Using Grassmann variables on fermionic theories

Question

I think the best way to put my question is the following: what (fermionic) theories make use of Grassmann variables?

Let me clarify my question a little further. I remember some discussions in quantum mechanics books about many-particle systems and, in particular, many-particle fermions. At least in the cases I know, these discussions are commonly superficial in the sense that they do not intend to solve anything, but rather they aim to introduce the mathematical settings of such systems, i.e. how to construct the Fock space, how to define the Hamiltonian etc. The discussion often ends there and usually no Grassmann algebra is used.

The first time I saw the use of Grassmann variables in a physics model was in QFT, where Dirac fields are quantized by using these variables.

However, the Grassmann algebra is a whole universe and it is certainly not just applicable in the Dirac field quantization. In fact, many references like this one make use of Grassmann variables alone to define a theory with a Hamiltonian, creation and operators and so on. This is a fermionic theory as the name suggests, but I cannot understand what it describes, that is, what physical systems are described by this formalism.

So, back to my question. What are these fermionic models described by Grassmann variables? Are they necessarily field theories? (Why is not used in quantum mechanics of many-particles systems otherwise?)

Remark: I obviously do not expect an answer which lists every possible fermionic theory which is described by Grassmann variables. Instead, I'd really like to know how one passes from a general physical theory to its description via Grassmann algebras. If possible, what are these Grassmann integrals trying to calculate.

Answer 1

The short answer is that Grassmann variables are needed when one needs to use the method of Path Integral Quantization (instead of Canonical Quantization) for Fermi fields. That applies for all theories of fermions.

All fermions must be described by anti-commuting fields and so apply the method of path integral, one will need to do integrals with anticommuting variables. So that's where the description of Grassmann variables comes in as the ordinary integral (of commuting variables like the complex numbers) obviously does not work.

For a long answer, you can follow Weinberg's development in his QFT Vol. 1 book. There he showed the full reasoning from why particles with half-integer spin must be described by anti-commuting fields (spin-statistics theorem) to why the path integral approach was needed and how it was formulated for fermions.

Answer 2

Second-quantized fermions (in any spacetime dimension) obey canonical anticommutation relations (CAR). Fermions are therefore (i) classically (i.e. when $\hbar$ is zero) or (ii) quantum mechanically in the path integral formalism described by Grassmann-odd supernumbers, while (iii) quantum mechanically in the operator formalism they are described by Grassmann-odd operators.

However, when considering an equation that is linear in $\psi$ (such as, e.g. the Dirac equation), it becomes agnostic to whether we treat $\psi$ as Grassmann-odd or Grassmann-even. Therefore it is often sufficient to consider Grassmann-even wavefunctions in first-quantized formulations.

For more information, see also e.g. this & this related Phys.SE posts.