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Given a partition function

$$ Z = \int \mathscr{D}(\bar{\psi}, \psi) \, \mathrm{exp} \left( \, - S[\bar{\psi}, \psi] \right) $$

with fermionic (Grassmann) fields. I seek to calculate the effective action $\Gamma[\bar{\Psi}, \Psi]$ within the one-loop approximation. The formula should read

$$ \Gamma [\bar{\Psi}, \Psi] \approx S [\bar{\Psi}, \Psi] - \operatorname{tr} \operatorname{log} S^{2} [\bar{\Psi}, \Psi] $$

Question: What is $ S^{2} [\bar{\Psi}, \Psi] $?

In my logic it should be

$$ S^{2} [\bar{\Psi}, \Psi] = \frac{\delta^2}{\delta \bar{\psi} \delta \psi} S[\bar{\psi}, \psi] \bigg \vert_{\psi = \Psi} $$

(with additional arguments added to the functional derivatives of the fields...) It confuses me that I saw a different formula for the bosonic counterpart; Can anybody recommend some literature with a derivation?

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    $\begingroup$ Saw where? Which page? $\endgroup$
    – Qmechanic
    Commented Jan 17, 2021 at 3:28

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The strategy is described in the book "Quantum Field Theory" by Itzykson and Zuber, Sec. 6-2, and particularly sec. 6-2-2 (pag. 289 of my edition).

The derivation can be generalized to fields of any statistic. Useful references include the lecture notes by K. Narain (from the Diploma in High Energy Physics, lectured at ICTP - Trieste), although I'm not sure these are one to the public (there are video recorded though); and "The global approach to Quantum Field Theory Vol. 1" by B. DeWitt, Chapter 23.

However, your logic is right.

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