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PS: I noticed that this is a level 5 vital priority article, but only B class rated. I have studied Yang-Mill gauge theories for over 15 years, and would like to try to contribute to raising this. [[User:PatentPhysicist|PatentPhysicist]] ([[User talk:PatentPhysicist|talk]]) 02:51, 22 February 2021 (UTC)
PS: I noticed that this is a level 5 vital priority article, but only B class rated. I have studied Yang-Mill gauge theories for over 15 years, and would like to try to contribute to raising this. [[User:PatentPhysicist|PatentPhysicist]] ([[User talk:PatentPhysicist|talk]]) 02:51, 22 February 2021 (UTC)


== References ==

Revision as of 02:52, 22 February 2021

Template:Vital article

Integrable solutions of classical Yang-Mills equations and QFT (Disputed section)

Disputed section

One of the main difficulties that one meets on managing Yang-Mills theory at low energies is that Hamiltonian homogeneous equations of the theory admit essentially chaotic solutions and nobody is able to formulate a quantum field theory starting from such solutions. But, beside chaotic solutions, this theory also admits integrable solutions that can be used for these aims. These solutions can be found with the so called Smilga's choice [1] that permits to fully map Yang-Mills theory on a massless field theory and, for this case, the infrared theory has been recently formulated [2]. So, one can compute the propagator and the spectrum [3]. The results appear to be in agreement with computations with lattice field theory [4] yielding a zero momentum propagator

being

,

and a mass spectrum

proper to an harmonic oscillator. The ghost field has the same propagator as a free particle, i.e. it decouples from the gluon field.

An example of such classical solutions can be yielded for SU(2) gauge group (but it is also easy to obtain it for SU(3))

being a Jacobi elliptic function, and integration constants and . This solution for Yang-Mills equations holds only if the following dispersion relation holds

showing in this way as a massless theory can have classical massive solutions.

Computation of the gluon propagator is essential because it permits to obtain, starting from quantum chromodynamics, a Nambu-Jona-Lasinio model and all hadronic phenomenology seen at lower energies can be derived. Research in this field is currently very active and important results are expected shortly.

Science or math?

Is Yang-Mills theory science or math? They are not the same thing.

The lead says, "Yang–Mills theory seeks to describe the behavior of elementary particles..." which suggests that its developers have intended to provide a mathematical model for this aspect of the physical world - that is, they intend to provide a tool for elucidating a scientific theory of the world.

After providing a précis list of the symmetry groups of the Standard Model, in the 'History and theoretical description' section I read "This may be the reason why confinement has not been theoretically proven, though it is a consistent experimental observation." While as a lay person, I understand that confinement is in fact what has been observed, I don't at all understand what - in the context of a scientific theory of the world - can even be meant by "theoretically proven".

I know that it is common to refer to large parts of mathematics as the 'theory' of this or that, as the Theory of Numbers' and so forth and though I think this is not nonsensical, I am uncomfortable with this sort of construction appearing here, if, as in the lead, 'YM Theory' is intended to be an article describing a scientific theory of the natural world. Theorems are objects of mathematics. Theories - which are always contingent and so are unprovable - are objects of science.

In 1979, during his noted series of popular lectures given at the University of Aukland, New Zealand, Richard Feynman said

