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\begin{document}
\title{Self/Anti-Self Charge Conjugate
States \linebreak for $j=1/2$ and $j=1$\thanks{Presented at the XXXIX
Congreso Nacional de F\'{\i}sica, Oaxaca, Oax., M\'exico, Oct. 14-18,
1996, the Primeras Jornadas de Investigaci\'on, Zacatecas, Nov. 18-19,
1996 and the V Reunion Anual de DGFM de la Sociedad Mexicana de
F\'{\i}sica, M\'exico, Apr. 17-18, 1997}}
\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\\
Email: VALERI@CANTERA.REDUAZ.MX\\
URL: http://cantera.reduaz.mx/\~\,valeri}
%\date{April 18, 1997}
\date{Received \qquad\qquad 1997; revised\qquad\qquad 1997}
\maketitle
\begin{abstract}
\baselineskip14pt
We briefly review recent achievements in the theory of neutral particles
(the Majorana-McLennan-Case-Ahluwalia construct for self/anti-self
charge conjugate states for $j=1/2$ and $j=1$ cases). Among new results
we
present a theoretical construct in which a fermion and an antifermion
have the same intrinsic parity; discuss phase transformations
and find relations between the Majorana-like field operator
$\nu$, given by Ahluwalia, and the Dirac field operator. Also we
give explicit forms of the $j=1$ ``spinors" in the Majorana
representation.
\end{abstract}
\pacs{11.30.Er, 12.10.Dm, 12.60.-i, 14.60.St}
\newpage
\baselineskip15pt
%\large{
The construct for self/anti-self charge conjugate states defined in the
momentum representation has been proposed in refs.~\cite{DVA,DVO}.
This is the straightforward development of the Majorana ideas~\cite{MAJ}
and the ideas of McLennan~\cite{MCL} and Case~\cite{Case}.
Let us present the previous results:
\begin{itemize}
\item
Self/anti-self charge conjugate spinors have been defined in the
$(1/2,0)\oplus (0,1/2)$ representation in the momentum space~[1c]:
\begin{eqnarray}
\lambda^{S,A} = \pmatrix{\pm i\Theta \phi_L^\ast (p^\mu)\cr
\phi_L (p^\mu)\cr}\quad,\quad
\rho^{S,A} = \pmatrix{\phi_R (p^\mu)\cr \mp i\Theta \phi_R^\ast
(p^\mu)\cr}
\quad
\end{eqnarray}
and have been named as the type-II spinors.
They are eigenstates of the charge conjugation operator:
\begin{eqnarray}
S^c_{[1/2]} = e^{i\theta_c} \pmatrix{0 & i\Theta\cr
-i\Theta & 0\cr} {\cal K}\quad,\quad \Theta \equiv -i\sigma_2 =
\pmatrix{0&-1\cr
1&0}\quad;
\end{eqnarray}
\begin{mathletters}
\begin{eqnarray}
S_{[1/2]}^c \lambda^{S,A} (p^\mu) &=& \pm \lambda^{S,A} (p^\mu)\quad,\\
S_{[1/2]}^c \rho^{S,A} (p^\mu) &=& \pm \rho^{S,A} (p^\mu)\quad.
\end{eqnarray}
\end{mathletters}
Similar (to a certain extent) states can be constructed in the higher
representations of the Lorentz group, e.g., in the $(j,0)\oplus (0,j)$
representation, $j>1/2$.
\item
The field operator
\begin{eqnarray}
\nu^{DL} (x^\mu) &=& \sum_\eta \int \frac{d^3 {\bf p}}{(2\pi)^3}
{1\over 2E_p} \left [ \lambda^S_\eta (p^\mu) a_\eta (p^\mu) \exp
(-ip\cdot
x) \right.\nonumber\\
&+&\left.\lambda^A_\eta (p^\mu) b_\eta^\dagger (p^\mu) \exp
(+ip\cdot x)\right ]\quad
\end{eqnarray}
has been proposed for this sort of states~[1c].
\item
$\lambda$ and $\rho$ spinors are {\it not} eigenspinors of the
$(j,0)\oplus (0,j)$ helicity operator
\begin{equation}
h= \pmatrix{{\bf J}\cdot \hat {\bf n} &0\cr
0& {\bf J}\cdot \hat {\bf n}\cr}
\end{equation}
(by the definition, indeed, because $\Theta_{[j]}
{\bf J} \Theta^{-1}_{[j]} = -{\bf J}^\ast$). The new quantum
number ({\it chiral helicity}) corresponding to the operator
$\eta=-\gamma^5 h$ has been introduced.
