0
$\begingroup$

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.

Improve this question
$\endgroup$
6
  • 2
    $\begingroup$ could you specifiy better what your problem is? the standard definitions apply for grassman fields too $\endgroup$
    – tbt
    Dec 1, 2020 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, 2020 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, 2020 at 22:53
  • $\begingroup$ Use grassmann derivatives to "extremise" the action. Taylor series work the same way etc @Antihero $\endgroup$
    – user21299
    Dec 2, 2020 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, 2020 at 1:42

2 Answers 2

Reset to default
1
$\begingroup$

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).
Improve this answer
$\endgroup$
0
$\begingroup$

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.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.