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?
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1$\begingroup$ related: physics.stackexchange.com/q/306426/84967 $\endgroup$– AccidentalFourierTransformCommented Oct 5, 2023 at 16:56
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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.
answered Oct 6, 2023 at 17:12
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$\begingroup$ Thank you for your answer! It is more clear to me now. I was indeed thinking about the Källén-Lehmann spectral decomposition as a way of identifying the particles of the interacting QFT (i.e. asymptotic free particles and bound states). However, in that case I would expect that one could find a new Fock space with new creation/annihilation operators acting over these new bound particles (e.g. in QCD a pion would have its own creation and an annihilation operator, and so on). However, I have never encountered this, which makes me think that I'm wrong about this thought. $\endgroup$ Commented Oct 6, 2023 at 17:34
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$\begingroup$ @TopoLynch As I say, you have to be careful not to mix the different notions: The Fock spaces are usually constructed from the modes of free fields, not from the K-L representation or something else. $\endgroup$– ACuriousMind ♦Commented Oct 6, 2023 at 17:44
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$\begingroup$ Yes, I understand that both notions must not be mixed. What I am still confused about is on whether the new particle content in the interacting theory, manifested as new poles in the Källén-Lehmann spectral function $\rho(s)$, define a new Fock space with associated creation/annihilation operators. E.g. if we find that there is a new bound state appearing as a pole for $k^2=m_b^2$, is there an operator $\hat a_b$ such that $\hat a_b^\dagger|0\rangle$ describe a state with only this new particle? Would these new operators define a new Fock space with the same properties as in the free case? $\endgroup$ Commented Oct 7, 2023 at 12:37
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$\begingroup$ @TopoLynch No, the interacting state space cannot be a Fock space like for free theories due to Haag's theorem. That's why we go to the asymptotic free fields to define the particles. Generally there is very little known about the true structure of the interacting state space. $\endgroup$– ACuriousMind ♦Commented Oct 7, 2023 at 13:09