%Paper: hep-th/9305141
%From: Valera <valera@teorica0.ifisicacu.unam.mx>
%Date: Tue, 25 May 93 20:32:11 -0600

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\begin{document}
\begin{titlepage}
\begin{center}
%{\large {\bf Universidad Nacional Aut\'{o}noma de M\'{e}xico,\\
%Instituto de F\'{\i}sica,\,Departamento de F\'{\i}sica Te\'{o}rica}}\\
%\end{center}
%\vspace*{10mm}
\hspace*{10cm} Preprint IFUNAM\\
\hspace*{10cm} FT\,93-016\\
\hspace*{10cm} May 1993\\
\vspace*{20mm}
%\begin{center}
{\large{\bf Lagrangian Formulation of the Joos-Weinberg's\\ $2(2S+1)$-- theory
and Its Connection\\ \vskip1mm with the Skew-Symmetric Tensor Description}}\\
\vspace{6mm}
{ {\bf \large V. V. Dvoeglazov}}$^{\,\sharp,\,\dagger}$\\

\vspace*{3mm}

{\it  Departamento de F\'{\i}sica Te\'{o}rica, \,Instituto de F\'{\i}sica,\\
Universidad Nacional Aut\'{o}noma de M\'{e}xico, \\
Apartado Postal 20-364, 01000 D.F. , MEXICO}\\
\end{center}
\vspace*{6mm}
\begin{abstract}
In the framework of the $2(2S+1)$-- theory of Joos-Weinberg for massless
particles,  the dynamical invariants have been derived from the Lagrangian
density which is considered to be a 4-- vector. A l\'a Majorana
interpretation of the 6-- component spinors, the field operators of
$S=1$ particles, as the left-- and right--circularly polarized radiation,
leads us to the conserved quantities which are analogous to ones obtained
by Lipkin and Sudbery.

The scalar Lagrangian of Joos-Weinberg theory is shown to be equivalent
to the Lagrangian of a free massless field, introduced by Hayashi. As a
consequence of a new "gauge" invariance this skew-symmetric field describes
physical particles with the longitudinal components only.
\end{abstract}
\vspace*{7mm}
\centerline{\bf Submitted to "Journal of Physics A"}
\vspace*{10mm}
\noindent
KEYWORDS: quantum electrodynamics, Lorentz group representation, string theory,
high-spin particles, bivector\\
PACS: 03.50.De, 11.10.Ef, 11.10.Qr, 11.17+y, 11.30.Cp\\
\noindent
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---------------------------------------------------------------------------------------------------------\\

$^{\sharp}$On leave from: {\it Dept.Theor.} \& {\it Nucl.Phys., Saratov State
University\\ and Sci.} \& {\it Tech. Center for  Control and Use of Physical
Fields and Radiations\\ Astrakhanskaya str. , 83,\, Saratov 410071 RUSSIA}

$^{\dagger}$ Email: VALERI@IFUNAM.IFISICACU.UNAM.MX\\
\hspace*{23mm}DVOEGLAZOV@MAIN1.JINR.DUBNA.SU\\
\end{titlepage}
\vspace*{15mm}


The investigation of relativistic high-spin fields is very important
in the connection of the present experimental situation when the particles
of spin up to 6 were found out~\cite{PDG}:\\


\vspace*{4mm}
\begin{center}
%Table I.\\
\begin{tabular}{||c|c|c|c||}
\hline
\hline
Meson&$f_6(2510)$&$I^G=0^+$&$J^{PC}=6^{++}$\\
\hline
Baryon&$\Delta(2420),H_{3,11}$&$I={3\over 2}$&$J^P={11 \over 2}^+$\\
\hline
Baryon&$N(2600),I_{1,11}$&$I={1 \over 2}$&$J^P={11 \over 2}^-$\\
\hline
\hline
\end{tabular}\\
\end{center}
\vspace*{5mm}

