Why can't the vector potential be seen as the wave function of a photon? When quantizing fermions in QFT we take the wave function of a spin $\frac{1}{2} $ particle and underpromote it to a classical field. We then quantize this very much classical field, giving rise to the "second quantization" (horrible expression in my opinion). We could go the other way around. For example if we wanted a theory in first quantization we could start with a classical spin $\frac{1}{2} $ field and promote it to represent the wave function of a fermion. Why can't we do the same for the electromagnetic field? When we quantize the electromagnetic theory we quantize the classical field $A^{\mu}$, which is almost the same as quantizing the classical fermion field. If we wanted a "photon wave function" in first quantization couldn't we just promote the classical $A^{\mu}$ field to be the wave function of a photon in first quantization? My intuition says this has something to do with the fact that photons have zero mass but I can't really grasp it.
 A: I think this is a very good question from the perspective of a student learning QFT. Your confusion (your question) appears because most textbooks or courses in QFT, it mixes the current understanding of QFT with the historical development of QFT (A very bad practice from my point of view).
In the current understanding, what we call spin is a pure relativistic effect associated with the different representations of the Lorentz group. From this perspective, we can say that the scalar field (Klein-Gordon field), the spinor field (Dirac/Majorana field) and the boson fields (Maxwell field and Yang-Mills fields) are classical fields, in the sense, they are no quantized yet. What's more, one can couple of all these fields with the gravitational field using the first-order formalism of General Relativity in a pure classical field setting. The punchline is there is nothing "quantum" in General Relativity,  then the photon field and the Dirac field can be considered as the same kind of beast (Classical beasts in opposite to the quantum beasts)
In the historical development, the photon field appeared firstly in the classical context of electromagnetism while the Dirac field altogether with the Klein-Gordon field appeared in the context of relativistic quantum mechanics. So, these fields were treated as different kinds of beasts, photon as a classical field, while Dirac/Klein-Gordon as wave functions, but this distinction is wrong. Even more, relativistic quantum mechanics theory is wrong, so the interpretation of the Dirac field a wave function is also wrong. In all these historical confusions is where the old-fashioned concept "second quantization" appeared (which is conceptually wrong).
So, answering your question directly. We do not try to interpret the photon field as a wave function because this interpretation is conceptually wrong (Not even mentioning potential mathematical problems).
As a note, technically speaking, the Dirac field was never a wave function, so we cannot underpromote it from a wave function to a classical field, and then, proceeding to quantize it (The whole concept of 2nd quantization is wrong).
