# Why am I wrong about how to view gauge theory?

Edit: I know there have been some similar questions but I don't think any had quite articulated my particular confusion.

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 (no interactions & no dynamics), but then imposes 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 the imposition of local symmetry 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 the mechanism for enforcing that symmetry. Beginning with no interactions, axiomatically or arbitrarily requiring gauge symmetry has the 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 and might not want to.

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. It also seems weird to call the very thing that defines a theory only a dumb redundancy in the theory.

So, where am I wrong in all this? Is this an untenable view in the context of gauge theory (neglecting ideas that are being researched that may or may not pan out)? Or is this a viable view to take, even if one you dislike it? I know very little about spinor-helicity methods but I gather they may have something to say about this. Does their success eliminate the possibility of my interpretation?

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The problem is that there is no reason or justification for imposing the U(1) local gauge symmetry. It would be nice if there was some fundamental reason why a U(1) gauge symmetry had to exist. For example various models of string compactification generate these local symmetries, though none of these models are a full description of the Standard model. Still, this is an example of what we mean by a hidden (i.e. hidden at Standard Model energies) degree of freedom. – John Rennie Mar 7 '14 at 7:59
Thank you for your comment! It helps. Is it incorrect that I have a choice of either 1) imposing the symmetry without justification or of 2) having no explanation for why the fundamental interactions take the form they do? I mean like how many gauge bosons there are or why some couple to themselves but the photon does not. Cheers! – gn0m0n Mar 7 '14 at 8:05
Well, we have to impose the symmetry because it works. To pretend it isn't there, or is a coincidence, as in your option 2 seems unnecessarily obstinate. I don't think the choice you present is appropriate. The choice is just to say either (1) we don't know why the symmetries are there or (2) to go looking for reasons why the symmetries are there. Obviously physicists prefer option (2). – John Rennie Mar 7 '14 at 8:16
@JohnRennie: isn't U(1) local gauge symmetry necessary if one wants local conservation of electric charge or am I misinterpreting what local means here (I don't use Noether theorem everyday sorry)? – gatsu Mar 7 '14 at 9:30
@gatsu: no, that's due to a global gauge symmetry – John Rennie Mar 7 '14 at 17:15

## 3 Answers

You are right, it is wrong to think that in gauge theory "gauge transformations are just a redundancy". This becomes true only if one abandons locality, ignores all boundary effects, all instanton effects, hence most of what is interesting about gauge theory. Of course forming gauge equivalence classes (say of observables) is something one wants to do every now and then, but considering only gauge equivalence classes means to kill gauge theory.

Examples:

1) Instantons: Every gauge bundle on a n-disk is equivalent to the trivial one. Yet there are non-trivial gauge bundles on the n-sphere -- the instanton sectors. If you think that only gauge equivalence classes count, then there is only the trivial gauge bundle on one hemisphere, the trivial gauge bundle on the other hemisphere, and you have to glue them trivially on the equator to get a global trivial gauge bundle. Instead, what really happens is that gauge transformations are not a redundancy, but are all that makes up the non-triviality of the instanton sector, by the clutching construction. Ignoring this means to have non-trivial global structures that are not obtained by gluing local structures, hence means to break the locality principle.

2) Boundary fields. The way that the WZW model appears on the boundary of Chern-Simons theory: the gauge transformations of the Chern-Simons theory on the boundary become the very fields of the WZW model.

3) higher codimension defects: Wilson loops. Similarly, The fields on the Wilson loop in Chern-Simons theory are entirely the gauge transformations of the ambient gauge field, restricted to the loop. See here for review and pointers to the literature.

4) generally: Locality in gauge theory -- It breaks if one disregards gauge transformaitons, see the pointers here.

[edit: a commenter below points out that this is all fine, but does not seem to address specifically the construction inquied about by the OP. In fact it does, here is how:

5) Gauge fields from local gauging: The traditional physics textbook way of deriving gauge fields from local gauge symmetry is an example of the local relevance of gauge transformations as follows.

Every fermion bundle is locally gauge equivalent to the trivial such, and so carries the trivial connection, given just by the derivative. But remembering that gauge transformations are a local reality, one observes that these take this trivial connection to one with a non-vanishing vector potential $A$. While this will still have vanishing field strength, already here something may happen globally: if a bunch of these $A$ are glued by gauge tansformations, we may still have globally a non-trivial instanton sector. Given this then we are led to allow general local vector potentials $A$ and find then that by gluing them across patches by gauge transformations, we find the full moduli space of all possible gauge fields.

