{(3)} - {\bf S}\cdot {\bf p} \right )
{\bbox\Psi}_{(3)}^\prime ={\bf p} \chi^\prime\\
&&\left ({\bf p}\cdot {\bbox \Psi}^\prime_{(3)}\right ) ={E\over c}
\chi^\prime\, .
\end{eqnarray}
\end{mathletters}
If $\chi^\prime = \chi^\ast$ then ${\bbox \Psi}^\prime_{(3)} = {\bf
E}+i {\bf B}$ and we recover the generalized Maxwell equations
(\ref{1}-\ref{4}). What is the physical interpretation of obtained
results?

\begin{itemize}

\item
We obtain ${\bf p}\cdot {\bbox \Psi}_{(3)} \neq 0$, therefore the free
photon may have a non-transverse component~\cite{ima};

\item
The $\chi$-fields may be function(al) of higher-rank tensor fields, thus
leading to equations which are non-linear in ${\bf E}$ and ${\bf B}$
(cf. ref.~\cite{donev});

\item
One can find possible relations to the
Ogievetski\u{\i}-Polubarinov-Kalb-Ramond field~\cite{op,kr,dvob}. After
performing the Bargmann-Wigner procedure for higher-spin fields we obtain
\begin{mathletters}
\begin{eqnarray}
&&\partial_\alpha F^{\alpha\mu} +{m\over 2} A^\mu =0\,,\label{p1}\\
&&2mF^{\mu\nu} = \partial^\mu A^\nu -\partial^\nu A^\mu\,,\label{p2}
\end{eqnarray}
\end{mathletters}
instead of the well-known Proca set:
\begin{mathletters}
\begin{eqnarray}
&&\partial_\alpha F^{\alpha\mu} +m^2 A^\mu =0\,,\\
&&F^{\mu\nu} = \partial^\mu A^\nu -\partial^\nu A^\mu\,,
\end{eqnarray}
\end{mathletters}

In fact, these sets are related to one another by a re-normalization
transformation:  $A^\mu \rightarrow 2mA^\mu$ or $F^{\mu\nu} \rightarrow
{1\over 2m} F^{\mu\nu}$. Ogievetski\u{\i} and Polubarinov~\cite{op} wrote:
``In the massless limit the system of $2s+1$ states is no longer
irreducible;
it decomposes and describes a set of different particles with zero mass
and helicities $\pm s,\pm (s-1), \pm 1, 0$ (for integer spin and if parity
is conserved); the situation is analogous for half-integer
spins."\footnote{One can compare this statement with
recent works of Dra. M. Kirchbach.}
One can also see this after performing the Lorentz
transformation in the $(1/2,1/2)$ representation, $A^\mu =
\Lambda^\mu_{\quad\nu} A^\nu_{basis}$ and looking for massless limit.
Thus, we find
\begin{equation}
u^\mu
({\bf p}, +1)= -{N\over \sqrt{2}m}\pmatrix{p_r\cr m+ {p_1 p_r \over
E_p+m}\cr im +{p_2 p_r \over E_p+m}\cr {p_3 p_r \over
E_p+m}\cr}\,,\quad  u^\mu ({\bf p}, -1)= {N\over
\sqrt{2}m}\pmatrix{p_l\cr m+ {p_1 p_l \over E_p+m}\cr -im +{p_2 p_l \over
E_p+m}\cr {p_3 p_l \over E_p+m}\cr}\,,\label{vp12}
\end{equation}
and
\begin{equation}
u^\mu ({\bf
p}, 0) = {N\over m}\pmatrix{p_3\cr {p_1 p_3 \over E_p+m}\cr {p_2 p_3
\over E_p+m}\cr m + {p_3^2 \over E_p+m}\cr}\,,\quad
u^\mu ({\bf p}, 0_t) = {N \over m} \pmatrix{E_p\cr p_1
\cr p_2\cr p_3\cr}\,.\label{vp3}
\end{equation}
Please note that for helicities $\sigma = \pm 1, 0$
one has $p_\mu u^\mu ({\bf p}, \sigma ) = 0$ (an analogue of the
Lorentz condition~\cite{dvob}). This is
not the case for ``time-like" photons.

In view of the fact that in the case $N=1$ we have divergent
behaviour of  certain parts of the 4-vector momentum-space functions in
$m\rightarrow 0$, the first degree of $m$ in the equations
(\ref{p1},\ref{p2}) can cancel this
divergent term in the denominators.  The massless limits of the
Proca-like equations are actually
\begin{equation}
\partial_\alpha F^{\alpha\mu} = -{m\over 2} A^\mu
\Longrightarrow \partial_\alpha F^{\alpha\mu} = \partial^\mu \chi\,.
\end{equation}

\item
Of course, when we pass over to the second quantization, the commutation
relations for $F^{\mu\nu}$ and $A^\mu$ may be changed in order to keep the
correct dimension of the fields and in order the action
to be {\it dimensionless}.

