〈 Berger | Dillon 〉

Isn't in QFT, the "classical" vacuum just the ground state in Hilbertspace of the Hamiltonian constructed from the fields with all interactions higher than quadratic set to zero? (I wish I understood that Hilbert space with an action of fields concretely, say as some L_2)

These operators always exist in any Poincare-invariant QFT. In an interacting gapped QFT, we say the vacuum has "virtual" particles because the Hamiltonian now doesn't commute at different times but we still require the vacuum to have a zero energy eigenvalue.

Mmmh let me brood on that. My fundamental problem is "the state" assumes a (projectivised) Hilbert space on which the field you want to quantise acts. A representation of the Poincaré group (or more generally the isometries of the Lorentzian manifold) should come for free.