Can Loll and Westra Extend Their 2D Topology Sum Over to 4D Quantum Gravity?

[marcus]

this poll refers to this other thread
https://www.physicsforums.com/showthread.php?t=81295

when I first read the "sum over topologies" papers linked there, i was very optimistic. Loll and Westra have been able to make the CDT path integral in 2D sum over topological variation

the topological variation allowed in the sum is carefully controlled, it is regulated using their causal layer structure. they are able to allow microscopic variation (like that envisioned by Wheeler in the "foam" image with its tiny wormholes) but to prevent large-scale weird stuff from happening.

this all sounds good, but can they make this work in 3D spacetime, and ultimately in 4D?

I have been thinking about this and now I am not sure that Loll etc. will be able to do this. Topo variation is inherently more controllable in 2D.

what do you think? I invite you to record your hunch about this in the poll here. It is public so we know who guesses what and check back. Frankly, right now I am undecided! I used to feel confident they could jack this result up to 4D and now I dont.

 

[marcus]

this poll refers to this other thread
https://www.physicsforums.com/showthread.php?t=81295
...

here are links from that other thread
http://arxiv.org/hep-th/0306183
http://arxiv.org/hep-th/0309012
http://arxiv.org/hep-th/0507012

A key quote is from the last paragraph of the third paper, right at the end
of "conclusions":

"A first step in establishing whether an analogue of our suppression mechanism also exists in higher dimensions would be to try and understand whether one can identify a class of causally preferred topology changes which still leaves the sum over geometries exponentially bounded. We will return to this issue in a future publication."

 

[marcus]

here are links from that other thread
...
...
http://arxiv.org/hep-th/0507012

In this paper the mathematics is nice. they explicitly calculate hamiltonian and partition function. they get closed formulas. the CATALAN numbers come up.

the catalan numbers count the number of distinct ways you can pinch a ring

if the ring is a circle of N dots, and you pinch it k times by identifying k pairs of dots, with 2k less than N
then the number of distinct ways to do this is CAT(k) the kth catalan number.
apparently there was a Mr. Catalan who liked pinching people's rings (you see it is very romantic)

In 2D spacetime, space is just like a ring, and triangulating it just puts some N points around in a circle
now WORMHOLES form by a process that is like pinching the ring by joining pairs of points. You can see pictures of this in
http://arxiv.org/hep-th/0309012

there arent all that many ways to pinch a ring and to riddle a spatial slice with briefly existing wormholes

BUT WHAT ABOUT PINCHING A BALLOON?

If you picture this you may see a process of combinatorial growth getting out of hand and becoming factorial instead of merely exponential. So I am worrying about what Loll et al will face when they try to extend this result to 3D spacetime.

 

[marcus]

I acknowledge that meteor was the first to take the step of saying it will extend to 4D, and that my response has elements of a "leap of faith".
I do not see how they can extend the path integral even to 3D. they must find some method to restrain topological variation, or the sum will blow up with too many topo possibilities (too many "handles" sticking out all ways, and handles sticking out of handles).
and they will be seeking a restraint on the topo variability which has a physical rationale.

anyway my better judgment still says they cannot extend it even to 3D
but I like a kind of "bold" streak that i sense in Loll. she is one of those people that is very hard to stop----it is different from intelligence which she also has.

and then I also liked very much the way they got the result (already not easy) for 2D. it seemed to me rather elegant, and many people had tried to do it before without success

and then meteor came in today and voted "Yes, they will be able to do it", so I took the chance and followed his example.

 

[Mike2]

I acknowledge that meteor was the first to take the step of saying it will extend to 4D, and that my response has elements of a "leap of faith".
I do not see how they can extend the path integral even to 3D. they must find some method to restrain topological variation, or the sum will blow up with too many topo possibilities (too many "handles" sticking out all ways, and handles sticking out of handles).
and they will be seeking a restraint on the topo variability which has a physical rationale.

Any chance that the topological changes are really particles emerging from the more stable background spacetime? You call them "baby universes", but could they be viewed as particles on a different scale?

 

[marcus]

Any chance that the topological changes are really particles emerging from the more stable background spacetime? You call them "baby universes", but could they be viewed as particles on a different scale?

