# Characterization of a gauge theory in terms of observables

1. Oct 23, 2015

### DrDu

Although I have a good understanding of how to do calculations in gauge field theory, I am still dissatisfied with my understanding of why we use them in the first place.
From a philosophical point, it should be possible to characterize a gauge theory in terms of observables only. I suppose one than has to work with nonlocal observables like Wilson loops and the introduction of non gauge invariant operators appears as a mathematical trick to restore locality.
However, I have never seen this worked out (besides not being sure whether it is correct at all).

2. Oct 23, 2015

### A. Neumaier

Quantum gauge theories are used mainly because they are renormalizable and match observations. No alternative theory is able to do that. In QED one can usually work with the field strength tensor (though not always, as seen in the Aharonov-Bohm effect); but in nonabelian field theories, the field strength tensor while covariant, is not an observable, and the observable structure is rather complicated. In dimension 1+1, the observable structure of pure Yang-Mills theory is fully nonlocal and fairly well understood; see papers by Rajeev, Landman & Wren, Hall. But in dimension 1+3, there are both local and nonlocal observables. For the local observables, see, e.g., my review of a failed solution of the Yang-Mills Millennium problem.

3. Oct 23, 2015

### DrDu

So let me reformulate my question: Given an algebra of observables, how do I know whether it corresponds to a gauge field theory.

PS: Could you please provide a more complete reference for Rajeev, Landman & Wren, Hall?

Thanks

4. Oct 23, 2015

### A. Neumaier

These are several papers:

Rajeev, S. G. (1988). Yang-Mills theory on a cylinder. Physics Letters B, 212(2), 203-205.

Mickelsson, J. (1990). On the quantization of a Yang-Mills system with fermions in (1+ 1) dimensions. Physics Letters B, 242(2), 217-224.

Landsman, N. P., & Wren, K. K. (1997). Constrained quantization and θ-angles. Nuclear Physics B, 502(3), 537-560.

Driver, B. K., & Hall, B. C. (1999). Yang–Mills theory and the Segal–Bargmann transform. Communications in mathematical physics, 201(2), 249-290.

Hall, B. C. (2001). Coherent states and the quantization of (1+ 1)-dimensional Yang–Mills theory. Reviews in Mathematical Physics, 13(10), 1281-1305.

5. Oct 23, 2015

### A. Neumaier

This is an open research question. In practice, one has to proceed in the reverse direction.

However, theories with massless vector bosons must be gauge theories; this follows from Weinberg's work.

6. Oct 23, 2015

### Urs Schreiber

This is a really good question that turns out to have been left pretty much open in the AQFT literature. A systematic analysis of the local net in abelian gauge theories was undertaken only rather recently in
where it was found that the traditional prescription of forming gauge invariant observables actually breaks locality
• Alexander Schenkel, On the problem of gauge theories in locally covariant QFT, talk at http://www.science.unitn.it/~moretti/convegno/convegno.html [Broken] Trento, 2014 (pdf)
I had then pointed out to the authors that this is to be expected on general grounds, and that one will need to keep the gauge transformations around in a groupoidal/stacky way if one is to preserve locality:
• Urs Schreiber, Higher field bundles for gauge fields, talk at http://www.science.unitn.it/~moretti/convegno/convegno.html [Broken] Trento, 2014 (web)
This was then actually carried out in
and it was found that this does recover locality: the authors consider the sheaf of groupoids (stacks) of gauge fields on spacetime and then form the correct cosimplicial algebras of observables on these groupoids.

So, while this does still not give sufficient conditions to answer the question for how to intrinsically read off from a local net of observables whether one is faced with a local gauge theory, it is a big step forward as it gives a nontrivial necessary condition: to have a local gauge theory at all, then it is necessary that it local net of observables does not take values in ordinary algebras, it has to take values in "higher algebras" in the sense of homotopy theory.

