Chisholm-Caianiello-Fubini Identities for $S=1$ Barut-Muzinich-Williams Matrices\thanks{This the modernized version of a EF-UAZ FT-95-14 unpublished preprint of 1995 of the second author. The modifications are due to the Thesis of the first author.}
%Trace theorems for the $S=1$ Barut-Muzinich-Williams matrices %are considered. The formulae of the relativistic products are found $S=1$ Barut-Muzinich-Williams matrices. They are analogs of the well-known Chisholm-Caianiello-Fubini identities. The obtained results can be useful in the higher-order calculations of the high-energy processes with $S=1$ particles in the framework of the $2(2S+1)$ Weinberg formalism, which recently attracted attention again. \noindent PACS numbers: 02.90.+p, 11.90.+t, 12.20.Ds
1307.2144.tex
\documentclass[]{article}

%\documentclass[]{revtex4}

\begin{document}

\title{Chisholm-Caianiello-Fubini Identities for $S=1$
Barut-Muzinich-Williams Matrices\thanks{This the modernized version of 
a EF-UAZ FT-95-14 unpublished preprint of 1995 of the second author. The modifications are due 
to the Thesis of the first author.}}

\author{{\bf M. de G. Caldera Cabral} and {\bf V. V. Dvoeglazov}\\
%\address{
UAF, Universidad Aut\'onoma de Zacatecas \\
Apartado Postal 636, Suc. 3 Cruces, Zacatecas 98068, Zac., M\'exico\\
E-mail:  valeri@fisica.uaz.edu.mx}

%\date{April 30, 1995}
\date{}

\maketitle

\begin{abstract}
%Trace theorems for the $S=1$ Barut-Muzinich-Williams matrices
%are considered. 
The formulae of the relativistic  products 
are found $S=1$ Barut-Muzinich-Williams matrices. They are analogs of the well-known Chisholm-Caianiello-Fubini
identities. The obtained results can be useful in
the higher-order calculations of the high-energy processes 
with $S=1$ particles  in the framework of the $2(2S+1)$ Weinberg formalism, 
which recently attracted attention again.

\noindent
PACS numbers: 02.90.+p, 11.90.+t, 12.20.Ds
\end{abstract}

%\pacs{PACS numbers: 02.90.+p, 11.90.+t, 12.20.Ds}

%\maketitle

\newpage

The attractive Weinberg $2(2S+1)$ component formalism for description
of higher spin particles~\cite{Weinberg} is based on the same principles
as the Dirac formalism for spin-1/2, Ref.~\cite{Ryder}. Further 
developments~\cite{Greiner,Dvoeglazov-old,Ahluwalia,Dvoeglazov,Dvoeglazov2}
showed that many interesting things can be found therein. For instance,
the connections with the modified Bargmann-Wigner formalism~\cite{Dvoeglazov2}
or the connections with the so-called Bargmann-Wightman-Wigner 
formalism (BWW)~\cite{Gelfand,BWW,Ahluwalia,Dvoeglazov}.
On the basis of the analysis of
the $(S,0)\oplus (0,S)$ representation space it was found there that
the intrinsic parities of boson and its antiboson can be opposite, see
also~\cite{Silagadze}.  ``If a neutrino is
identified with the self/anti-self charge-conjugate representation space,
then it may be coupled with the BWW bosons to generate physics beyond
the present day gauge theories.", see the above-cited references.
One more hint at the possible future application of these formalisms is the
tentative experimental evidence for a tensor coupling in the $\pi^- \rightarrow
e^- + \tilde\nu_e +\gamma$ decay, for instance~\cite{Bolotov}.
There exist  experimental opportunities to check the
existence of the ``unconventional"  bosons and fermions, and different types 
of interactions as well, beyond the Standard Model, e.~g., Ref.~\cite{DME}.

