How do we reconcile the fermionic wavefunction and fermion field?

Question

I know in the study of atoms and molecules we use quantum mechanics, where in the case of fermions, we have the property of wavefunction $$ \psi_{\alpha\beta\cdots \gamma}(t)=\psi_{[\alpha \beta \cdots \gamma]}(t) $$ where $\alpha,\beta,\gamma$ are abstract indices for both space and spinors. On the other hand in QFT we talk about Grassmann-valued fields and in the Lagrangian (path-integral) formalism we write down fields like $\Psi,\Phi$ with the property that $$ \Psi\Phi(\vec{x},t)=-\Phi\Psi(\vec{x},t). $$ I believe this is just another question confusing to many other grad students when they study these things. And my question is when I think of a bunch of fermions (identical or partial identity), what/which object should be in my mind? A function from spacetime $\mathbb{R}^{3+1}$ to what? (spinors or the mysterious object Grassmann number which is not even $\mathbb{C}$-valued). The former is a "quantum wavefunction" with norm squared interpreted as a probability for multi-particles, while the latter is a "classical field" with a more preference on the wave side in terms of the centuries-long confusion particle-wave duality.

I know that "coherent state is something like behaves classically" but even so I don't think there's even any classical limit s.t. some quantum state (a complex-valued function) becomes a Grassmann-valued spacetime function under this limit in a miracle way.

I have this confusion arise multiple times in my process of studying in various ways. But it never got solved completely. I wish there was someone that can answer my confusion once and for all. (Sorry for questioning such a beautiful treatment that is century-long and THE most precise theory throughout human civilization. It's just that to think of the behavior of free electrons floating in the air or space as a Grassmann number-valued function is too bizarre and counter-intuitive to me.)

Answer 1

The classical limit requires many particles to be in the same state. This allowed us to experience bosonic fields like the spin-1 electromagnetic field or spin-2 gravitational field as classical limits of quantum fields.

Fermions, by definition, cannot have multiple particles in the same state, and so there is no "classical field" limit of a fermion.

The fermionic field in QFT plays a formal role in defining the quantum theory. You should treat it at as a tool we use to calculate quantum amplitudes for processes involving fermions. You should not try to interpret it classically as a field we could directly measure or that could be in a coherent state like a bosonic field.

As you said, in the path integral formalism, fermions are represented by Grassman-valued fields in the path integral. The fact that a field appears in the path integral, does not imply that there is a limit where we can directly experience a classical version of this field as a coherent state. Instead, doing the path integral with a Grassman field allows us to answer typical questions we want to ask of a quantum field theory, like what is the amplitude for a given scattering process to occur.

In the operator formalism, the fermionic field operator transforms as a spinor and obeys canonical anti-commutation relationships; it is not a Grassman function. Much like a scalar field, there is a difference between the mathematical object that represents the fermionic field in the operator formalism, and in the path integral formalism.