# 1-Loop Approximation for Fermionic Effective Action

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?

• Saw where? Which page? Jan 17, 2021 at 3:28