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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 $ 



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\indent Evaluate the Fourier  Transform   in space and time of the following Greens functions.
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\def\psii{\psi(x_1 t_1) }
\def\arg{(1,2) }
\def\psiid{{\psi^\dagger}(x_2  t_2)}

%$\psii$
%$\psiid$

a)  $G^{t}\arg = -i<T(\psii \psiid)>$
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b)  $G^{\tilde{t}} \arg  = -i<{\tilde T}(\psii \psiid)>$
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c)  $G^{>} \arg = -i< \psii \psiid>$
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d)  $G^{<}\arg = i<\psiid \psii>$
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e)  $G^{K}\arg= -i<[ \psii , \psiid ]>$
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\def\tet{\theta({t_1} -{t_2})}

f)  $G^{R}\arg = -i \tet <\{\psii \psiid\}>$
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\def\tett{\theta({t_2} -{t_1})}
g)  $G^{A}\arg = i \tett <\{\psii, \psiid\}>$
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Compare them with the Fourrier transfomr of the Matsubara Greens function.
$ {g(\tau,x)}  = -< T( \psi(\tau,x) {\psi^{\dagger}}(0,0) ) >$.
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