# Quark confinement and meson confinement

tom.stoer
see my post #21

A. Neumaier
2019 Award
see my post #21

tom.stoer
Can you please show - in terms of post #21 - what goes wrong? It is nothing else but standard Dirac constraint quantization. I already said that the argument may by problematic in non-compact space; but it is strictly valid in compact space, e.g. for the 3-torus.

A. Neumaier
2019 Award
Can you please show - in terms of post #21 - what goes wrong? It is nothing else but standard Dirac constraint quantization. I already said that the argument may by problematic in non-compact space; but it is strictly valid in compact space, e.g. for the 3-torus.
Well, on the 3-torus there are no asymptotic states, so my argument against your conclusion becomes obsolete. So it is the non-compactness that must be the culprit.

Note that rigorous QFT on compact spaces is much easier since both Haag's theorem and the IR problem are absent. Indeed, the infinite-volume limit (which is not needed for compact space-time) is the hardest step in constructive field theory since it requires the most stringent estimates.

Arthur Jaffe and Edward Witten write in their description of the quantum Yang-Mills theory Millennium Problem (http://www.claymath.org/library/MPP.pdf#page=114): [Broken]
''So even having a detailed mathematical construction of Yang–Mills theory on a compact space would represent a major breakthrough. Yet, even if this were accomplished, no present ideas point the direction to establish the existence of a mass gap that is uniform in the volume. Nor do present methods suggest how to obtain the existence of the infinite volume limit.''

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tom.stoer
Well, on the 3-torus there are no asymptotic states, so my argument against your conclusion becomes obsolete. So it is the non-compactness that must be the culprit.

Note that rigorous QFT on compact spaces is much easier since both Haag's theorem and the IR problem are absent. Indeed, the infinite-volume limit (which is not needed for compact space-time) is the hardest step in constructive field theory since it requires the most stringent estimates.'
So we agree. The argument is fine for compact spaces but there's a loophole for non-compact spaces - unfortunately I don't know how to save the idea - but I think I do not have to be more clever than Witten and Jaffe :-)

A. Neumaier
2019 Award
So we agree. The argument is fine for compact spaces but there's a loophole for non-compact spaces - unfortunately I don't know how to save the idea - but I think I do not have to be more clever than Witten and Jaffe :-)
Yes, but since the argument fully breaks down for QED, I wouldn't call it a loophole but complete lack of argument.

I believe that a single quark in a sea of gluons is a valid sector of QCD, though not one realized in Nature, because the real universe contains many quarks and is colorless.

tom.stoer
Yes, but since the argument fully breaks down for QED, I wouldn't call it a loophole but complete lack of argument.
Certainly not. It is used in all canonical, non-perturbative approaches to QCD! There are numerous groups doing exactly that.

The appraoch is always identical:
- solve Gauss law = eliminate unphysical degrees of freedom
- solve QCD in the color-neutral sector

A. Neumaier
2019 Award
Certainly not. It is used in all canonical, non-perturbative approaches to QCD! There are numerous groups doing exactly that.

The appraoch is always identical:
- solve Gauss law = eliminate unphysical degrees of freedom
- solve QCD in the color-neutral sector
But the second step involves an additional assumption. One could instead solve QCD in a colored sector, and would get meaningful mathematical results at the same level of approximation as for the color-neutral case. But since this is not useful for phenomenology, it is not being done.

tom.stoer
That's not true. As I said G(x) ~ 0 which is (in the Dirac-formalism) translated into G(x)|phys> = 0 requires Q|phys> = 0. So if we restrict to localized states there should be no difference between the spectrum in R³ and T³ and therefore color-neutrality is not assumptions but strictly proven.

The only assumption I can see is that T³ and R³ have (approximately) the same physical content (for localized states / hadron spectroscopy, form factors, ...).

[But afaik there is not one single mathematically exact result b/c already at the classical level one cannot get rid of all gauge = Gribov ambiguities which is required for solving the Gauss law constraint. So there are always artefacts like isolated Gribov copies or ghosts or ... The only way out is lattice QCD where gauge fixing is not required b/c the path integral over the gauge group is finite and gauge fixing can safely be replaced by gauge averaging (it may slow down computations; I don't know]

A. Neumaier
2019 Award
That's not true. As I said G(x) ~ 0 which is (in the Dirac-formalism) translated into G(x)|phys> = 0 requires Q|phys> = 0. So if we restrict to localized states there should be no difference between the spectrum in R³ and T³ and therefore color-neutrality is not assumptions but strictly proven.
The unproved assumption used in your argument is that the states are localized.
But precisely this assumption is violated in charged stated, because of infrared issues, which are closely related to the infinite volume limit, i.e., noncompactness.
This can already be seen in QED, where the IR problem is tractable, and the physical charged states (involving a coherent admixture of photons) are associated to superselection rules coming from asymptotic conditions at infinity. There is no reason to believe that in QCD the situation should be much simpler than in QED.

tom.stoer
So you say one should throw away all QCD results based on T³

A. Neumaier
2019 Award
So you say one should throw away all QCD results based on T³
I never said that. One should view the QCD results based on the torus as results that may need modification in the infinite-volume limit, at least those modifications that are already needed in case of QED.

tom.stoer
There are of course known differences regarding topology, large gauge transformations / winding number, gauge field zero modes etc. It would not be a disaster to add some more topics to that list.

But: as long as we do not know how the topology of the universe looks like, this is academic; I can't believe (and I hope that I am not completely wrong) that the cosmological constant doe not affect the QCD spectrum :-)

A. Neumaier
2019 Award
There are of course known differences regarding topology, large gauge transformations / winding number, gauge field zero modes etc. It would not be a disaster to add some more topics to that list.

But: as long as we do not know how the topology of the universe looks like, this is academic; I can't believe (and I hope that I am not completely wrong) that the cosmological constant doe not affect the QCD spectrum :-)
You have here a double negation. Did you intend that not not x = x? I'd be surprised....

For the purposes here on earth, the universe can be regarded as being topologically
flat, and gravitation can be treated as an external field. Under these conditions, I believe that both QED and the standard model or its variants are consistent.

tom.stoer