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Why do gauge theories win out?

  1. Aug 28, 2013 #1

    marcus

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    "Gauge" is normally understood as "mathematically redundant"

    Why are gauge theories so prevalent? Why do they always seem to win out in the contest to describe the world?

    In a theory with some group of gauge symmetries, only the gauge-invariant quantities are considered physically meaningful.
    But the mathematics that people find most convenient or true to nature allows for other, gauge-dependent quantities and these are considered physically insignificant redundancy.
    So why not refine the math so as to eliminate all that meaningless "filler"? Isn't it inefficient, just "extra baggage" diluting the real physical content of the theory?

    But no: the successful theories are always gauge theories. Why is that?
     
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  3. Aug 28, 2013 #2
    The problem is, considering massless gauge bosons in [itex]3+1[/itex] dimensions, [itex]A_\mu[/itex] has 4 components, and there are only [itex]2[/itex] physical degrees of freedom. The little group [itex]ISO(2)[/itex] does not transform [itex]A_\mu[/itex] like a Lorentz vector (because of the "translation" part), so one has to connect [itex]A_\mu[/itex] to a conserved current [itex]j^\mu[/itex], and the fermionic origin of [itex]j^\mu[/itex] explains the so-called gauge-invariance, which is in fact just a mathematical redundancy.

    For massless particles, helicity formalism ([itex]p_{\alpha \dot \alpha}=\lambda_\alpha \tilde \lambda_ \dot \alpha[/itex]) is a first step (no need of a null constraint for [itex]p^2[/itex]). Unfortunately, we have always the [itex]U(1)[/itex]ambiguity : [itex]\lambda_\alpha \to e^{i\theta}\lambda_\alpha, \tilde \lambda_\dot\alpha \to e^{-i\theta}\tilde\lambda_\dot\alpha[/itex].

    I think that the work made by Nima Arkani-Hamed and others on gluons amplitudes (grasmannian, twistors, etc...), is very interesting. Certainly, it means that space-time has to be redesigned.
     
  4. Aug 28, 2013 #3

    tiny-tim

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    hi marcus! :smile:

    isn't it emmy noether's fault?

    isn't a gauge symmetry a geometrical symmetry just like any other, and so must have an associated physical "current"?
     
  5. Aug 28, 2013 #4

    marcus

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    Hi tim! :smile: Noether's theorem about how conserved quantities relate to global symmetries is beautiful. At the moment I cannot relate it to the diffeomorphism group of GR, which is something that relativists consider a gauge group.

    So that is an opportunity to stretch conceptually and broaden the discussion. In GR what they call a spacetime geometry is not just one metric tensor g on a given manifold. It is all the other metrics that g and accompanying matter could be mapped to by smooth maps (diffeomorphisms).

    So what relativists call a geometry is an equivalence class [g] under diffeos, of all the metrics that are equivalent to g by smooth mappings.

    The distinction between two metrics that only diffeo-differ (say merely by a smooth CHANGE OF COORDINATES) is not considered to be physically significant. It is just "gauge". It doesn't count.
    That is the essence of Einstein's principle of General Covariance and it kind of means that points of space and time do not have absolute meaning or objective reality, all that matters is the relationships among things.

    You know all this and I think it's fairly common knowledge. The reason I repeat it is to make the point that GENERAL RELATIVITY IS ALSO A GAUGE THEORY.
    It is just that the gauge group isn't linear transformations like in flat Minkowski space, the gauge group is mooshy morphy nonlinear transformations.

    So adding GR to the list, along with Maxwell Electrodynamics equation, and YangMill, or what Trimok was discussing, seems to suggest that ALL successful contemporary theories are gauge theories. Which if that's right does ask for an explanation.
     
    Last edited: Aug 28, 2013
  6. Aug 28, 2013 #5
    A really simple version of the argument I've heard is the following:

    Since [itex] [ A^\mu , A^\nu] [/itex] ~ [itex] \eta^{\mu\nu} [/itex], then [itex] \langle 0 |[ A^\mu , A^\nu] | 0 \rangle [/itex] has some negative values. But [itex] \langle 0 |[ A^\mu , A^\nu] | 0 \rangle [/itex] is a probability, which can't be negative. Therefore, you need some kind of symmetry to make the negative probability modes unphysical.

    This argument relies on two things: The first is Quantum mechanics, which is necessary for saying there is something wrong with the negative components. The second is assuming spin one particles exist, otherwise the above would likely imply that there shouldn't be spin one particles.

    I would also say that first part, [itex] [ A^\mu , A^\nu] [/itex] ~ [itex] \eta^{\mu\nu} [/itex], depends on a lot of the usual assumptions physicists make without stating. I think I read this argument in Peskin&Schroeder chapter 4.
     
  7. Aug 28, 2013 #6

    marcus

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    Hi DimReg, the question I'm asking is why does it seem that all successful theories--the ones that have won out as reasonably fundamental/accurate descriptions of the world--seem to be gauge theories?

    In the sense that General Relativity is a gauge theory, and Electrodynamics (say classical version with U(1) gauge symmetry), and other examples which you can give from particle theory. Is that a fair impression? If so, can you suggest some general reason for the overall prevalence?
     