In the beginning of the sixties Joos~\cite{Joos}, Weinberg~\cite{Weinberg}
and Weaver, Hammer, and Good~\cite{Weaver} have developed a description
of free particles with an arbitrary spin $S=0,{1 \over 2}, 1,{3 \over 2}
\cdots$ on the base of Wigner's ideas~\cite{Wigner} of constructing
quantum field theory. Following this description, the spin-one case
{}~\cite{Sankaranarayanan}-\cite{Tucker} as well as the spin-${3 \over 2}$
case~\cite{Shay} have been investigated in detail. The formulas for the
Hamiltonian for any spin have also been obtained~\cite{Mathews,
Williams}\footnote{ I would like to mention the recent articles
concerning this formalism~\cite{Santos}-\cite{Dvoeglaz}.}.
The wave functions in this approach form the basis
 of the $(S,0)\oplus(0,S)$ representation of the Lorentz group and are
presented by the $2(2S+1)$-- component spinor:
\begin{equation}\label{eq:psi}
\Psi=\left (\matrix{
\chi_\sigma\cr
\phi_\sigma\cr
}\right ),
\end{equation}
The transformation rules
\begin{eqnarray}
\cases{\chi_{\sigma}(\vec p)=exp\left (+\theta \hat{\vec p}\hat{\vec J}\right )
\chi_{\sigma}\left (0 \right ), & $ $\cr
\phi_{\sigma}\left (\vec p\right )=exp\left (-\theta\hat{\vec p} \hat {\vec
J}\right ) \phi_{\sigma}\left (0 \right ) & $ $}
\end{eqnarray}
(with $\theta$ is the boost parameter, $tanh \theta=\frac{\mid\vec p\mid}
{E}$, $\hat{\vec p}=\frac{\vec p}{\mid \vec p \mid}$, $\hat{\vec J}$ is the
angular momentum operator) represent generalization of the
well--known Lorentz boosts for the Dirac particle. It was noted in
Ref. [3b, p. 888] that the equation for this spinor:
\begin{equation}
(\gamma_{\mu\nu}p_\mu p_\nu + m^2)\Psi=0
\end{equation}
can be transformed to the equations for left-- and right--circularly
polarized radiation when massless $S=1$ field being considered.
Otherwords, we come to the Maxwell's free-space equations (Eqs. (4.21) and
(4.22) of Ref. [3b]):
\begin{eqnarray}\label{eq:Maxwell}
\cases{\vec\nabla\times[\vec E-i\vec H]+i(\partial/\partial t)[\vec E-i\vec
H]=0,& $ $\cr
\vec\nabla\times[\vec E+i\vec H]-i(\partial/\partial t)[\vec E+i\vec H]=0,& $
$}
\end{eqnarray}
if we consider (\ref{eq:psi}) as the Dirac "bivector"\,\footnote{See also
{}~\cite[p.149]{Ohnuki} for discussion about interpretation of components
of field transforming on the $(S,0)\oplus (0,S)$ representation of the
Lorentz group.} which can be decomposed in Pauli algebra as e.g.~\cite{Recami}:
\begin{eqnarray}\label{eq:EH}
\cases{
\chi=\quad\vec E+i\vec H,& $ $\cr
\phi=-\vec E+i\vec H& $ $}
\end{eqnarray}
($\vec E$ and $\vec H$ are Pauli vectors). In fact, this is the formulation a
l\'a Majorana~\cite{Recami2}-\cite{Gianetto}\footnote{See~\cite{Collins} for
discussion about connection of
$2(2S+1)$-- component multispinor $\Psi=(\Psi_{\alpha_1\ldots\alpha_{2S}})$,
submitted to the massless Bargmann-Wigner equation, with the antisymmetric
field tensor $F_{\mu_1\nu_1\ldots\mu_S\nu_S}$.}.

Attempts at describing the electromagnetic field in the terms of
electric and magnetic field vectors $\vec E, \vec H$ (but not a potential) as
independent
variables, or, equivalently, antisymmetric strength tensors, have been
undertaken
recently~\cite{Anderson}-\cite{Sudbery}. For example, in Ref.~\cite{Sudbery}
the 4--vector Lagrangian density:
\begin{equation}
{\cal
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L}_{\alpha}=^{*}F^{\mu\nu}\partial_{\nu}F_{\mu\alpha}-F^{\mu\nu}\partial_{\nu}\,^{*} F_{\mu\alpha}-2^* F_{\alpha\mu}j^{\mu}
\end{equation}
($F_{\mu\nu}$ is the electromagnetic field tensor, $^* F_{\mu\nu}=\epsilon
_{\mu\nu\rho\sigma}F^{\rho\sigma}$ is its
dual, $j^{\mu}$ is the electromagnetic current 4--vector) has been used to
determine the new conserved quantities analogous to the quantities deduced
from Lipkin's tensor~\cite{Lipkin}. The remarkable feature of this
formulation is that energy-momentum conservation is associated not
with translational invariance but with invariance under duality
rotations.

