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There're a bunch of models of fermions with quartic self-interactions. There's an introduction from this wikipedia page.

For example, one can construct the Soler model of self-interacting Dirac fermions:

$$\mathcal{L}=\bar{\psi}(i\displaystyle{\not}\partial-m)\psi+\frac{g}{2}(\bar{\psi}\psi)^{2}.$$

But these fermionic fields in the classical Lagrangian should be Grassmann number valued, i.e. $\left\{\bar{\psi},\psi\right\}=0$, and $\psi\psi=\bar{\psi}\bar{\psi}=0$. Then the quartic interaction automatically vanish. So in the partition function

$$\mathcal{Z}=\int\mathcal{D}\psi\int\mathcal{D}\bar{\psi}\,e^{-S\,[\bar{\psi},\psi]}$$

it seems that the quartic term makes no contribution.

Am I misunderstanding anything?

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The Dirac spinor in $d$ spacetime dimensions has $2^{[d/2]}$ complex Grassmann-odd components. Crossterms can survive in the quartic interaction term.

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  • $\begingroup$ Can you say something about the one dimensional case in the wikipedia page of the Soler model? It says in one dimension, this model is known as the massive Gross–Neveu model. In one dimensions, there's only one component. How does it make sense to have a quartic interaction? Thanks. $\endgroup$
    – Valac
    Mar 7, 2023 at 10:06
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    $\begingroup$ The Dirac spinor in 1+1D has 2 complex Grassmann-odd components. $\endgroup$
    – Qmechanic ♦
    Mar 7, 2023 at 10:25

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