Can Quantum Field Theory Be Applied in Loll's Fractal Universe?

[MTd2]

Let me be more rephrase what I wrote, to not sound rediculous as usualy I am. Loll's universe seems to mimick some futures of our universe, but I just can't see anywhere how a QFT can be defined over that space.

To begin with, the space we observe in this universe is not a single one, but an ensamble of spaces. One could say alright, since a sum-over-some parameters is what we always we expect from a quantum theory, but not in this case. In this situation, it doesn't end alright, because in some cases, the hausdorff dimension does not even converge to a value that is close to a unit, rather, you get "extreme" fractional numbers, like 2.5, 1.5. You can see that on this Article ( http://arxiv.org/PS_cache/hep-th/pdf/0404/0404156v4.pdf ) page 7, where the number of dimensions shows a continue convergence going from 0 to 4, depending on the V of a sample cell of that universe.

In the end, the best we'd get would be a highly amorphous, say fractal if you like, pseudo latice, in which we wouldn't get define a local continuous neighborhood, not even if we took a sample size N going to infinity. In such regimes of extreme fraction dimensions, I can't see how one could even locate a particle in space, because we wouldn't know if there is space, say universe, defined at that given place (supose we consider that space embeded in a higher dimensional space, so that we can define points even though they are not define at the original place). Much less we would be able to talke about the measure of fields, because the space would be really bad defined, highly irregular, between any 2 points of this universe.

Something else that I can´t see working, it is how mass could ever emerge from that. Usualy in the crazy theories around, mass has a relation with geometry and/or geometry. We expect a high curvature if we get close enough of a particle, but in that case, if you get close to anything, the dimenion number goes decreases. At a particle we wouldn't see anything, but a hole (dimension 0)! How you can define mass, or anything else, if you have a
hole. This is the universe, not a semiconductor. There's nothing outside it.

Maybe this is a toymodel, but if they don't put something else on that, I can't see how you could progress with that...

 

[Haelfix]

CDT is a lattice field theory. In lattice theories of gravity (but not necessarily only gravity), its rather common to have anomalous Hausdorff dimensions.

Also mathematically, field theories are better defined on a lattice than in the continuum limit.

 

[MTd2]

CDT is a lattice field theory. In lattice theories of gravity (but not necessarily only gravity), its rather common to have anomalous Hausdorff dimensions.

Could you give me a common example so that I could compare?

 

[marcus]

I don't know how the Utrecht group will include matter. They say that is what they are working on right now.

It might be good to have a look at that 2001 paper that describes the MOVES by which simplexes are shuffled around----in a pure gravity situation.

In pure 4D gravity there is a small set of local moves----which increase or decrease the number of simplexes meeting at a certain place, or reconnect and rearrange them without changing the number. You learn these 4 or 5 basic moves. It is how they randomize and convert one spacetime history to another, by repeated application.

I can only guess that however they include matter----say by coloring the simplexes with some colors, or chaning how they are glued, say by incorporating twists as you glue faces of the simplex together-----however they include matter, there will be a new set of MOVES which preserve the matter and allow it to propagate.

they will have moves which allow them to do as they have already been doing with pure gravity but which will allow 4D histories in which there are matter worldlines and where interaction is possible.

So the 4D history will be both a history of the geometry AND a history of the matter. and the path integral will be an average, as before, of all the 4D histories.

For me it is extremely hard to imagine how they can do this. The spinfoam people are also working on it. Freidel got some connection between spinfoams and Feynman diagrams in a simplifified case. We can only wait and see what comes out during the next few months about this. My guessing would not be of much help to you.

 

[Haelfix]

Could you give me a common example so that I could compare?

Yea the Ising model is a textbook example where there are anomalous scaling operators.

 

[marcus]

...Loll's universe seems to mimick some features of our universe, but I just can't see anywhere how a QFT can be defined over that space.

To begin with, the space we observe in this universe is not a single one, but an ensamble of spaces. One could say alright, since a sum-over-some parameters is what we always we expect from a quantum theory, but not in this case...

As I said, I can't guess how Ambjorn Loll and their group are going to include matter fields in with the geometry.

But each new paper by them that comes out usually has additional hints as to the directions they are taking.

So it might interest you that a new 37-page paper just came out today:

http://arxiv.org/abs/0807.4481
The Nonperturbative Quantum de Sitter Universe
J. Ambjorn, A. Goerlich, J. Jurkiewicz, R. Loll
37 pages, many figures
(Submitted on 28 Jul 2008)

"The dynamical generation of a four-dimensional classical universe from nothing but fundamental quantum excitations at the Planck scale is a long-standing challenge to theoretical physicists. A candidate theory of quantum gravity which achieves this goal without invoking exotic ingredients or excessive fine-tuning is based on the nonperturbative and background-independent technique of Causal Dynamical Triangulations. We demonstrate in detail how in this approach a macroscopic de Sitter universe, accompanied by small quantum fluctuations, emerges from the full gravitational path integral, and how the effective action determining its dynamics can be reconstructed uniquely from Monte Carlo data. We also provide evidence that it may be possible to penetrate to the sub-Planckian regime, where the Planck length is large compared to the lattice spacing of the underlying regularization of geometry." This repeats and summarizes much of what you have already read, but since it is the most up-to-date it may nevertheless be helpful

 

[MTd2]

Yea the Ising model is a textbook example where there are anomalous scaling operators.

But unlike there, here the avaraged dimension of the space does vary dynamicaly. For example, we know that the universe started with dimension 0 and grew until 4 in CDT. So, depending on how a particle is defined, it's surroundings will have close to 0 dimension. Such definition is not weird, because at near plank scale, a particle will tend to a singularity, and with an anology of the original singularity, it have a close to 0 dimension.

In pure 4D gravity there is a small set of local moves

I am not convinced about a local 4D gravity, because, well, at least I am not convinced about the dimensions of the neighborhood of the particle. If there was a set of moves induced a lower dimension, I would be happy.

 

[MTd2]

It seems 4D is very problematic. I will try to study exoctic smoothness in 4d. Whad do you think? Up to 5d, smoothness is equivalent to piecewise linear manifolds.

 

[Chronos]

I see no compelling reason to expect integer dimensions at all scales. It is merely a convenient approximation [not unlike a singularity] IMO. Quantum fuzziness [again IMO] is very suggestive of fractal dimensions in the Planck realm.