"We then have the following physical problem as I mentioned before - where does this number [1/α] come from? From experiment. I know. But a good theory would have that this thing is equal to one over two times pi times the cube root of three times six and so on, so that you know what it was, if you know what I mean.
"It's a number that has to be put in that nature has or so to speak, if you were religious you would say 'God has created that number.' But we would like to figure out if we can a little clue as to how He thinks, to make a number like this. For example, maybe that's... [pointing to a decimal on the chalkboard] why isn't that a four, there, you see?
[...]
"Now, that summarizes all of the problems associated with quantum electrodynamics. The most beautiful one is the coupling constant, one hundred thirty seven point... and so on and all good theoretical physicists put that up on their wall and worry about it. There is at the present time no idea of any utility for getting at that number. There have been from time to time suggestions but they didn't turn out to be useful. They would predict that the number was exactly a hundred and thirty seven... Well, the first idea was by Eddington, and experiments were very crude in those days... the number looked very close to a hundred and thirty six. So he proved, by pure logic, that it had to be a hundred and thirty six. Then it turned out that experiments showed that that was a little wrong - it was nearer a hundred and thirty seven. So he found a slight error in the logic and proved... 'pure logic' - it had to be exactly the integer, a hundred and thirty seven. It's not the integer. It's a hundred and thirty seven point oh three six oh. Every once in a while someone comes out and they find out that if they combine 'pi's and 'e's and twos and fives with the right powers and square roots, you can make that number.
"It seems to be a fact that's not fully appreciated by people who play with arithmetic that you'd be surprised how many numbers you can make by playing with 'pi's and twos and fives and so on. And if you haven't got anything to guide you except the answer, you can always make it come out even to several decimal places by suitable jiggling about. It's surprising how close you can make an arbitrary number by playing around with 'nice' numbers like 'pi' and 'e'.
"And therefore in the history of physics there [is] paper after paper [by] people who have noticed that certain specific combinations give answers which are very close in several decimal places to experiment except that the next decimal place of experiment disagrees with it. So it doesn't mean anything."
"Richard Feynman Video - The Douglas Robb Memorial Lectures - Part 4: New Queries"
Given at the University of Aukland, 1979
http://vega.org.uk/video/subseries/8
http://vega.org.uk/video/programme/48
~17:00

How is Feynman's discussion of the EM coupling constant not an example of the inappropriateness of imagining that proving a mathematical theorem or proving the consistency of some assertions in a symbolic calculus is the same thing as 'proving' an assertion about the world? Have I not argued sufficiently for the affinity between on the one hand claims for mathematical (geometric, topological etc.) derivations of α and on the other hand, hope for 'mathematical proof' that confinement is necessary?

Alternately, I understand that this observation is supported by a thirty-five-year-old perspective from a man with a complicated attitude toward 'villozovy' and so on. Is Feynman's view now considered dowdy? Or still just inconvenient? Rt3368 (talk) 17:03, 25 August 2015 (UTC)[reply]