If however, and that's the correct point the OP observes, one declares that all local gauge transformations are just redundancies, then that means to replace each local vector potential $A$ by its gauge equivalence class. Gluing these across patches never produces all global gauge field configurations (for instance if the gauge equivalence class was the 0-class, one never finds the global torsion class instanton sectors).

]

Mathematically what is going on here is the statement that gauge fields do not form a moduli space but a "moduli stack". The mathematical concept of stack is all about what it means to combine locality with the gauge principle. An exposition of this is in our arXiv:1301.2580.

For instance what govers examples 2) and 3) above, where gauge transformations in higher dimension become genuine fields in lower dimension is essentially an instance of the looping construction on stacks: the moduli stack $\mathbf{B}G$ of $G$-instanton sectors has a single component

$$\pi_0(\mathbf{B}G) \simeq \ast$$

(hence "there is only one gauge equivalence class") but it nevertheless remembers the full nature of gauge transformations

$$\pi_1(\mathbf{B}G) \simeq \pi_o(\Omega \mathbf{B}G)\simeq G \,.$$

Thinking that "gauge equivalence is a redundancy" means thinking that the moduli stack $\mathbf{B}G$ may just as well be replaced with its 0-truncation $\tau_0 \mathbf{B}G\simeq \ast$, which means thinking that local $G$-gauge theory is trivial.

The same kind of arguments apply to the full moduli stack $\mathbf{B}G_{conn}$ of $G$-gauge fields (instead of just their instanton sectors). Then $\pi_0(\mathbf{B}G_{conn})$ is the sheaf of gauge equivalence classes of $\mathfrak{g}$-valued differential 1-forms. That's more than just the point as before, but still just a faint shadow of what gauge theory is about.

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+1 for the informative reply. But I suspect this won't satisfy OP. OP seems to be amazed that imposing local phase symmetry to fermions automatically give you the gauge bosons, and it seems we are getting too much from too little, if gauge invariance is "just redundancy". You did point out that gauge field is more than redundancy, but non of the features you listed seems to directly address OP's amazement of "let local phase transformations on fermions be a symmetry, then there are gauge bosons and the dynamics!" – Jia Yiyang Mar 7 '14 at 11:00
@Jia Yiyang, I see, right, I have edited the reply above to include now a point (5) that makes the relation to the construction that the OP considers more explicit. – Urs Schreiber Mar 7 '14 at 11:36
@UrsSchreiber Within a down-to-the-lab electromagnetism, does your saying "we need gauge fields beyond equivalence classes" mean that one can't have a Lorentz-force description of the Aharonov-Bohm effect? – Slaviks Mar 7 '14 at 20:58
1) Are you saying that non-large gauge transformations (those that are continuously connected with the identity) change the physical state of a system? 2) The gauge transformations you say that they are not redundancies, do they tend to the identity in the space-time boundary? Are they the so-called "large gauge transformations"? Thanks. – Diego Mazón Mar 7 '14 at 21:18
@drake: locally every gauge transformation is small (for a connected gauge group), so already quotienting out small gauge transformations destroys locality. – Urs Schreiber Mar 7 '14 at 23:24

To be honest, I think that the route you describe (and which is also used in many textbooks) is not physically well motivated at all. You have begun with a theory of a fermion with a global symmetry which maps physical states to different physical states. This theory has the property that specifying initial conditions on a spacelike surface completely determines the evolution of the system. This is what you expect as a physicist.

Now for some reason you want to make this symmetry local. In doing so, you destroy this property. The local invariance means that a given set of initial conditions can result in any number of final states. Hopefully you agree this doesn't make physical sense. In order to fix this problem and regain the nice property, you have to go through some complicated gauge-fixing procedure which usually obscures other nice properties of the theory (Lorentz Invariance, unitarity, etc).

I agree it's cool that gauging the symmetry introduces new fields, but it's not that surprising since you introduced a new non-dynamical field by hand (the gauge parameter). Meanwhile, the price you pay is predictability until you gauge-fix.

