R1b

1. For an abelian theoery, from [math]\displaystyle{ U(y,x)\to e^{i\alpha(y)}U(y,x)e^{-i\alpha(x)} }[/math] and [math]\displaystyle{ n\cdot D\psi=\lim_{\epsilon\to 0}\frac{\psi(x+\epsilon n)-U(x+\epsilon n,x)\psi(x)}{\epsilon} }[/math], find the transformation laws of [math]\displaystyle{ A_\mu }[/math], and the formula for [math]\displaystyle{ D_\mu }[/math] as a function of [math]\displaystyle{ A_\mu }[/math] if [math]\displaystyle{ U(x+\epsilon n,x)=1-i\epsilon n\cdot A }[/math].
2. Derive the equation of motion of the field [math]\displaystyle{ A_\mu^{(l)} }[/math].
3. Derive the expression of the 3-gauge boson vertex in Yang-Mills SU(2).
4. Same question in Glashow's model.