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In my understanding of QM I expect the quantum state of the proton $| p^+ \rangle$, say in the rest frame, to be an eigenstate of the QCD Hamiltonian $H_{\text{QCD}}$, which describes the dynamics of quarks and gluons, with eigenvalue the proton mass $m_{p^+}$. $$ H_{\text{QCD}} | p^+ \rangle = m_{p^+} |p^+\rangle. $$ I am aware that in QFTs, particle number is not a conversed quantity, i.e. the particle number operator does not commute with $H_{\text{QCD}}$. Thus we can not expect $| p^+ \rangle$ to have definite particle number, which means that we can not have some multiparticle wave function $\psi(x_1,...,x_n)$ of the proton constituents.

My questions are:

  1. Are the above statements correct in the framework of QCD?
  2. The eigenvalue equation above must imply that $|p^+ \rangle$ is stationary, i.e. does not involve in time. This seems to imply that the proton is really a static configuration of quarks and gluons. However usually the interior of the proton is depicted as a complicated mess of interactions that is very dynamic in time. Is this picture wrong?
  3. Can one, at least in theory, write down the state $| p^+ \rangle$, say in terms quarks and gluon fields acting on the vacuum? If yes, is there some generic ansatz or form that is known?
  4. What are the obstacles in determining the state from the eigenvalue equation above? Is it just that it is too hard, or is it actually impossible for some reason?
  5. I know that there are a number of correlation functions that contain information about $| p^+ \rangle$, which are matrix elements of some operators between proton states, such as parton distributions functions. But would it be possible to somehow reconstruct $| p^+ \rangle$ from those functions?
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Oh, you have no idea how absolutely infernal your question is. We are so far away from solving such problems that we do not even know where to begin! If you do not wish to have difficulty counting the sleepless nights, avoid this rabbit hole at once.

Let me give you a teaser. We understand QED a lot better than QCD, but even there, we have difficulty finding the ground state of merely proton and electron. The basic step is the Bethe-Salpeter equation, which has an unphysical relative time. With some extra constraints, the two-body Dirac equation can be written in a way that eliminates the relative time, and also eliminates from the $4 \times 4 = 16$ components into 8 components. This is still unsolved in full QED.

Heck, it is worse than that. Because the photon is massless, QED has an infrared divergence. Really, an electron needs to be dressed by its Coulomb field. One way to dress it is by a method that generates infraparticles. The price to pay here is that the simple pole structure of the propagator turns into some other functional form, negating all the standard LSZ style analysis that QFT is currently built upon.

Another way to solve the infrared problem is to use coherent states, but then there is a new degree of freedom. Or we could manually do infrared renormalisation, cancelling term by term, as is the standard thing to do these days. However, I am really trying to avoid this last solution, because following Causal Perturbation Theory, we now have a much more sensible route to QFT: Namely, it begins with a careful treatment of stuff so that no UV infinities are calculated. It understands that QFT needs to be renormalised, i.e. some quantities like the mass and charge of the electron are input assumptions of the theory, and so will fix them and never change them. In this way, we would not need to do standard renormalisation, and we can provably always get results in agreement with standard renormalised QED stuff. As such, it is really distasteful to have to do standard IR renormalisation after having labouriously removed the UV divergences.

We are now in a better position to answer your questions.

  1. Is incredibly unlikely. Back before QCD and EW unification came along, there were many periods of time whereby people were seeing no hope in using QFT to compute results. So people had ideas similar to yours, and wanted to derive all the relevant physics by only studying the S-matrix. That turned out to be incredibly intractable. Important results were derived, and they have been incorporated into any proper treatment of QFT, but real progress really only happened after QCD and EW ideas were explored in the QFT framework, supplying the stuff that S-matrices alone seems to not be able to supply.

We have to detour again, and point out that the state of studying bound states in QFT is just woeful. This is to be contrasted with studying scattering states in QFT, which is in a much better shape. We really do not know what are the correct first ingredients of a study of bound states, so that we may expand in small correction parameters.

  1. There should be an energy eigenstate for us to study. It can be very complicated, maybe with coherent states of the gluons, or infraparticles of quarks. But we cannot be sure because we do not actually know how the actual final answer should even look like.

  2. and 4. seem to be some extension of the two-body Dirac equation with constraints kinda problem. There is a lot more work in figuring out even what constraints are sensible to impose. As a teaser of how diabolical this is, note that in the two-body Dirac equation case, we used the Poincaré spacetime symmetries to methodically reduce the possible things to depend upon, to just the relative time (in the Bethe-Salpeter equation), and then further derived a physical sensible constraint to get rid of the relative time and make the equations only care about the instantaneous coördinate displacement between the two particles. This problem is already so difficult as to not have been solved, and there would be entirely 4 more free coördinates, if we are even correct in the assumption that 3 free quarks is sufficient to make up the proton.

1.I will not even attempt to answer this. It will only embarrass myself.

I hope you keep your sanity, and help us in this battle some other day. Also, please look up things like your question. There are so much literature on how we all hate how little progress we have on such pressing issues.

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