# Fermionic Version of the effective Action

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.

• could you specifiy better what your problem is? the standard definitions apply for grassman fields too
– tbt
Dec 1, 2020 at 22:37
• So for Grassmann fields one also has $\Gamma [\varphi, \bar{\varphi}] = -F[ j[\varphi, \bar{\varphi}], j^*[\varphi, \bar{\varphi}] ] + \int j[\varphi, \bar{\varphi}] \varphi^* + j^*[\varphi, \bar{\varphi}] \varphi$ ??? Dec 1, 2020 at 22:47
• 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". Dec 1, 2020 at 22:53
• Use grassmann derivatives to "extremise" the action. Taylor series work the same way etc @Antihero
– user21299
Dec 2, 2020 at 0:45
• Try Chapters 7 and 8 of "Dynamical Symmetry Breaking in Quantum Field Theories" by V. A. Miransky Dec 2, 2020 at 1:42

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