# Renormalization - a dippy process - R. Feynman

## Main Question or Discussion Point

Feynman refers to "renormalization" as a dippy process on p.128 of his book "QED - The Strange Theory of Light and Matter".

His words are: "The shell game that we play to find n and j is technically called renormalization. But no matter how clever the word, it is what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving the theory of quantum electrodynamics is mathematically self-consistent. .... I suspect that renormalization is not mathematically legitimate."

If what Feynman wrote is true, then why do they still teach "renormalization" as being a useful method in universities?

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If what Feynman wrote is true, then why do they still teach "renormalization" as being a useful method in universities?
Because we right now have nothing better to actually make calculation. And they agree with experiment with a lot of significant digits. Those calculations are very reliable. Even though we do not have a common agreement on how to justify them rigorously.

Connes' work might change this however.

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It took centuries to rigorously justify Fourier's silly little manipulations. Very important and deep mathematics came out of the process. We still do the same silly meaningless manipulation BTW, they still make wonders, but now we know why it works.

Because we right now have nothing better to actually make calculation. And they agree with experiment with a lot of significant digits. Those calculations are very reliable. Even though we do not have a common agreement on how to justify them rigorously.

Connes' work might change this however.

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It took centuries to rigorously justify Fourier's silly little manipulations. Very important and deep mathematics came out of the process. We still do the same silly meaningless manipulation BTW, they still make wonders, but now we know why it works.

Sounds like we continue to teach that the world is flat because we don't know it is a sphere and even though a highly respected teacher told us we were being silly to continue teaching that the world is flat. Boy, are we humans foolish!

Sounds like we continue to teach that the world is flat because we don't know it is a sphere and even though a highly respected teacher told us we were being silly to continue teaching that the world is flat. Boy, are we humans foolish!
I am glad that you take it that way. I knew you were pulling my leg. Nice joke :rofl:

I am glad that you take it that way. I knew you were pulling my leg. Nice joke :rofl:
Sorry. I was not playing nice. I was being sarcastic. Sarcasm is hard to transfer in a subtle manner, and across a language difference. I'm still suggesting that renormalization should not be taught, just like we should not teach the world is flat.

Sorry. I was not playing nice. I was being sarcastic. Sarcasm is hard to transfer in a subtle manner, and across a language difference.
I know.
I'm still suggesting that renormalization should not be taught, just like we should not teach the world is flat.
The world is not flat, that is true. Not teaching renormalization is like not teaching trigonometry.

I knew you were being sarcastic, and I think your argument is rather ignorant. You don't have to know _everything_ about something in order to gain value from it. Renormalization is taught because it provides very accurate, very useful results and predictions. We know some other theory will supercede it, just as all theories are eventually superceded. You can't be so childishly dismissive of something that's been so successful if you don't have a better alternative. And your analogy to teaching that the earth is flat is completely wrong: in fact, a "flat" earth is immensly useful for certain types of cartography. Should we eliminate the use of mercator projections and the like because they're not perfect?

FEYNMAN has standing to call renormalization "dippy". No one on this forum does, however.

Feynman ignored a few new insights that came from statistical physics. It has turned out that the high energy physicists do their computations the wrong way around. In statistical physics you have some microscopic model and you can then write down an effective field theory. That effective field theory is constructed by integrating out some fraction of the degrees of freedom of the microscopic model.

In high energy physics, the standard model is actually precisely such an effective model that one could obtain by integrating out the microscopic degrees of freedom of the unknown theory of everything. This means that the integrals over momentum in Feynman diagrams do not really go to infinity and there are no real divergences.

But since we didn't derive the Standard Model from the theory of everything we don't know the "right way" to cut of the integrals. If we naively pretend that there is no cut-off then the intgrals will diverge as the high energy physicists in the early days found out. They also found the remedy: introduce a cut-off and then write all the observable of interest in terms of other observable quantities. You then find that the expression becomes insensitive to the cut off and you can let it go to infinity.

This means that the unknown details about the cut-off are irrelevant. These details get lost when you rewrite the quantity you are interested in in terms of the other observable quantities.

Haelfix
Renormalization was weird when it first came out back in the day, but the mystery was largely resolved in the 70s with the advent of the renormalization group and lattice gauge theory. It seems perfectly natural now and it would be weird if you *didn't* have to perform such a process.

malawi_glenn
Homework Helper
Einstein beleived in hidden variables in quantum mechanics and was agaist the probibalistic nature of it: "God does not play dice" is a famous quoting.

This is not the first time you qoute a historical physicist to state your points. It is of course always fun to read about what the grand ol masters of physics thougt, and get historical insights on how the theories and models of today have developed.

But I do think that one has to read these things through the eyes of contemporary physical insight to get things correct since theories and insight develops.

Lord Kelvin said in 1899 that "now has everyting that can be discovered been discovered, only some small details remains".. 6 yeas later, Einstein published his work on his special theory of relativity (much of that work was already known and worked out by guys like Lorentz and Poincare, Einstein did however see how to make the correct physical interpretations of it and so on). So by recalling what the great Kelvin said, should we believe that there was nothing more to discover? ;-)

All great men do misstakes sometimes, nobody can be perfect.