Last edited by a moderator: May 7, 2017
7. Oct 23, 2015

### DrDu

I hardly understand a word of what you are trying to tell me. However, I liked your example of the Dirac monopole. Could you explain in what sense the local net of observables does not take values in ordinary algebras? Maye you could name the observables explicitly and how they take values in this higher algebra?

8. Oct 23, 2015

### atyy

Is the sense of "local" in "local nets" the same as what physicists mean when they say that we use gauge redundancy in order to have manifest locality?

I had imagined that the "local" in "local nets" is closer to the idea of locality used when it is said that pure gravity does not have gauge-invariant local observables. I had also imagined that that sense of locality is not the same as that used to explain why we use gauge redundant variables.

The reason for my naive expectation is that we do imagine that Yang-Mills presumably has local gauge-invariant observables unlike gravity, but the gauge-invariant observables are not local in the sense of "manifest locality".

Last edited by a moderator: May 7, 2017
9. Oct 23, 2015

### Urs Schreiber

You should also look at the last article that I cited above, but here is a quick reply:

The algebra of observables is that of the functions on the (phase) space of field configurations. The key point now is that for a gauge theory, this space is not just like a (possibly infinite dimensional) manifold but is like a Lie groupoid. You find pictures of such Lie groupoids at Examples of prequantum fields I: Gauge fields .

So in gauge theory the algebra of local observables should be an algebra of functions on a Lie groupoid. To see what that might be, check out the pictures behind the above link where it is about the simplicial nerve of Lie groupoids. As you see there, these simplicial nerves are towers of ordinary spaces (manifolds) which are connected by lots of maps between them that satisfy some conditions.

So we know how to form algebras of observables in each simplicial degree of a Lie groupoid, these are just plain algebras of functions. But now the simplicial maps serve to pullback functions between different degrees. So as you consider the function algebra on a simplial nerve of a Lie groupoid, the result is a co-simplicial algebra. You may think of this as something a little bit more general than the more familiar dg-algebras.

This is then the "higher algbras" of local observables in gauge theories, as considered in the above article: over each patch of spacetime the cosimplicial algebra of functions on the smooth groupoid of gauge field configurations over that piece of spacetime

For more on this general idea of function algebras on gauge groupoids you could try to have a look at the entry function algebras on infinity-stacks.

But maybe first you should try to open that last article which I cited above. It's written by physicists for physicists.

Last edited: Oct 23, 2015
10. Oct 23, 2015

### atyy

To ask my question in post #7 a bit differently - is the definition of "local" in https://dl.dropboxusercontent.com/u/12630719/SchreiberTrento14.pdf [Broken] the same as that in http://www.science.unitn.it/~moretti/convegno/khavkine.pdf [Broken] ?

Last edited by a moderator: May 7, 2017
11. Oct 24, 2015

### Urs Schreiber

Right, these are two different aspects of locality. On the one hand there is locality of a single observable, and on the other hand there is locality of the algebra of observables as such. The first means that local observables are functions on fields which are obtained as integrals of functions of jets of fields over spacetime. The second refers to algebras of observables of global field configurations being obtainable by collecting algebras of observables of fields on local patches.

It is this latter sense that breaks in local gauge theory when one considers only gauge equivalence classes of fields: Since all gauge bundles are locally trivial, hence locally gauge equivalent to the trivial gauge bundle, we get that when quotienting out gauge equivalence then the global field configurations given by globally nontrivial gauge bundles may not be obtained by gluing local field configurations, because locally this forces all field configurations to be on trivial gauge bundles and gluing such in gauge equivalence classes only ever produces the trivial gauge bundle globally.

The reason that considering the groupoid of gauge bundles gets arounds this is because due to it remembering all its gauge transformations, when we glue the local trivial gauge bundles to a global gauge bundle, there is non-trivial information in the choices of gauge transformations used to glued on double intersection of patches, and the non-triviality of the global gauge bundle is all encoded in these gluing gauge transformations.