In present article the similar properties are shown out for the Lagrangian
density of Joos-Weinberg theory. Following ~\cite{Sudbery},
the Lagrangian is chosen as the 4--vector\footnote{See~\cite{Fushchich}
for the details of vector Lagrangian description.}:
\begin{equation}\label{eq:Lagr}
{\cal L}_{\alpha}=-i\bar\Psi\gamma_{\alpha\beta}\partial_{\beta}\Psi
+i(\partial_{\beta}\bar\Psi)\gamma_{\alpha\beta}\Psi.
\end{equation}

On the variational principle of the stationary action the above Lagrangian
leads to Euler--Lagrange equations:
\begin{eqnarray}\label{eq:LagEul}
\cases{\gamma_{\alpha\beta}\partial_{\beta}\Psi=0,& $ $\cr
(\partial_{\beta}\bar\Psi)\gamma_{\alpha\beta}=0,& $ $}
\end{eqnarray}
which are, in fact, the Eqs.($4, 4'$) of Ref.  [17b]. When $\alpha=4$
Eqs.(\ref{eq:LagEul}) are rewritten to  Eqs. (\ref{eq:Maxwell}), whereas when
$\alpha=i=1,2,3$ we come to:
\begin{eqnarray}
\cases{
\epsilon_{ikl}\frac{\partial E_l}{\partial t}+\partial_k H_i-
\partial_i H_k+(\partial_j H_j)\delta_{ik}=0,& $ $\cr
\epsilon_{ikl}\frac{\partial H_l}{\partial t}+\partial_i E_k-
\partial_k E_i - (\partial_j E_j)\delta_{ik}=0,& $ $}
\end{eqnarray}
whose symmetric and antisymmetric parts give the usual four Maxwell's
equations. Let us mark the coincidence of these equations with Eqs.
on p.L34 of  Ref.~\cite{Sudbery} as well as with the system of equations (17)
in Ref.~\cite[p.76]{Jancewicz}:
\begin{eqnarray}
\cases{
\frac{\partial \hat{H}}{\partial t}+\vec\nabla\wedge\vec E - (\vec\nabla\vec
E)\delta_{ik}=0,& $ $\cr
\frac{\partial\hat{E}}{\partial t}-\vec\nabla\wedge\vec H+
(\vec\nabla\vec H)\delta_{ik}=0.& $ $}
\end{eqnarray}
Here hats above $E$ and $H$ designate volutors.

The use of the proposed Lagrangian (\ref{eq:Lagr}) gives the opportunity to
receive dynamical invariants:

1)The energy-momentum tensor has the following form:
\begin{equation}
T^{\mu\nu}_{\alpha}={\cal L}_{\alpha}\delta_{\mu\nu}+i\bar\Psi\gamma_
{\alpha\nu}\partial_{\mu}\Psi-i(\partial_{\mu}\bar\Psi)\gamma_{\alpha\nu}
\Psi.
\end{equation}

2)The angular momentum tensor is
\begin{eqnarray}
\lefteqn{{\cal M}^{\nu ,\mu\beta}_{\alpha}=x_{\mu}T^{\nu\beta}_{\alpha}-
x_{\beta}T^{\nu\mu}_{\alpha}+}\nonumber\\
&+&i\bar\Psi\gamma_{\alpha\nu}A^{\Psi}_{\mu\beta}\Psi-i\bar\Psi A^{\bar\Psi}
_{\mu\beta}\gamma_{\alpha\nu}\Psi
\end{eqnarray}
(with $A^{\Psi}_{\mu\beta}$ and $A^{\bar\Psi}_{\mu\beta}$ are the generators
of the Lorentz transformations).\\
And, finally,

3)the current tensor is equal to
\begin{equation}
J^{\mu}_{\alpha}=-2\bar\Psi\gamma_{\alpha\mu}\Psi
\end{equation}
and is obtained as a consequence of gradient transformations:
\begin{eqnarray}
\cases{
\Psi=e^{i\theta}\Psi,& $ $\cr
\bar\Psi=\bar\Psi e^{-i\theta}& $ $}
\end{eqnarray}
(where $\bar\Psi=\Psi^{+}\gamma_{44}$),
which correspond to the duality rotations:
\begin{eqnarray}
\cases{
F_{\mu\nu}\rightarrow F_{\mu\nu}cos\theta+^{*}F_{\mu\nu}sin\theta,& $ $\cr
^{*}F_{\mu\nu}\rightarrow - F_{\mu\nu}sin\theta+^{*}F_{\mu\nu}cos\theta,& $ $}
\end{eqnarray}
implemented by Sudbery~\cite{Sudbery}.

