Renormalization Group for dummies

[bhobba]

In the Dirac Equation. The magnetic moment of the electron is calculated as 1. In the 4th term in the power series, it's equal to 1.0011596522. The interacting fields are the electron self magnetic field and the electron. What about the interactions of say two electrons, what would be the Dirac Equation counterpart of 1.0 in the magnetic moment of the electron calculation? Do you calculate the dirac equation of each electron by adding them or calculate both of them combined? And if the result is for example 3.0. After the fourth term in the power series, would the result only be 3.0111 or would it be 5.0 (I don't think it would just be a small 3.0111 isn't it because the electric field strength is bigger). Thanks.

Don't know the answer to that one - sorry.

Thanks
Bill

 

[waterfall]

Don't know the answer to that one - sorry.

Thanks
Bill

Anyone else know? atyy?

 

[A. Neumaier]

Do you mean that the full integral has a closed-form expression (involving modified Bessel functions) for any value of lambda, but the series expression (involving Gamma functions) has terms that only converge absolutely when lambda<1? So if we had lambda>1, and all we had was the series form, we might worry the integral doesn't exist, when in fact it does?

The series for this integral diverges for _any_ nonzero value of lambda. That's why it is called an asymptotic series.

 

[A. Neumaier]

I can't believe why no one is answer my simple question. I know in power series, one expand the terms. I'm asking if the techniques in Renormalizaton Group can be applied to Finance or Biology problem, not just in QFT. If one knows the answer to this. Please let me know. Thanks.

Look at
http://arnold-neumaier.at/ms/ren.pdf
to get a reasonably elementary explanation of renormalization and the renormalization group without any dirty tricks - is a simpler situation where everything can be understood explicitly.

 

[waterfall]

Look at
http://arnold-neumaier.at/ms/ren.pdf
to get a reasonably elementary explanation of renormalization and the renormalization group without any dirty tricks - is a simpler situation where everything can be understood explicitly.

I've got a bird's eye view of Renormalization Group already. It assumes the Laudau pole is non-existent.. which assumes there is a real length cutoff. So it is strings or something akin to preons where all particles are made of which avoid creation of the landau pole problem. What is your best guess of what it is. Since I think you believe that quantum fields are the primary and there are really no particles even as quanta or by principle. You are still left with charge being infinite say in the point focus on the quantum fields. Unless you are suggesting fields have natural cutoff and quantized? Please answer using plain words so we'd have an intuitive grasp of the situations. Thanks.

 

[waterfall]

Before landau pole which is inside the Planck length is reached.. one has to cross the Planck boundary and much prior before that.. one has to touch first the electroweak length scale.. I'm not describing about using 100 Gev probe to unite the EM and Weak force.. but in Renormalization Group which is sensitive to the landau pole, it is sensitive to the electroweak pole too.. in this context, what is the length scale of the electroweak pole? Hope my questions are clear. If not. Just intuit what I'm saying and explain the details. Thanks.

 

[waterfall]

The physical reason the coupling constant gets larger is the the shielding effect of the virtual particles around an electron. Landau showed that because of that for any finite charge at any distance the charge would measure zero. This created the famous zero charge problem. The only way around it is for the charge to actually be infinite and to grow to an infinite value the closer and closer you get to it - this is called the Landau pole. That is the reason for the infinities - the closer you get the higher the energy you must use. If I remember correctly this is not just theory at 90gv it measured 1/127 (OK to avoid egg on my face I checked it)

You talked as if the virtual particles were real and really there around electrons. These are just figurative right? The perturbation series terms are given fancy names like 2nd order or 3rd order virtual particles when these don't have factual existence. So the reason the coupling constant gets bigger is simply due to the perturbation series alone and not to any nonexistent virtual particles (unless you mean they really exist?).

As one other person posted the fact it blows up to infinity means the theory is sick and incorrect - but we already know that because it gets replaced by the electroweak theory which will probably also get replaced by something else someday and that something else will hopefully be free of these problems.

