# Haag's Theorem and the Poincare Group

• A
Gold Member
But there seems to be agreement that these states all describe composite objects....
True indeed. I guess they're composites without constituents or however one can express the structure of the QCD Hilbert space in English.

Were you in fact referring to direct sums of integrals of irreps as @A. Neumaier said.
Yes, I was using the word "sums" as including integrals. I didn't think the distinction would contribute to the discussion, but apparently this caused some confusion, so I apologize for that.
Ground state hydrogen will involve electrons which are infraparticles, so it's a least not clear to me that it's an irrep as opposed to an integral over irreps.
I chose the hydrogen atom specifically because it is electrically neutral and spherically symmetric. As long as it encounters no other particles, it shouldn't generate any photons, including the infrared ones that cause problems.
Anyway, the only way it could be an integral over irreps is if there is some scalar operator that takes different values for each irrep. Do you see any candidate for this? As far as I can tell, there is just one irrep, the same one that describes a free scalar particle. Please correct me if I am mistaken.
Just to say it is difficult to maintain this definition of composite when it comes to gauge theories like QCD. There none of the states in the Hilbert space have a field present in the Lagrangian. Using your definition all states in QCD are composite.
Indeed, the particles we usually call "elementary" in QCD are the quarks and gluons, which do not appear as states of their own.

DarMM
A. Neumaier
2019 Award
I chose the hydrogen atom specifically because it is electrically neutral and spherically symmetric.
Here is a paper on a relativistic model for the hydrogen atom:

Barut, Schneider, and Wilson, Quantum theory of infinite component fields, J. Math.Phys. 20, 2244 (1979).

Anyway, the only way it could be an integral over irreps
Because hydrogen has bound and scattering states, the mass spectrum is a combination of discrete and continuous. The discrete part leads to a direct sum; the continuous part ot an direct integral. The total Hilbert space is the direct sum of these two parts.

dextercioby
Because hydrogen has bound and scattering states, the mass spectrum is a combination of discrete and continuous. The discrete part leads to a direct sum; the continuous part ot an direct integral. The total Hilbert space is the direct sum of these two parts.
Well I was only ever discussing the ground state. The aroused states do not form simple irreps, even though they are discrete, because under time evolution they emit photons and decay.

Peter Morgan
Gold Member
I typed in a long comment that I again lost, as if it had never existed, because I tried to edit the preview. You all can be happy not to be regaled with my nonsense, but I'm really, really, really cross at that bug.