Hurkyl quote about ditching the manifold
I found that post, it is halfway down the second page of the
SO(3) plus Loop Quantum Gravity thread
"Hypothesis: Can we completely ditch the manifold at this point and abstract the idea of a connection merely to something that acts on edges to give rotations?"
I think this is a very contructive idea.
I do not at all see how to do it (which makes it all the more interesting) and I think Penrose was trying to do it when he
happened onto the idea of "spin networks"
(he had gotten fed up with the frustrations attendent on modeling
space with manifolds and wanted to break out of it)
this is not to say that Penrose's or anybodys first attempts have been successful. I just don't know of any successful attempts
Right now the motivation behind LQG is to achieve "background independence"
Earlier approaches, like stringtheory, were background dependent and perturbative-----involving a fixed initial commitment to a manifold with predetermined metric.
Background independence means "ditch the metric". Start with a manifold that has no built in arbitrarily chosen metric. So you start in some sense with no prejudices about the manifold. You start with a clean slate.
So LQG represents some progress. But it still starts with a manifold-----the manifold just has no preconceived curvature, no prior commitment to a metric, no gravity-----and you start off with a whole Hilbert space of possibilities for what the metric and curvautre and gravity could be. So gravity is quantized in LQG from the very outset.
Background independence seems clean and sound mathematically---how one would want it to be done----compared with the older fixed metric approaches. But it is still based on a smooth continuum or manifold. (just one with comparatively little structure)
Originally posted by meteor
According to loop quantum gravity, space is quantized, is made of tiny loops. It has been always believed that light is a wave with the property of travel through the vacuum, but after all, is possible that light is a vibration of these loops
In reading about LQG I haven't come across the idea that space is made of loops----tho the "labeled networks" may be closely related to loops. In LQG there is a 3D continuum, a manifold. It can have time incorporated into it making it 4D.
All the mainstream approaches, my impression is, even if they involve networks, loops, strings---extended objects of some kind---have an underlying continuum in which these things live.
that attractive lady, Fotini Markopoulos, was written up in Scientific American because of some ideas she seemed to be offering about a purely relational idea of space---but even her ideas, as I recall, had manifolds in them.
We humans, who are after all just moderately-evolved fish, do seem for the most part to need continuums to think with
I mean of course "continua"
