1. a) Let $\vec l_1=(dx,0)$ and $\vec l_2=(0,dy)$ be two infinitesimal vectors along the sides of a parallelogram of angle $\theta$, and $g_{ij}$ the corresponding metric tensor. What are the lengths $l_1$ and $l_2$? What is the relation between $\theta$ and $g_{12}$? Show that the infinitesimal area is given by $\sqrt{det(g)}dx dy$. b) Show that $\sqrt{-g} d^4 x$ is invariant under changes of variables.
2. Consider a Bogolyubov transformation for fermionic fields $a^-_k=\alpha_k b_k^-+\beta_k b_{-k}^+$, $a^+_k=\gamma_k b_k^+ +\delta_k b_{-k}^-$. What are the conditions on the coefficients?
3. Find the relation between the bosonic two vacua $|0;a\gt$ and $|O;b\gt$ that are related by the Bogolyubov transformation $a^-_k=\alpha^*_k b_k^-+\beta_k b_{-k}^+$, $a^+_k=\alpha_k b_k^+ +\beta_k b_{-k}^-$.
4. Show that $v_k(\eta)$ must have a Wronskian equal to $2i$ for the commutation relations of the fields to lead to a Hilbert space structure.