\item
$\lambda$ and $\rho$ spinors are {\it not} eigenspinors of the parity
operator, see formulas (36a,b) in ref.~[1c]. ``This is not related to
the
fact that $S_{[1/2]}^c$ and $S_{[1/2]}^s$ do not commute. Since
$S_{[1/2]}^c$ is {\it not} linear, it is possible to have a simultaneous
set of eigenspinors, but such a set does not have its eigenspinors of
type-II", in the opinion of D. V. Ahluwalia, ref.~[1c].
\item
The introduction of the interaction in the usual manner
(``covariantization" $\partial_\mu \rightarrow \nabla_\mu = \partial_\mu
-ieA_\mu$) was found to be impossible for these states because phase
transformations which correspond to this ``covariantization" would lead
to the consequence that the spinors would {\it not} keep their property
to
be self/anti-self charge conjugate spinors.
\item
Simple dynamical equations for $\lambda$ and $\rho$ spinors have been
obtained~[2d] on the basis of a new form of the Ryder-Burgard relation
(which connects the left- and right- parts of the bispinors in the frame
with zero momentum~\cite{Ryder,DVA1}). Here they are:
\begin{mathletters}
\begin{eqnarray}
i\gamma^\mu \partial_\mu \lambda^S (x) - m\rho^A (x) &=& 0\quad,\\
i\gamma^\mu \partial_\mu \rho^A (x) - m\lambda^S (x) &=& 0\quad,\\
i\gamma^\mu \partial_\mu \lambda^A (x) + m\rho^S (x) &=& 0\quad,\\
i\gamma^\mu \partial_\mu \rho^S (x) + m\lambda^A (x) &=& 0\quad.
\end{eqnarray}
\end{mathletters}
In fact they can be written in the eight-component form, see
also the old works~\cite{Markov,Belin} and the recent
works~\cite{Ziino,Robson}.
\item
The connection with the Dirac spinors has been found~[2a,b]. For
instance,
\begin{eqnarray}
\pmatrix{\lambda^S_\uparrow (p^\mu) \cr \lambda^S_\downarrow (p^\mu) \cr
\lambda^A_\uparrow (p^\mu) \cr \lambda^A_\downarrow (p^\mu)\cr} =
{1\over
2} \pmatrix{1 & i & -1 & i\cr -i & 1 & -i & -1\cr 1 & -i & -1 & -i\cr i&
1& i& -1\cr} \pmatrix{u_{+1/2} (p^\mu) \cr u_{-1/2} (p^\mu) \cr
v_{+1/2} (p^\mu) \cr v_{-1/2} (p^\mu)\cr}\quad.\label{connect}
\end{eqnarray}
See also ref.~\cite{Ziino}.
\item
The sets of $\lambda$ spinors and of $\rho$ spinors are claimed~[1c] to
be
{\it bi-orthonormal} sets each in the mathematical sense, provided
that overall phase factors of 2-spinors $\theta_1 +\theta_2 = 0$ or
$\pi$.
For instance, on the classical level $\bar \lambda^S_\uparrow
\lambda^S_\downarrow = 2iN^2 \cos ( \theta_1 + \theta_2 )$.
Corresponding commutation relations for this
type of states have also been proposed.
\item
The Lagrangian for $\lambda$- and $\rho$-type $j=1/2$ states was
given~[2d,formula(24)].
\item
While in the massive case there are four $\lambda$-type spinors, two
$\lambda^S$ and two $\lambda^A$ (the $\rho$ spinors are connected by
certain relations with the $\lambda$ spinors for any spin case), in a
massless case $\lambda^S_\uparrow$ and $\lambda^A_\uparrow$ identically
vanish, provided that one takes into account that $\phi_L^{\pm 1/2}$ are
eigenspinors of ${\bbox \sigma}\cdot \hat {\bf n}$, the
$2\times 2$ helicity operator.
\item
The possibility of the generalization of the concept of the
Fock space was noted. It leads to ``doubling" the Fock
space~\cite{Ziino}.
\item
There does not exist the self/anti-self charge conjugate ``spinors" in
the
$(1,0) \oplus (0,1)$ representation. Therefore, $\Gamma^5 S^c_{[1]}$
self/anti-self conjugate objects have been defined there.
\item
The {\it commutator} of the operations $U^s_{[1/2]}$ and $U^c_{[1/2]}$
in
the Fock space may be equal to zero when acting on the Majorana states.
The parity operator of the Fock space is the function of the charge
operator~\cite{Nigam}.