\item
Ogievetski\u{\i} and Polubarinov, Kalb and Ramond~\cite{op,kr}
analized the scalar Lagrangian of the antisymmetric tensor
field~\cite{dvoo,dvob,dvohpa} and ``gauge out" the {\it transverse}
components by means of
\begin{equation} F_{\mu\nu} \rightarrow F_{\mu\nu}
+\partial_\nu \Lambda_\mu - \partial_\mu \Lambda_\nu\,,
\end{equation} the
new ``gauge" transformation. Therefore, they obtained a {\it pure}
longitudinal field, the {\it notoph} (or, the {\it Kalb-Ramond field}, as
it is frequently called in the US literature).

\item
In~\cite{dvohpa,dvob} I have shown that a map exists between the
Ogievetski\u{\i}-Polubarinov formulation and the Weinberg $2(2s+1)$
theory~\cite{weinb}. In the latter case the Lagrangian is given by
\begin{equation}
{\cal L} = \partial_\mu \overline{\Psi}_{(6)} \gamma^{\mu\nu} \partial_\nu
\Psi_{(6)} \pm m^2 \overline{\Psi}_{(6)}\Psi_{(6)}\,,\label{lag}
\end{equation}
(or its analogues for fields of different dimensions).
The $\gamma^{\mu\nu}$ is a complete set of covariant matrices of the
$(1,0)\oplus(0,1)$ representation; $\Psi_{(6)}$ and
$\overline{\Psi}_{(6)}$ are bivectors. In general, various $\Psi$ can be
used, which differ by discrete symmetry transformations. The map
exists between the equations obtained from (\ref{lag})
\begin{equation}
[\gamma^{\mu\nu} \partial_\mu\partial_\nu \mp m^2 ] \Psi_{(6)} =0
\end{equation}
and
\begin{equation}
\partial_\mu \partial^\alpha
F_\alpha^{\quad \nu} -\partial^\nu \partial^\alpha F_{\alpha\mu} = - m^2
F_\mu^{\quad \nu} \end{equation}
\noindent (and its dual).

\end{itemize}

\bigskip

Next, we return to the van der Waerden-Sakurai derivation of the Dirac
equation:
\begin{equation}
(E^2 -c^2 \vec{\bf p}^{\,2}) I^{(2)}\Psi_{(2)} =
\left [E I^{(2)} - c {\bbox \sigma}\cdot {\bf p} \right ]
\left [E I^{(2)} + c {\bbox \sigma}\cdot {\bf p} \right ]
\Psi_{(2)} = m_2^2 c^4\Psi_{(2)}\,.\label{G1}
\end{equation}
If one denotes $\Psi_{(2)}=\eta$ one can define $\chi = {1\over m_1 c}
(i\hbar {\partial\over \partial x_0} - i\hbar {\bbox \sigma}\cdot
{\bbox\nabla}) \eta$. Please note that
we introduced the second mass parameter $m_1$.
The corresponding set of 2-component equations is
\begin{mathletters}
\begin{eqnarray}
&&(i\hbar {\partial \over \partial x_0}
-i\hbar {\bbox \sigma}\cdot {\bbox \nabla}) \eta
 =m_1 c\chi\,,\\
&&(i\hbar {\partial \over \partial x_0}
+i\hbar {\bbox \sigma}\cdot {\bbox \nabla}) \chi
={m_2^2 c\over m_1}\eta\,.
\end{eqnarray}
\end{mathletters}
In the 4-component form  we have
\begin{eqnarray}
&&\pmatrix{i\hbar (\partial/\partial x_0) &
i\hbar {\bbox \sigma}\cdot {\bbox \nabla}\cr
-i\hbar {\bbox \sigma}\cdot {\bbox \nabla}&
-i\hbar (\partial/\partial x_0)}
\pmatrix{\chi+\eta\cr\chi-\eta} = {c\over 2}
\pmatrix{(m_2^2/m_1
+m_1)&
(-m_2^2/m_1 +
m_1)\cr
(-m_2^2/m_1 +
m_1)& (m_2^2/m_1
+m_1)\cr}\pmatrix{\chi+\eta\cr\chi-\eta\cr}\,,\nonumber\\
&&
\end{eqnarray}
which results in
\begin{equation}
\left [i\hbar \gamma^\mu \partial_\mu - {m_2^2 c \over m_1}
{1-\gamma_5 \over 2} -m_1 c {1+\gamma_5 \over 2}\right ]
\Psi_{(4)} = 0\,.
\end{equation}
The ``new" massless equation is ($m_2\rightarrow 0$, $m_1 \neq 0$)
\begin{equation}
\left [ i\gamma^\mu \partial_\mu - {m_1 c\over \hbar} {1+\gamma_5
\over 2}\right ]\Psi_{(4)} =0\,.\label{nme}
\end{equation}
It is easy to check that dispersion relations are $E=\pm \vert {\bf
p}\vert$, that gives us the rights to call it {\it massless}, even though
there is a ``mass" parameter in (\ref{nme}).