  8. Aug 28, 2013 #7

    julian

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    When was the last time a new gauge principle was experimentally verified??? What happened to GUTS? Supersymmetry?? Higgs boson may have been discovered recently but that goes back to theory how many decades ago??
     
  9. Aug 28, 2013 #8

    atyy

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    In gauge theories, the "true" objects are loopy, but we usually like the non-loopy description, which is redundant.

    So why are the true objects loopy? Because of string theory:) Alternatively, the true objects are local, but they interact in such a way that the low energy degrees of freedom are loopy, like spin ice which has an emergent gauge field. http://arxiv.org/abs/1112.3793
     
  10. Aug 28, 2013 #9

    Pythagorean

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    Is the other stuff actually filler though? My limited understanding is that only the Lagrangian is invariant. The specific geometry will be frame-dependent (where a particular particle is at a particular time).
     
  11. Aug 28, 2013 #10

    julian

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    What emmy noether's found was that if you do the same experiment at the same time but at a different position but the dynamics was the same this implies via the Lagrangian that momentum is conserved, did the same experiment at the same place but at a different time and the dynamics was the same that the energy is conserved.

    How does this compare to gauge symmetries which are when the `state' can be changed without changing the `physical state' (mathematical redundancy)? You can move your apparatus from one place to another but is there an actual physical process by which you can make an U(1) transformation? Well maybe the Aharonov–Bohm effect, but aside from that.
     
    Last edited: Aug 28, 2013
  12. Aug 28, 2013 #11
    This is inspired by Rovelli's recent paper, I presume? I haven't yet had time to digest it, but I usually find these more conceptual papers of his very enjoyable (was a big fan of his discussion of entropy in his recent FQXi contest entry).
     
  13. Aug 28, 2013 #12

    julian

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    In non-general covariant theories the loopy description is a discrete approximation, in GR the loopy description is actually exact and well-defined, maybe describing truncated degrees of freedom though, precisely because of the non-local nature of diffeomorphisms - making it mathematically rigorous/finite. I think the non-loopy description is an approximation in the low energy limit - and relates to the r-Fock description type issues.
     
    Last edited: Aug 28, 2013
  14. Aug 28, 2013 #13
    I do think I answered your question to some extent though. Once you have lorentz invariance, quantum mechanics, and spin one particles, you're forced to have a gauge symmetry. Given that these are ingredients that we have a high degree of faith must be fundamental, it leads us to believe that the only promising fundamental theories also have these features, hence gauge theories are also the most successful theories we know of.

    If you agree with the argument I gave in my previous post, then the answer to your question would just be that spin one particles exist therefore gauge theories are necessary. (Off the top of my head, theories that have lorentz symmetry breaking could get around this, but those don't tend to do very well experimentally)

    General Relativity is a different story, since it doesn't require spin one particles. However it's symmetries are more inevitable: the gauge group of the standard model seems much more arbitrary than the "gauge group" of GR. For example, local lorentz invariance is a result of formulating GR as the theory of the metric on a semi-riemannian manifold -> gauging the lorentz group is something you would do even if you had never heard of gauge theory. Gauging SU(3) X SU(2) X U(1) on the other hand... For this reason, I'm hesitant to think that the gauge elements of GR and the gauge elements of the SM are of the same origin.

    My overall point is that gauge redundancies fit pretty well into things we know to be true. In my opinion, understanding "why gauge theories" would likely also include understanding "why quantum mechanics", which hopefully you agree would be biting off more than one can chew. I think there are some opinions of experts along the lines of "gauge symmetries show that we don't use the right degrees of freedom", but I don't know who those people might be.
     
  15. Aug 28, 2013 #14

    marcus

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    Exactly! I've been trying to understand his reasoning. It seems to be not about theories of separate phenomena in a stand-alone mode, but about how theories of separate phenomena couple together.

    When we couple a particle to geometry we don't couple it to the geometry itself, the equivalence class. We pick a representative of that class, a particular metric g on a particular manifold with particular coordinates. And we couple the particle to THAT with all its arbitrary and physically meaningless detail.
     
  16. Aug 28, 2013 #15

    marcus

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    Hi julian, DimReg, you have some good points. There's more than I can respond to right now. I'm also trying to read the august paper .5599 which is about this question of why gauge theories should be so ubiquitous. http://arxiv.org/abs/1308.5599.

    Note that when gr is considered as a gauge theory the gauge group is the diffeomorphisms which can really stir things around a lot on the manifold and even map from one manifold to another. they are not merely "nonlinear changes of coordinates" and not just local Lorentz symmetries.

    If you have two manifolds and a diffeo mapping between them and on one you have a solution of GR equation consisting of a metric g and a certain distribution of matter flowing around, then if you map all that over to the other manifold you get another metric and matter flow that is ALSO A SOLUTION of GR equation. Even if you imagine them looking quite different the two solutions represent the same reality. So that's general covariance---reality doesn't have a preferred structured layout---only relationships of stuff&geometry relative to stuff&geometry, and you can moosh it around as long as you moosh both the distance function and the matter consistently in the same manner. this may be a bad verbal translation, it's hard to express verbally (for me at least.)
     
