LQG: Gauge Theory, Holonomies, Ashtekar's GR, Loop Representation

[Coin]

So I'm trying to read through Baez&Muniain's "Gauge Fields, Knots and Gravity". One thing I was particularly hoping to get out of this was a specific understanding of what a "holonomy group" is. In the relevant section (p. 231-233 in the version I'm looking at), Baez& describe a holonomy by starting with the idea of parallel transport from one fiber of a bundle to another along a path: they note that it is possible to parallel-transport from a fiber back to that same fiber, by using a path that returns to its original point to form a loop; and since the result of such a parallel transport is path-dependent then with some paths it is possible for the contents of the vector to be transformed by the time it returns to its starting point. Therefore any loop on a connection defines a map from the fiber to itself, and they say that such a map is a "holonomy". Okay, so far so good; I imagine then that the holonomy group of a connection would be the group of all such maps.

However, then they say something I found very startling:

The notion of holonomy plays a basic role in gauge theories of physics. An example that we have already seen in electromagnetism is the phase a charged particle's wavefunction is multiplied by when the particle is moved around a loop. The phase is an element of the group U(1), the gauge group of electromagnetism. Recently there has been a lot of work on formulating gauge theories completely (or as much as possible) in terms of holonomies around loops. This approach, called the 'loop representation' of a gauge theory, has been used in quantum chromodynamics starting in the late 1970s. In 1990, Rovelli and Smolin published a paper in which they used Ashtekar's formulation of general relativity as an SL(2, C) gauge theory to construct a loop representation of quantum gravity...

Whoa! I've never heard either Ashtekar's formulation or the loop representation described this way-- Ashtekar's variables have always been described to me as something like reformulating GR "in terms of the connection rather than the metric", and I've never seen any link discussed between loop representations and gauges at all-- so I guess I'm just trying to figure out what to make of this.

Is Ashtekar's formulation of GR really a gauge theory? (And I almost hesitate to ask this, but didn't we just hear a bunch last month about how you can't trivially make a gauge theory of gravity because of the Coleman-Mandula theorem? Or does that not matter because GR is a theory of only spacetime/gravity and Coleman-Mandula is about unification theories..?)

Is there some reason I've never heard Ashtekar's variables described this way before? Am I just missing some kind of obvious thing where someone more familiar with gauge theories than me would find it obvious what the link is between Ashtekar's approach (which I had thought was in some abstract sense all about connections, but frankly I probably just misunderstand completely) and gauge symmetry?

And finally, if it is indeed accurate to describe the loop representation approach as "analyzing a gauge theory in terms of holonomies": so then can any gauge theory be loop representation-ified, the same way Ashtekar's GR was? Electromagnetism? The Standard Model? Lisi/E8?

Baez& eventually try to describe holonomy as really describing the curvature of the connection, but they don't bring up the gauge thing again in that section that I see and it's not clear to me from anything they say what if anything the relationship is which holonomy/curvature has to the presence of a gauge or gauge symmetry. If you loop-representation-ify a gauge theory does there turn out to be some sort of relationship between the resulting holonomy group and the original gauge group, or something? That's what their comment about phase shifting in electromagnetism seems to imply...

Sorry if these questions are a little messy, I'm just kind of confused here :)

 

[marcus]

Is Ashtekar's formulation of GR really a gauge theory?

Yes, at least that is my understanding. A background independent gauge theory (in the sense that the underlying manifold has no fixed metric, such as you might have in other cases of a gauge theory.)

...Or does that not matter because GR is a theory of only spacetime/gravity and Coleman-Mandula is about unification theories..?

I think that is right. these are good questions. I hope someone else will reply to you more knowledgeably and completely.

 

[f-h]

Yes, it is a gauge theory. A theory with a connection as basic ingredient is pretty much the definition of gauge theory as far as I can tell. In an even more general usage every theory with constraints is.

Yes you could try for a loop representation of other gauge theories (and this was tried before Rovelli/Smolin) but you'd get highly non-seperable hilbertspaces, because the background would allow you to distinguish between every infinitesimally different loop. You'd get different orthogonal states for every possible embedding of every possible graph into your manifold. The representation is far to singular to be made sense of physically.

Diffeomorphisms relate these different embeddings, thus the loop representation starts to become a viable tool.