My comments are:

\begin{itemize}

\item
Starting from a bit different definitions~\cite{dvon}, one can also obtain
the equation
\begin{equation} [i\gamma^\mu \partial_\mu - {m_3 c\over
\hbar} {1-\gamma_5 \over 2}]\Psi^\prime_{(4)} = 0 \end{equation} by means
of a similar procedure;

\item
Moreover, instead of $(1\pm \gamma^5)/2$ one can use any singular
$4\times 4$ matrix of appropriate physical dimension and still have
massless particles;

\item
I found relevant equations in the old~\cite{antec}
and new~\cite{rasp} literature; they
have been postulated therein;


\item
the terms $m(1\pm \gamma_5)/2$ in the massless equation violate chiral
invariance of the theory (due to $[\hat p,\gamma_5]_- \neq 0$).
The solutions are {\it not} eigenstates of $\gamma_5$ operator, but in the
case $m_{1,3} \rightarrow 0$ the chiral invariance is restored. So, they
have been called {\it measures of chirality}~[16b];

\item
In fact, corresponding $u_{\uparrow\downarrow}$, $v_{\uparrow\downarrow}$
of the Dirac massless equation are {\it mixed} and the ``new"
${\cal
U}_{\uparrow\downarrow}$, ${\cal V}_{\uparrow\downarrow}$,
${\cal
U}^\prime_{\uparrow\downarrow}$, ${\cal V}^\prime_{\uparrow\downarrow}$
represent a mixture of various polarization states (depending on
parameters $m_{1,3}$ and the signs of energy). This fact may be related to
the recent research of Majorana-like constructs ($m\neq 0$) when we also
mixed solutions of the Dirac equation in order to obtain self/anti-self
charge conjugate states~\cite{bz}.

\end{itemize}

The conclusions are:

\begin{itemize}

\item
There are a lot of speculations in the literature on the third state of
photon polarization. They present, in fact, since the 30s-50s. One should
finally check these claims in experiments;

\item
Dynamics of massless particles may differ from those derived from
the well-known Weyl and Maxwell equations.

\end{itemize}

\acknowledgments I greatly appreciate discussions with Profs. A.
Gersten, A. Raspini and  A. F. Pashkov. I thank Dra. M. Kirchbach
for discussions of the $(1/2,1/2)$ representation and of
the possible physical significance of the gauge in 1998-99.

This work has been partly supported by
the Mexican Sistema Nacional de Investigadores and
the Programa de Apoyo a la Carrera Docente.

\begin{references}
\baselineskip10pt
\footnotesize{
\baselineskip10pt

\bibitem{Ger} Gersten A 1999 {\it Found. Phys. Lett.} {\bf 12} 291;
2000 ibid. {\bf 13} 185

\bibitem{dvo1} Dvoeglazov V V 2000 {\it J. Phys A: Math. Gen.} {\bf 33}
5011

\bibitem{Ger2} Gersten A 2000 {\it Maxwell Equations -- the One-Photon
Quantum Equations.} Preprint, Ben-Gurion University, Israel

\bibitem{dvon} Dvoeglazov V V 2000 {\it Generalized Neutrino
Equations from the Einstein Postulate.}
Preprint EFUAZ FT-00-86, quant-ph/0012107, Zacatecas

\bibitem{Sak} Sakurai J J 1967 {\it Advanced Quantum Mechanics}
(Addison-Wesley), \S 3.2 (see also footnote on p. 91 and
Dvoeglazov V V 2000 {\it Investigaci\'on Cient\'{\i}fica} {\bf 2} 5)

