Homework Problems

1. Define:
   
   $$\Sigma_k=\pmatrix{\sigma_k & 0 \cr 0 & \sigma_k}.$$
   
   
   Prove that $[\gamma_1, \gamma_2] = 2i\Sigma_3$ and $\gamma_i \gamma_j = \delta_{ij} + i \epsilon_{ijk} \Sigma_k$. Prove that $\Sigma_k = i \gamma_4 \gamma_5 \gamma_k$ where $\gamma_5=\gamma_1\gamma_2\gamma_3\gamma_4$.

2. Prove that $\gamma_5$ is Hermitian and that $\{ \gamma_{\mu},\gamma_{5} \} = 2 \delta_{\mu 5}$ with $\mu = 1,5$, without using an explicit representation for $\gamma_5$. What do the operators ${1\pm\gamma_5\over 2}$ project from a Dirac spinor?

3. Find the ``Time Reversal'' operator for Dirac spinors. This operator should reverse both momentum and spin for an electron, leaving the charge and energy unchanged. Time reversal should be a symmetry of the Dirac equation. Show that this is true. We expect fields due to currents to change sign but fields due to static charges to remain unchanged.