Propagator for Dirac field The Feynman propagator for Dirac field is defined as:
$$S_{\alpha\beta} = \langle 0 | T \psi_\alpha(x)\bar{\psi}_\beta(y) | 0 \rangle.$$
It should, however, be a transition amplitude so that a particle at location $\mathbf{y}$ is generated at time $y_0$, moves to $\mathbf{x}$ and is destroyed at time $x_0$.
But how can a matrix be a transition amplitude?
I tried to find something about this on the internet, but I found nothing.
 A: 
It should, however, be a transition amplitude so that a pond at location $\mathbf{y}$ is generated at time $y_0$, moves to $\mathbf{x}$ and is destroyed at time $x_0$.

...pond? Anyway, the amplitude does not literally have this position-based interpretation. This is more of a rule-of-thumb type of interpretation that people use, and it is related to the way that such terms appear in Feynman diagrams - which also are suggestive, but not known to be literal interpretations of what is going on in the interaction region.

But how can a matrix be a transition amplitude?

The answer here is that with spin, you need a "matrix" of transition amplitudes. If we have spin 1/2, then there are 4 different amplitudes which each have their own respective probabilities:

*

*When starting with a spin $+1/2$ particle of momentum $p$, the probability to transition to a spin $+1/2$ particle of momentum $p'$

*When starting with a spin $+1/2$ particle of momentum $p$, the probability to transition to a spin $-1/2$ particle of momentum $p'$

*When starting with a spin $-1/2$ particle of momentum $p$, the probability to transition to a spin $+1/2$ particle of momentum $p'$

*When starting with a spin $-1/2$ particle of momentum $p$, the probability to transition to a spin $-1/2$ particle of momentum $p'$
Of course, these amplitudes will depend on momentum, unless you want the probability to get a particle at any momentum, in which case you integrate $\int d^3p$. On the other hand, often we don't care which spin, and so we just sum over all outgoing spin combinations and average over the incoming ones, which tend to be a 50/50 split.
