If gauge symmetries are really just redundancies in our description accounting for nonphysical degrees of freedom, then how does one explain the deep and powerful fact that if one begins with, say, just fermions and no gauge field in one's theory (and no interactions & essentially no dynamics) but then imposes the demand that the theory be invariant under local U(1) transformations, then one finds a vector field must be introduced? Note that I did not have any vector field in my theory before I demanded invariance under gauge symmetry. If one thinks of the vector field as being in the theory to begin with then I can see how one could see it as a necessary constraint to remove extra degrees of freedom - you've got an A_mu, that thing's got 4 degrees of freedom and it should only have 2. But if I imagine that I knew nothing about photons or the electromagnetic field, and I require my theory of fermions to have this U(1) symmetry, then the vector potential arises as way of enforcing that symmetry. Beginning with no interactions, axiomatically or arbitrarily requiring gauge symmetry has this amazing power to produce not only gauge fields in the theory, but the correct number of them and with the correct self-interactions (or lack of them)! I suppose one could say that SU(3) just happens to work because there just happen to be 8 gluons, similarly for SU(2) and U(1), but doesn't that seem awfully random and clunky (or... unnatural)? Doesn't it seem much more natural and coherent to say that there are 8 gluons precisely because there are 8 generators of SU(3), and so on? If I begin my theory without the gauge fields then it seems to me to make no sense to say that the shockingly powerful principle of requiring gauge invariance only accounts for a redundancy in a field that I have not even put in my theory yet! *I cannot get away with imposing gauge invariance without introducing exactly the kind of forces and interactions that we observe and that appear in the SM.* That statement seems way too powerful for a mere redundancy in our description. Again, maybe it's true that if you go about it from the other direction, ie, requiring one photon and three weak gauge bosons, etc., then you are forced to introduce the right gauge symmetry to account for the redundancies. But that seems much more ad-hoc to me - you have a lot of random things that happen to be true and a lot of coincidences that happen to work out - whereas if you think about the requirement of gauge symmetry as giving rise to these connections that tell you how to move around in your bundle, that sort of communicate the local transformation from one place to another, then you are only making one ad-hoc postulate, and it is a concise and elegant one with a ridiculous amount of explanatory power. So why is this not the conventional wisdom of how to look at gauge theories today? It seems to me like its become a fashion to be seen as unimpressed with gauge symmetry and to take it down a peg by not getting too excited and brushing it aside as "just a redundancy" in our description and not a mysterious and deep fact of nature, a symptom of being afraid to appear naive. So, where am I wrong in all this?
As far as I know, almost all great physicists were/are impressed with local gauge symmetry. To those who want to call it "redundancy"(whatever that means), I say you need to change the definition of symmetry first. Sam
I think your mistake is in your understanding of what the conventional wisdom is. My experience is that the conventional wisdom places gauge symmetry at the very heart of our understanding of the nature of our world.
I disagree. It's not right to think that there are 8 gluons because there are 8 generators of su3. Mathematically, symmetries are a good description of nature - not the other way around.
I'm not sure I agree, I don't really have a problem with calling it a redundancy of description. Essentially all we mean is that all physical states in the Hilbert space of a theory transform as gauge singlets. This is just a consequence of the validity of Gauss's law (note that this is a local statement only).
This does not change the fact that our definition of symmetries is valid for local gauge invariance. So, why should one be bothered and invent new names? Gauss’ law IS the GENERATOR of the local gauge group: it generates the correct gauge transformations on the physical (dynamical) fields, satisfies the Lie algebra of the gauge group and (as in any constrained dynamical systems) it restricts the allowed physical states of the system to those that solve the constraints: in QFT this means the following: a state is said to be physical if it is annihilated by all the generators (Gauss’ law) of the gauge group (i.e., gauge singlets). Sam
Yes, but note that this 'symmetry' here is completely trivial. The language is used as a distinction between a global symmetry, where we typically have a set of unitary operators that take states into new states, whereas here its more of an identification or equivalence class of the same state.
"Gauge symmetry" is analogous to "general covariance" in general relativity, and does not produce unique self-interactions, just as general relativity is not the unique theory consistent with "general covariance". The "gauge principle" is like the "equivalence principle" or "minimal coupling" principle of general relativity. The equivalence principle is not the unique principle consistent with general covariance - see the discussion following Eq 4.32 of http://arxiv.org/abs/gr-qc/9712019. It is Wilson's conception of renormalization that explains why quantum field theories should be "renormalizable" at low energies. Just like "general covariance" in general relativity, a "gauge symmetry" is a redundancy of our description. The true principle of general relativity is not "general covariance", but rather that it is the theory of a relativistic massless spin 2 particle. The gauge invariant quantities are nonlocal (they are things like Wilson loops or holonomies in GR), but the redundant description allows us to write equations using local fields and interactions.
There may be great danger in holding too rigid a view of requiring a universal application of any one gauge symmetry. The one we choose at any one moment may appear to apply to all cases we study. But with more experimentation we may eventually find many cases where it does not apply. One example is late 20th century confinement of Maxwell's electrodynamics into U(1) which was led by Heaviside. Heaviside was certainly brilliant and that endeavor helped greatly to simplify the expression of EM behavior. Maxwell had a broader vision that included SU(2) and he partly described that with the use of quaternions. In the last few decades we have found many different instances where U(1) cannot express the behavior of EM even in experimental situations which could be considered classical. And there lies a very interesting overlap between classical EM and QM as SU(2) topology resolves those situations.
Well, first off, you are wrong in not using any paragraphs in your very long question, so it makes a bit hard to follow and a pain to read! But other than that, it is a great question that I wondered about before, too. The thing with gauge invariance is, that is not observable, that it has no physical consequences. It is only introduced in order to describe the theory with a local Lagrangian and by that to make the computation and the study of field theory much easier. Global symmetries with conserved charges are physical, so are massless particles and the fact that they have two polarizations. I guess you all know that already. So you are completly right to point out: why is it that local, redundant and unobservable symmetries are such a help in understanding and doing calculations with field theories? Maybe when they find ways to do calculations in QFT without Lagrangians, it will become clearer what this whole gauge business means. As I (very little) understand the recursion relation approach goes in the direction, where S-matrices are computed without any Lagrangians.
Gauge symmetries are important to physicists because they are symmetries. And as everyone knows, we love using symmetries. However,you can't speak of general symmetries as the symmetries that describes your world. The color interactions belong to SU(3) because you have strong interactions described as they are... It is though, that you take physical symmetries (U1,SU2,SU3, etc) and derive everything, that helps you be amazed by their power... A general, let us say SU(6) symmetry, is totally meaningless (at least for particle physics)...