Considering Weinberg's spinor in accordance  with Eq. (\ref{eq:EH}) and
restricting
oneself by the first term of Lagrangian (\ref{eq:Lagr})
\footnote{It is possible because in terms of $\vec E$ and $\vec H$ the both of
Eqs. (\ref{eq:LagEul}), obtained from the first and second terms of
(\ref{eq:Lagr}), lead to the same motion equations. Moreover, the difference of
these Lagrangians is equal to the total derivative.}, we get the following
conserved quantities:
\begin{eqnarray}
T_{\{i}^{\quad 4\}4}&=&(\vec E\vec\nabla)\vec H-(\vec H\vec\nabla)\vec E+
\vec E(\nabla\vec H)-\vec H(\vec\nabla\vec E),\\
T_{\{4}^{\quad 4\}4}&=&\vec E [\vec\nabla\times\vec H]-\vec
H[\vec\nabla\times\vec E],\\
T_{\{i}^{\quad j\}4}&=&\vec\nabla\vee\left [\vec E\times\vec H\right ],\\
T_{[i}^{\quad 4]4}&=&-i[(\vec E\vec\nabla)\vec E+(\vec H\vec\nabla)\vec H+
\vec E(\vec\nabla\vec E)+\vec H(\vec\nabla\vec H)],\\
\tilde T_{i}\qquad&=&{1\over 2}\epsilon_{ijk}T_{[j}^{\quad k]4}=\left [(\vec
E\vec\nabla)\vec H-
(\vec H\vec\nabla)\vec E+\vec H(\vec\nabla\vec E)-\vec E(\vec\nabla\vec
H)\right ].
\end{eqnarray}

The value of $A^{\Psi}_{\mu\beta}$ is shown in~\cite{Sankaranarayanan}
to be $A^{\Psi}_{\mu\beta}=-{1 \over 6}\gamma_{5,\mu\beta}$ and,
correspondingly, $A^{\bar\Psi}_{\mu\beta}={1 \over 6}\gamma_{5,\mu\beta}$.
As opposed to ~\cite{Sudbery} we obtained
\begin{equation}
S^{4,ij}_{4}=0,
\end{equation}
but
\begin{equation}
S^{4,4i}_{4}=-4\left [\vec E\times\vec H\right ]_i.
\end{equation}
At last, we have the same expressions for $J^{\mu}_{\alpha}$ as in
Ref.~\cite{Sudbery}:
\begin{eqnarray}
J_{44}&=&-2(\vec E^2+\vec H^2),\\
J_{4i}&=&4i\epsilon_{ijk}E_j H_k,\\
J_{ij}&=&2[(\vec E^2+\vec H^2)\delta_{ij}-E_i E_j - H_i H_j],
\end{eqnarray}
which are the components of energy-momentum tensor in the common-used
formulation of QED.

Thus, the gauge transformations of the first kind lead to energy-momentum
conservation and the"charge" is identified with the energy density of the
field.

The scalar Lagrangian of the Joos-Weinberg's $2(2S+1)$-- theory was presented
in [12a, 14b] :
\begin{equation}\label{eq:Lagra}
{\cal
L}^{JW}=\partial_{\mu}\bar\Psi\gamma_{\mu\nu}\partial_\nu\Psi+m^2\bar\Psi\Psi.
\end{equation}