  17. Aug 28, 2013 #16

    atyy

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    I think you and trimok gave the textbook answers. Does it not extend to spin 2? The usual thing I've heard is that the special relativistic quantum theory of massless spin 2 is uniquely GR, automatically including the EP and general covariance, whereas the EP plus general covariance does not imply GR. Something like in Hinterbichler's review http://arxiv.org/abs/1105.3735.

    Maybe we can consider it inexact in GR also, since GR seems non-renormalizable - unless LQG or asymptotic safety work?
     
    Last edited: Aug 28, 2013
  18. Aug 28, 2013 #17

    marcus

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    Here's a key passage, from page 5. He distinguishes between gauge invariants on the one hand and gauge variables on the other. The former are observables in Dirac's sense of being predictable within the theory, and having a definite numerical value regardless of how we act on the system with the gauge symmetries. The latter, gauge variables are exactly that, their values depend, so to speak, on the circumstances (perhaps the presence of some other system brought into contact with the one under study.) There is an element of arbitrariness and variability not governed by the system itself--so the gauge variable is not a Dirac observable.
    ==quote http://arxiv.org/abs/1308.5599 ==
    That is, we can “measure” a gauge variable (below are several examples). But this number cannot be predicted by the dynamical equations of the system S. Quantities that can be measured but not predicted were denoted “partial observables” in [13].
    The conclusion that gauge-variables can be measured does not contradict Dirac’s original argument according to which considering a gauge variable as “observable” leads to indeterminism. In fact, it confirms it. Dirac used “observable” in the sense of predictable quantity. We can turn Dirac’s argument around: every time that the physics is such that a certain measurable quantity entering the description of a system is unpredictable in terms of the system’s dynamics alone (like the time, or the position of a single starship), it must be described in the formalism by a gauge variable. This is what gauge variables are.
    ==endquote==

    So if this is correct there must be gauge variables in physics (at least as long as physics must deal with two or more separate systems which, in some instances, need to be hooked together).
    A very simple example, or "parable", of this is given in the paper. It is of a line of spaceships which as far as we know is alone in the universe. Location-wise, all that matters are the, say, 9 numbers xi between the 10 spaceships. These suffice to tell the ships' locations (anything more would be redundant or meaningless). A ship's location is only known relative to the other ships because AFAWK there is nothing else in the universe. So that's like one selfcontained physical theory. Then another fleet appears with its separation distances yj. In order to relate the two we need one addtional datum, which might the distance between x10 and y1. Measurable, but not predictable within either of the two separate theories.
     
    Last edited: Aug 28, 2013
  19. Aug 28, 2013 #18
    I definitely gave the textbook answer, as I pointed out I read it in a textbook! I can't claim to have done too much original thinking...

    I don't think I agree with that review's characterization of the way Einstein derived GR, but I think his statements about spin 2 systems are correct.

    Edit: Rovelli's paper is more interesting than my regurgitation of textbook explanations of gauge theory, I now wish I had read that paper before posting...
     
  20. Aug 28, 2013 #19

    atyy

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    But is Rovelli right? His discussion is entirely classical. Usually we say that classical electromagnetism can be completely described by using the E and B fields which are "real", and we don't need the vector potential which is "gauge". Do classical E, B fields and charge count as "gauge" in Rovelli's language?

    How about a Turing machine? What is "gauge" in a Turing machine?
     
    Last edited: Aug 28, 2013
  21. Aug 28, 2013 #20

    marcus

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    No harm done! Your post contributed a lot to our understanding of one sector of the problem. I hope you comment on the recent paper as well!



    Well, he's an astute dude.:biggrin: Also these are fairly elementary matters. The electrodynamics vector potential is indeed "gauge", but look at equation (1) where interaction with a fermion is written using A. It seems to follow the pattern he's indicating.

    He's starting off by giving two classic examples: classical GR and classical ED. (Don't see the point of asking if E, B, and charge are "gauge". Classically I see no indication that anybody thinks they are gauge, E and B are what he calls the field F.)

    EDIT: I see I referred to the Maxwell potential as "vector potential"(which could be misunderstood). It's a four-vector and also sometimes called a "four-potential". To reduce chance of confusion I'll here quote from the paper and try to use its terminology consistently.
    ==quote 1308.5599 page 1==
    Electromagnetism can be expressed as a U(1) gauge theory for the Maxwell potential A. The gauge-invariant content of the theory, whose evolution is predicted by the Maxwell equations, is captured by the electromagnetic field F = dA. This leads us to say that what exists in Nature is only the gauge invariant content of A. However, we describe the coupling of the electromagnetic field with a fermion field ψ in terms of the interaction lagrangian density...
    ==endquote==

    EDIT FOR ATYY: Rovelli does not claim that gauge theory is necessary in any universe with at least two interacting objects. You must have misread. He gives an example of a universe with many objects described by a theory with no gauge component. Don't understand your saying "My skepticism is about the claim that gauge is necessary in all universes with at least two interacting objects"
     
    Last edited: Aug 29, 2013
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