The principal equation in this formalism is that of the ``$2s$"-order in the momentum
operators. The analogs of the Dirac $\gamma$-matrices have also ``$2s$"
vectorial indices:
\begin{equation}
[\gamma_{\mu_1 \mu_2...\mu_{2s}} \partial^{\mu_1}\partial^{\mu_2}...\partial^{\mu_2s} + m^{2s}]
\Psi (x) = 0.
\end{equation}
The covariant-defined $\Gamma$-~matrices for any spin have been
introduced by Barut, Muzinich and Williams~\cite{Barut}, see
also~\cite{Shay,TH}.  For the case of spin $S=1$ they have the following
form:\footnote{The Eiclidean metric is used.}
\begin{eqnarray}
&&\Gamma^{(1)} \equiv I = \pmatrix{ I & 0\cr 0 &
I} ,\nonumber\\
&&\Gamma^{(2)} \equiv \gamma_5 = \pmatrix { - I & 0\cr 0 &
I} , \nonumber\\
&&\Gamma^{(3)}_{\alpha\beta} \equiv \gamma_{\alpha\beta} = \pmatrix
{0 & - \tilde S_{\alpha\beta}^{\dagger}\cr - \tilde S_{\alpha\beta} & 0} , \nonumber\\
&&\Gamma^{(4)}_{\alpha\beta} \equiv \gamma_{4, \alpha\beta} = 
i\gamma_5 \gamma_{\alpha\beta} , \\
&&\Gamma^{(5)}_{\alpha\beta} \equiv \gamma_{5, \alpha\beta}
= i \left [\gamma_{\alpha\lambda},
\gamma_{\beta\lambda}\right ]_{-} , \nonumber\\
&&\Gamma^{(6)}_{\alpha\beta, \mu\nu} \equiv \gamma_{6, \alpha\beta, \mu\nu}
=
 \left [\gamma_{\alpha\mu}, \gamma_{\beta\nu} \right ]_{+}
+2\delta_{\alpha\mu} \delta_{\beta\nu} - \left [\gamma_{\alpha\nu},
\gamma_{\beta\mu}\right ]_{+} -2\delta_{\alpha\nu}\delta_{\beta\mu}= \nonumber\\
&&= - {1\over 12} \left [\gamma_{5, \alpha\beta}, \gamma_{5, \mu\nu}\right ]_{+}
+ 4\left (\delta_{\alpha\mu}\delta_{\beta\nu} -\delta_{\alpha\nu} \delta_{\beta\mu} \right )
- 4\epsilon_{\alpha\beta\mu\nu} \gamma_5 \,, \nonumber
\end{eqnarray}
where
\begin{eqnarray}
\tilde S_{44} &=& -I, \quad \tilde S_{i4}=\tilde S_{4i} = iS_i , \nonumber\\
\tilde S_{ij} &=& S_{ij} - \delta_{ij} = S_i S_j + S_j S_i - \delta_{ij} \,.
\end{eqnarray}
$S_i$ are the spin-1 matrices, and $\epsilon_{1234} = 1$.
They have the simmetry properties~\cite{Shay}:
\begin{eqnarray}
&&\gamma_{\alpha\beta} = \gamma_{\beta\alpha},\quad \sum_\alpha\,
\gamma_{\alpha\alpha} =0 , \nonumber\\
&&\gamma_{4, \alpha\beta} = \gamma_{4, \beta\alpha},\quad \sum_\alpha\,
\gamma_{4, \alpha\alpha} =0, \nonumber\\
&&\gamma_{5, \alpha\beta} = - \gamma_{5, \beta\alpha} , \\
&&\gamma_{6, \alpha\beta, \mu\nu} = - \gamma_{6, \beta\alpha, \mu\nu},\quad
\gamma_{6, \alpha\beta, \mu\nu} = \gamma_{6, \mu\nu, \alpha\beta} , \nonumber\\
&&\gamma_{6, \alpha\beta, \mu\nu} + \gamma_{6, \alpha\mu, \nu\beta}
+  \gamma_{6, \alpha\nu, \mu \beta} = 0 .\nonumber
\end{eqnarray}
The relativistic perturbation calculations of the processes
including the $S=1$ bosons will require the development technical methods
analogous to those which have been elaborated for the fermion-fermion
interaction, namely, reducing contracted products of the corresponding
$\Gamma$ matrices~\cite{Chisholm,Caianello,Good,Chisholm2}.
Our aim with this paper is to find the formulae of the relativistic
scalar products like that $\gamma_{\mu\alpha}\ldots \gamma_{\beta\mu}$.

