Equation for relativistic electron and two-component spinor Recently I heard that there is some "alternate" equation for the Dirac one. It can be introduced if we refuse some properties of the theory describes the electron, which Dirac used in his original article. Then we will get theory with spin 1/2-particle and with modified (in compare with Dirac equation) mass. 
I thought that it is connected with the invariance of theory under discrete Lorentz transformations. If we need it, we must to create theory which describes the sum of representations $\left(\frac{1}{2}, 0\right) + \left( 0, \frac{1}{2}\right)$ (because it is invariant under discrete transformations and as the consequence - under full Lorentz group transformations), so, finally, we will get the Dirac equation. But if we refuse the requirement of invariance under discrete transformations, we may describe two-component wave-function. Some analisys leads to the Klein-Gordon equation for it. But it has problems with propability density, so I think that the way of getting this alternate equation is another. 
Can you help me?
 A: Spinor algebra is very helpful in sorting out spinorial equations.
In chiral representation the (four-component) wave function of the fermion
field $\psi$ is considered as a formal sum of first rank spinor and first rank
co-spinor fields:
$\psi{}(x)=\left\{\xi{},\ \dot{\eta{}}\right\}=\ \left\{\xi{},\
0\right\}+\left\{0,\ \dot{\eta{}}\right\}=\ \left({\begin{array}{
cc}
{\xi{}}^1 \\
{\xi{}}^2\\
{\eta{}}_{\dot{1}} \\
{\eta{}}_{\dot{2}}
\end{array}}\right)$
Any quantities transforming like the products ${\xi{}}^{\mu{}}{\xi{}}^{\nu{}}$, 
${\eta{}}_{\dot{\mu{}}}{\eta{}}_{\dot{\nu{}}}$, 
${\xi{}}^{\mu{}}{\eta{}}_{\dot{\nu{}}}$ are called second rank spinors and denoted
by $a^{\mu{}\nu{}}$, $b_{\dot{\mu{}}\dot{\nu{}}}$, 
$c_{\dot{\nu{}}}^{\mu{}}$ correspondingly. Analogously one can define the spinors
of higher ranks.
There are 3 Lorentz-invariant constant second rank spinors ${\epsilon{}}_{\mu{}\nu{}}$,
 ${\epsilon{}}^{\dot{\mu{}}\dot{\nu{}}}$ and $c_{\mu{}\dot{\nu{}}}$ that play important role in spinor algebra.
1. Transition from subscript to superscript spinor indices
Spinors ${\epsilon{}}_{\mu{}\nu{}}$
and ${\epsilon{}}^{\dot{\mu{}}\dot{\nu{}}}$ are written as
$
{\epsilon{}}_{\mu{}\nu{}}=\left[\begin{array}{
cc}
0 & +1 \\
-1 & 0
\end{array}\right] \hspace{10mm}
{\epsilon{}}^{\dot{\mu{}}\dot{\nu{}}}=\left[\begin{array}{
cc}
0 & -1 \\
+1 & 0
\end{array}\right]
$
Transition from subscript to superscript spinor indices is established by means
of spinors ${\epsilon{}}_{\mu{}\nu{}}$
and ${\epsilon{}}^{\dot{\mu{}}\dot{\nu{}}}$:
$
{\xi{}}_{\mu{}}=\ {\epsilon{}}_{\mu{}\nu{}}{\xi{}}^{\nu{}},\ \hspace{10mm}
{\eta{}}^{\dot{\mu{}}}=\
{\epsilon{}}^{\dot{\mu{}}\dot{\nu{}}}{\eta{}}_{\dot{\nu{}}}
$
2. Complex conjugated spinors
Complex conjugates of spinors transform as co-spinors,
and vice versa, so that we can denote
\begin{array}{
cc}
{\xi{}}_{\dot{\mu{}}}=\bar{{\xi{}}}_{\mu{}} \\ \\
{\eta{}}_{\nu{}}=\bar{{\eta}}_{\dot\nu}
\end{array}
3. Charge conjugation
Second rank spinor $c_{\mu{}\dot{\nu{}}}$ (often denoted as $i\sigma_2$) has the form
$
c_{\mu{}\dot{\nu{}}}=\left[\begin{array}{
cc}
0 & +1 \\
-1 & 0
\end{array}\right] 
$
and can be used to transform first rank spinors to co-spinors and vice versa (charge conjugation):
$
{\chi{}}_{\dot{\mu{}}}=\ c_{\nu{}\dot{\mu{}}}{\xi{}}^{\nu{}} 
$
4. Majorana condition
Majorana (or neutrality) condition is a Lorentz-invariant property of electrically neutral spinors. It is expressed in the following form:
$
{\eta{}}_{\dot{\mu{}}}=\ c_{\nu{}\dot{\mu{}}}{\xi{}}^{\nu{}}
$ 
i.e. co-spinor $\dot{\eta{}}$ is charge conjugated to spinor $\xi{}$.
In spinor components this condition is written as follows:
\begin{array}{c}
{\xi{}}^1=- \ \bar{{\eta}}_{\dot 2} \\
\\
{\xi{}}^2=+ \ \bar{{\eta}}_{\dot 1}
\end{array}
5. Mass term
"Mass term" in spinorial equations appears when there is "mixing" of spinor and co-spinor components.
