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CSnowden

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CSnowden

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It does... and certainly, in condensed matter physics, this is the origin of the self-energy that causes the broadening of single-particle spectral function. But in a vacuum, these higher-order interactions are considerably weaker.

Zz.

- #3

CSnowden

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- #4

bhobba

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https://quantumfrontiers.com/2013/06/18/we-are-all-wilsonians-now/

Basically all QFT's need some sort of cutoff beyond which new physics is considered to happen. Not doing this leads to issues like the Landau pole:

https://en.wikipedia.org/wiki/Landau_pole

And indeed long before the Landau pole is reached the electroweak theory takes over.

This view has consequences for how fundamental theories are looked at these days eg gravity is no longer thought actually incompatible with QM:

https://arxiv.org/abs/1209.3511

Thanks

Bill

- #5

DarMM

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Although of course all QFTs reach a scale where they are probably incorrect physically, not all need a cutoff mathematically. For example in two dimensions Yang-Mills theories, Polynomial scalar theories and Yukawa theories can be defined without a cutoff. Similarly in three dimensions, although the allowed interactions for scalar and Yukawa theories are smaller.Basically all QFT's need some sort of cutoff beyond which new physics is considered to happen.

In four dimensions ##SU(N)## Yang-Mills doesn't require a cutoff and develops no Landau pole (proven by Balaban, Magnen, Rivasseau and Sénéor).

- #6

bhobba

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In four dimensions ##SU(N)## Yang-Mills doesn't require a cutoff and develops no Landau pole (proven by Balaban, Magnen, Rivasseau and Sénéor).

Now that I did not know - interesting. Of course Wilson's approach does not mandate a cutoff - it simply starts from assuming one and seeing what happens at a smaller cutoff.

Thanks

Bill

- #7

A. Neumaier

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In four dimensions ##SU(N)## Yang-Mills doesn't require a cutoff and develops no Landau pole (proven by Balaban, Magnen, Rivasseau and Sénéor).

I don't believe this. Please give references and clarify what you mean. How can there be a proof of this when not even the existence question is settled?

On the contrary, YM is likely to have a Landau pole at energies of the order of the mass of the lightest glueball; see the paper https://arxiv.org/abs/1311.6116 by Reinosa et al.:

Reinosa et al. said:It is commonly accepted that the description of the infrared behavior of the vacuum ghost and gluon correlators requires nonperturbative methods since standard perturbation theory, based on the Faddeev-Popov construction, predicts an infrared Landau pole at low momentum.

- #8

A. Neumaier

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Because the physical 1-electron state is already a self-consistent solution of QED; the e/m field field is the result of the self-interaction through the mediation of the e/m field.why doesn't the locally generated EM field loop back to self-interact with the associated electron matter field?

No. Renormalization issues are independent of the question of Landau poles.Basically all QFT's need some sort of cutoff beyond which new physics is considered to happen. Not doing this leads to issues like the Landau pole:

- #9

DarMM

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First to clarify I was answering in the context of what @bhobba said regarding a Wilsonian view and wasn't precise as I was only talking about the continuum limit Landau pole it prevents.I don't believe this. Please give references and clarify what you mean. How can there be a proof of this when not even the existence question is settled?

Being genuine, I don't know how much you'll actually want to read Balaban, Magnen, Rivasseau and Sénéor. Before I give the references, just so I know what you want want. "Existence" is a bit of an ambiguoug term in constructive field theory, do you mean the continuum limit exists, or do you mean a continuum limit and the infinite volume limit exists with all of the Osterwlader-Schrader or Wightman-Gårding satisfied.

The main examplar of what I meant are the seven papers by Balaban:

- T. Balaban, Propagators and renormalization transformations for lattice gauge theories I and II, Comm. Math. Phys. 95, 17 and 96, 223A984).
- Т. Balaban, Averaging operations for lattice gauge theories, Comm. Math. Phys. 98, 17 A985).
- T. Balaban, Spaces of regular gauge field configurations on a lattice and gauge fixing conditions, Comm. Math. Phys. 99, 75A985).
- T. Balaban, Propagators for lattice gauge theories in a background field, Comm. Math. Phys. 99, 389 A985). 322 References and Bibliography
- Т. Balaban, The variational problem and background fields in renormalization group method for lattice gauge theories, Comm. Math. Phys. 102, 277 A985).
- T. Balaban, Renormalization group approach to lattice gauge field theories, I: Generation of effective actions in a small fields approximation and a coupling constant renormalization in four dimensions, Comm. Math. Phys. 109, 249 A987).
- T. Balaban, Renormalization group approach to lattice gauge field theories, II: Cluster expansions, Comm. Math. Phys. 116, 1 A988); Convergent Renormalization Expansions for lattice gauge theories, Comm. Math. Phys. 119, 243 A988).
- T. Balaban, Large Field Renormalization I: The basic step of the R Operation, Comm. Math. Phys. 122, 175 A989); II Localization, Exponentiation and bounds for the R Operation, Comm. Math. Phys. 122, 355 A989).

Thus the theory develops no Landau poles when taking the continuum limit, the infrared limit is more difficult, but you have Yang-Mills existing on a Torus, remaining problems are uniqueness of the limit (though this might not be true, which would be interesting, there would be more quantum Yang-Mills than classical ones) and some of the Osterwalder-Schrader axioms do not hold.

