How does boson state operate on fermion state

I have a naive question on how Bosons interact with Fermions.

In quantum field theory, it is stated that the Fermions interact with each other by exchanging Bosons. That means, at a given time, a given Fermion state $psi_f$ interacts with a Boson state $psi_B$. My naive question is: is there a direct way to model the interaction between them? I mean: can we say something like: the Boson state operates on the Fermion state as: $psi_f’rightarrow{e^{ipsi_B}}psi_f$?

I have seen How does a boson interact with a fermion? and I am not asking that question. My question is whether it is possible to conceive the interaction by such a direct method. I understand that $psi_B$ may not be Hermitian.

Regards,
Kolahal