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I have a few questions about bound states in QFT and rigorous ways to picture them. If I had to summarise in one sentence the main question it would be something like:

Q: Do we know any example of a QFT (even a toy model, in low dimension, integrable if it helps …) where we have a well-defined Hilbert space, a well-defined Hamiltonian and actual eigenvectors of the Hamiltonian corresponding to bound states of the theory (or at least existence of these eigenvectors)?

Let me now try to detail a bit more what I mean. The original question I was thinking about, which is more physical but which is really unreasonable, would be whether in the standard model QFT, the hydrogen atom could theoretically be seen as an actual eigenvector of the Hamiltonian in the Hilbert space? I would imagine that this should then correspond to some bound state of the theory. With the standard intuition I have from QFT, I would naively picture this as a state in the Fock space with for instance a one-particle excitation of the electron field in a certain wave function and so on … A first problem in this picture is that the description of the proton is already a very difficult task in itself and a major question in QCD, I’m not trying to ask impossible questions., so maybe as a first simplification, one could forget about the internal structure of the proton and consider a model of QED with an electron field and another elementary field (more massive and positively charged) that would play the role of the proton. I am also wondering about bound states in QCD but even without the problems of strong-coupling, I am already wondering about the nature of bound states for QFT with weaker couplings, which is why I took this example of QED. But even there the question might be too hard.

I guess another problem with the Fock-state picture, which might be of a more conceptual nature, is that this somehow should not exist from Haag theorem, should it? Indeed, if I understood well, this theorem tells us that the Hilbert space of the theory is not a Fock space. So I imagine that to get a rigorous answer to the question, one should consider a QFT for which we have a more rigorous handling of the Hilbert space, maybe in one of the axiomatic approaches of QFT? Already here I guess we would have to consider toy models, maybe in lower dimensions. I am not so sure for which models we have a full understanding of the Hilbert space at the moment. This is somehow what brings me to the hopefully more reasonable question at the beginning of this thread: Q: Do we know any example of a QFT (even a toy model, in low dimension, integrable if it helps …) where we have a well-defined Hilbert space, a well-defined Hamiltonian and actual eigenvectors of the Hamiltonian corresponding to bound states of the theory (or at least existence of these eigenvectors)?

In this case, if the Hilbert space is not a Fock space, how should one interpret these states?

Another question is how renormalisation would enter in this picture, if it enters?

More generally, what would be the best way to think about bound states in QFT in a rather rigorous approach (at least conceptually, not only for computing purposes, even just for “existence”)? Is the idea of looking for normed eigenvectors of the Hamiltonian, that works in QM, not the right way to think about bound states in QFT? The poles of the S-matrix seem to be a nice way to access the bound states spectrum of the theory but I was wondering if there is a way to really “construct” the states. I have also looked at the Bethe-Salpeter equation but my understanding was that it does not construct the corresponding states in the Hilbert space.

Note: Searching if there were already answers to these questions on previous threads I have seen for instance Q: How are bound states handled in QFT? The question there is basically the same as mine above with a model of QED forgetting about the proton structure. But I also have the impression that the answers do not discuss directly the construction of states in the Hilbert space. More generally, in the various threads that I saw, I don’t think the questions related to the Hilbert space states were really answered and especially the ones concerning the Haag theorem. Please tell me if I missed a relevant thread.

Thanks for reading and for any answer / reference.

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  • $\begingroup$ You might be interested in the Schwinger model en.wikipedia.org/wiki/Schwinger_model $\endgroup$
    – kaylimekay
    Dec 27 '20 at 12:23
  • $\begingroup$ To clarify: You're asking about relativistic (Poincaré-symmetric) QFT, correct? (Nonrelativistic QFT has easier examples.) And does your definition of bound state require that the same theory also be capable of describing the process of forming that bound state from its constituents? The latter condition would rule out solvable examples of "bound states" whose constituents do not exist as separate particles nonperturbatively, even though they may appear to exist in a small-coupling expansion. $\endgroup$ Dec 27 '20 at 14:34
  • $\begingroup$ Thank you for your comments. Regarding the Schwinger model, do you know if "exactly solvable" in this case also means a direct access to the eigenvectors of the Hamiltonian? For the second comment: yes I had in mind relativistic QFT when asking but I would also be curious about non relativistic statements if they are easier to handle. I imagine there might be some results in condensed matter but I am less familiar with it. And indeed, I also had in mind a model which would built the bound state from elementary constituants but if there are relevant answers without, I am also interested. $\endgroup$
    – Sylvain L.
    Dec 27 '20 at 16:17
  • $\begingroup$ @SylvainL. I think so, but I have not taken the time to study that model to a level that I feel confident answering your question. If you discover something interesting there, I would be happy to learn about it from you. $\endgroup$
    – kaylimekay
    Dec 28 '20 at 13:53
  • $\begingroup$ Thanks for your answer. I will let you know if I find anything. $\endgroup$
    – Sylvain L.
    Dec 28 '20 at 20:10
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There are rigorous results on bound states in scalar $P(\phi)$ QFT's in two dimensions. See for example these two articles by Dimock and Eckmann.

  1. Dimock, J., and J-P. Eckmann. "Spectral properties and bound-state scattering for weakly coupled $\lambda P(\varphi)_2$ models." Annals of Physics 103, 2 (1977): 289-314.
  2. Dimock, J., and J-P. Eckmann. "On the Bound State in Weakly Coupled $\lambda(\phi^6-\phi^4)_2$." Communications in Mathematical Physics 51, 1 (1976): 41-54.
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  • $\begingroup$ Thank you for your interesting answer. I am looking at the articles and have a few questions. Do I understand well that they are based on the Bethe-Salpeter approach? As such, do they construct the bound states as eigenvectors? I see some reference in the paper to a Hilbert space A_\delta: is that connected to the well-defined space of these QFT in some axiomatic approach of QFT? Does this treatment includes a resolution of the problem posed by the Haag theorem? Sorry if my questions are naive ... $\endgroup$
    – Sylvain L.
    Dec 27 '20 at 16:28
  • $\begingroup$ ?1) Yes, the Bethe-Salpeter kernel is the typical conceptual tool used for this kind of problem. ?2) I believe so. Bound states should give eigenvectors of the mass operator $m^2=E^2-P^2$. ?3) I don't know, would have to read the paper in detail. The approach in these papers is constructive QFT instead of axiomatic QFT. ?4) Haag's thm is a waste of time.It just says that naive approaches to rigorously define QFT don't work. The constructive approach goes around that. $\endgroup$ Dec 28 '20 at 17:29
  • $\begingroup$ Thank you for your answers. I am slightly unsure of whether an hypothetical state for the hydrogen atom in QED should be an eigenvector of the mass operator or the Hamiltonian. If you have an eigenvector of the Hamiltonian and the momentum you can always boost and that stays an eigenvector of the mass operator with the same eigenvalue but does that then correspond to some continuous spectrum of the Hamiltonian? Regarding the last point could you develop on why Haag is a waste of time? Is it that it is true but constructive QFT build the right Hilbert space anyway so it is irrelevant? $\endgroup$
    – Sylvain L.
    Dec 28 '20 at 20:42

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