Note regarding signs and normalization used in these definitions: The signs of the Green functions have been chosen so that Fourier transform of the two-point ( n = 1 ) thermal Green function for a free particle is G ( k , ω ω n ) = 1 − − i ω ω n + ξ ξ k , and the retarded Green function is G R ( k , ω ω ) = 1 − − ( ω ω + i η η ) + ξ ξ k , where ω ω n = [ 2 n + θ θ ( − − ζ ζ ) ] π π β β is the Matsubara frequency.