Physics 570 Assignment #4 February 23, 2006
Due Thursday, March 9, 2006

1. Consider QCD, an $SU(3)$ gauge theory with fundamental representation fermions. Construct gauge-invariant local operators which will have non-zero amplitudes to create (i) a $\pi^+$ meson, (ii) a $\rho^+$ meson, or (iii) a proton, when acting on the vacuum. How can one extract the mass of these particles from the two-point correlators of the corresponding operators, in Euclidean space?
2. In an arbitrary non-Abelian gauge theory:
   1. A gauge field satisfying $\partial \cdot A = 0$ is said to be in Lorentz gauge. What Faddeev-Popov determinant must accompany a Lorentz-gauge gauge fixing term $\prod_x \delta(\partial \cdot A)$? If this functional delta function is represented a the limit of a Gaussian, $\exp(-\frac 1{2\xi} \int d^4x \> (\partial^\mu A^a_\mu)^2$, what is the resulting free gauge-field propagator?
   2. Let $\hat n$ be an arbitrary unit vector. Show that any gauge field configuration may be gauge transformed to ``axial gauge'' in which $\hat n \cdot A = 0$. What is the required gauge transformation? What is the appropriate Faddeev-Popov determinant for a gauge-fixing term $\prod_x \delta (\hat n \cdot A(x))$?
   3. Show that any gauge field configuration may be gauge transformed to ``radial gauge'' in which $x \cdot A(x) = 0$. What is the required gauge transformation?