\item
Several explicit constructs of the Bargmann-Wightman-Wigner-type
theories~\cite{BWW} have been presented in~\cite{DVA1,DVA,DVO,Ziino}.
\end{itemize}
We continue this research in the area of the physics of neutral
particles
because the present-day standard models do not provide any adequate
formalism for describing neutrino and photon. Among new results we now
present:
\begin{itemize}
\item
It is shown that the covariant derivative (and, hence, the
interaction) can be introduced in this construct in the following way:
\begin{equation}
\partial_\mu \rightarrow \nabla_\mu = \partial_\mu - ig \L^5 A_\mu\quad,
\end{equation}
where $\L^5 = \mbox{diag} (\gamma^5 \quad -\gamma^5)$, the $8\times 8$
matrix. With respect to the transformations
\begin{mathletters}
\begin{eqnarray}
\lambda^\prime (x)
\rightarrow (\cos \alpha -i\gamma^5 \sin\alpha) \lambda
(x)\quad,\label{g10}\\
\overline \lambda^{\,\prime} (x) \rightarrow
\overline \lambda (x) (\cos \alpha - i\gamma^5
\sin\alpha)\quad,\label{g20}\\
\rho^\prime (x) \rightarrow (\cos \alpha +
i\gamma^5 \sin\alpha) \rho (x) \quad,\label{g30}\\
\overline \rho^{\,\prime} (x) \rightarrow \overline \rho (x)
(\cos \alpha + i\gamma^5 \sin\alpha)\quad\label{g40}
\end{eqnarray}
\end{mathletters}
the spinors retain their properties to be self/anti-self charge
conjugate
spinors and the proposed Lagrangian~[2d, p. 1472] remains to be
invariant.
This tells us that while self/anti-self charge conjugate states has
zero eigenvalues of the ordinary (scalar) charge operator but they can
possess the axial charge (cf. with the discussion of~\cite{Ziino} and
the old idea of R. E. Marshak).
In fact, from this consideration one can recover the Feynman-Gell-Mann
equation (and its charge-conjugate equation). It is re-written in the
two-component form
\begin{eqnarray} \cases{\left [\pi_\mu^- \pi^{\mu\,-}
-m^2 -{g\over 2} \sigma^{\mu\nu} F_{\mu\nu} \right ] \chi (x)=0\quad,
&\cr
\left [\pi_\mu^+ \pi^{\mu\,+} -m^2
+{g\over 2} \widetilde\sigma^{\mu\nu} F_{\mu\nu} \right ] \phi (x)
=0\quad, &\cr}\label{iii}
\end{eqnarray}
where already one has $\pi_\mu^\pm =
i\partial_\mu \pm gA_\mu$, \, $\sigma^{0i} = -\widetilde\sigma^{0i} =
i\sigma^i$, $\sigma^{ij} = \widetilde\sigma^{ij} = \epsilon_{ijk}
\sigma^k$ and $\nu^{^{DL}} (x) =\mbox{column} (\chi \quad \phi )$.
\item
Next, because the transformations
\begin{mathletters}
\begin{eqnarray}
\lambda_S^\prime (p^\mu) &=& \pmatrix{\Xi &0\cr 0&\Xi} \lambda_S (p^\mu)
\equiv \lambda_A^\ast (p^\mu)\quad,\quad\\
\lambda_S^{\prime\prime} (p^\mu) &=& \pmatrix{i\Xi &0\cr 0&-i\Xi}
\lambda_S
(p^\mu) \equiv -i\lambda_S^\ast (p^\mu)\quad,\quad\\
\lambda_S^{\prime\prime\prime} (p^\mu) &=& \pmatrix{0& i\Xi\cr
i\Xi &0\cr} \lambda_S (p^\mu) \equiv i\gamma^0 \lambda_A^\ast
(p^\mu)\quad,\quad\\
\lambda_S^{IV} (p^\mu) &=& \pmatrix{0& \Xi\cr
-\Xi&0\cr} \lambda_S (p^\mu) \equiv \gamma^0\lambda_S^\ast
(p^\mu)\quad
\end{eqnarray}
\end{mathletters}
with the $2\times 2$ matrix $\Xi$ defined as ($\phi$ is the azimuthal
angle related to ${\bf p} \rightarrow {\bf 0}$)
\begin{equation}
\Xi = \pmatrix{e^{i\phi} & 0\cr 0 &
e^{-i\phi}\cr}\quad,\quad \Xi \Lambda_{R,L} (\overcirc{p}^\mu \leftarrow
p^\mu) \Xi^{-1} = \Lambda_{R,L}^\ast (\overcirc{p}^\mu \leftarrow
p^\mu)\,\,\, ,
\end{equation}
and corresponding transformations for
$\lambda^A$ do {\it not} change the properties of bispinors to be in the
self/anti-self charge conjugate spaces, the Majorana-like field operator
($b^\dagger \equiv a^\dagger$) admits additional phase (and, in general,
normalization) transformations:
\begin{equation} \nu^{ML\,\,\prime}
(x^\mu) = \left [ c_0 + i({\bbox \tau}\cdot {\bf c}) \right
]\nu^{ML\,\,\dagger} (x^\mu) \quad, \end{equation} where $c_\alpha$ are
arbitrary parameters. The ${\bbox \tau}$ matrices are defined over the
field of $2\times 2$ matrices\footnote{\baselineskip14pt
This concept is closely related
with the Wigner's concept of the {\it sign} spin, which was discussed
recently by M. Moshinsky~\cite{Mosh}. In general, this notation was used
extensively in the earlier works of many researchers.} and the Hermitian
conjugation operation is assumed to act on the $c$- numbers as the
complex
conjugation. One can parametrize $c_0 = \cos\phi$ and ${\bf c} = {\bf n}
\sin\phi$ and, thus, define the $SU(2)$ group of phase transformations.
