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I'm trying to understand QFT for the moment and have a question regarding the basic.

So we have a vectorspace (Hilbertspace) of our states. The operator \phi(x) measures the amplitude at point x, whereas the operator \pi(x) measures the momentum density..

The ladder operator

[itex]a^\dagger_p[/itex]

creates a state with momentum p. But it seems here that there are many different states with momentum p. Is the momentum density always homogenously distributed in this case?