Path integral expression for Dirac two-point function

Question

On page 302 of Peskin and Schroeder they state a path integral expression for the Dirac two-point function.

$$\langle0|T\psi_a(x_1)\bar{\psi}_b(x_2)|0\rangle=\frac{\int\mathcal{D}\bar{\psi}\int\mathcal{D}\psi \exp\left[i \int d^4x \bar{\psi}(i \displaystyle{\not}{\partial}-m)\psi\right] \psi_a(x_1)\bar{\psi}_b(x_2)}{\int\mathcal{D}\bar{\psi}\int\mathcal{D}\psi \exp\left[i \int d^4x \bar{\psi}(i \displaystyle{\not}{\partial}-m)\psi\right]}. \tag{p.302}$$

Where does this expression come from? Are they arguing by analogy with the scalar case? The path integral expression for correlation functions of scalar fields was worked out explicitly in pages 283-284 (eq. 9.18). Are P&S saying that it is possible to do something analogous for correlation functions of fermions that would then result in the above expression?

I would like to see for example how the anti-commuting nature of the integration variables arises.

Answer 1

I believe that one can in principle derive the right-hand path integral expression from the left-hand operator expression using Grassmann coherent states. This tends to be explained in much more detail in condensed matter texts than particle physics texts; see, for example, Negele and Orland.