Thanks
Bill

The message before this is due to your mentioning the "electroweak theory", what is it connection with the landau pole, is there such thing as electroweak pole? what are you talking about, pls refer to the message before this where I asked it in details. Thanks.

 

[bhobba]

You talked as if the virtual particles were real and really there around electrons. These are just figurative right?

They however have real effects - the shielding of the charge of an electron is one of them as well as things like the Casmir Effect.

Regarding the Landau pole although it is an issue it is calculated perturbatively and the region in blows up is one where perturbation theory will not work because it is so large. However re-normalisation group theory does and it shows it blows up in a different way - but blow up it does.

Thanks
Bill

 

[waterfall]

They however have real effects - the shielding of the charge of an electron is one of them as well as things like the Casmir Effect.

Prof. Neumaier mentioned how they are caused directly by the fields and not by any virtual particles which he refers to as Multivariate Integrals. Maybe he can give details when he comes here later.

Regarding the Landau pole although it is an issue it is calculated perturbatively and the region in blows up is one where perturbation theory will not work because it is so large. However re-normalisation group theory does and it shows it blows up in a different way - but blow up it does.

Thanks
Bill

Laudau pole is in the center of the Planck length. It may not have factual existence because the Planck scale may not even contain any spacetime so points lose its meaning.

Do you agree that we only do renormalization group calculations because the theory we have is only an effective field theory. Meaning when we come to the final true theory, we don't have to use any renormalization group, can anyone refute this?

 

[bhobba]

Do you agree that we only do renormalization group calculations because the theory we have is only an effective field theory. Meaning when we come to the final true theory, we don't have to use any renormalization group, can anyone refute this?

The re-normalisation group tells us how parameters such as the coupling constant varies with the regulator (conceptuality the cut-off - usually in energy) and also tells us where we would expect new physics to appear. If would be very unwise to push a theory beyond that IMHO - so yes I believe in the EFT approach as the only reasonable thing to do.

Thanks
Bill

 

[waterfall]

The re-normalisation group tells us how parameters such as the coupling constant varies with the regulator (conceptuality the cut-off - usually in energy) and also tells us where we would expect new physics to appear. If would be very unwise to push a theory beyond that IMHO - so yes I believe in the EFT approach as the only reasonable thing to do.

Thanks
Bill

Right. You mentioned below that the electroweak thing which I can't understand the context:

"As one other person posted the fact it blows up to infinity means the theory is sick and incorrect - but we already know that because it gets replaced by the electroweak theory which will probably also get replaced by something else someday and that something else will hopefully be free of these problems."

But QED is because of the broken symmetry where electromagnetism and the weak force is not united because of the low energy. Are you saying that near the Planck length, the electroweak force is active and this QED probe can still touch it? I know we need 100 Gev probe to touch it.. but in ordinary em field, can it probe the electroweak scale?

 

[waterfall]

There's this very interesting book called: "The Infinity Puzzle: Quantum Field Theory and the Hunt for an Orderly Universe"

https://www.amazon.com/dp/0465021441/?tag=pfamazon01-20

It's a laymen book and Renormalization Group was not mentioned however it seems to be related to the older Mass and Charge Renormalization used by Feynman, Schwinger, Tomonaga mentioned up to Chapter 3 which I just finished. Going to wiki.

"In theoretical physics, the renormalization group (RG) refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales."

My questions are how come the Renormalization Group was attributed to Kenneth Wilson when in fact even Feynman, Schwinger, Tomonaga used it called Mass and Charge Renormalization. And what would happen if you use the Kenneth Wilson's "Renormalization Group" on QED. Noting Wiki:

"They effectively devised the theory of mass and charge renormalization, in which the infinity in the momentum scale is cut-off by an ultra-large regulator, Λ"

Anyone has other perspective of this or can in a very sentences give some quick insight?
What I'm saying is that the infinity-infinity thing is related to mass and charge renormalization as mentioned by "The Story of Light" But wiki seems to be saying that this also used the regulator thing used in Wilson full Renormalization Group.