The following relations can be deduced by straightforward
calculations:\footnote{We have also used the Wolfram MATEMATICA programm to check them.}
\begin{equation}\label{eq:rel1}
\gamma_{\mu\alpha}\gamma_{\beta\mu} =3\delta_{\alpha\beta}
-{i\over 2} \gamma_{5,\alpha\beta} ,
\end{equation}

\begin{equation}
\gamma_{\mu\alpha}\gamma_5 \gamma_{\beta\mu} = -3\gamma_5 \delta_{\alpha\beta}
-{i\over 4} \epsilon_{\alpha\beta\sigma\tau} \gamma_{5,\sigma\tau} ,
\end{equation}

\begin{eqnarray}
\lefteqn{\gamma_{\mu\alpha}\gamma_{\sigma\tau}\gamma_{\beta\mu} =
2\gamma_{\sigma\tau}\delta_{\alpha\beta} +\gamma_{\alpha\beta} \delta_{\sigma\tau} -
\gamma_{\alpha\sigma}\delta_{\tau\beta} -\gamma_{\alpha\tau}\delta_{\sigma\beta} -}\nonumber\\
&-& \gamma_{\beta\sigma}\delta_{\alpha\tau} - \gamma_{\beta\tau} \delta_{\alpha\sigma} -
i\epsilon_{\alpha\beta\sigma\mu} \gamma_{4,\tau\mu} - i\epsilon_{\alpha\beta\tau\mu}
\gamma_{4,\sigma\mu} ,
\end{eqnarray}

\begin{eqnarray}
\lefteqn{\gamma_{\mu\alpha}\gamma_{4,\sigma\tau}\gamma_{\beta\mu} =
-2\gamma_{4,\sigma\tau}\delta_{\alpha\beta} -\gamma_{4,\alpha\beta} \delta_{\sigma\tau} +
\gamma_{4,\alpha\sigma}\delta_{\tau\beta} +\gamma_{4,\alpha\tau}\delta_{\sigma\beta} +} \nonumber\\
&+& \gamma_{4,\beta\sigma}\delta_{\alpha\tau} + \gamma_{4,\beta\tau} \delta_{\alpha\sigma} -
i\epsilon_{\alpha\beta\sigma\mu} \gamma_{\tau\mu} - i\epsilon_{\alpha\beta\tau\mu}
\gamma_{\sigma\mu} ,
\end{eqnarray}

\begin{eqnarray}
%\label{eq:rellas}
\lefteqn{\gamma_{\mu\alpha} \gamma_{5,\sigma\tau} \gamma_{\beta\mu} =
2\gamma_{5,\sigma\tau}\delta_{\alpha\beta} + 2\gamma_{5,\alpha\sigma} \delta_{\beta\tau} +
2\gamma_{5,\tau\beta} \delta_{\alpha\sigma}-} \nonumber\\
&-& 2\gamma_{5,\sigma\beta} \delta_{\alpha\tau} - 2\gamma_{5,\alpha\tau} 
\delta_{\sigma\beta} +12i \left ( \delta_{\alpha\sigma}\delta_{\tau\beta} - 
\delta_{\alpha\tau} \delta_{\sigma\beta}\right  )+ \\
&+& 12i \epsilon_{\alpha\sigma\tau\beta} \gamma_{5} , \nonumber
\end{eqnarray}

\begin{equation}\label{eq:rellas}
\gamma_{\mu\alpha} \gamma_{6, \sigma\tau , \rho\phi} \gamma_{\beta\mu} = 0 .
\end{equation}
The formulae for the $S=1$ matrices which have been used above
are presented in Appendix.
 




