Is the notion of particles dependent upon the Hamiltonian being quadratic?

Question

I am studying interacting QFT in the context of quantum fields in curved backgrounds, and I am getting some confussion about the concept of particles. To study some gravitational phenomena involving particles (e.g. Unruh effect, Hawking radiation, etc.), it is typically sufficient to deal with free fields, which are expanded in mode functions and particle/antiparticle operators (i.e. its energy eigenstates form a Fock space). This can be done, in general, because the Hamiltonian of free fields is quadratic in the field operators, and therefore one can calculate a single-particle Hamiltonian which can be diagonalized, giving rise to a band structure by means of which we describe this Fock space (eg. for the case of Dirac fermions, the vacuum state, prior to a particle-hole transformation, amounts to a filled lower band/Dirac sea). However, when one considers an interaction term, the Hamiltonian is no longer quadratic in the fields, and this band-structure cannot be obtained by direct diagonalization of the single-particle Hamiltonian (I am not even sure whether the notion of band structure remains). As a consequence, I do not understand if particles/anti-particles can only be defined when the Hamiltonian of the theory is quadratic, i.e., when the evolution of the theory preserves tha Gaussianity of the states. If this is the case, I imagine that a mean field approximation, which turns the Hamiltonian back to a quadratic one, would recover the notion of particles, is this the case?

Answer 1

Yes, the "default" particle states of QFT are only defined in the context of free theories/Hamiltonians. This is why the particle states in the scattering formalism live in the infinite (or "asymptotic") past and future, where the fields are supposed to be effectively free.

That doesn't mean it's completely impossible to talk about notions of "particle" in other contexts, but it does mean that you have to be careful about what you mean by "particle" in these contexts. For example, one way is looking at the Källén-Lehmann spectral representation of your interacting field theory and identifying certain properties of the spectrum there as signatures of particles and/or bound states in the interacting theory.