Evaluate the following Greens functions for a free Fermi gas. Take the average$< >$with respect to a state characterized by a distribution function $<{a_p}^{+} a_p> =n_p$. $<O>=\frac{tr e^{-\lambda H_0}O}{tr e^{-\lambda H_0}}$ with $H_0=\sum\limits_{p} \epsilon_p a^{\dagger}_{p}a_{p}$ and $n_p=\frac{1}{1+e^{\lambda\epsilon_p}}$

Evaluate the Fourier Transform in space and time of the following Greens functions.

a) $G^{t}(1,2) = -i<T(\psi(x_1 t_1) {\psi^\dagger}(x_2 t_2))>$

b) $G^{\tilde{t}} (1,2) = -i<{\tilde T}(\psi(x_1 t_1) {\psi^\dagger}(x_2 t_2))>$

c) $G^{>} (1,2) = -i< \psi(x_1 t_1) {\psi^\dagger}(x_2 t_2)>$

d) $G^{<}(1,2) = i<{\psi^\dagger}(x_2 t_2)\psi(x_1 t_1) >$

e) $G^{K}(1,2) = -i<[ \psi(x_1 t_1) , {\psi^\dagger}(x_2 t_2)]>$

f) $G^{R}(1,2) = -i \theta({t_1} -{t_2})<\{\psi(x_1 t_1) {\psi^\dagger}(x_2 t_2)\}>$

g) $G^{A}(1,2) = i \theta({t_2} -{t_1})<\{\psi(x_1 t_1) , {\psi^\dagger}(x_2 t_2)\}>$

Compare them with the Fourrier transfomr of the Matsubara Greens function. ${g(\tau,x)} = -< T( \psi(\tau,x) {\psi^{\dagger}}(0,0) ) >$.