The interplay between theoretical physics and mathematics has a long history and has its share of controversies. Mathematicians often work hard to rigorously justify analytic leaps made by theoretical physicists, who are often unconcerned with mathematical rigor. An earlier example where mathematicians and physicists struggled futilly for centuries to prove something observationally obvious is Stability of the Solar System. The give and take can be productive for both sides. Proof that Yang–Mills theory does not invariably lead to confinement might suggest a new direction for experiments, for example. And all too often that hard work gets us nowhere in terms of real world insights. But your argument is with the theoretical physics community, not Wikipedia, which only attempts to fairly summarize their work, and that community clearly considers a proof of confinement important. --agr (talk) 22:29, 25 August 2015 (UTC)[reply]
Very well. But I think that Wikipedians ought to be able to explain the meaning of phrases that appear in Wikipedia's articles. In the 'History and theoretical description' section I read
"This may be the reason why confinement has not been theoretically proven..."
What is meant by the phrase "theoretically proven"? Rt3368 (talk) 18:27, 28 August 2015 (UTC)[reply]
Physics is an experimental science and whenever people do experiments get back numbers. So, mathematics is the language used by nature and physics must use it. Theoretical physics is that part of physics that, by using mathematics, tries to explain what the outcomes of an experiments are and, sometimes, to provide other tests for experimental physicists. E.g., Higgs, using a mathematical model, postulated a new particle. On 2012, physicists at LHC observed it with the expected properties (except for the mass that was not possible to forecast but it will be with other mathematical models). So, also confinement is observed in all experiments where particles interacting through the strong force are observed, as at LHC or in cosmic rays. But, notwithstanding we have the correct mathematical model quantum chromodynamics that uses Yang-Mills theory, nobody was able to solve the equations to prove confinement so far. This is the meaning of "theoretically proven". I hope this is enough.--Pra1998 (talk) 20:58, 28 August 2015 (UTC)[reply]
No, it's not, actually.
As a lay person, I'm yet familiar with the narrative you describe, understand it at a lay person's level and don't dispute the events or tentative conclusions. I have no interest in unorthodox theoretical interpretations of these experimentalists' results.
In the context of the article, "theoretically proven" might mean
"proven by someone's theory of the word 'proof'"; or
"proven, using in some way some scientific theory of the world"; or
"theorematically proven"; or
"shown to be a consequence of a mathematical model, which model otherwise more or less informs a scientific theory of the world"; or
something else.
The first of these seems unlikely to have been intended. The second of these is impossible, since a theory of the world is always contingent and incapable of offering proof of anything. Something like the third or the fourth meaning above seems most likely but the matter is opaque.
Here's a pretty good proof, you'll allow:
Theorem: "There is NO largest prime number."
Proof:
Let S be a non-empty finite sized set of prime numbers.
Consider P, the product of all members of S.
Consider Q = P + 1.
Q must be evenly divisible by a prime number.
Q divided by any member of S must have remainder of 1.
Ergo, there must be a prime number not a member of S.
Q.E.D.
Proofs are objects of courts of law and of the tautological theorEMs of mathematics. They are never objects of the always contingent theorIEs of scientific inquiry. I say that "theoretically proven" joins other oxymoronic phrases such as "scientific proof" that at least in English have no meaning at all and convey no useful information or description. They're littered throughout discussions of science in popular media - including in articles in Wikipedia - and they obfuscate intended meaning. They are due to inattentive thinking and inattentive habits of expression and they proliferate because no one objects to their inadequacy. Rt3368 (talk) 15:05, 2 September 2015 (UTC)[reply]
Your points just say to me that you miss completely what physics is and what physicists do. I invite you to take a look at that article. This could help you before to cope with a highly specialistic article.--Pra1998 (talk) 16:23, 2 September 2015 (UTC)[reply]
That's a pretty empty response. I understand perfectly well what physics is and I understand also when an individual is utterly unfamiliar with the philosophy of science, to the extent that their expertise in a narrow field obscures the most straightforward ideas and trades enlightenment for obscurantism. It's a malady endemic in the sciences, and at Wikipedia. Rt3368 (talk) 01:05, 11 September 2015 (UTC)[reply]
No, you don't. You are utterly unfamiliar with physics and are lost in your obscure and empty analysis of what science is. Please, move on to a more proper place to discuss. Thanks.--Pra1998 (talk) 07:27, 11 September 2015 (UTC)[reply]
Would "This may be the reason why confinement has not been proven within Yang Mill theory..." make it more clear?--agr (talk) 19:57, 28 August 2015 (UTC)[reply]
That would be better. I think I would choose (trying to maintain some brevity) something like
"This may be the reason why confinement has not yet been proven to be a consequence of the Yang-Mills mathematical model..."
Alternately a simple change to "theorematically proven" - as long as this is a mathematically orthodox statement in the sense that a demonstration of the confinement requirement can be identified as a proof of some stated theorem - might convey the same meaning in a less explicit way.
In contrast, I believe a subsequent sentence,
"Proof that QCD confines at low energy is a mathematical problem of great relevance..."
is descriptive and correct as it stands, since it very rightly identifies that the truth status of a mathematical assertion that's part of QCD is - as it states - "of great relevance". Rt3368 (talk) 15:05, 2 September 2015 (UTC)[reply]

Scaling?

At this date (6/11/2017), it's written "{\displaystyle [A]=[L^{\frac {2-D}{2}}]} [A]=[L^{\frac {2-D}{2}}]" . I think it's 2-\frac{D}{2}, but I could be wrong (my own computation gives this power, and I also think it agrees with the next result, contrary to the given power) — Preceding unsigned comment added by 134.157.64.191 (talk) 18:23, 6 November 2017 (UTC)[reply]

Peter Woit is back with an edit war

I think that, in agreement with Woit's ideas we should remove all the section. The paper he is questioning is regularly published in a prestigious journal and is a collaboration with a reputable physicist.--Pra1998 (talk) 21:52, 27 April 2018 (UTC)[reply]

This page keeps on being vandalized by Peter Woit or some of his sockpuppets. This should be ended.--Pra1998 (talk) 14:09, 29 April 2018 (UTC)[reply]

Discussion, edit warring. Comment

Could we please discuss the passages and parts regarding known or unknown quantities which seem to inflame our anonymous editor so much? I'm afraid I don't know much about theoretical physics, but I'm willing to try to learn. — Javert2113 (talk) 15:05, 29 April 2018 (UTC)[reply]

Sorry, forgot to add @Pra1998: thoughts? — Javert2113 (talk) 15:11, 29 April 2018 (UTC)[reply]
It is quite simple. Wherever this guy reads "Frasca" he aims to remove the material without mercy. The point is that the cited paper is on the same foot of the others in that section and in the preceding one and so, based on his principles, we should remove both of them. The paper in question is published in one of the most important journals of particle physics, Physics Letters B (It is the one where CERN published its work on the discovery of the Higgs particle). Besides, it is a collaboration with one of the Editors of the European Physical Journal C that, as importance, is on the same foot of the other.--Pra1998 (talk) 15:13, 29 April 2018 (UTC)[reply]

