A How can quarks exist if they are confined?

[A. Neumaier]

you can take the Hadronic spectrum and compute a Quark 'mass'. Where 'mass' as you might expect is a bit of a fuzzy scheme dependent concept inside a strongly interacting composite object (it is certainly not the usual pole in the propagater, considering that there are strongly divergent infrared effects at play). In any event this is an active area of research (see the pdg section on this (p726))
http://pdg.lbl.gov/2015/download/rpp2014-Chin.Phys.C.38.090001.pdf

p.726, which you cite, says that the quark masses determined by lattice QCD are bare masses (i.e., parameters of the bare Lagrangians in a suitable approximation) and not physical masses (poles of the propagator). Trying to get the latter by fits to experiment produces values that violate causality: The Kallen-Lehmann decomposition of the propagator contains complex conjugate quark poles where the squared mass $m^2$ has a negative real part, while causality requires masses to be real and nonnegative. This implies that a quasi-free space based on the Fock construction but with the fitted Kallen-Lehmann decomposition leads to an indefinite inner product. Therefore it is not a Fock space in the sense in which the term is used by mathematical physicists (where a Hilbert space must result) but only an ''indefinite Fock space'' of the kind @samalhayat mentioned in a related thread.

As a consequence, the (non-free) field operators for quarks can also be defined only on a Krein space - i.e., a generaliziation of a Hilbert space obtained by replacing the definiteness condition of the inner product by the weaker nondegeneracy condition.

 

[A. Neumaier]

many observations (e.g. Deep Inelastic Scattering) look most natural in terms of quarks.

There is an interesting paper by Casher, Kogut, and Susskind in Phys. Rev. D19 (1974), 732-745. They discuss the exactly solvable toy model of QED_2 in 1+1 dimensional spacetime, which exhibits formal asymptotic freedom and confinement of electrons. They show that some sort of ''deep inelastic scattering of electrons'' occurs although the physical Hilbert space contains no electrons. They extend QED_2 to a model with 3 abelian quarks, leading to mesons and baryons.

 

[Demystifier]

Indeed! In our perception of physical reality there are neither Hilbert and Fock spaces, Lie and other groups in QT, nor configuration and phase spaces, no fiberbundles, Minkowksi and pseudo-Riemannian manifolds in classical physics. These are all description of our perceptions of Nature. It is an astonishing empirical fact that we can order our perceptions (at least the "objective" ones) using these mathematical entities.

Yes. In the formulation of QM I present in the paper linked in my signature, the main object of concern is neither observable (as in standard QM) nor beable (as in the usual formulation of Bohmian mechanics), but a perceptible.

 

[Demystifier]

In what sense?

At a more fundamental level, I think all particles of the Standard Model should be thought of as quasiparticles, in the same sense in which a phonon (the quantum of sound) is a quasiparticle. See the paper linked in my signature.

 

[king vitamin]

At a more fundamental level, I think all particles of the Standard Model should be thought of as quasiparticles, in the same sense in which a phonon (the quantum of sound) is a quasiparticle. See the paper linked in my signature.

I agree; in a strongly interacting quantum many-body system, whether it's a QFT or whether it's the Hubbard model, the excitations one sees at low energy will not have any simple relation to the "fundamental" degrees of freedom which are specified in the Hamiltonian.

 

[DarMM]

Yes sure, there are mathematical problems with (perturbative) gauge models which are solved with some mathematical tools (Faddeev-Popov quantization is more pragmatic, while the operator approach based on BRST is also very illuminating to understand some finer aspects). No matter which mathematical sophistication is necessary, one must not forget that these are all descriptions of nature, not nature itself!

Reading over this thread and noticed I never responded to this.

Although this is true, it's not really related to what was being discussed, which was the term "physical Hilbert space". This is a purely technical term in the quantisation of gauge theories, part of the BRST framework. Although "it's not nature" is true, there's no more need for it with this term than there is for "density matrix". It's just a technical term that has the word "physical" in it.