Mike let's explicitly broaden the group of people you are asking this to.
you can't just be asking this of me, what i think. I think John Wheeler used to speculate about spacetime foam and particles being topological tangles in it. He was major-league all through the latter half 20th century, as you surely know.

For my part, topological complexity down at Planck scale is a marginal concern. I am happy that Loll et al (last year) finally got to a 4D path integral that simply sums over possible lorentzian spacetime geometries with one given topology WITHOUT including different topologies.

it seems to me that a path integral (also called "sum over histories") that adds up a whole bunch of possible geometries in a kind of amplitude-weighted average, weighted in accordance with Einstein gen rel-----it seems to me that is finally the right way to do geometrical quantum gravity.

Hawking and friends were trying to do this all thru the 1980s, it has been recognized as a physically sensible way to go, but very hard, and finally they did it in 2004.

so I am happy. I don't know if I even WANT them to bother summing over different topologies as well, for now. I just want them to get the plain vanilla 4D case down perfect and work out all the details.

All Loll has done with allowing topo variation is she has done it in 2D.
I don't feel I should be looking ahead to far down that road. It worries me slightly that she is working on this now, instead of the vanilla 4D. I still have serious doubts that it is even possible to include topo varariation in the real thing. Anyway that's how I feel about it this morning

 

[selfAdjoint]

I think the Poincare conjecture and the Riemann Conjecture have been solved, and when this becomes internalized by the math community there will be an explosion of understanding of three-manifolds, combinatorics and all. The mathematical toolkit of the next generation is going to be much bigger, and things that look impossible today will become easy.

BTW, on how to pinch a balloon, see the Urs Schreiber-John Baez theory of 2-bundles, specifically designed to handle holonomy over surfaces.

 

[marcus]

I think the Poincare conjecture and the Riemann Conjecture have been solved, and when this becomes internalized by the math community there will be an explosion of understanding of three-manifolds, combinatorics and all. The mathematical toolkit of the next generation is going to be much bigger, and things that look impossible today will become easy...

a cheerful and exciting prospect. since the poll is off the screen, I will copy the results here----where we can glance at them at without scrolling:

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, and I: Yes they will extend their result to 4D spacetime path integral.

=========================
[EDIT: REPLY TO CHRONOS in next post]

Hi Chronos! thanks for the supportive comment. You are joining in the majority view here, I see from the poll. So I will update the 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---

There is room for more opinions here, and nobody has to stay strictly in bounds. selfAdjoint has stated reasons. I think I understand Spin Network's reasons. What mathematical intuition I have suggests they can't cope with topological variation in 4D so for me it is a leap of faith or confidence in Loll's momentum. I have to say that the MOST INTERESTING response so far is from Alejandro Rivero and I wish he would give some reasoning to support it. arivero usually or always has reasons for what he says. It could be enlightening to know why he thinks you can control topology sufficiently in 3D to define the CDT type quantum gravity path integral, but not in 4D. Or this may just be an astute guess of someone playing poker with nature.

 

[Chronos]

I am sympathetic to marcus's sentiment [assuming I interpretted it correctly]. I see nothing wrong with focusing [for now] on models that work then reverse engineering them.

 

[Spin_Network]

a cheerful and exciting prospect. since the poll is off the screen, I will copy the results here----where we can glance at them at without scrolling:

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, and I: Yes they will extend their result to 4D spacetime path integral.

=========================
[EDIT: REPLY TO CHRONOS in next post]

Hi Chronos! thanks for the supportive comment. You are joining in the majority view here, I see from the poll. So I will update the 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---

There is room for more opinions here, and nobody has to stay strictly in bounds. selfAdjoint has stated reasons. I think I understand Spin Network's reasons. What mathematical intuition I have suggests they can't cope with topological variation in 4D so for me it is a leap of faith or confidence in Loll's momentum.

So now that the cards are on the Table, how about everybody giving their reasons for why they voted above? Marcus I think it will be if everyone who voted, explains in detail WHY they think Loll et al will be successfull or un-successfull ?

I will take my opportunity to give reasons 'against', following voters reasons 'for'.

 

[arivero]

Well in this case I am sorry I can not offer reasons. I was on the 2D vote, because it is a very special case, where the timelike and spacelike sections are equally disconnected. Usually when such particular cases are studied, some trick is found to go up by one dimension; I believe to remember some parts of algebraic quantum field theory, and I voted 2+1 because of it, instead of 1+1.

