I am a mathematician self-studying physics, and a currently working on QFT with Srednicki's book.
One thing that bothers me is that for a scalar field (in the Hamiltonian version) there is a harmonic oscillator at every point of momentum space. These all commute with each other, which means that they can be observed simultaneously. This is an uncountable number of harmonic oscillators.
If a classical field is nonzero, then it is nonzero at uncountably many places. So it seems to me that a generic measurement of the field will yield uncountably many points with non-zero energy, corresponding to uncountably many bosons.
However, QFT seems to treat a field as having a countable number of particles. Each Feynmann diagram only involves a finite number of particles, and the LSZ reduction formula deals with a finite number of incoming/outgoing particles.
So my question is, Will a generic field state have countably many particles or uncountably many particles, and how does this reduce to a classical field in the classical limit?