A Toy Model of Renormalization and Reformulation

Question

Referee this paper: arXiv:1110.3702

Please use comments to point to previous work in this direction, and reviews to referee the accuracy of the paper. Feel free to edit this submission to summarise the paper (just click on edit, your summary will then appear under the horizontal line)

(Is this your paper?)

---

As you know, renormalization of equation coefficients is necessary when these (already good) coefficients acquire unnecessary perturbative corrections. Discarding those corrections is the essense of renormalization.

Perturbative corrections to the equation coefficients appear at a certain stage of theory development - when we are trying to make the phenomenological equations even more exact with including known to us, but missing in our equations, physical effects like radiation reaction force.

Our way of theory development is not always successful - we may unintentionally spoil our original equations, i.e., we may include something wrong in the missing term. That is why we encounter disagreement of new solutions with experiment. Sometimes those extra (bad) terms are so simple and evident that their discarding restores the original properties of our approximate equations. However, and this is a good news, we may formulate our theory correctly so that no coefficient corrections appear and the missing interaction is right from the very beginning, with no extra harmful terms. This approach may be called "Reformulation".

I chose simple mechanical equations on purpose to demonstrate unambiguously what is going on in QFT in reality. My model can be reformulated (=renormalized) exactly and perturbatively. I would like to see your opinions whether it is comprehensible and convincing or not.

Further explanation

There goes a discussion about what we should allow to the review section. My "reformulation" direction is called there a bullshit by dimension10.

Let me say a word in my defence. There is a formulation (a mainstream one) where the Lagrangian $L_{phys}$ only contains physical constants, but there is also a counter-term Lagrangian $L_{CT}$ next to it $L_{full}=L_{phys}+L_{CT}$. Conter-terms are supposed to subtract undesirable corrections "generated" with the original interaction Lagrangian $L_{int}$ containing in $L_{phys}$. Normally, the counter-term Lagrangian is joined with the interaction Lagrangian in one "renormalized" interaction and they are treated perturbatively. Subtractions are fulfilled in each order of perturbation theory starting from loop-like diagrams.

What if we could make the subtractions in the full Lagrangian exactly, before building the perturbation theory? Then we would not have to do any subtractions in our calculations. The perturbation theory would be different - with no harmful corrections to the equation coefficients, it would be just a regular, rootine calculation.

Currently we cannot carry out such subtractions in the full Lagrangian, unfortunately. So we do it perturbatively. In my toy model, however, it is possible. After that we arrive at a new expression of Lagrangian $L_{full}$ yielding a regular perturbation theory. In other words, reformulation is an exact renormalization of the full Lagrangian and it gives the same results, but directly. So it is not a bullshit, but a shortcut to the right results.

By the way, this new (exactly renormalized) Lagrangian contains a slightly different, but quite understandable and acceptable physics. That is why I believe that we physicists can figure out a realistic right Lagrangian from the physical reasonings.

Please, do not vote down this message. I already lost the right to vote answers up. Otherwise, I will quit your noble academy here.

Comments

@VladimirKalitvianski The problem is that reformulation is plain unnecessary, yet you have made inaccurate claims to justify it. Results in renormalisation are not flukes, they are perfectly justified. Arnold has already patiently explained to you what actually happens in renormalisation here.

@RonMaimon My downvote isn't a "philosophical prejudice against the author", I would cast the same downvote on accuracy (and upvote on originality) if it was written by someone else. 

---

@dimension10: "The problem is that reformulation is plain unnecessary, yet you have made inaccurate claims to justify it." Let us make a poll to learn what people prefer - a regular calculation or a calculation burdened with renormalization.

"Results in renormalisation are not flukes, they are perfectly justified." Justified by what? By our desire to transform our wrong results into the right ones? Isn't it better to transform (reformulate) our equations (our theory) instead of transforming our wrong results into the right ones?

---

@RonMaimon there might be specific situations, where one can use simpler calculations or physics intuiton and does not have to invoke the machinery of the renormalization procedure to "parameterize" or correctly take into account high energy effects in low energy equations. And if the paper would deal with just deriving and explaining such examples correctly and selfconsistently, it would probably be accurate. But wanting to generally dismiss the whole concept and idea of renormalization, which is very successfully applied and, thanks to Wilson now better understood, seems not accurate to me. The paper does not justify such broad far reaching claims.

