# How do we reconcile the fermionic wavefunction and fermion field?

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.)