Particle Physics in Curved Spacetime (Background Independent)

[fzero]

This all makes sense, and is certainly good to know, if only as historical basis.

But what interests me is that if you look at the new formulation, which has appeared only since 2010---say you look at the pedagogical review 1101.3660 that A. Neumaier just referenced---then where is the $$\mathbb{R}\times \Sigma$$ manifold?

Where are the Ashtekar variables? Where are the holonomies? Where is the old configuration space of pre-2010 Loop Gravity?

The metric operator in (22) of 1102.3660 is 3x3. This corresponds in some way to the metric that Ashtekar decomposes into the spin connection, possibly explained in the paper
http://arxiv.org/abs/gr-qc/9411005 where Rovelli and Smolin made the connection with spin networks.

I believe that the states in $$\mathcal{H}_\Gamma$$ are the holonomies, see the remark under equation (18) where this is identified with the Hilbert space of lattice gauge theory.

This is the point I was trying to make by the way I responded to Neumaier's question. It is actually very interesting. We have this new very concise formulation, with little or no "extra baggage". It is expressed in just a few equations---with Hilbertspace and operators defined in a rather direct transparent way.

In this new formulation, the question is very relevant---how do we know the dimensionality?

A. Neumaier refers to what it says right after equation (23) on page 4 of http://arxiv.org/abs/1102.3660 . Where paper [8] by Penrose is cited.
I think this may be the right place to look.

I think whatever connection this model has with canonical gravity is still related to the old one of Rovelli and Smolin. I don't know what makes it a new formalism, so I can't say what refinements of the old ideas are there. It may not start with the Ashtekar formalism, but the same type of formulas come up if you want to relate quantities in the $$SU(2)$$ variables to the metric. There could be differences in quantum theory, but the semiclassical physics probably agrees.

 

[marcus]

fzero, I think what you are getting at is right (if I understand you). If you put the extra baggage back in, you very likely can recover the old formulation that used a manifold RxSigma. At least if you do it right (put the baggage back in the right way.)

Of course when you start putting the extra stuff back in--to show historical continuity--you are putting stuff with dimensionality back into the picture. So you are in a sense putting the expected dimensionality in by hand.

The new formulation does not assume that a manifold exists. It does not talk about RxSigma and connections etc. It does not assume that space and spacetime are properly modeled by, for instance, smooth manifolds.

So therefore (I think) it really makes sense to ask! What do you mean by dimensionality!

How does the new manifoldless formulation, in its pure pristine condition, talk about dimensionality? Without hooking up to the historical past.

I think it is mathematically more interesting (fun) to approach it like that. Maybe.

What operators, defined on the new simple Hilbertspace, with the new simple setup, correspond to observing or experiencing the dimensionality of one's surroundings?

I think A. Neumaier may have pointed exactly to it. Some "angle" operators on page 4 around equation (26).

(On the other hand it is clear you have a point---there probably is this solid connection with the past that one can establish if one puts the past baggage and machinery back in. Have to go to lunch, back later.)

 

[marcus]

...There could be differences in quantum theory, but the semiclassical physics probably agrees.

Yes! I suspect you are right. That's important too, otherwise going to the manifoldless formulation would be a bad move. Also think that for example the area and volume operators agree, new with old. Except that in the new formulation a region you want to measure volume of (since there is no manifold) is defined as a set of nodes. A surface you want to measure area of is defined as a set of links---which you imagine the surface cuts.

To establish the agreement all one needs to do (I think) is set up the right dictionary of correspondences between the manifoldless setup and the manifoldy one.

I believe that the states in $$\mathcal{H}_\Gamma$$ are the holonomies, see the remark under equation (18) where this is identified with the Hilbert space of lattice gauge theory.

Yes! that is a helpful remark, helpful to me anyway. You already have assimilated the dictionary of correspondences between continuum gauge theory and lattice gauge theory, so that it is second nature to you. So it is more immediate to recognize what is going on.

For me, the states are functions from L-tuples of group elements to the complex numbers.
You make an assignment of a group element h_l to every link l = 1,...,L.
Let's call this assignment of group elements to links in the graph {h_l}.
For every such assignment, the state gives you a complex number.

For me, because no manifold exists and space is not considered to be a manifold, I can't imagine that such a thing as a connection exists, or an holonomy either.
But you might call this assignment of group elements {h_l} a lattice connection and then the state maps from connections to complex numbers and you recognize it as a lattice holonomy.

It seems obvious now. But I still don't see that these close analogies establish the dimensionality. I still think to get the dimension we probably have to invoke that "Penrose angle observable" thing that A. Neumaier mentioned.