\setcounter{equation}{0}


\section*{Appendix}
\appendix

Here we present the set of algebraic relations for $S=1$ spin
matrices, cf.~\cite{Varshalovich,Weaver}. We imply a summation on the repeated indices.
\begin{eqnarray}
&& S_k S_i S_k = S_i ,\\
&& S_k S_i S_j S_k = 2\delta_{ij} - S_j S_i ,\\
&& S_k S_i S_j S_l S_k = S_l S_i S_j + S_j S_l S_i - S_j \delta_{il} ,\\
&& S_k S_i S_j S_l S_m S_k =
\delta_{ij}\delta_{lm} + \delta_{im}\delta_{jl} - S_m S_l S_j S_i ,
\end{eqnarray}
and
\begin{eqnarray}
&& S_{ij} S_k = \delta_{ij} S_k +{1\over 2} \delta_{jk} S_i +{1\over 2}
\delta_{ik} S_j +{i\over 2} \left ( \epsilon_{ikl} S_{jl} + \epsilon_{jkl}
S_{il} \right ) ,\\
&& S_{ik} S_{jl} + S_{jl} S_{ik} = 2\delta_{ik} S_{jl}
+2\delta_{jl} S_{ik}+\left ( \epsilon_{ilm} \epsilon_{jkn}
-\epsilon_{ijm} \epsilon_{kln}\right ) S_{mn} ,\\
&& S_l S_{ij} S_m =
2\delta_{ij} \delta_{lm} -\delta_{im}\delta_{jl} -\delta_{jm}\delta_{il}
-\delta_{lm} S_{ij} +\nonumber\\
&& \qquad +\delta_{im}S_l S_j +\delta_{jm} S_l S_i +\delta_{il} S_j S_m +\delta_{jl} S_i S_m ,\\
&& S_l S_{ij} S_m - S_m S_{ij} S_l = \delta_{il} (S_j S_m - S_m S_j ) + \delta_{jl} (S_i S_m - S_m S_i )+\nonumber\\
&& \qquad + \delta_{im} ( S_l S_j - S_j S_l) +\delta_{jm} (S_l S_i - S_i S_l) ,\\
&& {\mbox or}\qquad \qquad \qquad \quad = -i\epsilon_{ilm} S_j - i\epsilon_{jlm} S_i
- 2\delta_{ij} (S_m S_l - S_l S_m) ,\\
&& S_i S_j S_k  + S_j S_k S_i + S_k S_i S_j = S_i \delta_{jk} + S_k \delta_{ij}
+ S_j \delta_{ik} +\nonumber\\
&& \qquad + {i\over 4} \left ( \epsilon_{ijl} S_{lk} +\epsilon_{kil} S_{jl}
+\epsilon_{jkl} S_{il}\right ) .
\end{eqnarray}
This set supplies the known formulae for $S=1$ spin matrices, e.~g.
presented in~\cite{Weaver}:
\begin{eqnarray}
&& S_i S_j S_k + S_k S_j S_i = \delta_{ij} S_k + \delta_{jk} S_i , \\
&& \tilde S_{ik} S_j =-{i\over 2} \left [ \delta_{ij} \tilde S_{4k}
+\delta_{jk} \tilde S_{4i} +\epsilon_{jil} \tilde S_{lk} +\epsilon_{jkl}
\tilde S_{il}\right ]\\
&&\Sigma_i^2 = S_3^2 ,\, \, {\mbox no \,\, summation},\\
&& \Sigma_i \Sigma_j + \Sigma_j \Sigma_i = 2\delta_{ij} S_3^2 , \\
&& \Sigma_i \Sigma_j - \Sigma_j \Sigma_i = 2i\epsilon_{ijk} \Sigma_k ,
\end{eqnarray}
where
\begin{equation}
\Sigma_1 \equiv S_1^2 -S_2^2 ,  \quad \Sigma_2 \equiv {\tilde S}_{12} = S_1 S_2 + S_2 S_1 ,
\quad \Sigma_3 \equiv S_3 .
\end{equation}

\begin{thebibliography}{99}
\footnotesize{
\bibitem{Weinberg} S. Weinberg, {\it Phys. Rev.} {\bf 133}, B1318 (1964).