There is no need to deal with the complex scientific issues here. Pra1998=Marco Frasca, and my understanding is that Wikipedia policy does not allow people to add references to their own work to Wikipedia pages. Peterwoit (talk) 14:50, 30 April 2018 (UTC)[reply]

Commuting versus non-commuting gauge theories

I would like to add a new section to this article, with the above section title, following the section "Quantization." My proposed section is below. Are there any objections or suggestions, prior to my doing so?

The physics study of Yang-Mills gauge theory (for compact simple gauge groups) entails understanding what happens to Maxwell’s electrodynamics, and U(1) quantum electrodynamics (QED), when Maxwell’s commuting (abelian) gauge fields become non-commuting (nonabelian) gauge fields covariantly transforming under the compact simple Yang-Mills gauge group SU(N) with NxN Hermitian generators and a commutator typically normalized such that for each .

In flat spacetime, in classical electrodynamics, a gauge-invariant field strength is related to the gauge fields by:

.

This may also be written more generally as using the gauge-covariant derivative , because the commutator . With and Coulomb constant , the classical Maxwell equation for electric charge strength is:

,

which spacetime-covariantly includes Gauss’ electricity and Ampere’s current laws. The classical equation for magnetic charge strength is

,

which spacetime-covariantly includes Gauss’ magnetism and Faraday’s induction laws. The zero in the monopole equation and thus the non-existence of magnetic monopoles (setting aside possible Dirac charge quantization) arises from the flat spacetime commutator of ordinary derivatives being . In integral form, the Gauss’ magnetism law component of the above becomes \oiint , whereby there is no net flux of magnetic fields across closed spatial surfaces. (Note: The point of various “bag models” of QCD quark confinement, is that there is similarly no net flux of color charge across the closed spatial surfaces of color-neutral baryons.)

In quantum electrodynamics, the charge density becomes related to the Dirac wavefunctions for individual fermions by where is the electric charge strength related to the running "fine structure" coupling by , and for the electron, up and down fermions, and their higher-generational counterparts. Meanwhile the propagators for the individual photons which form the gauge fields are obtained by inverting the electric charge equation and converting from configuration into momentum space using the substitution and the prescription. Because the charge equation is not invertible without taking some further steps, it is customary to utilize the gauge condition to obtain

which includes the photon propagator up to a factor of . Alternatively, one can introduce a Proca mass by hand into the charge equation. Then, is no longer a gauge condition but a requirement to maintain continuity (charge conservation), and with we arrive at the inverse:

which includes a massive vector boson propagator up to . Of course, adding a mass by hand destroys renormalizability, so it is necessary to find a way that this can be restored.

In Yang-Mills Gauge Theory, becomes a non-commuting gauge field, , and the field strength therefore graduates to the gauge-covariant, not gauge-invariant:

.

With A replaced by G, it will be seen that this contains the equation from the Mathematical overview above. Using differential forms, this may be written as the curvature arising from the gauge connection. Yang-Mills gauge theory differs from the abelian gauge theory of U(1) electrodynamics, by the mathematical and physical consequences of what happens when the gauge fields go from commuting to non-commuting in this way. PatentPhysicist (talk) 17:34, 21 February 2021 (UTC)[reply]

I think this could be a valuable addition to the article. Chanacya (talk) 18:41, 21 February 2021 (UTC)[reply]

PS: I noticed that this is a level 5 vital priority article, but only B class rated. I have studied Yang-Mill gauge theories for over 15 years, and would like to try to contribute to raising this. PatentPhysicist (talk) 02:51, 22 February 2021 (UTC)[reply]


References

  1. ^ A. V. Smilga, Lectures on Quantum Chromodynamics, World Scientific (2001).
  2. ^ M. Frasca, Strongly coupled quantum field theory, Phys. Rev. D 73, 027701 (2006).
  3. ^ M. Frasca, Infrared gluon and ghost propagators, Phys. Lett. B 670, 73 (2008).
  4. ^ A. Cucchieri, T. Mendes, What's up with IR gluon and ghost propagators in Landau gauge? A puzzling answer from huge lattices, PoS (LATTICE 2007) 297