On the other hand, generically, four dimensions feels always as a very special case, and I do not believe on general applicability of lower dimensional methods (I am surprised that dimensional regularisation works after all, but it is only a 4-epsilon). In topology there are some constructions having obstructions just in D=4. So the wording "to extend" in the poll reinforced my feeling against the "Yes to 4D".

After all, we *want* D=4 to be special.

 

[marcus]

So now that the cards are on the Table, how about everybody giving their reasons for why they voted above? Marcus I think it will be if everyone who voted, explains in detail WHY they think Loll et al will be successfull or un-successfull ?

I will take my opportunity to give reasons 'against', following voters reasons 'for'.

I agree it would be cool to know some of people's reasons, but I actually myself guessed WITHOUT reasons (and even a little bit against my rational assessment of the mathematical situation)

I actually think you get a better spectrum of guesses if you allow people freedom to guess entirely as they please with or without logical scaffolding. But I am not an expert in this kind of thing!

What happened with me is I guessed irrationally (copying what Meteor said, as it happens) and then the next day I thought of a reason that it might be right. funny business isn't it

[EDIT after seeing preceding arivero post] thanks Alejandro for explicating your guess!

 

[marcus]

Our most recent respondent is Kea, who joins Spin Network in predicting that Loll will be unable to sum over topologies in spacetime dimension higher than 2.

To update the results:

Will Loll etc be able to extend sum over topologies to dim > 2?

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

Arivero, Pietjuh, Random Guy: Yes for 3D but not 4D.

Meteor, selfAdjoint, Chronos and I: Yes they will extend their result to 4D spacetime path integral.

 

[marcus]

BTW I had some thoughts about the quest to include sum over topologies in the spacetime path integral----about Loll's program to include some controlled microscopic topo variation, if not the whole smorgasbord of topo variation

one was the hopeful thought that what they found in "Reconstructing the Universe" was that the thick slices had dimension 4D at large scale (as they should) and dimension 2D at small scale. So the regular CDT quantum spacetime, which on macro level looks 4D in some sense ACTS LIKE IT IS 2D down at the micro level. And Loll Westra found they could include brief, causally harmless, topo variation in the 2D case. So maybe they can get it to work in the regular quantum spacetime. that is already pretty vague tenuous reasoning---anyway I'll be interested to see their next paper.

the other was a thought about the BOJOWALD BOUNCE. When Bojowald made a symmetry reduction and developed LQC, the cosmology offshoot of LQG, he found that when matter gets extemely dense it has a repulsion effect. So at the pit of a black hole (using LQC techniques) there is a kind of bounce where what is collapsing stops trying to pull itself together tighter and starts to drive itself apart. contraction changes over to expansion. And he found that happening around the Big Bang, which had a prior contraction phase and was really a Big Bounce.

Remember that LQC has a Hamiltonian time-evolution operator with a good semiclassical limit. To some extent it can be viewed as its own independent theory. It uses the scale factor as a clock, to time the other stuff, and it is fairly tractable. The Hamiltonian leads to a discrete-time difference equation. Far enough away from the classical singularity it blends in with the good old classical model. And it generically predicts inflation without gimmicks. (a recent paper by Singh with references to
papers about these matters is http://arxiv.org/abs/gr-qc/0507029)

So LQC is interesting and one wants to know if features of it carry over to Loll CDT.

Now I got this suspicion today that when and if Loll includes topologies in the 4D spacetime path integral, at this time a kind of "Bojowald Bounce" behavior might creep in.

For example suppose collapse by some mechanism squeezes out wormholes, then by the analysis of http://arxiv.org/abs/hep-th/0507012 this would increase the effective cosmological constant Lambda(the expansive dark energy)

I remember in the recent Loll Westra paper there is a relation between the holefulness of spacetime---its tendency to be riddled with wormholes, the density of wormholes----and the effective cosmological constant Lambda! And so if Loll is expecting to do black hole collapse WITH "sum over topologies" included in the path integral, then she might be expecting to find the negative pressure behavior associated with Lambda and dark energy. And something analogous about the classical Big Bang singularity.

Well that is pretty speculative and I haven't had time to consider if it even makes the slightest bit of sense!

But now it does seem to me that it is not such a waste of time for them to be considering "sum over topologies".

