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

[A. Neumaier]

In a last couple of days I was reading about a whole new line of research about IR physics, reviewed in https://arxiv.org/abs/1703.05448. In particular, it is argued that large gauge transformations can map physical states into new physically inequivalent states.

The part of Strominger's work you refer to is actually not so new.

It is known at least since the 1960s that relativistic quantum field theories with zero mass particles in the defining Fock space (which includes QED and QCD) have a nontrivial superselection structure, in which the superselection sectors define physically inequivalent Hilbert spaces, and that large gauge transformations provide (because of the inequivalence somewhat ill-defined) maps between these sectors. Ignoring this superselection structure is the root of the infrared problems.

For a reasonably rigorous discussion see the 1968 papers by Kibble.

In the context of this thread, it is argued that, in the absence of confinement, quarks in a color singlet state may physically turn into a color non-singlet state. https://arxiv.org/abs/1707.08016

This doesn't help your thesis that single quarks at rest might exist, since the physical Hilbert space of QCD only consists of uncolored (gauge invariant) states.

 

[Demystifier]

... the physical Hilbert space of QCD only consists of uncolored (gauge invariant) states.

Does this statement depend on the assumption of confinement? Or is it a direct consequence of gauge invariance as such? If the latter is the case, can it be proved rigorously?

 

[A. Neumaier]

Does this statement depend on the assumption of confinement?

No. It is an assumption that goes into the canonical construction of QCD. There is no sensible canonical Hilbert space with colored states. See the literature cited earlier.

Or is it a direct consequence of gauge invariance as such?

It is the only known way to make exact nonabelian gauge invariance work in a Hilbert space setting.

If the latter is the case, can it be proved rigorously?

See the literature cited earlier.

 

[Demystifier]

No. It is an assumption that goes into the canonical construction of QCD. There is no sensible canonical Hilbert space with colored states. See the literature cited earlier.

It is the only known way to make nonabelian gauge invariance work in a Hilbert space setting.

See the literature cited earlier.

But then the same should also work for gauge group SU(2) of weak interactions. In particular, electrons should not exist in states with SU(2) charge. Any yet, single electrons exist. Isn't it a contradiction?

 

[A. Neumaier]

But then the same should also work for gauge group SU(2) of weak interactions. In particular, electrons should not exist in states with SU(2) charge. Any yet, single electrons exist. Isn't it a contradiction?

This is because the SU(2) symmetry is broken at the vacuum level, and hence for single particle states that are interpreted on that level. Renormalization destroys the gauge symmetry and with it its Hlbert space structure.

Let me suggest that you properly learn quantum gauge field theory rather than making claims about something where you don't understand the foundations. The latter are much more than writing down a discretized action and putting it on a lattice, so that Bohmian mechanics can be applied!

 

[vanhees71]

Renormalization does not destroy the gauge symmetry of the standard model. That discovery was worth a Nobel prize for 't Hooft and Veltman in 1999.

It's a somewhat complicated issue that the naive way to calculate S-matrix elements with "asymptotic free single electrons" carrying weak (and of course electric) charges lead to correct results in perturbation theory. Observable are of course also here only gauge-invariant quantities. For a concise treatment, see the following manuscript by Axel Maas (Uni Graz):

https://static.uni-graz.at/fileadmin/_Persoenliche_Webseite/maas_axel/ew2021.pdf

 

[Demystifier]

Let me suggest that you properly learn quantum gauge field theory rather than making claims about something where you don't understand the foundations. The latter are much more than writing down a discretized action and putting it on a lattice, so that Bohmian mechanics can be applied!

Here we are discussing questions the answers to which cannot easily be found in textbooks. I am confused with this stuff, and I want to understand it better. My claims are not definite statements, but provisional counterarguments that should motivate more careful thinking. Even though I do think that some (not all) of QFT can be better understood from the lattice point of view, my main motivation for this does not come from Bohmian mechanics. Perhaps both my views about gauge theories and my views about Bohmian mechanics originate from a third prejudice, which roughly could be stated as a moto "Symmetries are overrated".

 

[A. Neumaier]

Renormalization does not destroy the gauge symmetry of the standard model. That discovery was worth a Nobel prize for 't Hooft and Veltman in 1999.

I'd rather say that they got the Nobel prize for the converse statement, namely the proof that gauge symmetry does not destroy the renormalizability of the standard model.

I was referring to the fact that after renormalization of a gauge theory with broken gauge symmetry at $T=0$ you have lost the gauge symmetry in the renormalized Hilbert space of the vacuum sector.

 

[A. Neumaier]

Here we are discussing questions the answers to which cannot easily be found in textbooks.

