Undergrad Single Quark at Rest: The Mystery of Dark Matter?

[Demystifier]

What is the construction of this wall/container? Does it contain atomic nuclei? If yes, there will be strong/color force interactions between the wall/container and the plasma.

In principle, I guess one could use Tokamak https://en.wikipedia.org/wiki/Tokamak

 

[malawi_glenn]

In principle, I guess one could use Tokamak https://en.wikipedia.org/wiki/Tokamak

Magnetic fields then, generated by some material, still non zero probablity for a quark to be near enough the material and interact?

 

[Demystifier]

Magnetic fields then, generated by some material, still non zero probablity for a quark to be near enough the material and interact?

In principle, the probability for that can be made sufficiently small.

 

[malawi_glenn]

In principle, the probability for that can be made sufficiently small.

What about the gluons? They are not affected by the magnetic field

 

[Demystifier]

What about the gluons? They are not affected by the magnetic field

Here "gluons" are not free particles, but a Coulomb-like field attached to the quarks.

 

[A. Neumaier]

Why do Wightman axioms not describe the charged sector? (I want a theoretical answer not depending on experimental data.)

Because they describe the sector of a QFT containing the vacuum, which is neutral by definition.

 

[A. Neumaier]

One of the consequences of Wightman axioms is the CPT theorem. What does C stand for in this theorem, if Wightman axioms do not describe the charged sector?

Charge conjugation is a formal operation that applies to all charges, not specifically electric charge.
https://en.wikipedia.org/wiki/C-symmetry#In_quantum_theory
In nonrelativistic theories, CPT symmetry covers electric charge. But in QED, states with different charge quantum number live in different superselection sectors, and the vacuum sector is uncharged.

But the vacuum sector contains (at least perturbatively) states made from any number of electron-positron pairs, which are neutral but on which the perturbative CPT symmetry acts nontrivially because it interchanges electrons and positrons.

 

[Demystifier]

But in QED, states with different charge quantum number live in different superselection sectors, and the vacuum sector is uncharged.

But the vacuum sector contains (at least perturbatively) states made from any number of electron-positron pairs, which are neutral but on which the perturbative CPT symmetry acts nontrivially because it interchanges electrons and positrons.

Interesting! From this point of view, would you say that the electric charged sector of the Standard Model is totally irrelevant for physics, because it can never be realized in experiment? When we "isolate" electron here, we always have a proton (or something else with a positive charge) somewhere else.

 

[malawi_glenn]

Here "gluons" are not free particles, but a Coulomb-like field attached to the quarks.

Yes, but there can still be interaction.

 

[A. Neumaier]

Now going back to the "at rest" part.
What does it mean for a quantum particle to be at rest? Is it even possible?

It means - a particle considered in its rest frame, which exists in the massive case (only).

Interesting! From this point of view, would you say that the electric charged sector of the Standard Model is totally irrelevant for physics, because it can never be realized in experiment? When we "isolate" electron here, we always have a proton (or something else with a positive charge) somewhere else.

Nonsense. QED (and the standard model) has many physically relevant superselection sectors, including charged ones. This is no contradiction with the fact that only its vacuum sector is described by Wightman's axioms.

 

[Demystifier]

Yes, but there can still be interaction.

Yes, but not much different from electromagnetic (EM) interactions. Since we can make walls made of protons and electrons that shield from EM interactions, it seems plausible that something similar might be possible for colored Yang-Mills interactions.

 

[Demystifier]

QED has many physically relevant superselection sectors, including charged ones.

But charged ones cannot be realized experimentally, right? For example, in ion collision experiments, the ions themselves are produced by starting from neutral atoms.

 

[A. Neumaier]

But charged ones cannot be realized experimentally, right? For example, in ion collision experiments, the ions themselves are produced by starting from neutral atoms.

Of course they can. Though they appear in pairs (or highr multiples). But absorbing one charged particle prepares the other oppositely charged particle for further experimentation. This is a standard case of collapse.

 

[malawi_glenn]

Yes, but not much different from electromagnetic (EM) interactions. Since we can make walls made of protons and electrons that shield from EM interactions, it seems plausible that something similar might be possible for colored Yang-Mills interactions.

There will still be color exchange with the wall/container/thing.

Maybe if you could build solid objects with WIMPs or similar :)

 

[vanhees71]

But "collapse" must not be taken too literally, particularly not in relativistic local QFT, where a collapse in the literal sense can never happen, because there cannot be faster-than-light effects of a local measurement (and all measurements we do are local).

 

[A. Neumaier]

But "collapse" must not be taken too literally, particularly not in relativistic local QFT, where a collapse in the literal sense can never happen, because there cannot be faster-than-light effects of a local measurement (and all measurements we do are local).

Well, the collapse is an effective description, projecting away the part irrelevant for the subsequent experiments. Once one measures (and absorption counts as a measurement), relativistic invariance is broken anyway...

 

[vanhees71]

Yes, but it's not something that violates causality and locality. It's an update of the description after a preparation/measurement procedure, but I think that's off-topic in this very interesting thread about what's observable in gauge theories.