\bibitem{sud} Sudbery A. 1986 {\it J. Phys. A: Math. Gen.}  {\bf 19} L33

\bibitem{dvoo} Dvoeglazov V V 1993 {\it Hadronic J.} {\bf 16} 459

\bibitem{Hott} Abreu E M C and Hott M 2000 {\it Phys. Rev. D} {\bf 62}
027702

\bibitem{ima} Imaeda K 1950 {\it Prog. Theor. Phys.} {\bf 5} 133;
Ohmura T (Kikuta) 1956 {\it Prog. Theor. Phys.} {\bf 16}  684; {\it ibid}
685

\bibitem{donev} Donev S and Tashkova M 1998 {\it Ann. Fond. L de Broglie}
{\bf 23} 89; {\it ibid} 103; {\it ibid} 109; Donev S 2000 Extended Electrodynamics: Basic Equations and
Photon-like (3+1)-Soliton Solutions in Vacuum.  {\it Photon: Old Problems
in Light of New Ideas.} Ed. V. V.  Dvoeglazov, pp. 32-56. Series {\it
Contemporary Fundamental Physics} (Huntington, NY: Nova Science
Publishers)

\bibitem{op} Ogievetski\u{\i} V I and Polubarinov I V
1966 {\it Yadern. Fiz.} {\bf 4} 216 [English translation:
1967 {\it Sov. J. Nucl. Phys.} {\bf 4} 156]

\bibitem{kr} Hayashi K 1973  {\it Phys. Lett.} {\bf B44}  497;
Kalb M and Ramond P. 1974 {\it Phys. Rev.} {\bf D9} 2273

\bibitem{dvob} Dvoeglazov V V 2000 Longitudinal Nature of Antisymmetric
Tensor Field After Quantization and Importance of the Normalization.
{\it Photon: Old Problems in Light of New Ideas.} Ed. V. V.
Dvoeglazov, pp. 319-337. Series {\it Contemporary Fundamental
Physics} (Huntington, NY: Nova Science Publishers)

\bibitem{dvohpa} Dvoeglazov V V 1997 {\it Helv. Phys. Acta} {\bf 70}
677; {\it ibid} 686; {\it ibid} 697;
1998 {\it Ann. Fond. L. de Broglie} {\bf 23} 116;
1998 {\it Int. J. Theor. Phys.} {\bf 37} 1915

\bibitem{weinb} Weinberg S 1964 {\it Phys. Rev.} {\bf B133}  1318;
{\it ibid} {\bf B134} 882; {\it ibid} {\bf 181} 1893

\bibitem{antec} Tokuoka Z 1967 {\it Prog. Theor. Phys.}
{\bf 37}  603; Gupta N D S 1967 {\it Nucl. Phys.} {\bf B4}
147

\bibitem{rasp} Raspini A 1994 {\it Int. J. Theor. Phys.}
{\bf 33} 1503; 1996 {\it Fizika B} {\bf 5} 159;
1997 {\it Fizika B} {\bf 6} 123; 1998 {\it Fizika B} {\bf 7} 83;
1999 A Review of Some Alternative
Descriptions of Neutrino. {\it Photon and Poincar\'e Group.}
Ed. V. V. Dvoeglazov. Series {\it Contemporary Fundamental
Physics} (Commack, NY: Nova Science), p. 181-188

\bibitem{bz} Barut A O and Ziino G 1993 {\it Mod. Phys. Lett.} {\bf
A8} 1011; Ziino G 1996 {\it Int. J. Mod. Phys.} {\bf A11} 2081;
Dvoeglazov V V 1996 {\it Nuovo Cim.} {\bf B111}
483; 1997 {\it Fizika} {\bf B6} 111     ; 1997 {\it Hadronic J.} {\bf 20} 435;
Ahluwalia D V 1996 {\it Int. J. Mod. Phys.} {\bf A11} 1855

}
\end{references}

\end{document}





math-ph0102001.extracted_bib
\bibitem{Ger} Gersten A 1999 {\it Found. Phys. Lett.} {\bf 12} 291;
2000 ibid. {\bf 13} 185


\bibitem{dvo1} Dvoeglazov V V 2000 {\it J. Phys A: Math. Gen.} {\bf 33}
5011


\bibitem{Ger2} Gersten A 2000 {\it Maxwell Equations -- the One-Photon
Quantum Equations.} Preprint, Ben-Gurion University, Israel


\bibitem{dvon} Dvoeglazov V V 2000 {\it Generalized Neutrino
Equations from the Einstein Postulate.}
Preprint EFUAZ FT-00-86, quant-ph/0012107, Zacatecas


\bibitem{Sak} Sakurai J J 1967 {\it Advanced Quantum Mechanics}
(Addison-Wesley), \S 3.2 (see also footnote on p. 91 and
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