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A What does gauge theory explain?

  1. Dec 13, 2017 #1
    This is a bit of a philosophical/conceptual question. I've done tons of reading on it, of course, but haven't found anything that makes me go 'ah ha'!

    I am working steadily through the mathematical formalism of differential geometry, but am struggling to grasp how the things we say in this language actually explain the physics we observe.

    Everywhere I go, 'gauge symmetry' and 'gauge invariance' seem to be at the heart of things. I often read statements along the lines of 'gauge symmetries give rise to forces' or 'fundamental interactions are mediated by gauge fields' etc. And a lot of hand-wavey stuff about how the fundamental forces serve to 'reconcile' the different gauge (phase) settings at different points.

    I'm at the edge of my understanding with this stuff, and the unsatisfactory nature of talk like this is probably mostly an artifact of my reading level. I am rapidly learning the mathematical formalism of differential geometry. I'm fairly comfortable with group theory, Lie derivative, covariant derivative, and I am now learning about fiber bundles. I have not internalized anything about connections yet, which I understand to be important.

    Anyhow, my pointed question is:

    1. What does 'gauge theory' actually explain? In other words, how would nature be different if we couldn't use gauge theory to describe it? Would the fundamental forces just not exist? I've read that these local symmetries 'constrain' or 'determine' the laws of nature. In what sense? What limitations, specifically, does the demand of local symmetry put on our formulation of natural laws? I haven't been able to find an answer to this that makes sense to me. (Just because I'm still learning.)

    My dim understanding is that there is a symmetry group parameter that we can shift by a different amount for each point in the space. But global invariance needs to be maintained, so we need a way to reconcile or compensate for the fact that the parameter was shifted by different amounts at two points. Somehow (?) the forces of nature are the manifestation, to our experience, of nature doing this reconciliation.

    But unpacking this, 1) what is the motivation for going from global to local symmetry? Is it theoretical, experimental, both? 2) what is the motivation for demanding global invariance? Why, physically, do the varying shifts in the parameter (applying different group elements to different points in the space) have to be reconciled? What would the big disaster be if we didn't?

    Another, related but even more philosophical, question:

    2. Do gauge theories/symmetry principles actually explain anything? Or are they just more elegant and satisfying ways of labeling the same phenomena? In other words, I suspect that the forces of nature don't 'arise from' symmetry principles causally but that they are just equivalent to symmetry principles. So you could work from symmetry to forces of nature or forces of nature to symmetry.

    So is saying the forces happen 'because' of symmetry pretty much the same as saying that you flipped heads 'because' you didn't flip tails? That isn't an explanation as much as a reformulation. Is it the same with gauge principles? Or is there substantive explanation to be found there which satisfies the yearning for explanation, in the way that the kinetic theory satisfyingly explained the Second Law of Thermodynamics or GR satisfyingly explains gravity as spacetime curvature?

    I'm really just trying to fill in gaps in my understanding and thought asking specific questions might accelerate that. Thanks.
  2. jcsd
  3. Dec 16, 2017 #2


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    I don't think there is any rule saying that a global symmetry MUST have a corresponding local symmetry. A famous counter-example is Weyl's first gauge theory, based on scale. A global change of scale has no observable effect (pre-quantum, anyway). If you assume that it's a gauge symmetry, it has weird consequences: An object making a trip around the universe might return smaller than when it left. So the world doesn't seem to obey such a symmetry.

    I think that there are two points to gauge theories:
    1. It's a heuristic way to come up with new interactions: Notice a global symmetry. Propose a local version. Voila! A new type of interaction.
    2. Gauge symmetry in lots of cases is necessary for the theory to be consistent. The gauge symmetry results in the magical cancellation of terms in perturbation theory that would make the theory blow up, otherwise.
  4. Dec 18, 2017 #3

    Urs Schreiber

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    These are good questions to ask.

    Indeed, much of the talk about how symmetry principles give rise to gauge theory only describes the ingredients of gauge theory, without really deriving it.

    What most texts really only say when they speak about "local gauging" etc is the simple fact (simple with some diff geometry in place) that as soon as you pick any fiber bundle and regard it as a field bundle for some field species (as in 3. Fields) then you are immediately entitled to considering another type of fields given by connections on that field bundle.

    Here one needs to face a terminology issue: There is the (very) general concept of gauge symmetry in field theory (chapter 10. Gauge symmetries) and then there is the special case of Yang-Mills theory, whose fields are connections on fiber bundles, which is what many people and texts really mean when they say "gauge theory". But not every gauge symmetry is as in a Yang-Mills theory.

    You can tell that people don't really think that the "local gauging" argument alone "derives" Yang-Mills gauge theory from first principles by the interest generated by schemes which actually do derive YM-theory from other (presumably deeper) principles: This happened originally with Kaluza-Klein theory, which showed that gauge theory in the incarnation of Yang-Mills theory follows from gravity, then with string theory, which showed that both the structure of gravity and Yang-Mills theory appear from certain deeper assumptions. The reason some people are interested in this is precisely because it picks out (gravity and) gauge theory (in the form of Yang-Mills theory) from the otherwise vast space of imaginable local Lagrangian field theories that one could consistently write down.

    The principles of differential geometry and of Lagrangian field theory alone are not sufficient to "derive" that the world is described by Yang-Mills gauge theory. It remains a striking experimental fact that in the vast space of possible local Lagrangian densities that one could write down, nature picks those of Einstein-Yang-Mills-Dirac-Higgs form. Just saying "gauge symmetry" alone does not explain this, even if after the fact it looks very "natural".
  5. Dec 30, 2017 #4
    If you have a free fermion theory with a global U(1) symmetry and you, by fiat, replace a global exp(V) transformation (V=const) by a local one - exp(V(x)), the theory obviously breaks.

    The gist of the above statements is: the theory can be repaired (its equations will again be valid) by adding a vector field and changing derivatives to gauge covariant ones. The resulting modified theory now has interactions.

    Physical theories generally strive to build mathematical models which can be used to calculate correct predictions for experimental data.
    It was found out that gauge theories are the only ones (AFAIK) which succeed to do that for particle physics. That's why we use them.

    Then physicists would work on finding theories which do describe nature. We would use any type of theory as long as it is able to produce correct predictions.
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