Let us note,  implying the interpretation of the Weinberg's 6-spinor  as in
(\ref{eq:EH}),
we can rewrite the Lagrangian (\ref{eq:Lagra}) as following:
\begin{equation}\label{eq:Lagran}
{\cal L}^{JW}=(\partial_\mu F_{\nu\alpha})(\partial_\mu F_{\nu\alpha}) -
2(\partial_\mu F_{\mu\alpha})(\partial_\nu F_{\nu\alpha}) + 2(\partial_\mu
F_{\nu\alpha})(\partial_\nu F_{\alpha\mu}).
\end{equation}
It leads to the Euler-Lagrange equation:
\begin{equation}
{\,\lower0.9pt\vbox{\hrule \hbox{\vrule height 0.2 cm \hskip 0.2 cm \vrule
height 0.2
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cm}\hrule}\,}F_{\alpha\beta}-2(\partial_{\beta}F_{\alpha\mu,\mu}-\partial_{\alpha}F_{\beta\mu,\mu})=0,
\end{equation}
where ${\,\lower0.9pt\vbox{\hrule \hbox{\vrule height 0.2 cm \hskip 0.2 cm
\vrule height 0.2 cm}\hrule}\,}=\partial_{\nu}\partial_{\nu}$.
The Lagrangian (\ref{eq:Lagran}) is found out here to be
equivalent to the Lagrangian of free massless skew-symmetric field
given in~\cite{Hayashi}\,\footnote{See also description of closed strings
on the base of this Lagrangian in~\cite{KalbRamond,Love}.}:
\begin{equation}\label{eq:LagHa}
{\cal L}^{H}=\frac{1}{8}F_k F_k,
\end{equation}
with $F_k=i\epsilon_{kjmn}F_{jm,n}$. It can be rewritten
\begin{eqnarray}
{\cal L}^{H}&=&{1\over 4}(\partial_{\mu} F_{\nu\alpha})(\partial_{\mu}
F_{\nu\alpha})-
{1\over 2} (\partial_{\mu} F_{\nu\alpha})(\partial_{\nu}
F_{\alpha\mu})=\nonumber\\
&=&-{1 \over 4}{\cal L}^{JW}-{1\over 2}(\partial_{\mu}
F_{\alpha\mu})(\partial_{\nu} F_{\alpha\nu}),
\end{eqnarray}
which confirms the above statement, taking into account the possibility of use
the Fermi method {\it mutatis mutandis} as in Ref.~\cite{Hayashi}. The second
term in (\ref{eq:Lagran}) can be excluded by means of generalized Lorentz
condition (which is just well-known Maxwell's motion equations)\footnote{Let us
mention some analogy with the potential formulation of QED. In some sense the
Lagrangian (\ref{eq:Lagran}) corresponds to the choice of
"gauge-fixing" parameter $\xi=-1$, ${\cal L}^{H}$  of  Ref.~\cite[formula
(5)]{Hayashi} corresponds to the "Landau gauge" ($\xi =0$), and ${\cal L}^{H}$
(formula (9) of cited paper) does to the "Feynman gauge"($\xi=1$).}.

In  turn the Lagrangian (\ref{eq:LagHa}) is invariant under new "gauge"
transformations:
\begin{equation}\label{eq:gauge}
F_{\mu\nu}\rightarrow
F_{\mu\nu}+A_{[\mu\nu]}=F_{\mu\nu}+\partial_{\nu}\Lambda_{\mu}-
\partial_{\mu}\Lambda_{\nu}
\end{equation}
The cited paper~\cite{Hayashi} proves the Lagrangian describes massless
particles
having the longitudinal physical components only. The transversal components
are removed
by means of the "gauge" transformation (\ref{eq:gauge}).

If we implement this "gauge" transformations to the Dirac "bivector"
\footnote{See Ref.~\cite[p.244]{Jancewicz} for discussion of Clifford algebra
in the Minkowsky space.}:
\begin{equation}
F\rightarrow F+ e_4\wedge A_{[4k]}e_k+{i \over 2} A_{[jk]} e_j\wedge
e_k=F+{1\over 2} A_{[\mu\nu]}e_{\mu}\wedge e_{\nu}
\end{equation}
we can obtain the same result, very surprising in the point of view of
Weinberg's theorem about connection between the helicity $\lambda$ and the
Lorentz group representation $(A,B)$,\, $B-A=\lambda$.

{\bf Acknowledgements.} I would like to express my sincere gratitude to V G
Kadyshevsky, R N Faustov, I G Kaplan, M Moreno, Yu F Smirnov, M Torres, Yu N
Tyukhtyaev,  E Bautista, S V Khudyakov, A B Klimov and  G Loyola for interest
in the works and most helpful discussions.
Also, I  greatly appreciate the technical assistance of A S Rodin and A Wong.

I am most grateful to Prof. A M Cetto, Head of  the Departamento de F\'{\i}sica
Te\'{o}rica at the IFUNAM,  for the creation of  excellent conditions
for work.

Finally, it  should be mentioned that this work has been financially supported
by the CONACYT ( Mexico ) under contract No. 920193.

\newpage
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\end{document}








