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*though in a way that it diverges when the cutoff is removed, so that nothing physical remains.*" This is for an interacting theory. There have also been short, relevant previous discussions here and here, also regarding interacting theories.

In non-interacting theory, the vacuum state an eigenstate of the particle number operator (eigenvalue zero). Ok, all good. Now, for an interacting theory, you introduce a cutoff allowing you to work in Fock space, and end up with a complicated superposition of bare-particle Fock states... so, now if I were to apply the particle number operator repeatedly in this case, I would get random numbers of bare particles ... but we don't observe this (if we did, it would imply the vacuum contained particles).

Here are my questions:

1. As @A. Neumaier alludes to above - if we remove the cutoff and apply renormalization, does the complex multiparticle state disappear? It seems yes, but in another post @Avodyne noted that this my not possible be in 3 + 1D?

2. Does the

*physical vacuum*of interacting QFT then

*always*contain zero physical particles?

**That is, is the physical vacuum state an eigenstate of the particle number operator, with eigenvalue zero**? (using the interacting Hamiltonian of course)