Steve Carlip: Dimensional Reduction at Small Scale

[atyy]

"In view of the results obtained here, we expect that a FP with three attractive directions will be maintained." http://arxiv.org/abs/0705.1769

But that's the dimension of the attracting critical surface, which I'm not sure is the same as a spectral dimension. I do remember that AS used to have the critical surface dimension 2 at a lower truncation, but it's 3 in the Codello et al work.

 

[marcus]

"In view of the results obtained here, we expect that a FP with three attractive directions will be maintained." http://arxiv.org/abs/0705.1769

But that's the dimension of the attracting critical surface, which I'm not sure is the same as a spectral dimension.

You can be sure it is not the same. The UV critical surface exists in theory space. The presumably infinite dimensional space of all theories of gravity based on choices of parameters at all orders.

The spectral dimension referred to is the dimension of spacetime, or in some cases space.
There are several ways to measure spacetime (or space) dimensionality. The Hausdorff method compares radius to volume. If the H. dimension of some space is 1.7 at a particular point that means that the volume of a small ball at that point grows as
r^1.7 ---as the 1.7 power of the radius.

Obviously the H. dimension can depend on the scale at which you are measuring.

The spectral dimension is just another way to gauge dimensionality. You run a random walk and see how fast the walker gets lost. How likely is he to accidentally return to start. The higher the dimension the less likely. After all, on a line the walker is almost sure to return to the origin eventually.

Loll and friends generate small random quantum universes in the computer and study them. So they have plenty of opportunity to run random walks both in the full universe and in a spatial slice. So they can measure the spectral dimension of the spacetimes and of the space---always measured or observed at a certain point and at a certain scale.
And they can take averages over many points, and many universes, and see how the dimensionality depends on the scale (how long the random walker is allowed to wander).
Have to go, back later.

 

[atyy]

You can be sure it is not the same. The UV critical surface exists in theory space. The presumably infinite dimensional space of all theories of gravity based on choices of parameters at all orders.

OK, I understand that better I think.

The spectral dimension in CDT is related to the anomalous dimension in AS, which is 2 in order to have a continuum limit.

The dimension of the critical surface is a separate quantity that must be "small" in order to have a "small" number of theories with a continuum limit, otherwise we will have too many possibilities to choose from, and we have to do an infinite number of experiments to choose a right theory (is this really a problem, wouldn't experimentalists be happier knowing that there will always be more experiments to perform)?

 

[marcus]

..(...wouldn't experimentalists be happier knowing that there will always be more experiments to perform)?

heh heh
Yes that might make some experimentalists happy, job security for themselves and their descendents for all eternity .
But I think you and I are assuming standard Baconian science rules, ethics, values. An unstated assumption is that a scientific theory is not only explanatory
but must also be predictive.

To be predictive, the number of free parameters must be finite. Like 3 or like 23. So you do a finite number of experiments to determine the values and then you are go for prediction all the rest of the way.

In this scheme you keep the experimentalists happy by telling them that they can continue to test your predictions (based on the first 3 or 23 measured inputs) and if they find a discrepancy then the theory goes out the window. The theory is dead long live the new theory.

But if a theory has an infinite number of adjustables, then you can never predict in that way, and never test. As soon as you run into a discrepancy you simply adjust the next parameter---every failure of prediction is just a "measurement" of something new. The theory is mush!

I think I am not telling you anything new Atyy, and you were kidding about keeping the experimentalists happy by having an endless supply of parameters for them to measure. But I still had to say something about this.

It would be really cool if AsymSafe QG only needs a small number of parameters to be measured. Like Percacci-Codello-Rahmede suggests just three!

Have you looked at any of Reuter's lecture slides or papers where he plots computer output as a graphic picture of the renormalization group flow? The flow spiraling in round the UV fixed point? And sailing off in a beeline for infinity in the IR?

AsymSafe QG is very pictorial. If you haven't seen examples of Reuter's graphic flow plots (he used to put them in almost every paper) and if you want to see some, let us know and I or someone else will dig up links to old Reuter papers that have the graphics.

 

[atyy]

I think I am not telling you anything new Atyy, and you were kidding about keeping the experimentalists happy by having an endless supply of parameters for them to measure. But I still had to say something about this.

Half kidding

What I don't understand is this: suppose the metric field has a UV fixed point, but the critical dimension is infinite - then wouldn't that mean that we have in fact been able to make only a finite number measurements to get predictivity at low energies using GR? So would it be possible that there are an infinite number of parameters for arbitrarily high energies, but we only need to measure a few more parameters each time to gain predictivity whenever we want to step up the energy scale?

 

[atyy]

Another question whether we really need the critical dimension to be "small" - suppose AS is correct with infinite critical dimension and CDT is also correct with respect to its two current predictions (de Sitter universe, spectral dimension 2) - then wouldn't that mean that AS, CDT had predicted some things - just not everything? Or would the correct predictions of CDT imply a "small" critical dimension?

 

[marcus]

... So would it be possible that there are an infinite number of parameters for arbitrarily high energies, but we only need to measure a few more parameters each time to gain predictivity whenever we want to step up the energy scale?

Yes I think that is right. I am not an expert but I believe that situation would NOT be what Weinberg was thinking of in 1976 and gave the name "asymptotic safe".
That would a series of "effective theories" getting better and better.
But no one theory would be "fundamental" in the sense that you can take it all the way, as high energy as you like, and it stays applicable.

And what you describe sounds like a practical and reasonably satisfactory situation.

But Weinberg, and Reuter and Percacci after him, and all the others are not talking about that. They want something that is what they call "nonperturbatively renormalizable" where it is renormalizable in the sense that you only have to determine a finite number of parameters experimentally and then you are good to go all the way.
You never have to adjust and plug in another number.

That is what Weinberg meant when he said it's possible that something like string theory is not needed, and is not how the world is. It's possible that the way the world is is just what we are used to. Geometric general relativity and QFT combined in the effective unification we already have, or something like that, and it turns out to be predictive/fundamental in other words (nonperturbatively) renormalizable after all. You remember where he was saying things like that in his talk.

The november Perimeter workshop on that will probably be interesting and I bet they post videos of some or all of the talks. Loll will be there, Weinberg, Smolin, Percacci, Litim, Reuter. Quite a group!

+++++++
About your post #53, I am not ready to assume that any of these preliminary results are right, or anything more than an accident if they are right. I don't have the trained intuition and vision of someone who actually researches this stuff. You are thinking some interesting "what if" stuff, but it is too complex for me right now. I think they are clearly on to something that should be pursued full force, and get funding and workshop support and all that. I hope they attract grad student and postdoc brainpower, and I think they will. But I can't assume they are right and project ahead like that. Have to wait and see.

Also this is a big challenge to Rovelli to see if there is any scale-dependence in LQG (I mean spinfoam of course). Another thing to wait-see about.

 

[atyy]

Yeah, perhaps the predictability is most important - more important than having a continuum limit - because I think electroweak theory doesn't have a continuum limit, so we can't have a "final theory" even if the metric field has a continuum limit (ie. asymptotically safe gravity will be an effective field theory that gives no clue when it fails, just like QCD). So perhaps even if AS, CDT are right, this will leave the door open for string theory, or LQG plus matter?

 

[MTd2]

I guess there is something obvious, but interesting, being overlooked here: all these theories approach the dimension of a string at plank scale. Perhaps, there is a kind of non-supersymmetrical, maybe even critical, string theory yet to be invented.