For instance, the free Dirac equation in spinorial form can be written as
\begin{array}{columns}
{\partial{}}^{\mu{}\dot{\nu{}}}  {\eta{}}_{\dot{\nu{}}}\ =\ -im \ {\xi{}}^{\mu{}} \\ \\ {\partial{}}_{\mu{}\dot{\nu{}}}  {\xi{}}^{\mu{}}=\ -im \ {\eta{}}_{\dot{\nu{}}}
\end{array}
The most general form of manifestly Lorentz-invariant spinorial equation with "mixing" is as follows:
\begin{array}{cols}
{\partial{}}^{\mu{}\dot{\nu{}}}{\eta{}}_{\dot{\nu{}}}=\
 \ f_{\nu{}}^{\mu{}} \ {\xi{}}^{\nu{}} \\ \\
{\partial{}}_{\mu{}\dot{\nu{}}}{\xi{}}^{\mu{}}=\
 \ {\dot{f}}_{\dot{\nu{}}}^{\dot{\mu{}}} \ {\eta{}}_{\dot{\mu{}}}
\end{array}
where $f_{\nu{}}^{\mu{}}$ and $\dot{f}^{\dot \mu}_{\dot \nu}$  are second rank spinor and second rank co-spinor correspondingly.
Let us demonstrate that free Dirac equation is just a special case of the more general equation presented above.
The most general form of spinorial equation can be made similar to free Dirac equation, if we require that spinor $\xi{}$ and co-spinor $\dot{\eta{}}$ are eigenvectors of second rank spinor matrices $f_{\nu{}}^{\mu{}}$ and $\dot{f}^{\dot \mu}_{\dot \nu}$:
\begin{array}{
ccc}
f_{\nu{}}^{\mu{}} \ {\xi{}}^{\nu{}}=\ \lambda{}\ {\xi{}}^{\mu{}} \\ \\
\\
{\dot{f}}_{\dot{\nu{}}}^{\dot{\mu{}}} \ {\eta{}}_{\dot{\mu{}}}= {\lambda{}}\
{\eta{}}_{\dot{\nu{}}}
\end{array}
Here $\lambda{}$ is an eigenvalue. It is important to note that this "eigenvector" condition is Lorentz-invariant.
Now our spinorial equation has the form:
\begin{array}{cols}
{\partial{}}^{\mu{}\dot{\nu{}}}{\eta{}}_{\dot{\nu{}}}= \lambda{}\
{\xi{}}^{\mu{}}\\ \\ {\partial{}}_{\mu{}\dot{\nu{}}}{\xi{}}^{\mu{}}=\
{\lambda{}}\ {\eta{}}_{\dot{\nu{}}}
\end{array}
which is very similar to free Dirac equation.
Now the "type" of equation (i.e. Dirac, Majorana or Weyl) will only depend on the special choice of second rank spinor matrices $f_{\nu{}}^{\mu{}}$ and $\dot{f}^{\dot \mu}_{\dot \nu}$.
In particular, we can choose $f_{\nu{}}^{\mu{}}$ and $\dot{f}^{\dot \mu}_{\dot \nu}$ as
$f_{\nu{}}^{\mu{}}=\left[
\begin{array}{cc}
0 & m \\ 
-m & 0
\end{array}\right] $
$\dot{f}^{\dot \mu}_{\dot \nu}=\left[
\begin{array}{cc}
0 & m \\
-m & 0
\end{array}\right]$
the eigenvectors corresponding to the eigenvalue ($\lambda = - im$) will be:
$
\xi_D = \left[
\begin{array}{c}
1 \\ \\ -i
\end{array}\right]\phi(x)
$
$ 
\dot\eta_D = \left[
\begin{array}{c}
1 \\ \\ -i
\end{array}\right]\phi(x)  
$
This case corresponds to Dirac equation.
Alternatively, we can choose $f_{\nu{}}^{\mu{}}$ and $\dot{f}^{\dot \mu}_{\dot \nu}$ as
$
f_{\nu{}}^{\mu{}}=\left[
\begin{array}{cc}
im & 0 \\
0 & -im
\end{array}\right]
$
$
\dot{f}^{\dot \mu}_{\dot \nu}=\left[
\begin{array}{cc}
-im & 0 \\
0 & im
\end{array}\right]
$
and the eigenvectors corresponding to the eigenvalue ($\lambda = - im$) will be:
$
\xi_M = \left[
\begin{array}{c}
0 \\ \\ 1
\end{array}\right]\phi(x)
$
$ 
\dot\eta_M = \left[
\begin{array}{c}
1 \\ \\ 0
\end{array}\right]\phi(x)
$
It is easy to check that spinors $\xi_M$ and $\dot\eta_M$ automatically satisfy Majorana condition. Hence, Majorana equation is also the special case of the most general spinorial equation.
Both Dirac and Majorana spinors belong to $\left(\frac{1}{2}, 0\right) + \left( 0, \frac{1}{2}\right)$ representations of $SL(2,C)$, but they are only subspaces in the entire space of $\left(\frac{1}{2}, 0\right) + \left( 0, \frac{1}{2}\right)$ representation.
Here you can read more about the most general form of spinorial equation. You will see that it can be used to develop the concept of electromagnetic mass and charge.
To read more about spinors and spinorial algebra, you can read:
Laporte, O. and G. E. Uhlenbeck, Phys. Rev. 37, 1380 (1931)
Rumer, Yu.B. and A.I. Fet. Group theory and quantum fields. Moscow, USSR: Nauka publisher, 1977
