I Visualizing the Attached EM Field for Free Electron in QFT

[CSnowden]

My understanding of the QFT model of a free electron is that there is a localized higher energy level in the electron matter field which couples to the EM field in two ways: (1) the coupling allows the electron matter field to 'feel' a force from an outside EM field and accelerate in response and (2) the electron matter field excitation generates a localized higher energy level in the surrounding EM field which can be interpreted as an 'attached' EM field (really just a local higher energy level in the single EM field pervading all space-time). Does this characterization sound correct? If so, why doesn't the locally generated EM field loop back to self-interact with the associated electron matter field?

 

[ZapperZ]

My understanding of the QFT model of a free electron is that there is a localized higher energy level in the electron matter field which couples to the EM field in two ways: (1) the coupling allows the electron matter field to 'feel' a force from an outside EM field and accelerate in response and (2) the electron matter field excitation generates a localized higher energy level in the surrounding EM field which can be interpreted as an 'attached' EM field (really just a local higher energy level in the single EM field pervading all space-time). Does this characterization sound correct? If so, why doesn't the locally generated EM field loop back to self-interact with the associated electron matter field?

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.

 

[CSnowden]

Thanks for the clarification - is this the source of the re-normalization issue which required the empirical adjustment to cancel in QFT calculations or is that a separate issue?

 

[bhobba]

Wilson sorted renormalization out:
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

 

[DarMM]

Basically all QFT's need some sort of cutoff beyond which new physics is considered to happen.

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.

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

 

[bhobba]

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

 

[A. Neumaier]

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.:

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.

 

[A. Neumaier]

why doesn't the locally generated EM field loop back to self-interact with the associated electron matter field?

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.

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:

No. Renormalization issues are independent of the question of Landau poles.

 

[DarMM]

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?

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.

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:

1. T. Balaban, Propagators and renormalization transformations for lattice gauge theories I and II, Comm. Math. Phys. 95, 17 and 96, 223A984).
2. Т. Balaban, Averaging operations for lattice gauge theories, Comm. Math. Phys. 98, 17 A985).
3. T. Balaban, Spaces of regular gauge field configurations on a lattice and gauge fixing conditions, Comm. Math. Phys. 99, 75A985).
4. T. Balaban, Propagators for lattice gauge theories in a background field, Comm. Math. Phys. 99, 389 A985). 322 References and Bibliography
5. Т. Balaban, The variational problem and background fields in renormalization group method for lattice gauge theories, Comm. Math. Phys. 102, 277 A985).
6. 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).
7. 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).
8. 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).

These will take months to read and understand. These show that the continuum limit action exists, is bounded and that the continuum limit of gauge invariant observables exists.

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

 

[DarMM]

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

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 $\Lambda \rightarrow 0$.

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.

 

[CSnowden]

I follow the general outline of the discussion and appreciate the exposure to the Wilson references which are quite interesting. I didn't quite follow the reference 'the e/m field field is the result of the self-interaction through the mediation of the e/m field.' which achieves an admirable level of recursion appropriate the topic :) - would appreciate any further insight you could provide.

 

[A. Neumaier]

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.

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.

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.

 

[A. Neumaier]

I follow the general outline of the discussion and appreciate the exposure to the Wilson references which are quite interesting. I didn't quite follow the reference 'the e/m field field is the result of the self-interaction through the mediation of the e/m field.' which achieves an admirable level of recursion appropriate the topic :) - would appreciate any further insight you could provide.

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.

 

[DarMM]

So you say that quantum Yang-Mills on a torus (in which version - 3D? 4D? Euclidean?) is proved to exist.

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.

 

[CSnowden]

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.

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?

 

[A. Neumaier]

that the electron matter field raises the potential energy level in the local EM field

These terms are not really well-defined, so your question is too vague to be answered.

 

[A. Neumaier]

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?

 

[CSnowden]

These terms are not really well-defined, so your question is too vague to be answered.

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.

 

[CSnowden]

Perhaps I should say something like 'an excitation of the electron matter field couples to raise a local excitation of the EM field'

 

[CSnowden]

I'll like to state a core question I have regarding QFT field coupling in more detail. There have been two effects of EM field coupling to the electron mater field described: (1) acceleration of the electron (matter field) in response to an external EM field and (2) an interaction of the self-generated EM field with the electron matter field, resulting a subtle change in the electron matter field mass. I'm wondering what causes the difference between these two effects?