I want to throw in a simple way to think about quantum field theory, for people who understand the quantization of the simple harmonic oscillator.
You think of the field’s Fourier modes as independent harmonic oscillators. The quantum field is therefore a tensor product of uncountably many quantized harmonic oscillators. Call the energy levels of a single oscillator |0>, |1>, |2>, etc. In QFT 101, you say that one increment of energy level in one mode corresponds to one particle with momentum p = hbar.k, where k is the wave vector of the Fourier mode. So the ground state of the whole field is Π_p |0>_p (by which I mean the direct product of ground states |0> for all momenta p, i.e. for all modes), and the state of the field corresponding to one particle with momentum p0 and nothing else is |1>p0 x Π(p!=p0) |0>_p.
If you want to represent, in field terms, the presence of a particle not in a momentum eigenstate, first you represent the single-particle wavefunction as a superposition of single-particle momentum eigenstates, and then you write down the sum of the corresponding field states, each as above. That sum is the quantum field state you were after.
You can do the analogous thing for multi-particle wavefunctions. Also, this is all for bosons. For fermions, the only states available to each mode are |0> and |1>.
I just wanted to spell this out so (a few) people can see what it takes to represent quantum field states as states of a Hilbert space. The Hilbert space in question looks like a tensor product of uncountably many harmonic-oscillator Hilbert spaces, each of which is countably infinite in size.
Usually, instead of talking about Π_p |0>_p, you just talk about “|0>_vac”, and particle states are described in terms of “creation and annihilation operators” which add or subtract the presence of a particle. As komponisto says, you never have uncountably many particles there at once, so it seems like there is excess mathematical structure here. But that is the structure you get if you just straightforwardly quantize a field.
I want to throw in a simple way to think about quantum field theory, for people who understand the quantization of the simple harmonic oscillator.
You think of the field’s Fourier modes as independent harmonic oscillators. The quantum field is therefore a tensor product of uncountably many quantized harmonic oscillators. Call the energy levels of a single oscillator |0>, |1>, |2>, etc. In QFT 101, you say that one increment of energy level in one mode corresponds to one particle with momentum p = hbar.k, where k is the wave vector of the Fourier mode. So the ground state of the whole field is Π_p |0>_p (by which I mean the direct product of ground states |0> for all momenta p, i.e. for all modes), and the state of the field corresponding to one particle with momentum p0 and nothing else is |1>p0 x Π(p!=p0) |0>_p.
If you want to represent, in field terms, the presence of a particle not in a momentum eigenstate, first you represent the single-particle wavefunction as a superposition of single-particle momentum eigenstates, and then you write down the sum of the corresponding field states, each as above. That sum is the quantum field state you were after.
You can do the analogous thing for multi-particle wavefunctions. Also, this is all for bosons. For fermions, the only states available to each mode are |0> and |1>.
I just wanted to spell this out so (a few) people can see what it takes to represent quantum field states as states of a Hilbert space. The Hilbert space in question looks like a tensor product of uncountably many harmonic-oscillator Hilbert spaces, each of which is countably infinite in size.
Usually, instead of talking about Π_p |0>_p, you just talk about “|0>_vac”, and particle states are described in terms of “creation and annihilation operators” which add or subtract the presence of a particle. As komponisto says, you never have uncountably many particles there at once, so it seems like there is excess mathematical structure here. But that is the structure you get if you just straightforwardly quantize a field.