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\title{On the existence of additional solutions
of the equations in~the~$(1/2,0)\oplus (0,1/2)$ representation
space\thanks{Submitted to ``Fizika B"}}
\author{{\bf Valeri V. Dvoeglazov}}
\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}
%\date{July 15, 1996}
\maketitle
\bigskip
\begin{abstract}
We analyze dispersion relations of the equations recently proposed
by Ahluwalia for describing neutrino. Equations for type-II spinors are
deduced on the basis of the Wigner rules for left- and right- 2-spinors
and the Ryder-Burgard relation. It is shown that equations contain
acausal solutions which are similar to those of the Dirac-like
second-order equation. The latter is obtained in a similar way, provided
that we do not apply to any constraints in the
calculation process.
\end{abstract}
\pacs{PACS numbers: 03.65.Pm, 12.90.+b}
%\newpage
Recently, Ahluwalia proposed a new wave equation for describing
self/anti-self charge conjugate states $\lambda^{S,A} (p^\mu)$
of any spin~\cite{DVA96}:
\begin{eqnarray}
&&{\cal D} \lambda (p^\mu) =\label{genweq1}\\
&&\pmatrix{-\,\openone & \zeta_\lambda\,\exp\left(
{\bf J}\,\cdot \bbox{\varphi}\right )
\,\Theta_{[j]}\,\mit{\Xi}_{[j]}\, \exp\left( {\bf J}\,\cdot
\bbox{\varphi} \right )\cr \zeta_\lambda\,\exp\left(-\, {\bf
J}\,\cdot\bbox{\varphi}\right)
\,\mit{\Xi}^{-1}_{[j]}\,\Theta_{[j]}\, \exp\left(- \,{\bf
J}\,\cdot\bbox{\varphi} \right) & -\,\openone}\,\lambda
(p^\mu)\,=\,0\,.\nonumber
\end{eqnarray}
Analogous equations for $\rho^{S,A} (p^\mu)$ bispinors have been
derived in ref.~[2d]. In the $j=1/2$ case spin matrices ${\bf
J}$ are chosen to be the Pauli matrices $\bbox{\sigma}/2$; in the
$j=1$ case, the Barut-Muzinich-Williams matrices;
$\bbox{\varphi}$ are the parameters of the Lorentz boost. The notation
coincides with that of refs.~\cite{DVA96,DVO95a}.
$\Theta_{[j]}$ is the Wigner's operator defined as
$(\Theta_{[j]})_{\sigma , \, {\sigma^\prime}} = (-1)^{j+\sigma}
\delta_{\sigma^\prime , \, -\sigma}$ with $\sigma$ and $\sigma^\prime$
as eigenvalues of ${\bf J}$. $\mit{\Xi}_{[j]}$ is a $(2j+1)\times
(2j+1)$
matrix which connects the ``$2j+1$ spinor" and its complex conjugate
such
that $ [\phi_{_L}^{\sigma} (\overcirc{p}^\mu )]^\ast =
\mit{\Xi}_{[j]} \phi_{_L}^{\sigma} (\overcirc{p}^\mu)$. And
$\zeta_\lambda$ are phase factors which are fixed by conditions of
self/anti-self charge conjugacy imposed on the type-II spinors
$\lambda$.
While formally the $j=1/2$ equation ``may be put in the form
$(\Gamma^{\mu\nu} p_\mu p_\nu +m\Gamma^\mu p_\mu -2 m^2 \openone)
\lambda
(p^\mu)=0$ ... it turns out that $\Gamma^{\mu\nu}$ and $\Gamma^\mu$ do
not
transform as Poincar\'e tensors." Other forms of neutrino equations have
been presented in refs.~\cite{Ziino,DVO95a,DVO95b} and gauge
interactions
have been introduced there.\footnote{The question of equivalence of
these
equations still deserves further elaboration and this paper presents a
certain part of this analysis.} These constructs give alternative
insights
in neutrino dynamics, which is different from that based on the
common-used Weyl massless equations. Indications that neutrino may not
be
a Dirac particle and may have different dynamical features have appeared
in analyses of the present experimental situation~\cite{NO}. Earlier
considerations of this problem can be found in
refs.~\cite{Maj,Markov,MLC,Fush}.
Both the equations (\ref{genweq1}) and the equations of
ref.~\cite{DVO95a,DVO95b,DVO95c} have been obtained by using different
forms of the Ryder-Burgard
relation~\cite{RB,BWW,DVA96,DVO95a,DVO95b,DVO95c} which connects
zero-momentum $(0,j)$ left- and $(j,0)$ right- spinors, and the Wigner
rules for their transformations to the frame with the momentum ${\bf
p}$.