One can select the Lagrangian which is composed from the both field
operators (with $\lambda$ spinors and $\rho$ spinors)
and which remains to be
invariant with respect to this kind of transformations. The conclusion
is: it is permitted a non-Abelian construct which is based on
the spinors of the Lorentz group only (cf. with the old ideas of T. W.
Kibble and R. Utiyama) . This is not surprising because both the
$SU(2)$
group and $U(1)$ group are the sub-groups of the extended Poincar\'e
group
(cf.~\cite{Ryder}). Another non-Abelian model was proposed in the
$(1,0)\oplus (0,1)\oplus (1/2,1/2)$ by T. Barrett (e.g.,
ref.~\cite{Barrett}) and, recently,
by M. W. Evans and
J.-P. Vigier~\cite{Evans}.\footnote{\baselineskip14pt
The Evans-Vigier
construct got strong criticism, which is mainly reasonable. I want
to add that one can find algebraic errors in their works.
Nevertheless, nobody seems to be able to doubt the ${\bf B}$- cyclic
relations.}
\item
The new construct has been presented in which the fermion and its
antifermion may have the same intrinsic parities~[1c,2f]. We can deduce
the following properties of creation (annihilation) operators in the
Fock
space: \begin{mathletters} \begin{eqnarray} U^s_{[1/2]} a_\uparrow
({\bf
p}) (U^s_{[1/2]})^{-1} &=& - ia_\downarrow (- {\bf p})\, ,\quad
U^s_{[1/2]} a_\downarrow ({\bf p}) (U^s_{[1/2]})^{-1} = + ia_\uparrow
(- {\bf p})\, ,\nonumber\\
&&\\
U^s_{[1/2]} b_\uparrow^\dagger ({\bf p}) (U^s_{[1/2]})^{-1} &=&
+ i b_\downarrow^\dagger (- {\bf p})\, ,\quad
U^s_{[1/2]} b_\downarrow^\dagger ({\bf p}) (U^s_{[1/2]})^{-1} =
- i b_\uparrow (- {\bf p})\, ,\nonumber\\
&&
\end{eqnarray} \end{mathletters}
what signifies that the states created by the operators $a^\dagger
({\bf p})$ and $b^\dagger ({\bf p})$ have very different properties
with respect to the space inversion operation, comparing to
the Dirac states ($\pm$ stand for denoting the positive- (negative)
energy states). Namely,
\begin{mathletters} \begin{eqnarray} U^s_{[1/2]} \vert {\bf
p},\,\uparrow >^+ &=& + i \vert -{\bf p},\, \downarrow >^+\, ,\quad
U^s_{[1/2]} \vert {\bf p},\,\uparrow >^- = + i
\vert -{\bf p},\, \downarrow >^-\, ,\\
U^s_{[1/2]} \vert {\bf p},\,\downarrow >^+ &=& - i \vert -{\bf p},\,
\uparrow >^+\, ,\quad
U^s_{[1/2]} \vert {\bf p},\,\downarrow >^- = - i
\vert -{\bf p},\, \uparrow >^-\, .