The present non-topo version that you get in "Reconstructing the Universe" does not have a bojowald bounce feature, for example. Putting in topological variation might just provide an opportunity for the two theories CDT and LQC to come closer together.

BTW the paper that this thread is about "Taming the Cosmological Constant with Sum over Topologies in 2D" or whatever the title is, this paper is essentially what they are delivering in Paris at the Einstein 2005, and guess who is scheduled to deliver the paper!

Loll herself? no. Willem Westra who has coauthored "sum over topologies" papers with Loll since 2003? No. the third author, someone we didnt hear of earlier, named Stefan Zohren. This is part of how she puts her grad students through Basic Training. grow them up fast because there is a lot of work to do

 

[Spin_Network]

BTW I had some thoughts about the quest to include sum over topologies in the spacetime path integral----about Loll's program to include some controlled microscopic topo variation, if not the whole smorgasbord of topo variation

This will actually be included in my reply as to 'why' I see the problems, without going into details just yet, the initial path integral, cannot,also be a 'Partial', it is fundamental at source

But I will give a full explination at a later date, I have been working on a concise reply, but I have been sidetracked elswhere for now.

 

[marcus]

Ohwilleke has joined the Ariveristas.

To update the results:

Will Loll etc be able to extend sum over topologies to dim > 2?

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

Arivero, Ohwilleke, Pietjuh, Random Guy: Yes for 3D but not 4D.

Meteor, selfAdjoint, Chronos and I: Yes they will extend their result to 4D spacetime path integral.

 

[Chronos]

Embedding tiny, transient worm holes into Planck space is what sold me on the prospects of a 4D solution. It very much looks like the right way to produce a slight cosmological constant.

 

[Hans de Vries]

I voted for No, 2D is the max. Although they may achieve something in
the direction the question for me is what will it be worth, that is, what are
all the restrictions needed on the input to get a sensible output.

The idea is that light wiggles its way trough infinitely curved space-time to
end up moving at straight lines and constant speed after a path-integral.

First problem seems to be that path-lengths tend to infinity in all practical
mathematical curves which exhibit a curvature which is proportional to 1/r.
There are many examples of such curves. In fact most of the curves found
in books on fractals have exactly this 1/r behavior and they all tend to be
of infinite length. (e.g. Peano curve)

This 1/r problem starts in 2 or more space dimensions. The current 2D
work refers to one space/one time dimension.
(http://arxiv.org/PS_cache/hep-th/pdf/0306/0306183.pdf )

The 1/r behavior being the result of Heisenbergs Uncertainty Principle.
That is, in the interpretation that chaos is reversely proportional to scale,
(Which is something entirely different than the Fourier Transform interpretation
that I prefer personally)

Also, such an 1/r behavior would seem to make it problematic for things to
cancel out on a larger scale as it does in liquids and gasses where there's
randomness at small scale and strait-line propagation at large scale.

I seems to me that much more restrictions are needed than simple causality.


Well, we'll see.


Regards, Hans.

 

[marcus]

Hans de Vries agrees with Kea that sum over topologies will only work in dim = 2, and gives reasons for this. (I also can only imagine how it would work in 2D but predicted 4D in a fit of optimism.)

To update the results:

Will Loll etc be able to extend sum over topologies to dim > 2?

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

Arivero, Ohwilleke, Pietjuh, Random Guy: Yes for 3D but not 4D.

Meteor, selfAdjoint, Chronos and I: Yes they will extend their result to 4D spacetime path integral.

 

[Chronos]

I expect a solution something along the lines of d1^2 + d2^2 + d3^2 - d4^2 = 0.

 

[marcus]

I expect a solution something along the lines of d1^2 + d2^2 + d3^2 - d4^2 = 0.

well tomorrow (20 July) Loll's grad student Stefan Zohren will present their "sum over topologies" paper at the Paris "Einstein 2005" conference


« Nonperturbative sum over topologies in 2D Lorentzian Quantum Gravity »
by Loll, Westra, and Zohren

to find the Wednesday parallel session on "the nature of spacetime" go here:

http://einstein2005.obspm.fr/indexr.php

then click on CONFERENCE to get the sidebar menu
then you click on PARALLEL PROGRAM
and then on NATURE OF SPACE-TIME. it should come up