In context what I meant is that we know Yang-Mills doesn't need a Wilsonian style cut-off effective action treatment, it's remaining problems are outside that framework.

See the following post to @bhobba

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DarMM

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That's true, Balaban et al's proofs use Wilsonian ideas. If you start with an action at scale ##\Lambda## they flow to ##\Lambda^{'}##. They then basically show that the ##\Lambda^{'}## action remains well defined as ##\Lambda \rightarrow 0##.Now that I did not know - interesting. Of course Wilson's approach does not mandate a cutoff - it simply starts from assuming one and seeing what happens at a smaller cutoff.

Thanks

Bill

This is basically what I meant @A. Neumaier , in a Wilsonian view we know Yang-Mills doesn't need a cutoff. Or at least the continuum limit works as well as any cut-off action. What's missing is if it has the extra properties expected of the continuum limit and especially, the infrared limit.

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CSnowden

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- #12

A. Neumaier

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I was referring to quantum Yang-Mills as defined by the Clay Millennium problem. So you say that quantum Yang-Mills on a torus (in which version - 3D? 4D? Euclidean?) is proved to exist.I don't know how much you'll actually want to read Balaban, Magnen, Rivasseau and Sénéor. Before I give the references, just so I know what you want want. "Existence" is a bit of an ambiguoug term in constructive field theory, do you mean the continuum limit exists, or do you mean a continuum limit and the infinite volume limit exists with all of the Osterwalder-Schrader or Wightman-Gårding satisfied.

Already from a rigorous renormalization group point of view, asymptotically free theories don't have a Landau pole close to infinite energies, and on a compact space they are not expected to have any Landau pole since infrared problems are absent. But on Minkowski space one expects an infrared Landau pole at physically realizable energies - which is much worse than the ultraviolet Landau pole of QED which is at physically irrelevant energies. Thus for asymptotically free theories the difficulties are in the experimentally accessible infrared, while for QED and the weak interactions, the difficulties are in the far ultraviolet.

- #13

A. Neumaier

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Think of a simple coupled classical mechanical system consisting of two free parts, with an interaction mediating forces between them. Essentially you have two equations of motion ##F(x)=0## and ##G(y)=0##, which upon coupling become ##F(x)=\epsilon C(x,y)## and ##G(x)=\epsilon D(x,y)##. The effective self-coupling is obtained by using the second equation to solve for ##y=y(x)## and substituting it into the first equation, and the renormalized first particle (with self-interactions included) is the solution of ##F(x)=\epsilon C(x,y(x))##, which is an equation for ##x## alone. In QFT one has essentially the same situation, but complicated by an infinite number of degrees of freedom and the resulting technical problems in the renormalization.

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DarMM

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4D Euclidean, in the sense that the limit exists and has finite expectation values for gauge invariant observables, though not all of the Osterwalder-Schrader axioms have been shown to hold.So you say that quantum Yang-Mills on a torus (in which version - 3D? 4D? Euclidean?) is proved to exist.

- #15

CSnowden

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Thank you for the very clear explanation, that is helpful. The coupling explanation raises a point from an earlier post, namely that the electron matter field raises the potential energy level in the local EM field (the 'attached' field) but can it be said that the EM field does the same in reverse (ie. raises the potential energy of the matter field)? I believe the answer was yes and that the self-interaction did impact the matter field energy. It seems that the matter field induces quite a substantial local EM field (electric charge of a free electron is quite observable) - is the symmetric impact of the EM field on the matter field of a lower order of magnitude?Think of a simple coupled classical mechanical system consisting of two free parts, with an interaction mediating forces between them. Eseentially you have two equations of motion ##F(x)=0## and ##G(y)=0##, which upon coupling become ##F(x)=\epsilon C(x,y)## and ##G(x)=\epsilon D(x,y)##. The effective self-coupling is obtained by using the second equation to solve for ##y=y(x)## and substituting it into the first equation, and the renormalized first particle (with selfinteractions included) is the solution of ##F(x)=\epsilon C(x,y(x))##, which is an equation for ##x## alone. In QFT one has essentially the same situation, but complicated by an infinite number of degrees of freedom and the resulting technical problems in the renormalization.

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A. Neumaier

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These terms are not really well-defined, so your question is too vague to be answered.that the electron matter field raises the potential energy level in the local EM field

- #17

A. Neumaier

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4D Euclidean, in the sense that the limit exists and has finite expectation values for gauge invariant observables, though not all of the Osterwalder-Schrader axioms have been shown to hold.

Is there a version of Osterwalder-Schrader for the 4-dimensional torus?

- #18

CSnowden

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I am a lay person with an interest in QFT and trying to learn (although I do have bachelors degrees in mathematics and physics), so would appreciate any coaching toward terminology that may be more standard within the field of study. I was trying to communicate that the two fundamental QFT fields constituting the electron (say a single free electron for specificity) were coupled and by virtue of this coupling the electron matter field induced a local EM field. Perhaps as a starting point I should ask for the proper terminology to use for this description. I used the phrase 'raised the potential energy level in the local EM field' previously as I was trying to stress that there is really just a single pervasive EM field and not a new, separate attached EM field created. Perhaps the same question on this as to the appropriate terminology that should be used. Thanks for any assistance with these very basic, foundational questions.These terms are not really well-defined, so your question is too vague to be answered.

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CSnowden

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CSnowden

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