Problem Set 1

1) 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}

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

a) G^{t}(1, 2) = −i < T (ψ(x_{1}t_{1})ψ^{†}(x_{2}t_{2})) >

b) Gt̃(1, 2) = −i < T̃(ψ(x_{1}t_{1})ψ^{†}(x_{2}t_{2})) >

c) G^{>}(1, 2) = −i < ψ(x_{1}t_{1})ψ^{†}(x_{2}t_{2}) >

d) G^{<}(1, 2) = i < ψ^{†}(x_{2}t_{2})ψ(x_{1}t_{1}) >

e) G^{K}(1, 2) = −i < [ψ(x_{1}t_{1}), ψ^{†}(x_{2}t_{2})] >

f) G^{R}(1, 2) = −iθ(t_{1} − t_{2}) < {ψ(x_{1}t_{1})ψ^{†}(x_{2}t_{2})} >

g) G^{A}(1, 2) = iθ(t_{2} − t_{1}) < {ψ(x_{1}t_{1}), ψ^{†}(x_{2}t_{2})} > Compare them with the Fourrier transfomr of the Matsubara Greens func-

tion. g(τ, x) = − < T (ψ(τ, x)ψ^{†}(0, 0)) >.