I'm a bit sceptic towards the axiomatic-field-theory approach since it hasn't brought us closer to the goal of formulating relativistic (interacting) QFTs mathematically rigorously. If I understand your remarks in this thread right, the Wightman system of axioms can't even describe electron-electron scattering within its scheme since this is a state with non-zero net charge. I think then we have to admit that we don't have a mathematically satisfactory description yet and have to live with the physicists' pragmatic approach of declaring perturbative, maybe also resummed/unitarizing approaches (e.g., in the Schwinger-Dyson approach to QCD) as the best that can be done with the mathematical tools at hand. What then you can "prove" (at a physicists' level of rigor) is the gauge-independence of the S-matrix ("vacuum QFT") or also observables defined within "finite-temperature QFT" as, e.g., black-body radiation or the electromagnetic radiation from a heavy-ion collision from a fireball of locally equilibrated hot and dense medium (QGP-HRG fireball).

 

[A. Neumaier]

If I understand your remarks in this thread right, the Wightman system of axioms can't even describe electron-electron scattering within its scheme since this is a state with non-zero net charge.

The Haag-Kastler framework is more permissive in this respect. since it is not bound to the vacuum sector. In the Wightman framework you need to describe electron-electron scattering by keeping some far away positrons around (in the AQFT parlance a 'charge behind the moon'), which is a bit ugly.

In spite of these shortcomings, the Clay Millennium problem is about existence of pure Yang-Mills QFTs in the Wightman framework, where glueballs are bound states in the vacuum sector.

Things are different if one admits indefinite scalar products; then there is a well-developed and predictive perturbative algebraic theory in Krein spaces.

 

[vanhees71]

I guess this is the formalization of Gupta-Bleuler quantization for the Abelian (QED) case and "covariant operator formalism" for non-Abelian gauge theories a la Kugo, Ojima et al?

 

[Demystifier]

Also in electroweak theory, as in any gauge theory, only gauge-invariant self-adjoint local operators can represent observables.

But in electroweak theory, a physical state can have a non-zero SU(2) charge, am I right? I don't fully understand why physical states can have a non-zero SU(2) charge, but cannot have a non-zero SU(3) (color) charge; where does the difference come from, exactly?

 

[malawi_glenn]

But in electroweak theory, a physical state can have a non-zero SU(2) charge, am I right? I don't fully understand why physical states can have a non-zero SU(2) charge, but cannot have a non-zero SU(3) (color) charge; where does the difference come from, exactly?

Isn't it entirely up to if the model has asymptotic freedom or not? Quarks and SU(3) is not enough, you need to specify representation and number of states for the sign of the beta function

 

[vanhees71]

This is pretty subtle. A standard reference concerning the gauge invariance of observables is

https://doi.org/10.1016/0550-3213(81)90448-X

 

[malawi_glenn]

This is pretty subtle. A standard reference concerning the gauge invariance of observables is

https://doi.org/10.1016/0550-3213(81)90448-X

I wish I had access
...

 

[Demystifier]

This is pretty subtle.

So if I'm wrong in this thread, I believe I'm wrong in a non-trivial and interesting way.

 

[vanhees71]

That's for sure right. It's a highly non-trivial question, how to justify the standard perturbative calculuations in electroweak theory, using non-gauge invariant quantities.

 

[A. Neumaier]

I guess this is the formalization of Gupta-Bleuler quantization for the Abelian (QED) case and "covariant operator formalism" for non-Abelian gauge theories a la Kugo, Ojima et al?

yes.

 

[Demystifier]

I found an interesting analogy in condensed matter physics. For that matter I present two excerpts from the book Guidry and Sun "Symmetry, Broken Symmetry and Topology in Modern Physics".

The idea that colored states of quarks should exist in interacting QCD because they exist in free QCD is analogous to Fermi liquids, where interacting states of electrons are very similar to free electrons, related to them by adiabatic change of the interaction. The opposite idea, that such states don't exist in interacting QCD, is analogous to Cooper pairs (which are analogous to QCD mesons) the energy of which is non-analytic in the coupling constant and diverging for zero coupling.

 

[A. Neumaier]

The idea that colored states of quarks should exist in interacting QCD because they exist in free QCD is analogous to Fermi liquids, where interacting states of electrons are very similar to free electrons, related to them by adiabatic change of the interaction. The opposite idea, that such states don't exist in interacting QCD, is analogous to Cooper pairs (which are analogous to QCD mesons) the energy of which is non-analytic in the coupling constant and diverging for zero coupling.

One problem with your analogies is that they don't show that the first is not realized in QCD.

Another problem is that condensed matter theory is about quasiparticles in a nonrelativistic QFT whose vacuum is a thermal state with a structure very different from the QCD vacuum.

 

[vanhees71]

In QCD colored states are not "realized", because they don't specify a gauge-independent observable. They are simply not interpretable in any physical way.

 

[Demystifier]

In QCD colored states are not "realized", because they don't specify a gauge-independent observable. They are simply not interpretable in any physical way.

A purely theoretical question (not realized in actual nature). What if we extend the Standard Model by adding couplings between Higgs and gluons, analogous to the couplings between Higgs and weak gauge bosons? In that case gluons become massive due to the Higgs mechanism and gauge invariance becomes spontaneously broken. In that theory colored states could be realized, right? And yet the theory is still gauge invariant at a more fundamental level.