Problems QFT 3

1. For a massless particle, calculate the retarded propagator [math]\displaystyle{ [\Phi^*(x),\Phi(y)] }[/math] in [math]\displaystyle{ x }[/math] space.
2. Establish the transformation properties of [math]\displaystyle{ \bar \psi\gamma^\mu\psi }[/math] and [math]\displaystyle{ \bar \psi\gamma_5\gamma^\mu\psi }[/math] under the parity transformation.
3. Find the expression of the Dirac hamiltonian corresponding to the Dirac equation for 1-particle states. Show that it commutes with [math]\displaystyle{ \overrightarrow L+\overrightarrow{\Sigma\over 2} }[/math] and [math]\displaystyle{ \overrightarrow p\cdot\overrightarrow\Sigma }[/math].
4. Using the Dirac algebra, simplify the following: [math]\displaystyle{ \gamma_\mu\gamma^\mu }[/math],[math]\displaystyle{ \gamma_\mu\gamma_\nu\gamma^\mu }[/math], [math]\displaystyle{ \gamma_\mu\gamma_\nu\gamma_\tau\gamma^\mu }[/math], and [math]\displaystyle{ \gamma_\mu\gamma_\nu\gamma_\tau\gamma_\sigma\gamma^\mu }[/math].
5. Consider the Weyl equations and find the helicities [math]\displaystyle{ \overrightarrow p\cdot\overrightarrow\sigma\over 2 |\overrightarrow p| }[/math] of the particles described by these equations.