Maybe this is why I can't seem to understand the first thing about QFT. In the article on second quantization, they say that the name shouldn't really be "second quantization", because:
One is not quantizing "again", as the term "second" might suggest; the field that is being quantized is not a Schrödinger wave function that was produced as the result of quantizing a particle, but is a classical field (such as the electromagnetic field or Dirac spinor field), essentially an assembly of coupled oscillators, that was not previously quantized.
WHAT??? I thought the Dirac equation was just the relativistic Schrodinger equation, albeit with the added benefit of spin and relativistic corrections. It still gives discrete energy states, involves complex numbers, etc. etc. Whereas, or so I thought, a classical field is something where you can directly measure the value at any point, like you can measure the electric/magnetic field by putting a stationary/moving charge there.
So this must be why the QFT wavefunction is now a functional of Dirac+EM field configurations, right? But I'm still missing that key conceptual link: how does a given Dirac configuration correspond to a single physical reality (such that it is amenable to quantization!), and not a probability distribution like the Schrodinger wavefunction? How do you measure the Dirac field? Or if you can't, then why doesn't that matter, and how do the structures of QFT connect to experiment?