\end{eqnarray}
\end{mathletters}
For the charge conjugation operation in the Fock space we have
two physically different possibilities. The first one
\begin{mathletters}
\begin{eqnarray}
U^c_{[1/2]} a_\uparrow ({\bf p}) (U^c_{[1/2]})^{-1} &=& + b_\uparrow
({\bf p})\, ,\quad
U^c_{[1/2]} a_\downarrow ({\bf p}) (U^c_{[1/2]})^{-1} = + b_\downarrow
({\bf p})\, ,\\
U^c_{[1/2]} b_\uparrow^\dagger ({\bf p}) (U^c_{[1/2]})^{-1} &=&
-a_\uparrow^\dagger ({\bf p})\, ,\quad
U^c_{[1/2]} b_\downarrow^\dagger ({\bf p})
(U^c_{[1/2]})^{-1} = -a_\downarrow^\dagger ({\bf p})\,
\end{eqnarray}
\end{mathletters}
is, in fact, reminiscent with the Dirac construct.
The action of this operator on the physical states are
\begin{mathletters}
\begin{eqnarray}
U^c_{[1/2]} \vert {\bf p}, \, \uparrow >^+ &=& + \,\vert {\bf p},\,
\uparrow >^- \quad,\quad
U^c_{[1/2]} \vert {\bf p}, \, \downarrow >^+ = + \, \vert {\bf p},\,
\downarrow >^- \quad,\quad\\
U^c_{[1/2]} \vert {\bf p}, \, \uparrow >^-
&=& - \, \vert {\bf p},\, \uparrow >^+ \quad,\quad
U^c_{[1/2]} \vert
{\bf p}, \, \downarrow >^- = - \, \vert {\bf p},\, \downarrow >^+ \quad.
\end{eqnarray} \end{mathletters}
But, one can also build the charge conjugation operator in the
Fock space which acts, {\it e.g.}, in the following manner:
\begin{mathletters}
\begin{eqnarray}
\widetilde U^c_{[1/2]} a_\uparrow ({\bf p}) (\widetilde
U^c_{[1/2]})^{-1}
&=& - b_\downarrow ({\bf p})\, ,\quad \widetilde U^c_{[1/2]}
a_\downarrow ({\bf p}) (\widetilde U^c_{[1/2]})^{-1} = - b_\uparrow
({\bf p})\, ,\\
\widetilde U^c_{[1/2]} b_\uparrow^\dagger ({\bf p})
(\widetilde U^c_{[1/2]})^{-1} &=& + a_\downarrow^\dagger ({\bf
p})\, ,\quad
\widetilde U^c_{[1/2]} b_\downarrow^\dagger ({\bf p})
(\widetilde U^c_{[1/2]})^{-1} = + a_\uparrow^\dagger ({\bf p})\, ,
\end{eqnarray}
\end{mathletters}
and, therefore,
\begin{mathletters}
\begin{eqnarray}
\widetilde U^c_{[1/2]} \vert {\bf p}, \, \uparrow >^+ &=& - \,\vert {\bf
p},\, \downarrow >^- \quad,\quad
\widetilde U^c_{[1/2]} \vert {\bf p}, \, \downarrow
>^+ = - \, \vert {\bf p},\, \uparrow >^- \quad,\quad\\
\widetilde U^c_{[1/2]} \vert
{\bf p}, \, \uparrow >^- &=& + \, \vert {\bf p},\, \downarrow >^+
\quad,\quad
\widetilde U^c_{[1/2]} \vert {\bf p}, \, \downarrow >^- = + \, \vert
{\bf
p},\, \uparrow >^+ \quad.
\end{eqnarray}
\end{mathletters}
One can convince ourselves by straightforward
verification in the correctness of the
assertions made in~\cite{DVA} (see also the old paper~\cite{Nigam}) that
the situation is possible when the operators of the space inversion and
the charge conjugation commute in the Fock space. For instance,
\begin{mathletters}
\begin{eqnarray}
U^c_{[1/2]} U^s_{[1/2]} \vert {\bf
p},\, \uparrow >^+ &=& + i U^c_{[1/2]}\vert -{\bf p},\, \downarrow >^+ =
+ i \vert -{\bf p},\, \downarrow >^- \quad,\\
U^s_{[1/2]} U^c_{[1/2]} \vert {\bf
p},\, \uparrow >^+ &=& U^s_{[1/2]}\vert {\bf p},\, \uparrow >^- = + i
\vert -{\bf p},\, \downarrow >^- \quad.
