Greens functions from path integral

[aaaa202]

Let me post this question again in a slightly modified form. On the attached picture the path integral for the partion function: Z = Tr(exp(-βH))
Now according to what it says on the picture it should be easy from this to get the Green's function in the path integral formalism. The Green's function is given by:
G(xx',τ-τ') = 1/ZTr[exp(-H(β-τ)c_xexp(-H(τ-τ'))c_x'exp(-Hτ')]
But how exactly does this trivally allow us to apply the formula for the partion function path integral?

 

[stevendaryl]

Let me post this question again in a slightly modified form. On the attached picture the path integral for the partion function: Z = Tr(exp(-βH))
Now according to what it says on the picture it should be easy from this to get the Green's function in the path integral formalism. The Green's function is given by:
G(xx',τ-τ') = 1/ZTr[exp(-H(β-τ)c_xexp(-H(τ-τ'))c_x'exp(-Hτ')]
But how exactly does this trivally allow us to apply the formula for the partion function path integral?

I'm not sure what you're asking. Are you asking how to get $G$ from $Z$, or how to derive the path-integral expression for $G$?

 

[aaaa202]

I'm asking how you end up with the equation 2.7 given that we know the path integral representation of Z.