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For a scalar field theory one introduces the partition function with external sources

$$ Z[j] = \int \mathscr{D} \varphi \, \exp \left( -S[\varphi] + \int j \, \varphi \right) \text{,} $$

the analogon of the free energy

$$ F[j] = \ln Z[j] \text{,} $$

and for the mapping

\begin{alignat}{2} C^{\infty}(\mathbb{R}^D) & \longrightarrow \, & C^{\infty}(\mathbb{R}^D) \\ j(\bullet) & \longmapsto & \frac{\delta}{\delta \, j(\bullet)} F[j] = \langle \varphi (\bullet) \rangle =: \phi[j] (\bullet) \end{alignat}

we denote with $j[\phi]$ the formal inverse mapping and define the effective action as

$$ \Gamma[\phi] = - F[j[\phi]] + \int \phi \, j[\phi] \text{.} $$

I seek to calculate the effective action $\Gamma^{4}$ (within first order perturbation theory) for a fermionic field theory with an action that is quartic in the (Grassmann-)fields.

Peskin, Schwartz, Altland and Coleman (my "standard-literature") don't seem to help.

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    $\begingroup$ could you specifiy better what your problem is? the standard definitions apply for grassman fields too $\endgroup$ – tbt Dec 1 '20 at 22:37
  • $\begingroup$ So for Grassmann fields one also has $\Gamma [\varphi, \bar{\varphi}] = -F[ j[\varphi, \bar{\varphi}], j^*[\varphi, \bar{\varphi}] ] + \int j[\varphi, \bar{\varphi}] \varphi^* + j^*[\varphi, \bar{\varphi}] \varphi $ ??? $\endgroup$ – Antihero Dec 1 '20 at 22:47
  • $\begingroup$ My problem is that I seek to do a one-loop-approximation and arrive at the fermionic version of the formula $\Gamma_{\text{one loop}} [\varphi, \varphi^*] = S [\varphi, \varphi^*] + \frac{1}{2} \operatorname{tr} \, \operatorname{log} S^{(2)} [\varphi, \varphi^*]$. But this formula is derived under the assumption that the pair $(\varphi, \varphi^*)$ extremizes the action. Now as one is working with Grassmann variables I am unsure of how to interprete something like "extremizing the action". $\endgroup$ – Antihero Dec 1 '20 at 22:53
  • $\begingroup$ Use grassmann derivatives to "extremise" the action. Taylor series work the same way etc @Antihero $\endgroup$ – alexarvanitakis Dec 2 '20 at 0:45
  • $\begingroup$ Try Chapters 7 and 8 of "Dynamical Symmetry Breaking in Quantum Field Theories" by V. A. Miransky $\endgroup$ – MadMax Dec 2 '20 at 1:42
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The standard definition of the 1PI effective action applies to Grassmann-odd fields as well (up to sign-conventions), cf. above comment by user tbt. E.g. Ref. 1 defines in QED

$$ \Gamma[A_{\rm cl},\bar{\psi}_{\rm cl},\psi_{\rm cl}]~=~W_c[J,\eta,\bar{\eta}]-\int\! d^4x (J^{\mu} A_{\mu} +\bar{\psi}_{\rm cl}\eta+ \bar{\eta}\psi_{\rm cl}).\tag{8.1.76}$$

References:

  1. L.S. Brown, QFT, 1992; eq. (8.1.76).
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Okay so the consensus is that it is usual to define the Grassmann analogon of the effective action in a similar way as one does for scalar field theories. For example: It's done in the book from @Qmechanic or here in chapter 1.6.2.

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