\bibitem{Ryder} L. H. Ryder, {\it Quantum Field Theory.} (Cambridge Univ. Press, 1985).

\bibitem{Greiner} W. Greiner, {\it Relativistic Quantum Mechanics.} (Springer-Verlag,
Berlin-Heidelberg, 1990).

\bibitem{Dvoeglazov-old} V. V. Dvoeglazov and N. B. Skachkov, {\it Yad. Fiz.}
{\bf 48}, 1770 [English translation: {\it Sov. J. Nucl. Phys.},
1065] (1988);  JINR Communications R2-87-882 (1987); V. V. Dvoeglazov and S. V. Khudyakov,
{\it Hadronic J.} {\bf  21}, 507 (1998); V. V. Dvoeglazov , {\it Rev. Mex. Fis. Suppl.}
{\bf 40}, 352 (1994), hep-th/9401043.
%V. V. Dvoeglazov, {\it Hadronic J.}, {\bf 16}(5), 423-428
%(1993); {\it ibid}, {\bf 16}(6), 459-467 (1993)

\bibitem{Ahluwalia} D. V. Ahluwalia, M. B. Johnson and
T.  Goldman, {\it Phys.  Lett.} {\bf B316}, 102 (1993).

\bibitem{Dvoeglazov} V. V. Dvoeglazov, {\it Int. J. Theor. Phys.} 
{\bf 37}, 1915 (1998).

\bibitem{Dvoeglazov2} V. V. Dvoeglazov, {\it Int. J. Mod. Phys.} {\bf B20}, 1317 (2006);
{\it J. Phys. CS} {\bf 91}, 012009 (2007); {\it  Int. J. Mod. Phys. CS} {\bf 03}, 121 (2011).

\bibitem{Gelfand} I. M. Gelfand and M. L. Tsetlin, ZhETF {\bf 31}, 1107 (1956); 
G. A. Sokolik, ZhETF {\bf 33}, 1515 (1957).

\bibitem{BWW} E. P. Wigner, in {\it Group Theoretical Concepts
and Methods in Elementary Particle Physics -- Lectures of the Istanbul
Summer School of Theoretical Physics, 1962.} Ed. by F. G\"ursey (Gordon and Breach, New York-London-Paris, 1964).

\bibitem{Silagadze} Z. K. Silagadze, {\it Yad. Fiz.} {\bf 55}, 707
[English translation: {\it Sov. J. Nucl. Phys.}, 392] (1992).

\bibitem{Bolotov} V. N. Bolotov {\it et al.}, {\it Phys. Lett.}
{\bf B243}, 308 (1990).

\bibitem{DME} D. N. Casta\~no, {\it Dark Matter Constrains from High Energy Astrophysical Observations.} Ph. D. Thesis (2012), http://eprints.ucm.es/15300/1/T33772.pdf . 

\bibitem{Barut} A. Barut, I. Muzinich and D. N. Williams,
{\it Phys. Rev.} {\bf 130}, 442 (1963).

\bibitem{Shay} A. Sankaranarayanan and R. H. Good, {\it Nuovo Cimento}
{\bf 36}, 1303 (1965); D. Shay and R. H. Good, Jr., {\it Phys. Rev.}
{\bf 179}, 1410 (1969).

\bibitem{TH} R. H. Tucker and C. L. Hammer, Phys. Rev. {\bf D3}, 2448 (1971).

\bibitem{Chisholm} J. S. R. Chisholm, {\it Proc. Cambridge
Phil. Soc.} {\bf 48}, 300 (1952).

\bibitem{Caianello} E. R. Caianiello and S. Fubini, {\it Nuovo Cimento}
{\bf 9}, 1218 (1952).

\bibitem{Good} R. H. Good, Jr., {\it Rev. Mod. Phys.} {\bf 27}, 187 (1955).

\bibitem{Chisholm2} J. S. R. Chisholm, {\it Nuovo Cimento} {\bf 30}, 426
(1963); J. Kahane, {\it J. Math. Phys.} {\bf 9}, 1732 (1968);
J. S. R. Chisholm, {\it Comp. Phys. Comm.} {\bf 4}, 205 (1972).