The Dirac equation may also be obtained in such
a way~\cite[footnote \# 1]{DVA96}. The detailed
discussion of this techniques can be found in~\cite{DVO95d}.
It was claimed in ref.~\cite{DVA96} that $\lambda^S (p^\mu)$ spinors
answer for ``{\it positive} energy solutions, \ldots [meanwhile],
$\lambda^A (p^\mu)$ are the {\it negative} energy solutions". We, in
fact, used this interpretation in~\cite{DVO95a}. Let us now check by
straightforward calculations, which dispersion relations has the
equation
(\ref{genweq1}) in the case of $j=1/2$? Rewriting it to the form (31) of
ref.~\cite{DVA96} yields the equation of the second order in $p_0$ and
the matrix in the left side has the dimension four. So, one should have
eight solutions. The system for analytical calculation MATEMATICA
2.2 enables us to found that the determinant of the matrix ${\cal D}$ is
equal to
\begin{equation} \mbox{Det} \left [{\cal D} \right ]=\left ( p_0^2 -
p_1^2
- p_2^2 - p_3^2 -m^2 \right )^2 \frac{( p_0^2 -p_1^2 -p_2^2 -p_3^2 +3m^2
+4mp_0 )^2}{16m^4 (p_0 +m)^4} \quad.
\end{equation}
As a result of
equating the determinant to zero we deduce that, indeed, the equation
(\ref{genweq1}) has eight solutions in total with
\begin{equation} p_0 =
\pm \sqrt{{\bf p}^{\,2} +m^2}\quad,
\end{equation}
each two times; and
with the acausal dispersion relations: \begin{equation} p_0 = -2m \mp
\sqrt{{\bf p}^{\,2} +m^2}\quad,\label{adr2} \end{equation} each two
times.
We come across the same situation when deriving the Dirac
equation by the Ryder-Burgard-Ahluwalia technique provided that we do
{\it
not} apply to the constraint $p_0^{\,2} -{\bf p}^{\,2} =m^2$ from the
beginning. Indeed, one has
\begin{mathletters} \begin{eqnarray}
\Lambda_{_R} (p^\mu \leftarrow \overcirc{p}^\mu) \Lambda_{_L}^{-1}
(p^\mu
\leftarrow \overcirc{p}^\mu) &=& \frac{p_0^2 + 2mp_0 + {\bf p}^2 +m^2 +
2(p_0+m) (\bbox{\sigma}\cdot {\bf p})}{2m(p_0+m)}\quad,\\ \Lambda_{_L}
(p^\mu \leftarrow \overcirc{p}^\mu) \Lambda_{_R}^{-1} (p^\mu \leftarrow
\overcirc{p}^\mu) &=& \frac{p_0^2 + 2mp_0 + {\bf p}^2 +m^2 - 2(p_0+m)
(\bbox{\sigma}\cdot {\bf p})}{2m(p_0+m)}\quad. \end{eqnarray}
\end{mathletters}
Thus, the second-order momentum-representation ``Dirac" equation can be
written:
\begin{equation}
{1\over 2m(p_0+m)} \left [ (\gamma^\mu p_\mu
\mp m)\gamma^0 +2m \right ] (\gamma^\nu p_\nu \mp m) \Psi_{\pm} (p^\mu)
=0
\quad,\label{deq1}
\end{equation}
or
\begin{equation}
{1\over 2m(p_0+m)} (\gamma^\nu p_\nu \mp m) \left [ \gamma^0
(\gamma^\mu p_\mu
\mp m) +2m \right ] \Psi_{\pm} (p^\mu) =0\quad.\label{deq2}
\end{equation}
The corresponding coordinate-representation of these equations
($m\neq 0$ and $p_0 \neq -m$) is
\begin{equation}
\left [ (i\gamma^\mu \partial_\mu
-m)\gamma^0 + 2\wp_{u,v} m \right ] (i\gamma^\nu \partial_\nu - m) \Psi
(x^\mu) =0\quad,
\end{equation}
or
\begin{equation}
(i\gamma^\mu \partial_\mu -m)
\left [ \gamma^0 (i\gamma^\mu \partial_\mu
-m) +2\wp_{u,v} m \right ] \Psi (x^\mu) =0\quad,
\end{equation}
where $\wp_{u,v} =\pm 1$ depending on what solutions, with either
positive
or negative energies, are considered.
