Can Loll's Team Extend Sum Over Topologies to Higher Dimensions?

[marcus]

I just had a hopeful thought, actually it was last night on the couch but it was already bedtime.

this is about this thread
https://www.physicsforums.com/showthread.php?t=81626
which so far has these results:

---who forecasts what, on the question of---
Will Loll etc be able to extend sum over topologies to dim > 2?

Spin Network: No, 2D is the max. Sum will blow up in higher dimension

Arivero: Yes for 3D but not 4D.

Meteor, selfAdjoint, Chronos, and I: Yes they will extend their result to 4D spacetime path integral.
---that's all so far---

It is really great that, at least in 2D spacetime, Loll Westra managed to include brief microscopic topological variation in their path integral. But there is so much MORE possible topo variation in 3D and 4D that I cannot see how they could include it in those higherdimension cases without the sum blowing up.

BUT WAIT! Their recent paper "Reconstructing the Universe"
http://arxiv.org/hep-th/0505154
provided some evidence that our 4D spacetime, at very microscopic scale, MAY ACT LIKE IT IS 2D !

so the topo variation in 4D, as long as you keep it brief and microscopic and causally harmless, as they do in their recent 2D paper, may actually be
controllable down at that tiny scale.

because at that tiny scale the universe is acting like a 2D spacetime

 

[marcus]

here is a thread about the Loll Westra sum over topologies papers

https://www.physicsforums.com/showthread.php?t=81295

here are their 3 papers

this thread is about these three papers
http://arxiv.org/hep-th/0306183
http://arxiv.org/hep-th/0309012
http://arxiv.org/hep-th/0507012

the latest one, the one this year, I think is elegant. it has that business about how many ways can you pinch a ring, and the answer is the Catalan numbers
that sequence grows only exponentially with the number of points on the ring, and it counts the possible brief small wormholes in a time-slice.
so the wormholes are controlled and you can toss them in with the geometries you are summing over

basically so far Loll CDT does not include topo variation. Its idea is to sum over all possible spacetime GEOMETRIES for some fixed spacetime topology. So the usual CDT "sum over histories" or "path integral" does not include any spacetimes with handles or wormholes or whatever. It usually fixes the topology to be Time Cross the 3-Sphere----that is R x S³---a very usual sort of spacetime. and then makes this huge amplitude weighted average of every possible geometric shape the thing could have.

For me, that seems quite satisfactory, and they can simulate it in a computer and start right away to get results about quantum expectation values of things measured on this path integral "sum over histories". Great. Why worry about including sum over topology?

I don't know. but respected people like Sidney Coleman and Stephen Hawking have been talking about including topo variation for some decades, and John Archibald Wheeler. the elders of the tribe want this. they think nature does it this way.

and maybe tiny topological tangles or zits or dinguses have something to do with where particles come from, but that seems like giddy speculation to me. so apparently one wants to be able to include microscopic topological variation in the quantum spacetime sum. Go for it Loll. Go for it Westra.

 

[R0man]

, which is what they know how to sum over.

I am excited by this possibility and hopeful that Loll and her team will be able to extend their sum over topologies to higher dimensions. It would be a major breakthrough in our understanding of quantum gravity and the structure of the universe. And the fact that their recent paper provides evidence that the universe may behave like a 2D spacetime at a microscopic level is a promising sign that this extension may be possible. I can't wait to see what further research and developments will come from this thread and the work of Loll and her team. The potential implications for our understanding of the universe are truly exciting. Here's to hoping for more progress in this fascinating field!