Why is the Jacobian factor for fermionic variables different from that for bosonic ones?

Question

In Srednicki's textbook Quantum Field Theory, Section 77 discusses anomalies and the path integral for fermions. The path integral over the Dirac field is defined to be

\begin{equation} Z(A) \equiv \int \mathcal{D}\Psi\mathcal{D}\overline{\Psi} e^{iS(A)} \tag{77.8} \end{equation} where \begin{equation} S(A) \equiv \int d^{4}x \overline{\Psi}i\displaystyle{\not}{D}\Psi \tag{77.9} \end{equation} is the Dirac action, $i\displaystyle{\not}{D}= i\gamma^{\mu}D_{\mu}$ is the Dirac wave operator, and \begin{equation} D_{\mu} = \partial_{\mu} -igA_{\mu} \end{equation} is the covariant derivative.

Now consider an axial U(1) transformation of the Dirac field, but with a spacetime dependent parameter $\alpha(x)$: \begin{equation} \Psi(x) \rightarrow e^{-i\alpha (x)\gamma_{5}} \Psi(x) \tag{77.12} \end{equation} \begin{equation} \overline{\Psi}(x) \rightarrow \overline{\Psi}(x)e^{-i\alpha (x)\gamma_{5}} \tag{77.13} \end{equation} We can think of eqs. (77.12) and (77.13) as a change of integration variable in eq.(77.8); ... ...

The change of variable in eqs.(77.12) and (77.13) is implemented by the functional matrix \begin{equation} J(x, y) = \delta^{4}(x-y) e^{-i\alpha (x)\gamma_{5}} \tag{77.17} \end{equation} Because the path integral is over fermionic variables (rather than bosonic), we get a $(\det J)^{-1}$ (rather than $\det J$) for each of the transformations in eqs. (77.12) and (77.13), so that we have
\begin{equation} \mathcal{D}\Psi \mathcal{D}\overline{\Psi}\rightarrow (\det J)^{-2}\mathcal{D}\Psi \mathcal{D}\overline{\Psi} \tag{77.18} \end{equation}

I don't understand this. Why is the Jacobian factor for fermionic variables $(\det J)^{-1}$ while that for bosonic ones $det J$?

Answer 1

Well, the full explanation comes from Berezin integration of Grassmann-odd variables.

1. One argument is that the Jacobian matrix is a supermatrix, so that the Jacobian determinant is given by the superdeterminant.

2. Another argument is that Grassmann-odd integration is the same as Grassmann-odd differentiation! The Jacobian factor can therefore be understood from the chain rule of differentiation (rather than integration by substitution).

Both arguments lead to that the Jacobian appears inverted.