Loop gravity Hamiltonian-for Jeff and/or Eigenguy

[marcus]

Originally posted by selfAdjoint
Marcus, in the Svetlichny notes, look especially at page 29, paragraph beginning "Let now G be a Lie group", where he introduces the connection one form, with values in the Lie algebra. Remind you of anything?...

yeah I remember you talking about vertical and horizontal and so on, Svetlichny is what we should have had handy then.

I haven't yet gotten around to putting a link to Svetlichny's "Preparation" in the differential geometry forum, tho I did start an appropriate thread there

 

[marcus]

Originally posted by Ambitwistor
The Svetlichny paper is definitely closer to what is needed to understand the Ashtekar variables...

I would like to, so its the right thing. I think I said thank you already, but if not, thanks.

Maybe we have two different constructive things we could do. One is to compile an annotated bibliograph to post here so that someone like selfAdjoint or me can teach himself efficiently about loop gravity to the point of following current developments with some degree of understanding

The other is a riskier agenda which is to patch together essays by different people to make a kind of online Loop Gravity for the People textbook----actually the obvious style model is John Baez (X for bears of little brain,...many examples) and its not impossible that he will do an intuitive quantum gravity for the people set of webpages at some point. But until that happens there is an obvious niche because people are interested in it and there is no basic entry-level text, not that I know of anyway.

I don't want either of these two agendas to get in the way of good person-to-person conversations you might have with selfAdjoint or nonunitary. So I might start a separate thread to accumulate explanations and ideas for these projects

 

[marcus]

there is something almost obvious about loop gravity, I hope it won't turn people off if i put it in a really dirt-common way. A connection (the way a tangent vector swirves while being transported along a path) is a visual intuitive idea. Likewise the little animated film that shows that starting on the equator and going up to the north pole and so on, precisely because of the curvature it comes back different from how it started.

these are bedrock accessible intuitions you can build on and develop an idea of quantizing geometry

because if the soup of all possible connections is a reasonalble facsimile of the soup of all possible geometries (on the underlying manifold you want to study)

then why not take the soup, sorry i mean "space", of all possible connections as the -------what you build wavefunctions on.

and then if A is configuration space how else can you imagine defining a quantum state

f:A --> C

besides with things like loops and networks? Really. If you want to get a complex number out of a connection A that is how you are going to go about it---with some kind of path to run the connection on. I never bothered ot find out who Wilson was who first proposed taking traces of holonomies and it must be a time-honored practice almost a reflex by now.

Maybe this has already been covered in "Three Roads" or in some past "This Week's Finds". In that case let's cite our sources and paraphrase the gist of it. Anyway one way or another post a rock-bottom conceptual version of loop gravity.

I am trying to figure out what the catch is. What would turn out to be prohibitively difficult? The next thing would have to be the measure on A so you can integrate and actually have a hilbertspace. And then there is the business of modding out morphisms. (so that the network is an abstract knotwork rather than specific thing embedded some particular way). It would be so nice if all this could be said in plain language, conveying a sense of familiarity. Well, enough wishful raving at least for now.