Can the equation (\ref{genweq1}) be put in a more convenient form? The
eight-component form, we proposed recently~[2d,Eqs.(17,18)],
does not have acausal solutions. In the process of its deriving we
assumed
certain relations\footnote{See, {\it e.g.}, formulas (48) of
ref.~\cite{DVA96}.} between $\lambda^{S,A} (p^\mu)$ and $\rho^{S,A}
(p^\mu)$. In the present article we are not going to apply them.
Following the procedure of deriving the equations
(\ref{deq1},\ref{deq2})
one can arrive at a rather complicated equation:
\begin{eqnarray}
\lefteqn{{1\over 4m (p_0 +m)}
\left \{ (\gamma^\mu p_\mu + m\gamma^0)\left [ {\cal S}
(\gamma^\nu p_\nu +m\gamma^0)
-2m\gamma^0\right ] + \right.}\nonumber\\ &+&\left. \left [
(\gamma^\mu p_\mu
+m\gamma^0) {\cal S} -2m\gamma^0 \right ] (
\gamma^\nu p_\nu +m\gamma^0 )\right \}
\lambda^{S,A} (p^\mu) =0 \quad,\label{td}
\end{eqnarray}
where
\begin{equation} {\cal S} =
\pmatrix{0 & \zeta_\lambda \Theta \Xi\cr \zeta_\lambda \Xi^{-1} \Theta &
0\cr}\quad.
\end{equation}
But, as mentioned in~\cite{DVA96}, one may
consider that $\phi$, which presents in the generalized Ryder-Burgard
relation (see Eq. (27) of ref.~\cite{DVA96} or Eq. (38) in~[2c]), is
the
azimuthal angle {\it associated with} ${\bf p}$, the 3-momentum of a
particle. In this case one can find commutation relations between $\hat
p
\equiv \gamma^\mu p_\mu$, matrices $\gamma^5$, $\gamma^0$ and ${\cal
S}$.
\begin{equation}
\left [ \hat p , {\cal S} \right
]_{-} =0\quad,\quad \left [\gamma^0 , {\cal S}\right ]_- =0\quad,\quad
\left [ \gamma^5 , {\cal S} \right ]_+ = 0\quad, \label{cr}
\end{equation}
and
\begin{equation} {\cal S}
\lambda^{S,A} (p^\mu) = \lambda^{S,A} (p^\mu) \quad,\label{sa}
\end{equation}
because
in this case
\begin{equation}
\Lambda_{_{L,R}}^\ast = \mit{\Xi}
\Lambda_{_{L,R}} \mit{\Xi}^{-1}\quad.
\end{equation}
We finally arrive at
\begin{equation} \left [\hat p^2 -m^2 \right ] \openone_{4\times
4}\,\,\lambda^{S,A} (p^\mu) = 0\quad,
\end{equation}
{\it i.e.}, at the
Klein-Gordon equation for each component of $\lambda^{S,A} (p^\mu)$. Why
did acausal solutions fall out? It appears bispinors $\lambda^A
(p^\mu) \equiv -\gamma^5 \lambda^{S} (p^\mu)$ can satisfy the
positive-energy equation ($\zeta_\lambda =i$) and bispinors
$\lambda^S \equiv -\gamma^5 \lambda^A (p^\mu)$, the negative-energy one
($\zeta_\lambda =-i$), but dispersion relations
will be acausal, Eq. (\ref{adr2}), in this unusual case.\footnote{In
the process of the proof one should take into account commutation
relations (\ref{cr}) and hence that ${\cal S}^{^+}
\gamma^5 \lambda^S (p^\mu) = \lambda^A (p^\mu)$ and ${\cal S}^{^-}
\gamma^5 \lambda^A (p^\mu) = \lambda^S (p^\mu)$.} So, assuming that
in the equations (\ref{td}) one should take $\zeta_\lambda
=i$ for describing $\lambda^S$ and $\zeta_\lambda = -i$, for $\lambda^A$
we, in fact, implicitly impose mass-shell constraints.
The same situation is for the equations (\ref{deq1},\ref{deq2}),
$u (p^\mu)$ and $v (p^\mu)\equiv \gamma^5 u (p^\mu)$ can satisfy both
the
positive- and the negative-energy equations, but the dispersion
relations
could be unusual.
>From a mathematical viewpoint the origin of appearance of these
solutions seems to be related with the properties
of the Lorentz transformation operators with respect to
herimitian conjugation operation, see~\cite[p.404]{DVAT} for
discussion.
One should further note that the problem of acausal solutions have
interrelations with a mathematical possible situation when operators of
the
continuous Lorentz transformations are combined with other
transformations of the Poincar\'e group to give $\Lambda_{_R} = -
\Lambda_{_L}^{-1}$. Thus, the question, whether these solutions would
have some physical significance, should be solved on the basis of the
rigorous analysis of the general structure of the Poincar\'e
transformation group and of the experimental situation in neutrino
physics.
