Posted: Aug 12, 2010 8:49 am
newolder wrote:I've got a bit confused (again) here:twistor59 wrote:(Note in most of the mathematical literature the term “holonomy” is used for closed curves only. In LQG papers, it’s conventional to abuse the term). … where P and Q are the endpoints of the curve C.
So, if P<>Q it's not strictly a holonomy? Also, if P<>Q how is it describing a 'loop'?
Nope, it's not strictly a holonomy as mathematicians use the term, but it seems to have crept into the papers on LQG. And no, it's not describing a loop.
The quantization (which hopefully I'll get to when I come back from my holidays Wheeeeeee!) is based on spin networks, which are graphs of curves, each of which has holonomy information. These graphs, of course, contain lots of closed loops.
The very first attempts at doing this stuff I believe used a "loop transform" to convert from the connection representation to the loop representation (like Wilson loops in gauge theories), but in the modern approach, it's all built up from spin networks. (Unfortunately the original papers aren't on the arxiv and I don't have copies).