I'm currently busy with a course in QFT and am completely baffled by Spinors. In particular there are two parts, that while I mostly understand the theory, struggle to show mathematically (especially as we have not covered group-theory).
Lorentz Invariance
The first problem I have is showing a set of equations is Lorentz invariant. I understand that this means that 'the physics is the same regardless of which inertial frame is being used'. Am I correct in saying that to show that a set of equations are Lorentz invariant one has to perform a change of variables from x,t into x' and t'. Then if the equations have the same form they are Lorentz invariant?
Take for example the following coupled equations (in $\psi$ and $\phi$):
$$ i\left( \psi_{t}(x,t) - \psi_{x}(x,t) \right) - m\phi(x,t) + \psi(x,t) \overline{\phi}(x,t) = 0 \\ i\left( \phi_{t}(x,t) + \phi_{x}(x,t) \right) - m\psi(x,t) + \phi(x,t) \overline{\psi}(x,t) =0 $$
These are Lorentz invariant if under the transformation x,t into x',t' the equations become:
$$ i\left( \psi_{t}'(x',t') - \psi_{x}'(x',t') \right) - m\phi'(x',t') + \psi'(x',t') \overline{\phi}'(x',t') = 0 \\ i\left( \phi_{t}'(x',t') + \phi_{x}'(x',t') \right) - m\psi'(x',t') + \phi'(x',t') \overline{\psi}'(x',t') =0 $$
Spinors
The second and more serious problem I have is showing a set of equations are (Lorentz Invariant) Spinors. From what I understand a set of equations are Spinors if they transform like Spinors. Using the previous example, these are spinors if they transform as follows:
$$ i \left(\psi_{t}(x,t) - \psi_{x}(x,t) \right) - m\phi(x,t) + \psi(x,t) \overline{\phi}(x,t) = 0 \\ i\left( \phi_{t}(x,t) + \phi_{x}(x,t) \right) - m\psi(x,t) + \phi(x,t) \overline{\psi}(x,t) =0 $$
becoming
$$ i\left( S \psi_{t}(x',t') - S \psi_{x}(x',t') \right) - m S \phi(x',t') + S \psi(x',t') S \overline{\phi}(x',t') = 0 \\ i\left( S \phi_{t}'(x',t') + S \phi_{x}(x',t') \right) - m S \psi(x',t') + S \phi(x',t') S \overline{\psi}(x',t') = 0 $$
where $\Lambda$ is the usual Lorentz transform and $S$ is the spinor transform $e^{\frac{1}{2}\Omega_{\rho \sigma} M^{\rho \sigma}$}$ and $\psi'(\Lambda x) = S \psi(x)$.
If this is incorrect, how does one prove that it is a set of spinor equations (at least for the linear part of the set of equations)