Finally, let us mention that another second-order equation in the
$(1/2,0)\oplus (0,1/2)$ representation space has been investigated
in~\cite{Barut} and relations with the problem of the lepton mass
spectrum have been revealed (see also~\cite{Markov,Nigam}).
\medskip
{\it Acknowledgments.}
I acknowledge the help of Profs. D. V. Ahluwalia, A. E. Chubykalo,
M. W. Evans, A.~F. Pashkov and S. Roy. I am grateful to Zacatecas
University, M\'exico, for a professor position.
This work has been partially supported by the Mexican Sistema
Nacional de Investigadores, the Programa de Apoyo a la Carrera Docente
and by the CONACyT, M\'exico under the research project 0270P-E.
\bigskip
\bigskip
\bigskip
\begin{references}
\footnotesize{
\baselineskip13pt
\bibitem{DVA96} D. V. Ahluwalia, Int. J. Mod. Phys. {\bf 11} (1996) 1855
\bibitem{DVO95a} V. V. Dvoeglazov, Rev. Mex. Fis. Suppl. {\bf 41}
(1995) 159; Bol. Soc. Mex. Fis. Suppl. {\bf 9} (1995) 28;
Int. J. Theor. Phys. {\bf 34} (1995) 2467; Nuovo Cim. {\bf
108}A (1995) 1467
\bibitem{Ziino} G. Ziino, Ann. Fond. L. de Broglie {\bf 14} (1989) 427;
ibid {\bf 16} (1991) 343; Int. J. Mod. Phys. {\bf 11} (1996) 2081;
A. O. Barut and G. Ziino, Mod. Phys. Lett.
A{\bf 8} (1993) 1011
\bibitem{DVO95b} V. V. Dvoeglazov, Nuovo Cim. {\bf 111}B (1996) 483
\bibitem{NO} C. Athanassopoulos {\it et al.}, Phys. Rev. Lett. {\bf 75}
(1995) 2650; S. M. Bilen'ky {\it et al.}, Phys. Lett. B{\bf 356} (1995)
273
\bibitem{Maj} E. Majorana, Nuovo Cim. {\bf 14} (1937) 171
\bibitem{Markov} M. A. Markov, ZhETF {\bf 7} (1937) 579, 603;
Preprint JINR D-1345, Dubna, 1963
\bibitem{MLC} J. A. McLennan, Phys. Rev. {\bf 106} (1957) 821;
K. M. Case, Phys. Rev. {\bf 107} (1957) 307
\bibitem{Fush} N. D. Sen Gupta, Nucl. Phys.
B{\bf 4} (1967) 147; Z. Tokuoka, Prog. Theor. Phys. {\bf 37} (1967)
603; V. I. Fushchich and A. L. Grishchenko, Lett. Nuovo Cim. {\bf
4}
(1970) 927; V. I. Fushchich, Nucl. Phys. B{\bf 21} (1970) 321; Theor.
Math. Phys. {\bf 7} (1971) 3; Lett. Nuovo Cim. {\bf 4} (1972) 344;
M.
T. Simon, ibid {\bf 2} (1971) 616; T. S. Santhanam and A. R.
Tekumala,
ibid {\bf 3} (1972) 190; M. Seetharaman, M. T. Simon and P. M.
Mathews,
Nuovo Cim. {\bf 12}A (1972) 788
\bibitem{DVO95c} V. V. Dvoeglazov, Hadronic J. Suppl. {\bf 10} (1995)
349
\bibitem{RB} L. H. Ryder, {\it Quantum Field Theory.} (Cambridge
University Press, 1987)
\bibitem{BWW} D. V. Ahluwalia, M. B. Johnson and T. Goldman, Phys. Lett.
B{\bf 316} (1993) 102
\bibitem{DVO95d} V. V. Dvoeglazov, {\it De Dirac a Maxwell: Un Camino
Con
Grupo de Lorentz.} Investigaci\'on Cient\'{\i}fica, in press
\bibitem{DVAT} D. V. Ahluwalia and D. J. Ernst, Int. J. Mod. Phys. E{\bf
2} (1993) 397
\bibitem{Barut} A. O. Barut, Phys. Lett. {\bf 73}B (1978) 310
\bibitem{Nigam} R. Acharya and B. P. Nigam, Lett. Nuovo Cim. {\bf 31}
(1981) 437; B. P. Nigam, J. Phys. G{\bf 16} (1990) 1553
}
\end{references}
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