Neat question. The short answer is **no, the real spinor does not turn complex under time evolution.** That's at the very heart of the "big deal".

To see this explicitly, go to the rest frame, as you may always Lorentz-transform yourself back. So, the Dirac equation reduces to just
$$
(i\tilde \gamma ^0 \partial_t - m)\psi =0,
$$
hence
$$
(\partial_t +im\tilde \gamma^0)\psi=0 .
$$
Note in this representation $i\tilde \gamma ^0$ is real and antisymmetric, and thus antihermitean, as it should be!

Since *it does not mix up the real and imaginary parts of the Dirac spinor* $\psi$, we may consistently take the imaginary part to vanish, so the spinor is real: A Majorana spinor in this, Majorana, representation. (In the somewhat off-mainstream basis you display, you may take $C=-i\tilde \gamma^0=-C^T=-C^\dagger=-C^{-1} $.)

The evident solution, then, is
$$
\psi (t) = \exp (-itm\tilde \gamma ^0 ) ~~\psi(0)=\exp (-itm ~\tilde \sigma_2\otimes\sigma_1 ) ~~\psi(0)~~,
$$
where, as seen, the exponential is *real*, so the time-evolved spinor is real, as well, forever and ever. Thus,
$$
\psi (t) = (\cos (tm)-i\tilde \gamma^0 \sin (tm) ) ~\psi(0) .
$$

Now, thruth be told, this is an "existence proof" of the consistency of the split. Few of my friends actually use the Majorana representation in their daily lives. It just reminds you that a Dirac spinor is resolvable to a Majorana spinor plus i times another Majorana spinor.

The properties of the Dirac equation are the same in both first and second quantization, so all the moves and points made also hold for field theory as well, without departure from the standard textbook transition of the Dirac equation.

*Edit in response to @annie marie hart 's question.*

In effect, the complex nature of Schroedinger's equation is replicated with real matrix quantities, given $\Gamma\equiv i\tilde \gamma^0 ~\leadsto ~\Gamma^2=-I$. As a result, all complex unitary propagation features of Schroedinger's wavefunctions are paralleled here by the *real unitary matrices* acting on spinor wave functions, normalized in the technically analogous sense.

This post imported from StackExchange Physics at 2020-11-06 18:50 (UTC), posted by SE-user Cosmas Zachos