I am having some basic questions about how to interpret Lagrangians, lets start with Dirac:
$L = \bar{\Psi} (i \gamma^{\mu} \partial_{\mu} -m) \Psi$,
where $\Psi$ is a Dirac-Spinor, $m$ is the mass, $\gamma^{\mu}$ is a gamma matrix and $\partial_{\mu}$ is the derivative.
1) Is $m$ a vector or a matrix or a scalar? I always thought that it is a scalar, but for some reason $\bar{\Psi} m \Psi$ is allowed in the Lagrangian, but $m \bar{\Psi} \Psi$ isn't, so it can't be a scalar! EDIT m is a scalar and can be anywhere in the Dirac equation, but these terms are not valid for the Standard model!
2) What exactly is violated so that $m \bar{\Psi} \Psi$ is not allowed? Is it not invariant under something? EDIT these terms are both not valid for the Standard model, because they are not gauge invariant!
3) I always thought that a Dirac spinor contains all the possible for a fermion, the two spin states for a particle and the two spin states for the antiparticle. Is this assumption correct?
4) Why do we need $\bar{\Psi}$? What does it represent? Do the particles and anti-particles trade places? Or is my interpretation in 3) wrong and $\Psi$ represents the particle and $\bar{\Psi}$ and antiparticle.
5) The Dirac equation describes a free massive fermion moving through space and time, does the $\bar{\Psi}$ indicate an interaction?
I was trying to understand it from wikipedia, but I failed. Any answer to any of the questions above, will be appreciated.