%M. Moreno, {\it J. Math. Phys.} {\bf 26}(4), 576-584 (1985)

\bibitem{Varshalovich} D. A. Varshalovich {\it et al.}, {\it Quantum Theory of
Angular Momentum.} (World Scientific, Singapore, 1988).

\bibitem{Weaver} D. L. Weaver, {\it Am. J. Phys.} {\bf 46}, 721 (1978); {\it idem.}, {\it J. Math. Phys.} {\bf 19}, 88 (1978).

}



\end{thebibliography}

\end{document}
For the sake of completeness let us also reproduce the trace theorems
for the $S=1$ Barut-Muzinich-Williams matrices comparing with the ones
for the Dirac case.

\bigskip

{\it Theorem 1:}  For $i\neq 1$; $Tr\, \Gamma_a^{(i)} = 0$,
where $Tr\, (A)$ means the trace of $A$.

\medskip

{\it Theorem 2:}  $(\Gamma^{(i)}_a)^2 = k  I$ (summation is implied), i.~e. the sum
is proportional to the unit matrix. One has the difference with
the set of Dirac matrices, which may be defined in the way when 
squared  each of them is a unit matrix.

\medskip

{\it Theorem 3:} The 36 matrices $\Gamma_a^{(i)}$ are linearly independent
and complete. They therefore form a basis for the algebra, and
any element $A$ of the algebra can be expanded as
\begin{equation}\label{eq:expan}
A= \sum_{i,a} A_a^{(i)} \Gamma^{(i)}_a .
\end{equation}

\medskip

{\it Theorem 4:} The traces of a product of an odd number of
$\gamma_{\mu\nu}$ or  $\gamma_{4,\mu\nu}$  are zero. That is,
\begin{eqnarray}
Tr\, (\gamma_{\alpha_1 \beta_1} \ldots \gamma_{\alpha_{2n+1}\beta_{2n+1}}) =0\\
Tr\, (\gamma_{4, \alpha_1 \beta_1} \ldots \gamma_{4, \alpha_{2n+1}\beta_{2n+1}}) =0
\end{eqnarray}
with $n=0,1,\ldots $.

\medskip

{\it Theorem 5:} In the expansion of
\begin{equation}
\gamma_{\alpha_1 \beta_1} \ldots \gamma_{\alpha_n \beta_n} =
\sum_{i,a} A^{(i)}_a \Gamma^{(i)}_a ,
\end{equation}
we have
\begin{equation}
A_a^{(1)} = A^{(2)}_a = A^{(5)}_a = A^{(6)}_a = 0 ,
\quad {\mbox for}\quad n \quad {\mbox odd},
\end{equation}
and
\begin{equation}
A^{(3)}_a = A^{(4)}_a = 0, \quad {\mbox for}\quad n \quad {\mbox even}.
\end{equation}

\medskip

{\it Theorem 6:}  There exists a $6\times 6$ nonsingular
matrix $C$, such that $C\gamma_{\alpha\beta} C^{-1} = \gamma_{\alpha\beta}^T$,
where $\gamma_{\alpha\beta}^T$ is the transpose of $\gamma_{\alpha\beta}$, and
\begin{equation}
C \Gamma^{(i)}_a = \pm (\Gamma^{(i)}_a )^T ,
\end{equation}

the sign ``$+$" is for $i=1,2,3,6$, the sign ``$-$" is for $i=4,5$.