\end{eqnarray} \end{mathletters}
The second choice of the charge conjugation operator answers for the
case
when the $\widetilde U^c_{[1/2]}$ and $U^s_{[1/2]}$ operations
anticommute:
\begin{mathletters} \begin{eqnarray}
\widetilde U^c_{[1/2]} U^s_{[1/2]} \vert {\bf p},\, \uparrow >^+ &=&
+ i \widetilde U^c_{[1/2]}\vert -{\bf
p},\, \downarrow >^+ = -i \, \vert -{\bf p},\, \uparrow >^- \quad,\\
U^s_{[1/2]} \widetilde U^c_{[1/2]} \vert {\bf p},\, \uparrow >^+ &=& -
U^s_{[1/2]}\vert {\bf p},\, \downarrow >^- = + i \, \vert -{\bf p},\,
\uparrow >^- \quad.
\end{eqnarray} \end{mathletters}
Next, one can compose states which would have somewhat similar
properties to those which we have become accustomed.
The states $\vert {\bf p}, \,\uparrow >^+ \pm
i\vert {\bf p},\, \downarrow >^+$ correspond to the positive (negative)
parity, respectively. But, what is important, {\it the antiparticle
states} (moving backward in time) have the same properties with respect
to
the operation of space inversion as the corresponding {\it particle
states} (as opposed to $j=1/2$ Dirac particles). This is again in
accordance with the analysis of Nigam and Foldy~\cite{Nigam}, and
Ahluwalia~[1c]. The states which are eigenstates of the charge
conjugation operator in the Fock space are \begin{equation} U^c_{[1/2]}
\left ( \vert {\bf p},\, \uparrow >^+ \pm i\, \vert {\bf p},\, \uparrow
>^- \right ) = \mp i\, \left ( \vert {\bf p},\, \uparrow >^+ \pm i\,
\vert
{\bf p},\, \uparrow >^- \right ) \quad. \end{equation} There is no a
simultaneous set of states which were ``eigenstates" of the operator of
the space inversion and of the charge conjugation $U^c_{[1/2]}$.
\item
We found the Majorana representation of the Barut-Muzinich-Williams
matrices and the spinors of the (modified) Weinberg formulation (the
momentum-space functions in the $(1,0)\oplus (0,1)$ representation
space).
In this representation the $\gamma_{\mu\nu}$
matrices are the real matrices; the $\gamma_5$ matrix is the pure
imaginary matrix. The
matrix of the unitary transformation is:
\begin{mathletters}
\begin{eqnarray} U &=& {1\over 2\sqrt{2}}\pmatrix{(1-i) +(1+i) \Theta&
-(1-i) +(1+i) \Theta\cr (1+i) +(1-i)\Theta& -(1+i) +
(1-i)\Theta\cr}\quad,\\ U^\dagger &=& {1\over
2\sqrt{2}}\pmatrix{(1+i)+(1-i)\Theta& (1-i)+(1+i)\Theta\cr -(1+i)
+(1-i)\Theta& -(1-i) +(1+i) \Theta\cr}\quad.
\end{eqnarray}
\end{mathletters}
As a result we arrive, $\gamma_{\mu\nu}^{^{MR}}=U\gamma_{\mu\nu}^{^{CR}}
U^\dagger$:
\begin{mathletters} \begin{eqnarray}
\gamma_{00}^{^{MR}} &=& \pmatrix{0&\Theta\cr \Theta
&0\cr}\, ,\quad
\gamma_{01}^{^{MR}} = \gamma_{10}^{^{MR}} =\pmatrix{0&-J_1
\Theta\cr -J_1 \Theta&0\cr}\, ,\\
\gamma_{02}^{^{MR}} &=& \gamma_{20}^{^{MR}} =\pmatrix{iJ_2 \Theta& 0\cr
0 & -iJ_2 \Theta\cr}\, ,\quad
\gamma_{03}^{^{MR}} = \gamma_{30}^{^{MR}} =\pmatrix{0&-J_3
\Theta\cr -J_3 \Theta&0\cr}\, , \nonumber\\
&&\\
\gamma_{ij}^{^{MR}} &=& \gamma_{ji}^{^{MR}} = {1\over 2}
\pmatrix{i (J_{ij}^\ast -J_{ij} ) \Theta & (J_{ij}^\ast +
J_{ij})\Theta\cr
(J_{ij}^\ast +J_{ij} )\Theta & -i (J_{ij}^\ast -J_{ij})
\Theta\cr}\, , \\
&& \mbox{and}\quad \gamma_5^{^{MR}} = \pmatrix{0&i\openone\cr
-i\openone & 0\cr}\, .
