# What is physical in the principle of local gauge invariance? [closed]

Modern theories of interactions in particle physics are gauge ones. I know how the gauge fields are introduced in equations ($D = \partial + A$). I just do not see any physical motivation in it. I am afraid it is done by analogy with QED and that's it. I conclude that it is not the only possible way of description of interactions. But maybe there is something essential that I am missing?

EDIT: As the main failure of gauge way of "introducing" interactions I can point out its intrinsically perturbative character. One cannot switch off permanent interaction without severe problems.

dbrane answered to that in comments "... you can treat gauge field theories non-perturbatively and still gather valuable insights ..." Let us see. First question: With what does $A$ interact - with bare or real electron?

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## closed as not constructive by Luboš Motl, Moshe, Colin K, dbrane, Deepak Vaid Feb 20 '11 at 8:44

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If the gauge bosons predicted by insisting on having local gauge invariance are consistently observed, why would you complain? More importantly, general relativity too is based on a local gauge invariance - do you find that unphysical too? –  dbrane Feb 18 '11 at 20:54
Gauge theories have severe conceptual and mathematical problems, why close eyes on it? "Let me advance as bad theory as a previous one"- is not a good motivation. –  Vladimir Kalitvianski Feb 18 '11 at 21:01
Can you list these conceptual and mathematical problems? And in each case please mention why you think these problems are a result of imposing local gauge invariance (which is natural if you accept the Equivalence Principle) and not because of something else. –  dbrane Feb 18 '11 at 21:05
@Vladimir: you are again discussing renormalization. This has nothing to do with gauge theory per se. I suggest you either ask this as a separate question or else drop the subject altogether. –  Marek Feb 18 '11 at 21:51
@Vladimir: You didn't answer any of my questions. Like Marek said, you keep going into renormalization. I asked whether you find the local GI of relativity to be unphysical. I also asked you to mention conceptual problems arising from imposing local gauge invariance. –  dbrane Feb 18 '11 at 22:01

You sure are having battles here, Vladimir! I find myself, however, having moderate sympathy with this particular question.

I think your comment, “I just do not see any physical motivation in it. I am afraid it is done by analogy with QED and that's it.” can be seen as a large part of its own answer. When we say that something is "Physically Motivated", I take this to mean that a plausible argument can be given for using, in a new situation, a generalization of a mathematical model that has previously been used successfully as a description for Physical phenomena. The form of a "plausible argument" is not given a priori, it's just a question of what Physicists as a group find plausible. Plausibility has an acid test, which is whether a given Physicist thinks an idea for a new class of mathematical models has enough promise that they spend their own time developing the mathematics and its relationships with experiment. All that said, QED is physically successful, and enough Physicists found it plausible to consider generalization to non-Abelian gauge fields that, over the course of 15 years, from the mid-1950s to 1972, say, with perhaps a few hundred people working on them, a new, moderately empirically successful class of mathematical models was constructed. The analogy with QED is significant, but it's the arguments for why someone in 1955 might think intensively about such models that I think you're not paying enough attention to. Those arguments are still known in the community, and they play out in various ways in the comments on your question and on your comments, but I think it's fair to say that they are not very clearly elaborated. There is no axiomatic QCD, that lays out both the mathematics and why it's especially natural as a Physical model, for a Physicist to point to, for example.

Underlying all your questions, answers, and comments, however, is your railing, as I see it, against renormalization. 40 years ago, you would have been in company with very eminent Physicists, but today the game has largely moved on. People do talk loosely about bare and real electrons as you do, but the principal modes of discussion are now in terms of the renormalization group and the surrounding mathematics, which is well enough constructed as mathematics that as far as I can tell most Physicists are content with it, and almost all Physicists are content to calculate with it. As far as I've seen, models for real, shielded particles are relatively ad-hoc, and in any case, and insofar as they are not ad-hoc, are ultimately so grounded in the Taylor series mathematics of Feynman diagrams and the renormalization group approach that they are essentially not a new approach.

All sorts of moves are being made, both in the system and on the periphery, to tighten up the mathematics more, or to construct new non-perturbative methods, but they will remain peripheral for most Physicists until an approach is constructed that is significantly better as mathematics than the renormalization group approach, which seems more explanatory, and which is, additionally, more usable in an engineering sense than the renormalization group approach. I think it's significant that the renormalization group approach is more-or-less usable as engineering, but I think it's clearly enough not easy enough to use the renormalization group approach as engineering that a replacement for this way of thinking and constructing models will emerge in due course. I conclude, therefore, that I agree with you when you write “I conclude that it is not the only possible way of description of interactions“, but the hard issue is how to construct something that is better, preferably significantly better. Gotta do the math, and it's gotta be good math, but, harder, because it requires a certain kind of simplicity that we'll only recognize when we see it, it's gotta be good for use as engineering as well.

Peter.