\smallskip

{\it Corollary 1:} If even and odd products of $\gamma_{\alpha_i \beta_i}$ matrices are expanded
as in (\ref{eq:expan}), then
\begin{equation}
\gamma_{\alpha_{2n+1}\beta_{2n+1}} \ldots \gamma_{\alpha_1 \beta_1} = A^{(3)}_a \Gamma^{(3)}_a - A^{(4)}_a \Gamma^{(4)}_a ,
\end{equation}
and
\begin{equation}
\gamma_{\alpha_{2n}\beta_{2n}} \ldots \gamma_{\alpha_1 \beta_1 } = A^{(1)}_a \Gamma^{(1)}_a
+A^{(2)}_a \Gamma^{(2)}_a - A^{(5)}_a \Gamma^{(5)}_a  + A^{(6)}_a \Gamma^{(6)}_a .
\end{equation}
The proof is analogous to the spin-$1/2 $ case (see, e.~g., ref.~[15b]). Just it is necessary
to take into account that
\begin{equation}
\gamma_{\alpha_1\beta_1 } \ldots \gamma_{\alpha_{2n+1} \beta_{2n+1} } = 
A^{(3)}_a \Gamma^{(3)}_a + A^{(4)}_a \Gamma^{(4)}_a ,
\end{equation}
and
\begin{equation}
\gamma_{\alpha_{1}\beta_{1} } \ldots \gamma_{\alpha_{2n} \beta_{2n} } = A^{(1)}_a \Gamma^{(1)}_a
+A^{(2)}_a \Gamma^{(2)}_a + A^{(5)}_a \Gamma^{(5)}_a  + A^{(6)}_a \Gamma^{(6)}_a .
\end{equation}
and apply the {\it Theorem 6} two times.

\bigskip

Then, 


Using the relations (\ref{eq:rel1} -\ref{eq:rellas}) one can prove the following
formulae which are the results of the paper:

\medskip

{\it Corollary 2:}
\begin{equation}
\sum_{\mu=1}^{4} \gamma_{\mu\alpha} \gamma_{\alpha_1 \beta_1}\ldots
\gamma_{\alpha_{2n+1}\beta_{2n+1}}\gamma_{\beta\mu} = ,
\end{equation}
what gives for  $\alpha=\beta $
\begin{equation}
\sum_{\mu, \alpha=1}^{4} \gamma_{\mu\alpha} 
\gamma_{\alpha_1 \beta_1}\ldots
\gamma_{\alpha_{2n+1}\beta_{2n+1}}\gamma_{\alpha\mu} =  
4 \gamma_{\alpha_{2n+1}\beta_{2n+1}}\ldots \gamma_{\alpha_1 \beta_1} .
\end{equation}

\smallskip

{\it Corollary 3:}
\begin{equation}
\sum_{\mu=1}^{4} \gamma_{\mu\alpha} \gamma_{\alpha_1 \beta_1}\ldots
\gamma_{\alpha_{2n}\beta_{2n}}\gamma_{\beta\mu} =
\end{equation}
what gives  for $\alpha=\beta$
\begin{equation}
.
\end{equation}
\smallskip

\begin{eqnarray}
\lefteqn{\left [ \gamma_{\mu\rho}, \gamma_{\nu\lambda} \right  ] _+ + \left [ \gamma_{\mu\nu},
\gamma_{\rho\lambda}\right ] _+ +\left [\gamma_{\mu\lambda}, \gamma_{\rho\nu}\right  ] _+ =}\nonumber\\
&=& 2\left ( \delta_{\mu\nu}\delta_{\rho\lambda} +\delta_{\mu\rho} \delta_{\nu\lambda} + \delta_{\mu\lambda} \delta_{\nu\rho}\right )
\end{eqnarray}
(see, e.~g.,~\cite[p.B1332]{Weinberg}).

1307.2144.extracted_bib
\begin{thebibliography}{99}
\footnotesize{
\bibitem{Weinberg} S. Weinberg, {\it Phys. Rev.} {\bf 133}, B1318 (1964).

\bibitem{Ryder} L. H. Ryder, {\it Quantum Field Theory.} (Cambridge Univ. Press, 1985).

\bibitem{Greiner} W. Greiner, {\it Relativistic Quantum Mechanics.} (Springer-Verlag,
Berlin-Heidelberg, 1990).