But much of it can be found in the standard reference already given in my post #67.

Perhaps both my views about gauge theories and my views about Bohmian mechanics originate from a third prejudice, which roughly could be stated as a motto "Symmetries are overrated".

This is quite a limiting prejudice. In fact, symmetries are underrated. They are the main stuff that makes complex situations tractable and hence understandable.

I have the opposite motto: quantum mechanics (and quantum field theory) is essentially applied Lie groups (i.e., applied symmetries). This is a very fruitful motto that allowed me to understand a huge amount of details in terms of a single principle.

 

[Demystifier]

I have the opposite motto: quantum mechanics (and quantum field theory) is essentially applied Lie groups (i.e., applied symmetries).

Now we know where our disagreements come from.

 

[Demystifier]

I'd rather say that they got the Nobel prize for the converse statement, namely the proof that gauge symmetry does not destroy the renormalizability of the standard model.

That's wrong. As you stated it looks as if renormalizability would be more obvious if there was no gauge symmetry, which is wrong. It is exactly the opposite, it looked as if the Standard Model was not renormalizable, but then it turned out that gauge symmetry makes it renormalizable, essentially because some problematic terms cancel out due to the symmetry.

I was referring to the fact that after renormalization of a gauge theory with broken gauge symmetry at $T=0$ you have lost the gauge symmetry in the renormalized Hilbert space of the vacuum sector.

Spontaneous breaking of gauge symmetry is a misnomer. Gauge symmetry cannot be spontaneously broken, which @vanhees71 will explain to you better than I would.

 

[Demystifier]

But much of it can be found in the standard reference already given in my post #67.

In #69 I explained that the paper only proves non-existence of localized colored states, where localized means not having an infinite Coulomb-like tail.

 

[A. Neumaier]

In #69 I explained that the paper only proves non-existence of localized colored states, where localized means not having an infinite Coulomb-like tail.

This is enough since nonlocalized states are not square integrable. Hence they cannot lead to associated probabilities, which makes them unobservable.

As you stated it looks as if renormalizability would be more obvious if there was no gauge symmetry, which is wrong. It is exactly the opposite, it looked as if the Standard Model was not renormalizable, but then it turned out that gauge symmetry makes it renormalizable, essentially because some problematic terms cancel out due to the symmetry.

Before their work it looked as if nonabelian gauge symmetry (obviously present in the standard model) did destroy renormalizability, because it can only be achieved with interactions that seemed to be nonrenormalizable based on power counting. They showed that in fact it does not destroy renormalizability, essentially because some problematic terms cancel out due to the symmetry. Thus ...

they got the Nobel prize for the converse statement, namely the proof that gauge symmetry does not destroy the renormalizability of the standard model.

... but in fact helps to reduce the degree of renormalizability.

Spontaneous breaking of gauge symmetry is a misnomer.

Misnomer or not, it is the term that is generally used.

 

[Demystifier]

This is enough since nonlocalized states are not square integrable. Hence they cannot lead to associated probabilities, which makes them unobservable.

Are you forgetting that we are talking about classical and quantum field theory, and not about quantum mechanics of particles? In field theory, the field is not a probability amplitude. The Coulomb field around an electric charge is not a probability amplitude and does not need to be square integrable. (If I made such a trivial error, you would probably argue that it's because I'm a Bohmian who thinks that everything needs to be explained in terms of pointlike particles. It seems that deep in your bones you are more Bohmian than I am. )

 

[A. Neumaier]

Are you forgetting that we are talking about classical and quantum field theory, and not about quantum mechanics of particles?

No.

In field theory, the field is not a probability amplitude.

But probabilities are still taken between states, and these must be normalizable!
Don't forget that your statement was not about colored fields but about colored states!

 

[vanhees71]

That's wrong. As you stated it looks as if renormalizability would be more obvious if there was no gauge symmetry, which is wrong. It is exactly the opposite, it looked as if the Standard Model was not renormalizable, but then it turned out that gauge symmetry makes it renormalizable, essentially because some problematic terms cancel out due to the symmetry.

Spontaneous breaking of gauge symmetry is a misnomer. Gauge symmetry cannot be spontaneously broken, which @vanhees71 will explain to you better than I would.

Indeed, the Ward-Takashi/Slavnov-Taylor identities save the day concerning renormalizability. Historically, this became clear when 't Hooft discovered exactly this, when Veltman told him that there was a term most probably destroying renormalizability of non-Ablelian gauge theories, and 't Hooft figured out that this is not the case. It's a very exciting story, written up here:

F. Close, The infinity puzzle, Basic Books (2011)

Be warned! It's a page turner. I lost at least one night of sleep, because I couldn't stop reading on ;-).