However, I think in the Reviews section it would be good to explain votes by a corresponding review or a short comment at least (or upvote an already existing explanation). Doing this, people are free to cast their votes in accordance with their best knowledge about the topic at hand, which should not be negligable of course ... ;-)

---

@Dilaton: Thanks a lot!

---

@Dilaton: I don't think you understand Vladimir's position fully. He is not a crackpot, he is simply an ornery guy who understands the elephant's tail while everyone else is busy studying the trunk. He doesn't believe it has a trunk, he thinks the reports of a trunk are mangled reports of the tail.

But it doesn't matter. His general bullshit stuff in introductions and conclusions are peripheral. You need to look at the model itself, and what it does, following the details. You don't look at the sentence structure of the introduction and conclusion, and check whether it agrees with what other introductions and conclusions say, that is total politics.

---

@dilaton have you managed to make a start in doing a review of the paper?

---

This has not yet gone too far, as I was invited to the wedding of my nicest colleague and good friend last Saturday ...
 

Answer 1

The paper (of which I review version 19 dated August 4, 2013) presents equations (19), (20) coupling a single particle with position vector $r_p$ to a 3-dimensional oscillator with position vector $r_{osc}$. 

The main work is spent on the derivation of the equations (in the first 7 of 11 pages); no computations are done after the equation is derived, except to mention that the model has a straightforward perturbation theory, so that ''we may now calculate the results of collision of any such compound bodies, Doppler and retardation  effects at any distance $S$ without conceptual and mathematical difficulties''. No such computations are actually done.

Instead, the last three pages are spent with a general discussion, relating the derivation to philosophical musings about an alternative approach to quantum field theory in which no renormalization would be needed. No attention is given to the question whether such an alternative would produce results in accordance with experiment.

To those who find no serious fault with existing quantum field theory, the paper, though it seems technically correct in its mathematical (rather than philosophical) considerations, is devoid of useful content.

Comments

I hadn't claimed that one couldn't do these computation, just informed the reader that they were not done.

The purpose of a review is to orient the reader what to expect. Those interested in the derivation can read the paper to find out how things were derived. You wrote enough in your exposition at the top of the review page.

 I didn't condemn you but gave you +1 for accuracy and abstained from a vote on originality.

---

I didn't vote on originality. Nobody is obliged to vote. Solving a toy problem that nobody considered before is in some sense original, but in another sense just another application of the same well-known technique to a problem invented by you, so what.

---

Your toy-model may work fine in exactly one context: in the context of your toy-model.

Any broad overreaching negative claims about other people's work are not scientifically backed up or justified in this paper or elsewhere, neither experimentally nor theoretically (mathematically).

So I agree with Arnold, and the paper would be much better without these broad unjustified overreaching claims.

---

I do not think the methods in the papers are so original, now that I understand them (I just couldn't figure out exactly what he was doing before). This method can be used in the far infrared to rewrite the interaction, because there you can mix up the kinetic term of the electron and the electromagnetic field with a rotation, like VK's toy model does. When the electromagnetic interaction can be approximated as "p-eA(0)", so that A is evaluated at a fixed point i.e. when the matrix elements of A(x)-A(0) is small, which is for long wavelength radiation compared to any length scale in the electron motion.

(wrong paragraph about paper with unrelated material removed, I was just stupid, and brainwashed by reading too much VK)

The general overreaching claims I don't care about, because such claims are just insignificant, except for getting attention and shouting. The only content of a paper is what is supported by the actual methods or calculations.

---

Take your time, Ron, do not hurry to judge. Maybe in the end you will find my construction not only non-original, but also plainly banal.

---

I have not don so many things in this article, but why to list them? What the readers of your review will think of it?

Concerning perturbation theory, it was done (expansion of the force in a small parameter). It is so elementary and natural i this problem setup.

And you think there is nothing original? How about an error in coupling? How about renormalization, which works perfectly? One can even turn my paper upside down and use it as a "proof" of renormalization being unavoidable!

---

There is no originality, nor creativity. I just demonstrated how one can make an error in coupling equations and how renormalization can luckily "work". Some people deny any error and you do not admit a coincidence in the renormalization success. For them my toy model may become a sobering example.

As well, it shows that my reformulation program is not "plain unnecessary" or a "bullshit". It is about the same system and it may give the same results, but directly (a short-cut). But it's nothing.