I think the other route that you mention, which is advocated in Weinberg's books and supported by recent research on on-shell approaches to QFT, makes more physical sense. You are doing quantum mechanics. Hilbert Space organizes itself into representations of global symmetries. Do the little group analysis. Discover you cannot put helicity $\pm1$ in a local field built out of creation and annihilation operators without gauge invariance. Once you are at this point (committed to a local description) then you get all of Dr. Schreiber's important points, many of which can actually be related to measurable physical quantities (for example instantons, the axial anomaly and $\pi^0 \to \gamma \gamma$ ). Please note that when doing gauge theory, you still make a choice on which states to identify through boundary conditions on the fields.

Since you asked specifically about the relevance of spinor-helicity methods, I'll say a few words. It is true that in recent years there has been tremendous progress in calculating scattering amplitudes on-shell, where there is no need for gauge invariance, the action of the little group is simple and manifest, and the calculations are much simpler. Some people like to say this demonstrates that we don't need any local formulation and that gauge theory isn't fundamental. However, you should keep in mind that almost all of these advances have been made in computing perturbative quantities. A lot of the most interesting effects in gauge theory are actually non-perturbative, and the amplitudes program doesn't really have a way of capturing this information. If they figure out how to calculate an instanton contribution to scattering in the on-shell formalism, then I think ultimately the answer will be that gauge-invariance is totally unnecessary and unphysical. Until then, the most you can say is that the local gauge-invariant picture allows you to calculate more things. In physics, the only real evaluation of a formalism is how it reproduces measurements.

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Thanks, especially for the information on on-shell methods. I will also be referring back to this. – gn0m0n Mar 13 '14 at 7:28

This is how I understand this issue.

First, I believe you may agree that imposing gauge invariance is a sensible thing to do. If we want our fields to be invariant under some kind of transformation it better be local, since two separate space-time points shouldn't be related in any unnecessary way, otherwise we may violate causality. A different issue is what symmetry we must impose but I believe this is not what you are asking.

Second, once we have imposed the symmetry we want to keep our derivative concept, the problem is that we have introduced a new object in addition to the dynamical fields which varies point to point. Now, this is better formulated in the theory of principal bundles, but I don't know how familiar you are with these. The point is the following and it is very similar to the fact that in GR we have separate vector spaces tangent to each point and we have to introduce a connection (covariant derivative) to relate the tangent spaces of different points. Now we have fields at different points and a transformation which is different at each point. So we have to introduce a connection to be able to relate the separate points. Gauge fields are these connections (on principal bundles), and its gauge symmetry is just the transformation law of a connection, and so a consequence of their nature.

If one looks at the transformation law of the Christoffel symbols in GR they transform in the same way as the gauge fields. The principal difference is that in GR we impose the condition that this connection is metric compatible, but in fact you can leave it as an independent degree of freedom. The beautiful thing is that if you do that and than you vary the GR action you get an equation for the connection which says that it is metric compatible!! This is not the case in modified gravity theories and you have to consider the connection as a separate degree of freedom.

Edit: Sorry for getting into the GR stuff but I think the analogy is useful.

This behaviour is the same in gauge theories, we leave the connections degrees of freedom and interpret them as dynamical fields.

I don't know if this answers you question or if it's too mathematical (or if you want it more mathematical) but I believe is the easiest way to see it.

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" If we want our fields to be invariant under some kind of transformation it better be local, since two separate space-time points shouldn't be related in any unnecessary way, otherwise we may violate causality. " This is argument is weak, what about all the global transformations like rotations, translations etc.? They don't seem to obstruct causality at all. – Jia Yiyang Mar 7 '14 at 10:49
@JiaYiyang Maybe I phrased it wrong, there is no reason why fields at two different points should undergo the same transformation, and then be related in some way. This is even stronger if the points can't have a causal relationship. On the other hand global phase transformations are just a particular case of local ones. In fact, the actual way of thinking about it is asking: if the parameters of the transformations are unobservable, why should we restrict them to be equal everywhere? – Julio Parra Mar 7 '14 at 13:15
Thank you for your answer - and the GR is fine. I am somewhat familiar with the idea of bundles and connections (I actually slipped in a little reference to them in my question). It is interesting to me that your answer and Dan's answer seem to contradict each other. He makes it sound like gauging the symmetry is physically unmotivated, while you make it sound like the most natural thing in the world. It's this seeming disagreement among knowledgeable people that makes me think the basis of gauge theory is not universally agreed upon and partially motivated my question. – gn0m0n Mar 13 '14 at 7:34