But I do think that one has to read these things through the eyes of contemporary physical insight to get things correct since theories and insight develops.
The issue goes much deeper in my opinion. If a colleague and you had a scientific disagreement on (say) general relativity for instance, you would never go back to Einstein's papers to settle it. You would sit together and make calculations. That's one difference between science and religion. As Weinberg put it in a talk on this (reporting approximatly) "We do have heroes in the world of Science, but we don't have Prophets".

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I am refering to the methodology of the OP.

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malawi_glenn
Homework Helper
The issue goes much deeper in my opinion. If a colleague and you had a scientific disagreement on (say) general relativity for instance, you would never go back to Einstein's papers to settle it. You would sit together and make calculations. That's one difference between science and religion. As Weinberg put it in a talk on this (reporting approximatly) "We do have heroes in the world of Science, but we don't have Prophets".

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I am refering to the methodology of the OP.
Yes that is of course reality too. Very good quoting: "Science have heroes, but no prophets" ;) :D

<off topic>
Yes that is of course reality too. Very good quoting: "Science have heroes, but no prophets" ;) :D
Quite a few interesting talks about science and religion.
</off topic>

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malawi_glenn
Homework Helper
Dawkins LOL sorry but he is a joke according to me :)

I am still waiting to hear buckey's response on renormalization altough;)

He asked about electrons tunneling through the atomic nucleus yesterday.. and today he refuses renormalization. Quite fun.

<still off topic (where are the mods?)>
Dawkins LOL sorry but he is a joke according to me
I don't know him. I understand he organized this event. Weinberg's talk is very interesting. Other people talked there as well.
</off topic>

Einstein beleived in hidden variables in quantum mechanics and was agaist the probibalistic nature of it: "God does not play dice" is a famous quoting.

This is not the first time you qoute a historical physicist to state your points. It is of course always fun to read about what the grand ol masters of physics thougt, and get historical insights on how the theories and models of today have developed.

But I do think that one has to read these things through the eyes of contemporary physical insight to get things correct since theories and insight develops.

Lord Kelvin said in 1899 that "now has everyting that can be discovered been discovered, only some small details remains".. 6 yeas later, Einstein published his work on his special theory of relativity (much of that work was already known and worked out by guys like Lorentz and Poincare, Einstein did however see how to make the correct physical interpretations of it and so on). So by recalling what the great Kelvin said, should we believe that there was nothing more to discover? ;-)

All great men do misstakes sometimes, nobody can be perfect.
Does that include you and me?

Dawkins LOL sorry but he is a joke according to me :)

I am still waiting to hear buckey's response on renormalization altough;)

He asked about electrons tunneling through the atomic nucleus yesterday.. and today he refuses renormalization. Quite fun.

Sorry, had to do some work. As you know, at 55, I can not be a grad student.
It seems you've caught on that I'm poking around at todays physics looking for holes and errors, of which, there seem to be a good number. Yes, I don't have the math experience which is actually a good thing. I've not been brainwashed or brainwashed myself into accepting the many hard to accept concepts that many PF members accept as reasonable or without question. I'm still able to look at the base of the concept, the kinematics. If those don't make sense and have no physical evidence then, in my book, they are suspect.

Theories are still theories, they are not laws. This group likes to regurgitate the many successes of the standard model, but what are those successes. All I see is math and ideas that are used as evidence of the successes.

If QM, QT, QFT, QED and QCD are so great, then why don't we see true quantum based electric devices, photon devices and the like. All current devices, claimed to be quantum, can easily be called single electron or single photon. If the use of a single particle justifies quantum calling, then I can call light and electricity multi-quantum phenomena.

How's that for an answer. And, if you'd be so kind, try to be civilized and write without insinuation or trying to demean. I have this bad habit of jumping on the band wagon once someone starts the name calling. Deal?

It seems you've caught on that I'm poking around at todays physics looking for holes and errors, of which, there seem to be a good number.
Name one
Yes, I don't have the math experience which is actually a good thing. I've not been brainwashed or brainwashed myself into accepting the many hard to accept concepts that many PF members accept as reasonable or without question.
PLEASE STOP INSOLTING US. This is annoying. Because we know more than you do and you are frustrated about it does not give you the supreme right to have public contempt towards us
This group likes to regurgitate the many successes of the standard model, but what are those successes. All I see is math and ideas that are used as evidence of the successes.
Please search for yourself on google, you will find plenty of references. Or use the 80 books you personnally have on your bookshelf.

why don't we see true quantum based electric devices, photon devices and the like.
But we do. Your computer uses transistors and the like.

malawi_glenn
Homework Helper
PLEASE STOP INSOLTING US. This is annoying. Because we know more than you do and you are frustrated about it does not give you the supreme right to have public contempt towards us
Please search for yourself on google, you will find plenty of references. Or use the 80 books you personnally have on your bookshelf.
I will report that post of his.

We are also more updated and know math better, we can see whats going on behind the math.

