Fermions and Grassmann Integration

The generating functional built the path integral for bosonic fields, where the integration variables are ordinary numbers. Fermions anticommute — the Dirac field is quantized with anticommutators — so their path-integral variables cannot be ordinary numbers. They are Grassmann numbers, and the machinery of Berezin integration reproduces the fermion propagator, the closed-loop minus sign, and the fermionic determinant.

Conventions: $\hslash =c=1$, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

Grassmann numbers

A set of Grassmann (anticommuting) variables ${\theta }_i$ satisfies

$${\theta }_i{\theta }_j=-{\theta }_j{\theta }_i,\text{so in particular}{\theta }_i^2=0.$$

Nilpotency (${\theta }_i^2=0$) means any function of a single Grassmann variable terminates at linear order: $f(\theta )=a+b\theta$. This is the algebraic image of the Pauli exclusion principle — a fermionic mode can be occupied zero or one times, never more. The classical "field" $\psi (x)$ inside the fermionic path integral is therefore a Grassmann-valued field, not a c-number function.

Berezin integration (Definition)

Integration over Grassmann variables is defined (there is no measure-theoretic integral to recover) by the Berezin rules, chosen so that integration equals differentiation:

$$\int d\theta 1=0,\int d\theta \theta =1.$$

The single non-trivial consequence — a Gaussian Berezin integral — runs opposite to the bosonic case. For a pair $\theta ,\bar{\theta }$,

$$\int d\bar{\theta }d\theta e^{-\bar{\theta }a\theta }=a,$$

and for $n$ pairs with a matrix $A$,

$$\boxed{\int \prod _{i}d{\bar{\theta }}_{i}d{\theta }_i{e}^{-{\bar{\theta }}_i{A}_{ij}{\theta }_j}=\det A.}$$

Contrast the bosonic Gaussian $\int \prod dx{e}^{-x^{\top }Ax}\propto (\det A{)}^{-1/2}$: the determinant appears with a positive power for fermions, $+1$ instead of $-\frac{1}{2}$. This single sign flip is the source of every fermionic minus sign in perturbation theory.

The Dirac generating functional

For the Dirac field with action $S=\int d^{4}x\bar{\psi }(i/\partial -m)\psi$, couple Grassmann sources $\eta ,\bar{\eta }$ and integrate:

$$Z[\eta ,\bar{\eta }]=\int \mathcal{D}\bar{\psi }\mathcal{D}\psi \exp \left(i\int d^{4}x[\bar{\psi }(i/\partial -m)\psi +\bar{\eta }\psi +\bar{\psi }\eta ]\right).$$

The Gaussian Berezin integral gives $Z_0=\det (i/\partial -m)$ times the source term, and two source derivatives extract the fermion propagator — exactly the $S_F$ from canonical quantization:

$$⟨0|T\psi (x)\bar{\psi }(y)|0⟩=S_F(x-y)=\left(\frac{i}{i/\partial -m}\right)(x-y).$$

The propagator is the inverse of the Dirac operator, just as the bosonic propagator is the inverse of $(\square +m^2)$; the difference is entirely in the Grassmann bookkeeping.

The two functional signatures of fermions

Both notorious fermionic Feynman rules from the Dirac page drop out of the Berezin calculus without further input:

• Closed-loop minus sign. A closed fermion loop is a trace over a cycle of propagators, and the anticommuting nature of the sources supplies an overall $(-1)$. Diagrammatically this is the $\det A=e^{\mathrm{Tr}\ln A}$ expansion: each loop is a term in $\mathrm{Tr}\ln$, carrying the sign.
• The fermionic determinant $\det (i/\partial -m)$. Integrating out the fermions in a background field leaves this determinant in the effective action — the origin of vacuum-polarization effects, the chiral anomaly (via the non-invariance of $\mathcal{D}\bar{\psi }\mathcal{D}\psi$ under chiral rotations — the Fujikawa mechanism), and the Coleman–Weinberg potential.

Summary

Where this leads

• Effective action from integrating out fields, including the fermion determinant: the effective action.
• The chiral anomaly from the non-invariant fermion measure: chiral anomaly.
• Lattice fermions and the doubling problem, where the Grassmann determinant is the computational bottleneck: lattice field theory.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 9.5.
• Berezin, The Method of Second Quantization.
• Zee, Quantum Field Theory in a Nutshell, Ch. II.5.
• Srednicki, Quantum Field Theory, Ch. 44.