1. The problem statement, all variables and given/known data

2. Relevant equations
Please see below.

3. The attempt at a solution
No idea about part (a).

Trying to work out part (b), I asked my tutor and he said:

I think it's this. We have the Dirac Equation $$\left ( i \hbar \gamma ^{\mu } \partial_{\mu }-mc\right )\Psi =0$$ but for a local phase shift we have to let $$\Psi \rightarrow \Psi ^{'}= e^{i\alpha (x)}\Psi $$

If we do this we get an unwanted term, i.e. invarience is lost.

Therfore we put $$\partial\mu \rightarrow D\mu= \partial\mu - \frac{iq}{\hbar}A_{\mu}$$

From this we can define $$A_{\mu}^{'}=A_{\mu}+ \frac{\hbar}{q}\partial_{\mu}\alpha (x)$$

Which would then make the thing varient.

My question is how does this relate to the (b) question? is this what it is asking, I have no idea.

Any suggestions more than welcome!

for part (c) havent I just done that? not sure what it wants.

Think I have this. You derive the A mu dash term and then this says that there must be some form of EM interaction for the dirac equation to be satisfied.

edit

Also for part (A) ... QED with Abelian gauge theory with group U(1)