J D Christensen's Detailed Paper Discussed by Marcus: A Fascinating Read

[Spin_Network]

detailed by marcus in another thread, paper here:
http://arxiv.org/abs/gr-qc/0512004
this is interesting, has anyone else read through it?

 

[Spin_Network]

detailed by marcus in another thread, paper here:
http://arxiv.org/abs/gr-qc/0512004
this is interesting, has anyone else read through it?

The interest that I have is based on some intuitive concepts I had on another forum some..actually many years ago. This forum although still online:http://www.superstringtheory.com/forum/forums_i.html

no longer allows entry into threads, not even to read.

But, the JDC paper above has rekindled my interest, with regards to Compactification of Dimensional domains.

Basically I stated in previous forums that, Compacting a 3-D "object-of-anything" into that of 2-D "comprable-object", is like the Geometric transformation/deformation of a Square, into a Circle.

On page 7 of the Christensen paper, he has a number of labels, specifically one that is 3/4 (three quarter complete) of this image:

http://groups.msn.com/RelativityandtheMind/shoebox.msnw?action=ShowPhoto&PhotoID=10

but at a slightly different angle, I used the concept of "squaring the circle" to progress into manifold embeddings and inverted compactification, of dimensional geometrics.

Basically, to transform form one space to another using defined and structures with limits, as for instance the Loll transformations of 4-D to 2-D, you have to NOT maintain the geometric parimiters, the logical simplistic form for dimensionally reducing a 4-D cube say, is you have to end up with a 2-D NoN cube.

Loll does straight transformations, and ends up with 1.8, a close approximation of 2, transforming from 4-D.

So there is a "loss" of product in dimensional reduction, and if one reverse's the action as detailed by Loll Dynamical Triangulations, transforming from 2-D up to 4-D, one has to add products to maintain the 4-D dimensionality, so thus the Loll model has to "Glue" or add products.

Again, its not that there is a 3+1 spacetime, it is that there is 3-Dimensions "in" TIME!..any action involving reductionism, from a higher dimension to one of lower status, must forfeit a product, thus it is the +1, 'time' product that dissapears in Macro to Quantum transformations, or from Relativity to Quantum domains, time does not have a continueous path.

It is by no co-incidence that when one fixes a gaze from 'macro' Relativity frame, upwards and out into the cosmos, the time-horizon is consequently increasing with non-dimensional scale? it is projected by observer 'default'!

 

[marcus]

Hi SN,
I relate this Christensen paper to Cherrington's
http://arxiv.org/gr-qc/0508088 which originally appeared in August and which cited this one while it was still in preparation, for a key step.

So that provides the context. Chr--paper is one key step (the finiteness of a certain integral) needed for the overall finiteness result proved in the Che--paper.

the reference in the August paper is [12], mentioned in the first paragraph of the Conclusions section on page 11.

"an essential ingredient of this proof is a recent finiteness result due to Christensen for a large class of 10j like integrals"

Superficially, to me, this is the kind of result that is referred to as "technical"-----few people actually go thru it. A few people check it out to see if it is OK and if there is no challenge then people just accept the technical result as true and go ahead and use it.

but even tho it is technical I think it is a significant step forward. it says that YES you can do certain integrals that are an essential part of the spinfoam approach. People have been considering only the Euclidean (compact group) case because they didnt know if it was rigorous to consider the Lorentzian (noncompact group). This has hampered progress. Now it is better. The spinfoam approach looks more rigorous because things are welldefined in the Lorentzian case now.

I have to go to a thing this morning---so just leave this quick reaction. maybe someone else will check what I said to make sure I haven't missed some important detail.

I believe you are right to take note of these papers and start a thread looking at them in more depth. there will probably be more to follow, in the Lorentzian spinfoam department

 

[marcus]

I guess the thing to point out is that the group reps in the compact case are labeled by (half) integers---much easier to handle---and in the noncompact case the main series of reps is labeled by a continuous parameter.

so in the simpler case the partition function was an admittedly complicated sum, but it was still a sum over a discrete set of possibilities. but in the Lorentzian case which they just did the computations were not discrete sums, they were integrals----and one wasnt quite sure that they were convergent!

so the finiteness result of these two papers by Cherrington and by Christensen is really critical, even tho it appears to be just a technical detail (or that's how it looks to me, maybe you see more and other significance)

wish I didnt have to go to this meeting----I should be gone already