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I am self-studying quantum field theory, and I've been struggling to understand the nature of the states that emerge in quantum field theories.

After thinking about it, what I think one has in the state is like having a wavefunction at each point in space (or in momentum space). Basically, we are substituting a variable with a wavefunction, and we had a variable at each point (a classical field). This picture agrees with the standard QM being a zero-dimensional field.

For example, I've been watching a lecture in which phonons were explained by quantizing a field. The classical fields were the separation from the equilibrium position of each atom and its momentum. Fourier-transforming these we get a superposition of harmonic oscillators. Now, we promote these to operators. Ok, now what I imagine is that these quantum-harmonic-like operators should act on a quantum-harmonic-like space. Also, following the analogy, the one particle state for a specific p would look spread out in the basis of the eigenstates of the operator that corresponds to the amplitude of this Fourier mode.

So, to summarize, am I right in saying that the states in the Fock states are not just plane waves, but plane waves with a fuzzy amplitude? Does this fuzziness have any implications?

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