Thanks.

 

[waterfall]

There's this very interesting book called: "The Infinity Puzzle: Quantum Field Theory and the Hunt for an Orderly Universe"

https://www.amazon.com/dp/0465021441/?tag=pfamazon01-20

It's a laymen book and Renormalization Group was not mentioned however it seems to be related to the older Mass and Charge Renormalization used by Feynman, Schwinger, Tomonaga mentioned up to Chapter 3 which I just finished. Going to wiki.

"In theoretical physics, the renormalization group (RG) refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales."

My questions are how come the Renormalization Group was attributed to Kenneth Wilson when in fact even Feynman, Schwinger, Tomonaga used it called Mass and Charge Renormalization. And what would happen if you use the Kenneth Wilson's "Renormalization Group" on QED. Noting Wiki:

"They effectively devised the theory of mass and charge renormalization, in which the infinity in the momentum scale is cut-off by an ultra-large regulator, Λ"

Anyone has other perspective of this or can in a very sentences give some quick insight?
What I'm saying is that the infinity-infinity thing is related to mass and charge renormalization as mentioned by "The Story of Light" But wiki seems to be saying that this also used the regulator thing used in Wilson full Renormalization Group.

Thanks.

Reading the archives at PF to see the difference between Feynman Renormalization and modern day Wilson Renormalization, I came across the following by Eugene (I heard this guy has his own theories so I can't differentiate if what he is saying is mainstream or his own.. or rather.. what topics where he created his own ideas versus that of mainstream so I can avoid topics where he made it up.. this is because the archives has a lot of his contributions):

https://www.physicsforums.com/showthread.php?t=183903 msg #2 by Eugene:

"In the end of 1920's Dirac, Pauli, Weiskopff, and Jordan formulated a quantum theory of interactions between electrons and photons in a loose analogy with Maxwell's classical electrodynamics. This early quantum electrodynamics (QED) was very successful in calculations of various scattering processes in lowest orders of the perturbation theory. Unfortunately, all contributions to the S-matrix in higher orders came out infinite.

In late 1940's Tomonaga, Schwinger and Feynman found the way to fix this problem of infinities by renormalization. The renormalization basically adds certain infinite counterterms to the Hamiltonian of the early QED. The form of these counterterms was selected such that the resulting theory satisfied two physical principles. First, the calculated electron's mass should be equal to the measured electron's mass. Second, the calculated interaction energy between two electrons at large distances should be equal to the classical expression e2/r. These two requirements lead to two types of renormalization counterterms in the Hamiltonian - the mass and charge renormalization counterterms."

And https://www.physicsforums.com/showthread.php?p=3628984 in msg #20 by atyy:

"Probably the chief conceptual advance since Feynman's original work (not the book, which is late), is why renormalization works. This is provided by "renormalization group" and "effective field theory ideas". The basic idea is that we don't need theories that are consistent at all energies. They just need to work at low energies. Renormalization flow is the process of seeing what a theory given only its symmetries "looks like" at lower and lower energies. A "renormalizable theory" like QED is one in which the flow converges to a fixed point, about which perturbation theory can be done."

My questions. What happens if you apply the modern Renormalization Group" idea to the Feynman era QED problem? How would it differ then to the mass and charge renormalization techniques?

 

[bhobba]

My questions. What happens if you apply the modern Renormalization Group" idea to the Feynman era QED problem? How would it differ then to the mass and charge renormalization techniques?

The difference would be in understanding what's going on - that's all. The early users of re-normalization did not understand what was going on - this led to jokes about looking for infinities under physicists rugs. Now we know the answer - some parameters like the coupling constant depend on the cut-off. In order for the perturbation method to work you have to use a parameter to perturb about that is small. The re-normalization group allows you to see exactly how it depends on the cutoff and and at what energies you can reasonably be confident the theory is valid - that is the EFT approach. It not hard really.

