1. In the case of two scalar hermitian fields $\Phi_a(x),\ a=1,2$ with the same mass, show that the lagrangian density ${\mathcal L}=(1/2)\sum_a(\partial_\mu \Phi_a)(\partial^\mu\Phi_a)-m^2\Phi_a^2$ has a continuous symmetry. Calculate the associated charge. Compare with the case of a complex scalar field.
2. In the case of two complex scalar fields $\Phi_a(x),\ a=1,2$ with the same mass, show that the lagrangian density ${\mathcal L}=\sum_a(\partial_\mu \Phi_a^*)(\partial^\mu\Phi_a)-m^2|\Phi_a|^2$ has several continuous symmetries. Calculatethe corresponding charges.
3. Establishthe transformation properties of $\bar \psi\gamma^\mu\psi$ and $\bar \psi\gamma_5\gamma^\mu\psi$ under the parity transformation.
4. Using the Dirac algebra, simplify te following: $\gamma_\mu\gamma^\mu$,$\gamma_\mu\gamma_\nu\gamma^\mu$, $\gamma_\mu\gamma_\nu\gamma_\tau\gamma^\mu$, and $\gamma_\mu\gamma_\nu\gamma_\tau\gamma_\sigma\gamma^\mu$.