\end{eqnarray} \end{mathletters}
The $3\times 3$ matrix $\Theta$ corresponds to the Wigner operator in
the
spin-1 representation
\begin{equation}
\Theta = \pmatrix{0&0&1\cr
0&-1&0\cr
1&0&0\cr}\quad.
\end{equation}
If one writes
\begin{mathletters}
\begin{eqnarray}
u^{^{MR}} (p^\mu) &=& {1\over 2} \pmatrix{\phi_{_L} +
\Theta \phi_{_R}\cr \phi_{_L} +\Theta \phi_{_R}\cr} +{i\over 2}
\pmatrix{-\phi_{_L} +\Theta \phi_{_R}\cr \phi_{_L} -\Theta\phi_{_R}\cr}
=
{\cal U}^+ +i{\cal V}^+\, ,\label{usp}\\
v^{^{MR}} (p^\mu) &=& {1\over
2} \pmatrix{-\phi_{_L} + \Theta \phi_{_R}\cr -\phi_{_L} +\Theta
\phi_{_R}\cr} +{i\over 2} \pmatrix{\phi_{_L} +\Theta \phi_{_R}\cr
-\phi_{_L} -\Theta\phi_{_R}\cr}= {\cal U}^- +i{\cal V}^-\, . \label{vsp}
\end{eqnarray} \end{mathletters}
(with ${\cal U}$ are the {\it real} parts and ${\cal V}$, the imaginary
parts of the ``bispinors") one can see that
\begin{equation}
v^{^{MR}}
(p^\mu) = \gamma_5^{^{MR}} u^{^{MR}} (p^\mu) =
i\gamma_5^{^{WR}}\gamma_0^{^{WR}} u^{^{MR}} (p^\mu)
=\pmatrix{0&i\openone
\cr -i\openone &0\cr} u^{^{MR}} (p^\mu)\,. \label{connect1}
\end{equation}
Surprisingly, we have
\begin{mathletters} \begin{eqnarray}
&&{\cal U}_\uparrow^+ (p^\mu) = {\cal U}_\downarrow^+ (p^\mu)\, , \quad
{\cal V}_\uparrow^+ (p^\mu) = -{\cal V}_\downarrow^+ (p^\mu)\quad,\\
&&\mbox{but}\quad {\cal U}_\rightarrow^+ (p^\mu) = 0\, ,\quad {\cal
V}_\rightarrow^+ (p^\mu) \neq 0\, .
\end{eqnarray} \end{mathletters}
While the ``longitudinal" bispinor $u_\rightarrow (p^\mu)$ has only the
imaginary part in this representation, the negative-energy bispinor
$v_\rightarrow (p^\mu)$ has only the real part.
Finally, it is interesting to note that the $\lambda^{S(A)} (p^\mu)$ and
$\rho^{S(A)} (p^\mu)$ spinors become the pure real (pure imaginary)
spinors in the momentum space representation for both $j=1/2$ and $j=1$
case.
\item
Furthermore, we have found some connections between the Dirac field
operator and the Majorana-like operator composed of $\lambda^{S,A}$
spinors. If one uses relations (\ref{connect}) between the
self/anti-self
charge conjugate spinors and the Dirac spinors (together with the
identities between $\lambda$ and $\rho$ spinors) one can deduce:
\begin{equation}
\Psi^{Dirac} (x^\mu) = (1 +{ i\gamma^\mu \partial_\mu \over m} )
\nu^{^{ML}} (x^\mu) \quad.
\end{equation}
The commutation relations (for the creation/annihilation
operators of self/anti-self charge conjugate states) may be slightly
different comparing to those presented in~[1c] but the set of the states
is still {\it bi-orthonormal}.
Finally, it is interesting to note that
\begin{mathletters}
\begin{eqnarray}
\left [ \nu^{^{ML}} (x^\mu) + {\cal C} \nu^{^{ML\,\dagger}} (x^\mu)
\right ]/2 &=& \int {d^3 {\bf p} \over (2\pi)^3 } {1\over 2E_p}
\sum_\eta
\left [\pmatrix{i\Theta \phi_{_L}^{\ast \, \eta} (p^\mu) \cr 0\cr}
a_\eta
(p^\mu) e^{-ip x} \right .\nonumber\\
&+& \left.\pmatrix{0\cr
\phi_L^\eta (p^\mu)\cr } a_\eta^\dagger (p^\mu) e^{ip x} \right ]\, , \\
\left [\nu^{^{ML}} (x^\mu) - {\cal C} \nu^{^{ML\,\dagger}} (x^\mu)
\right
]/2 &=&\int {d^3 {\bf p} \over (2\pi)^3 } {1\over 2E_p} \sum_\eta
\left [\pmatrix{0\cr \phi_{_L}^\eta (p^\mu) \cr } a_\eta (p^\mu)
e^{-ip x} \right .\nonumber\\
&+&\left. \pmatrix{-i\Theta
\phi_{_L}^{\ast\, \eta} (p^\mu)\cr 0 \cr } a_\eta^\dagger (p^\mu)
e^{ip x} \right ] \, .
