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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.

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  • $\begingroup$ Yes, they are arguing by analogy. $\endgroup$
    – Qmechanic
    Dec 12, 2023 at 19:23
  • $\begingroup$ Thank you @Qmechanic. I'm trying to understand exactly where the need for anti commuting integration variables comes from. After revisiting the scalar case in page 283, it seems like the anti commuting nature of the integration variables must necessarily be assumed from the get go for the fermion analogue of eq. 9.15 so as to end up with the correct definition of time ordering for fermionic operators in the fermion analogues of eq. 9.17 and 9.18. Would you agree with this explanation? $\endgroup$
    – Function
    Dec 12, 2023 at 21:45
  • $\begingroup$ For what it's worth, the anti-commutation of the Dirac field is discussed in P&S section 3.5. $\endgroup$
    – Qmechanic
    Dec 13, 2023 at 10:52

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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.

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