He probably needs books that are younger than the ones he have, I dont think he have books younger than 1960 hehe.

vanesch
Staff Emeritus
Gold Member
Sounds like we continue to teach that the world is flat because we don't know it is a sphere and even though a highly respected teacher told us we were being silly to continue teaching that the world is flat. Boy, are we humans foolish!
Apart from several answers here that indicated that renormalization is now better understood than the handwaving trick it seemed to be when it was first discovered (caricaturing, it was something like 4 + infinity = infinity + x hence x = 4), you seem to have a totally wrong view of what is science about. If we have a "theory" (a set of calculational rules) that can spit out numbers that agree with experiment, well, then that theory is a good one. And it turns out that quantum field theory, using renormalisation, just does that. Even though we may not completely understand WHY it does so.

As to your "the world is flat" statement, in fact, as long as you look at a small part of the world (say, a town), the hypothesis of "the world is flat" is an extremely good one. It means that you can use 2-dim Euclidean geometry when you read a city plan, which is exactly what you do. So teaching people to read a map "as if the earth were flat" is not something stupid to do: it is extremely practical. You only need to know what are the limits of its applicability.

When you lay out plans for a house, you use flat geometry. You take it that the sum of the angles of the 4 walls of a room must make up 360 degrees. Well, that is "working with the assumption that the world is flat". You don't do spherical trigonometry when you look at your house (otherwise the sum would not be 360 degrees). You "assume that the earth is flat".

In fact, it is very very well possible that renormalization is something of the kind: we look at the "flat" (low-energy) part of the world here. If the technique works, meaning, cranks out numbers in agreement with measurement, that's all science is supposed to do.

Of course, often science also provides a "picture" or an ontological view of things, where one assigns "reality" to the objects in the theory. In as much as this is enlightening, and gives one a "coherent world picture", then that's fine. But it is not the aim of science to give a "world picture" or "an ontology". The aim of science is to set up theories that make predictions that correspond to observations. As far as QFT is concerned, that seems to work out up to now for those cases in which the computations are tractable.

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There is kind of two ways of viewing renormalization. First, the effective field theory way, which has been given here and secondly the mathematical way.

I'll take a theory which has been mathematically proven to exist by Glimm and Jaffe, that is $$\phi^{4}$$ in three spacetime dimensions.
Now $$\phi^{4}$$ is an interacting theory and what physicists normally do is approximate calculations in the interacting theory by using a series in the free field theory. However many terms in this series diverge and must be repaired by absorbing the divergences into the constants of the theory.

Glimm and Jaffe however found that free Klein-Gordon and interacting $$\phi^{4}$$ live in two different Hilbert spaces and the perturbation series we use to calculate quantities in $$\phi^{4}$$ is ill-defined, since we are using operators from the wrong space. This is where the infinities of renormalization come from in a mathematical sense. Renormalization is then just a process of repairing this. It results in a well defined series, which agrees terms by term with the correct, axiomatically derived, series Glimm and Jaffe got by working in the correct $$\phi^{4}$$ Hilbert space the entire time. Nothing suspicous at all. For a long time physicists didn't have the mathematics to deal correctly with interacting field theories (even now the mathematics of Glimm and Jaffe is quite hard) and simply had a bunch of "free field tricks" to get around this, one of which is renormalization.

malawi_glenn
Homework Helper
Does that include you and me?
Of course, that is why confirmation of results must be done independently so a concensus can be established. Discoveries must be verified, articles must be peer reviewed and so on.

There is kind of two ways of viewing renormalization. First, the effective field theory way, which has been given here and secondly the mathematical way.

I'll take a theory which has been mathematically proven to exist by Glimm and Jaffe, that is $$\phi^{4}$$ in three spacetime dimensions.

...

There is:
Baez, Segal, Zhou "Introduction to algebraic and constructive quantum field theory".
Glimm and Jaffe's "Quantum Physics, a functional integral point of view"

The texts are pedagogical, but the maths is quite dense and advance. (The mathematics in Glimm and Jaffe didn't really exist until Glimm and Jaffe!)

A good warm up is Reed and Simon "Methods of Modern Mathematical Physics: IV Analysis of Operators". They don't look at field theory, but rather the anharmonic oscillator from regular quantum mechanics. Here the free theory is the harmonic oscillator and they calculate anharmonic quantities as a power series. They get a divergent series whose individual terms need to be renormalized. Basically all of the problems we associate with QFT show up in a QM problem. The renormalization here is not infinite because all quantum mechanical systems live in the same Hilbert Space, due to the Stone-VonNeumann theorem.
It might be useful to take a look at this theorem. See the excellent paper here: http://www.math.umd.edu/~jmr/StoneVNart.pdf" [Broken]. Problems in QFT appear to be due to a break down of this theorem.

Also possibly look into the Wightman axioms for QFT given in:
Streater and Wightman "PCT, Spin, Statistics and all that".

The main additional difficulty in quantum field theory is proving that a given theory exists, which is exceptionally difficult. If it is unclear as to what I mean by "exist" I will give more detail.

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Gokul43201
Staff Emeritus