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Peter, I have a toy model with a very simple math. One can use practically the same equations as in QED but with other their meaning. In my model the real electron is "large and soft" (quantum mechanically smeared) and photons are just its elementary excitations. The most "difficult" thing is a concept of a "compound" electron. But as soon as we agree that the electron is permanently coupled with its quantized electromagnetic field (the exact QED equations say so), it is easy to accept a slightly different physical construction (in which the electron is coupled non perturbatively). –  Vladimir Kalitvianski Feb 19 '11 at 16:49
To me, a toy model isn't OK to face off with a big mathematical system like QFT, even if it might give a few people some feeling for how to construct a big enough alternative mathematical system. It needs a class of models that people can more-or-less comprehend as a whole system of thought, but that is still large enough that we feel able to model almost anything we might come across. We have to believe it will work. It must give enough insight to allow us to construct a first model, and we have to construct methods for refining those models. I think it needs very carefully chosen math. –  Peter Morgan Feb 19 '11 at 17:56
To go at the content of your comment, if you're using the same equations as QED (or nearly), that sounds no different from other people's ways of talking about and modeling shielded particles. There are such ideas in the literature. To me, the problem is to construct new math, not to construct new interpretation. "Quantum mechanically smeared" is too vague --you need more precision--; if it's QED math, is it a quantized field? QED can be interpreted to be about an electron field interacting with an EM field, no particles, but with subtle discrete structure. No new math in that, though. –  Peter Morgan Feb 19 '11 at 18:13
My electron is not "shielded" but "smeared". Somewhat like in atom - you have a "cloud" instead of a "point". Yes, we need correct interpretation, then the math is easy to apply. In my model one cannot "switch off" the quantized EMF generated by the electron - it is its essential feature and is provided by construction. An external field (sourced with the other charges) can be switched on like $D = \partial + A_{ext}$ but not the proper EMF. That's the difference. Of course, it is not in the state of the art yet but in course of "slow developing". The main difficulty is to find funding for it. –  Vladimir Kalitvianski Feb 20 '11 at 0:10

Strange, I thought that a perfect example and motivation for gauge fields is the electromagnetic field (the classical one). Isn't that a very good physical motivation!

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CED (as well as QED) is not self-consistent, don't forget it. –  Vladimir Kalitvianski Feb 18 '11 at 20:56
So! CED (as well as QED) is very successful as a physics theory. That seems enough to me for motivation. –  MBN Feb 18 '11 at 21:04
No, you do not use runaway solutions. You use an approximation because the exact solution is not physical. And that approximation is not the original theory. So you do not have a theory. Or you have unsuccessful theory. –  Vladimir Kalitvianski Feb 18 '11 at 21:13
When you say that classical electrodynamics is not self-consistent, are you referring specifically to issues surrounding point particles? Classical dynamics with nonsingular charge distributions is self-consistent, as far as I know. –  Ted Bunn Feb 18 '11 at 21:33
To Ted Bunn: yes, all plasma physics uses smooth charge distributions, I know that. Yes, I speak of a single point-like charge model issues. –  Vladimir Kalitvianski Feb 18 '11 at 22:44

Of course it was done by analogy with QED. If QED works that way why shouldn't other theories too? Naturally it's not this simple and it took time to make sense of other gauge theories but nevertheless it turns out that it works which is actually the only thing that matters.

Now, there is certainly no reason why it should be so. Also, by the very definition of gauge invariance of redundant degrees of freedom, you are free to fix the gauge and forget about gauge theories altogether if you are able to write down your Lagrangian in some other way.

So, in my opinion, there are only two reasons why one would introduce gauge invariance:

1. metaphysical one: it's a beautiful theory and the requirement of locality is definitely appealing as well (in the same way people liked moving from instant-action theories to local theories)

2. practical: it simplifies things a lot; both conceptually and computationally. Instead of looking for interaction given by arbitrary terms, you can just specify your field content and the group it's transforming under and you're done. The rest is routine (haha) work of verifying whether theory works.

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Thanks, Marek, for your answer. So there is no real physical motivation to do so, it is just a blind guess. That's what is not good. –  Vladimir Kalitvianski Feb 18 '11 at 21:04
Vladimir, I wouldn't call that a "blind guess". My be a very educated guess? –  MBN Feb 18 '11 at 21:07
Without physical motivation? No, it is a blind guess for sure! Or artificial intelligence guess, if you like ;-). –  Vladimir Kalitvianski Feb 18 '11 at 21:09
Yes, a very educated guess. And one that turned out very nicely too. Although, truth be said, the theory was first tried as $SU(2)$ for strong interactions which didn't work at all. Only much later it was realized that $SU(3)$ was the right group. Also, there are lots of other "problems", e.g. with introducing mass, that needed to be overcome. But in conclusion I'd say that it was very nice theoretical work that paid off. I don't understand @Vladimir's sentiment at all. Perhaps he's not familiar with all the successes of Standard Model yet... –  Marek Feb 18 '11 at 21:14
SU(3) turned out better for classification of particles, I agree, but to introduce interactions? No, it is a different topic. I see one patch on another and a fanatic belief in our intellectual power whereas the physics task is different - follow experiments first of all. –  Vladimir Kalitvianski Feb 18 '11 at 21:19