You seem to stuck on the re-normalization group - its not required to do re-normalization - simply to better understand what's going on. It does not change the method one whit which is a way of adding counter-terms to ensure what you get is finite. But the math of that is quite complicated - good luck in finding someone to explain it - you really need a textbook.

Thanks
Bill

 

[waterfall]

The difference would be in understanding what's going on - that's all. The early users of re-normalization did not understand what was going on - this led to jokes about looking for infinities under physicists rugs. Now we know the answer - some parameters like the coupling constant depend on the cut-off. In order for the perturbation method to work you have to use a parameter to perturb about that is small. The re-normalization group allows you to see exactly how it depends on the cutoff and and at what energies you can reasonably be confident the theory is valid - that is the EFT approach. It not hard really.

You seem to stuck on the re-normalization group - its not required to do re-normalization - simply to better understand what's going on. It does not change the method one whit which is a way of adding counter-terms to ensure what you get is finite. But the math of that is quite complicated - good luck in finding someone to explain it - you really need a textbook.

Thanks
Bill

Yes Renormalization Group is easy to understand as it is related to Effective Field Theory, taking the words of nrqed when he replied to Eugene in msg # 20 of the thread https://www.physicsforums.com/showthread.php?t=183903&page=2:

"I may be wrong but it sounds as if you imply that an effective field theory approach implies the assumption of granularity of spacetime (I may have misinterpreted your words, if so I apologize). Saying that a theory is an eft does not imply that. It just implies that at some scale "new physics" arises. The nature of this new physics is quite arbitrary, it could be granularity of spacetime but it could be a new force, inner structure to the particles (including string-like structure) etc etc etc. So in that sense it is quite general."

There is another possibility, related to the foundations of QM. If one take the view of Bohmian Mechanics, there may be no virtual particles taking all paths and the infinity problem resulting from all this amplitude path-integral approach like stuff because in the world of Bohmian Mechanics, particles are always particles. Hope a Bohmian for example Demystifier can confirm whether this is true.

Btw.. Feynman being a genius that he was. How come he didn't arrive at the Renormalization Group idea himself (about coupling constant depending on the cutoff) while mulling about it. Could there be a development in physics later on that popularize the idea which Feynman didn't dare think about? Anyone got a clue?

 

[atyy]

My questions. What happens if you apply the modern Renormalization Group" idea to the Feynman era QED problem? How would it differ then to the mass and charge renormalization techniques?

http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf p 193
The final results, at an energy scale E well below the initial cutoff λ0, are the same as we would predict via renormalized perturbation theory, up to small corrections by powers of E/λ0.
"
The advantage of the Wilson scheme is that it gives a nonperturbative definition of the theory which is applicable even if the theory is not weakly coupled. ..."

"The Wilson scheme also allows us to give physical meaning to nonrenormalizable theories. Given an action for a nonrenormalizable theory, we can regard it as an effective action. ..."

 

[bhobba]

Btw.. Feynman being a genius that he was. How come he didn't arrive at the Renormalization Group idea himself (about coupling constant depending on the cutoff) while mulling about it. Could there be a development in physics later on that popularize the idea which Feynman didn't dare think about? Anyone got a clue?

Maybe because he was busy doing other stuff such as superconductivity and the theory of Partons.

Thanks
Bill

 

[strangerep]

Prof. Neumaier mentioned how [supposed effects of virtual particles] are caused directly by the fields and not by any virtual particles which he refers to as Multivariate Integrals. Maybe he can give details when he comes here later.

He already talked a lot about this sort of thing. See these threads:

https://www.physicsforums.com/showthread.php?t=75307

https://www.physicsforums.com/showthread.php?t=460685