\bibitem{Dvoeglazov-old} V. V. Dvoeglazov and N. B. Skachkov, {\it Yad. Fiz.}
{\bf 48}, 1770 [English translation: {\it Sov. J. Nucl. Phys.},
1065] (1988);  JINR Communications R2-87-882 (1987); V. V. Dvoeglazov and S. V. Khudyakov,
{\it Hadronic J.} {\bf  21}, 507 (1998); V. V. Dvoeglazov , {\it Rev. Mex. Fis. Suppl.}
{\bf 40}, 352 (1994), hep-th/9401043.
%V. V. Dvoeglazov, {\it Hadronic J.}, {\bf 16}(5), 423-428
%(1993); {\it ibid}, {\bf 16}(6), 459-467 (1993)

\bibitem{Ahluwalia} D. V. Ahluwalia, M. B. Johnson and
T.  Goldman, {\it Phys.  Lett.} {\bf B316}, 102 (1993).

\bibitem{Dvoeglazov} V. V. Dvoeglazov, {\it Int. J. Theor. Phys.} 
{\bf 37}, 1915 (1998).

\bibitem{Dvoeglazov2} V. V. Dvoeglazov, {\it Int. J. Mod. Phys.} {\bf B20}, 1317 (2006);
{\it J. Phys. CS} {\bf 91}, 012009 (2007); {\it  Int. J. Mod. Phys. CS} {\bf 03}, 121 (2011).

\bibitem{Gelfand} I. M. Gelfand and M. L. Tsetlin, ZhETF {\bf 31}, 1107 (1956); 
G. A. Sokolik, ZhETF {\bf 33}, 1515 (1957).

\bibitem{BWW} E. P. Wigner, in {\it Group Theoretical Concepts
and Methods in Elementary Particle Physics -- Lectures of the Istanbul
Summer School of Theoretical Physics, 1962.} Ed. by F. G\"ursey (Gordon and Breach, New York-London-Paris, 1964).

\bibitem{Silagadze} Z. K. Silagadze, {\it Yad. Fiz.} {\bf 55}, 707
[English translation: {\it Sov. J. Nucl. Phys.}, 392] (1992).

\bibitem{Bolotov} V. N. Bolotov {\it et al.}, {\it Phys. Lett.}
{\bf B243}, 308 (1990).

\bibitem{DME} D. N. Casta\~no, {\it Dark Matter Constrains from High Energy Astrophysical Observations.} Ph. D. Thesis (2012), http://eprints.ucm.es/15300/1/T33772.pdf . 

\bibitem{Barut} A. Barut, I. Muzinich and D. N. Williams,
{\it Phys. Rev.} {\bf 130}, 442 (1963).

\bibitem{Shay} A. Sankaranarayanan and R. H. Good, {\it Nuovo Cimento}
{\bf 36}, 1303 (1965); D. Shay and R. H. Good, Jr., {\it Phys. Rev.}
{\bf 179}, 1410 (1969).

\bibitem{TH} R. H. Tucker and C. L. Hammer, Phys. Rev. {\bf D3}, 2448 (1971).

\bibitem{Chisholm} J. S. R. Chisholm, {\it Proc. Cambridge
Phil. Soc.} {\bf 48}, 300 (1952).

\bibitem{Caianello} E. R. Caianiello and S. Fubini, {\it Nuovo Cimento}
{\bf 9}, 1218 (1952).

\bibitem{Good} R. H. Good, Jr., {\it Rev. Mod. Phys.} {\bf 27}, 187 (1955).

\bibitem{Chisholm2} J. S. R. Chisholm, {\it Nuovo Cimento} {\bf 30}, 426
(1963); J. Kahane, {\it J. Math. Phys.} {\bf 9}, 1732 (1968);
J. S. R. Chisholm, {\it Comp. Phys. Comm.} {\bf 4}, 205 (1972).

%M. Moreno, {\it J. Math. Phys.} {\bf 26}(4), 576-584 (1985)

\bibitem{Varshalovich} D. A. Varshalovich {\it et al.}, {\it Quantum Theory of
Angular Momentum.} (World Scientific, Singapore, 1988).

\bibitem{Weaver} D. L. Weaver, {\it Am. J. Phys.} {\bf 46}, 721 (1978); {\it idem.}, {\it J. Math. Phys.} {\bf 19}, 88 (1978).

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\end{thebibliography}