\end{eqnarray} \end{mathletters}
thus
naturally leading to the Ziino-Barut scheme of massive chiral fields,
ref.~\cite{Ziino}.
The conclusion is: a lot of work is still required to make certain
conclusions about the relevance of the presented construct to describing
the physical world and to the present situation in the neutrino physics.
But, it is important that this construct is permitted by the
requirements
of the extended Poincar\'e group symmetry; it is based on the very
viable postulates: in fact, after imposing the conditions of
the self/anti-self charge conjugacy we derived all consequences only on
the
basis of the Wigner rules for transformations of left- and right- handed
2-spinors and on the relations between these spinors in the frame with
zero momentum.
Thus, as I was taught in the Gorbachev's epoch: ``everything is
permitted unless forbidden".
\end{itemize}
{\bf Note Added.}
I would like to express my gratitude to the anonymous referee of
``Hadronic Journal" for the useful suggestion to include the
discussion of the Santilli {\it isodual} conjugation [ R. M. Santilli,
Comm. Theor. Phys. {\bf 3} (1994) 153; R. M. Santilli, Hyperfine
Interactions {\bf 109} (1997) 63; {\it Elements of Hadronic Mechanics,
Vols. I and II} (Ukraine Academy of Sciences, Kiev, 1995);
J. V. Kadeisvili, Math. Methods Appl. Sciences {\bf 19} (1996) 1349]
into this paper. The Santilli {\it isoduality} is characterized by the
map of conventional unit $$ +1 \rightarrow 1^d = -1^\dagger = -1\, .$$
Hence, the {\it isodual} images of the known fields are introduced. They
are considered to be of the crucial importance in the description of the
antimatter on the {\it first quantization level}. The need for new
mathematics was also argued by F. Antonuccio [ gr-qc/9311032,
hep-th/9408166], who substantially developed new mathematical
construct\footnote{\baselineskip14pt
This branch of mathematics is now known as the
{\it semi-complex} analysis. It is based on the some sort of
``splitting"
of the unit, $j^2 =1$. The norm of the semicomplex numbers is {\it real}
but ``unlike the complex case, it may take on {\it negative} values".}
in applications to the Poincar\'e group and to the differential
calculus.
The author of the {\it isodual} mathematics writes ``that isoduality on
a
Hilbert space is equivalent to charge conjugation", [ R. M. Santilli,
Hyperfine Interactions {\bf 109}, 63 ]; ``the proof \ldots first
appeared in paper of 1994 [ R. M. Santilli, Comm. Theor. Phys. {\bf 3},
153 ]" and in subsequent monographs. Since the classical representation
of
the antiparticles is {\it not} yet elaborated in the completely
consistent form in the framework of the ordinary ideas (cf. refs.~[ E.
C.
G. Stueckelberg, Helv. Phys. Acta {\bf 14} (1941) 372, 588; ibid {\bf
15}
(1942) 23; J. P. Costella, B. H. McKellar and A. A. Rawlinson, Am. J.
Phys. {\bf 65} (1997) 835 ]), the Santilli's approach has certain
logical bases and, seems, may serve as the alternative to the
introduction of the common-used Fock representation space (the secondary
quantization scheme). Particularly, some advantages of this approach
manifest themselves in the ``axiomatically consistent inclusion of
gravitation in the unified gauge theories of electroweak interactions".
Furthermore, the author of the {\it isodual} mathematics made several
predictions (particularly, the concept of {\it isodual} light), which
can
distinguish his theory from the `old physics of antimatter'.
With respect to our ideas (and apart from the discussion of the Fock
space) the Santilli approach may be useful in the construction of
the self/anti-self charge conjugate states for spin $j=1$. The
difficulties of the construction of such $(1,0)\oplus (0,1)$
``bispinors"
were discussed in the previous works~[1c,2c] and above.
The complete investigation of possible
connections between the Santilli's approach and the theories based on
the
{\it extended} Poincar\'e group, particularly, the comparison with the
Barut-Ziino concept of the {\it doubling} of the Fock space~\cite{Ziino}
will be presented in future works.
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\end{document}