Stationary phase method for theories with both bosons and fermions

Question

I am wondering if there exists a method to compute path integrals using the stationary phase method for theories with both bosons and fermions. (I am aware of such a method for theories with bosons only.)

One of the most basic problems which bothers me is how to interpret stationary points of the action (or, in other words, solutions of the equations of motion). For examples let us consider the Lagrangian of QED: $$\mathcal{L}=-\frac{1}{4} F_{\mu\nu}F^{\mu\nu}+\bar\psi(i\gamma^\mu D_\mu-m)\psi.\tag{1}$$ The equations of motions are \begin{eqnarray*} \partial_\nu F^{\nu\mu}=e\bar\psi\gamma^\mu\psi,\tag{2}\\ (i\gamma^\mu D_\mu-m)\psi=0.\tag{3} \end{eqnarray*}

I am not sure how to interpret the first equation: the left hand side is bosonic, while the right hand side is fermionic (however it is even as a product of two fermions).

Answer 1

Superized$^1$ theories are theories where its variables are supernumbers, typically of definite Grassmann-parity.

Much mathematics can be generalized to supermathematics, such as, e.g. the stationary phase approximation.

Each supernumber consists of a body and a soul. A Grassmann-odd variables can only have a soul, while a Grassmann-even variable in principle can have both a body and a soul. OP's eq. (2) implies that the Grassmann-even variable $F_{\mu\nu}$ classically$^2$ has no body at the stationary point. See also this and this related Phys.SE posts.

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$^1$ Superized theories are not to confused with supersymmetric theories, which are superized theories with a supersymmetry.

$^2$ At the quantum-mechanical level, contractions of operators can produce a non-zero body.