## Friday, May 17, 2013

### Dimensional Reduction

 Dimensionally reduced scientist.
“Science is the only news,” Steward Brand wrote. My reading of this sentence is that science, the exploration of nature and natural law, is the ultimate source of inspiration. Developing a model and studying its properties can be like discovering a new world, and the discoveries that are the most fascinating are the ones that are surprising and unintuitive.

Probability amplitudes and wavefunctions are examples of such surprising and unintuitive properties, examples that are now a century old and that have changed the way we think about the world. Holography is a more recent example. And, gathering momentum in the quantum gravity community right now, is dimensional reduction.

Dimensional reduction means that on short distances the dimension of space-time decreases. To quantify what this means one has to be very careful with defining “dimension.”

The way we normally think about the dimension of space is to picture how lines spread out from a point. How quickly the lines dilute into their environment tells us something about the spheres we can draw around the point. The dimension of these spheres can be used to define the “Hausdorff dimension” of a space. The faster the lines dilute with distance, the larger the Hausdorff dimension.

The notion of dimension that is relevant for the effect of dimensional reduction is not the Hausdorff dimension, but instead the “spectral dimension.” The spectral dimension can be found by first getting rid of the Lorentzian signature and going to Euclidean space. And then to watch a random walker who starts at one point, and measure the probability for him to return to that point. The smaller the average return probability, the higher the probability he’ll get lost, and the higher the number of dimensions. One can define the spectral dimension from the average return probability.

Normally, for a flat, classical space, both notions of dimension are identical. However, there have been several approaches toward quantum geometry that found that the spectral dimension at short distances goes down from four to two. The return probability for short walks is larger than expected. One says that the spectral dimension “runs”, meaning it depends on the distance at which space-time is probed.

Surprising. Unintuitive.

This strange behavior was first found in Causal Dynamical Triangulations (hep-th/0505113), where one does a numerical simulation of an actual random walk in Euclidean space. But in other approaches one does not need a numerical simulation; it is possible to study the spectral dimension analytically as follows.

The behavior of the random walk is governed by a differential equation, the diffusion equation, in which there enters the metric of the background space-time. In approaches to quantum gravity in which the metric is quantized, it is then the expectation value of the operator that the metric has become which enters the diffusion equation. From the diffusion equation one calculates the return probability for the random walk.

This way, one can then infer the spectral dimension also in Asymptotically Safe Gravity (hep-th/0508202). Interestingly, one finds the same drop from four to two spectral dimensions. Yet another indication comes from Loop Quantum Gravity, where the scaling of the area operator with length changes at short distances. It is somewhat questionable whether the notion of a metric makes sense at all in this regime, but if one nevertheless constructs the diffusion equation from this scaling, one again finds that the spectral dimension drops from four to two (0812.2214). And Horava-Lifshitz gravity is maybe the best studied case where one finds dimensional reduction (0902.3657).

Surprising. Unintuitive. It is difficult to interpret this behavior. Maybe a good way to picture it, as Calcagni, Eichhorn and Saueressig suggested, is to think of the quantum fluctuations of space-time hindering a particle’s random walk and slowing it down. It wouldn’t have to be that way. Quantum fluctuations could also be kicking the particle around wildly, thus increasing the spectral dimension rather than decreasing it. But that’s not what the theory tells us. One shouldn’t take this picture too seriously though, because we’re talking about a random walk in Euclidean space, so it’s not an actual physical process.

It seems strange that such entirely different approaches to quantum gravity would share a behavior like this. Maybe our theories are trying to teach us a lesson about a very general property of quantum space-time. But then again, the spectral dimension does not say all that much about the theory. There are many different types of random walks that give rise to the same spectral dimension. And while these different approaches to quantum gravity share the same scaling behavior for the spectral dimension, they differ in the type of random walk that produces this scaling (1304.7247).

So far, this is an entirely theoretical observation. It is interesting to speculate whether one can find experimental evidence for this scaling behavior. In fact, this recent paper by Amelino-Camelia et al aims to “explore the cosmological implications” of running spectral dimensions. At least that is what the first sentence of the abstract says. If you read the second sentence though you’ll notice that what they actually explore are modified dispersion relations. And while modified dispersion relations lead to a running spectral dimension, the opposite is not necessarily the case. But is there any better indication for a topic being hot than that people use it in the first sentence of an abstract to draw the readers interest?

1. The fact that quantum paths have spectral dimension 2 was known to Feynmann himself (I believe). It has also been remarked by L. Nottale in his approach (sometimes "crackpottized" by mainstreamers) of Scale relativity.

What is interesting the most is the fact that while people has focused on searches large/small extra dimensions in colliders or experiments, this parallel approach of spectral dimension seems to point out that the spacetime dimension is in fact variable but not larger than 4 (as XD physicists claim) but quite to the contrary, effective dimensionality is less than four! And the holographic picture is also in the air in some way yet to be completely understood.

Good blog entry Sabine!

2. The fact that quantum paths have spectral dimension 2 was known to Feynmann himself (I believe). It has also been remarked by L. Nottale in his approach (sometimes "crackpottized" by mainstreamers) of Scale relativity.

What is interesting the most is the fact that while people has focused on searches large/small extra dimensions in colliders or experiments, this parallel approach of spectral dimension seems to point out that the spacetime dimension is in fact variable but not larger than 4 (as XD physicists claim) but quite to the contrary, effective dimensionality is less than four! And the holographic picture is also in the air in some way yet to be completely understood.

Good blog entry Sabine!

3. Very interesting post, but IMO you slide from model to realism too easily. I suggest, for example, that you would be better to write "Dimensional reduction means that on short distances the dimension of [the] space-time [model] decreases", which follows the pattern of "Developing a model and studying its properties can be like discovering a new world" in the first paragraph. On the other hand, perhaps you intend realism?

4. Peter:

Accuracy frequently conflicts with readability. This is such a case. This is a blog, not a journal on philosophy. I want people to understand what I write, and though oversimplification may be a sin, it's means to an end. Best,

B.

5. The whole of organic chemistry reduces to flat non-crossing 2-D Schlegel diagrams. Everybody looks at a small subset of possible things in a fashionably insular way. A very few K_5 molecules (Kuratowski's theorem) cannot be flattened given connectivity. We know of one flat molecule that spontaneously trimerizes into a topological trefoil knot,

http://www.ncbi.nlm.nih.gov/pubmed/23139329
http://cen.acs.org/content/cen/articles/90/i46/Molecular-Building-Blocks-Self-Assemble/_jcr_content/articlebody/subpar/articlemedia_0.img.jpg/1352418006764.jpg
http://www.sciencemag.org/content/338/6108/783/F2.medium.gif

God is a geometer, the Devil is a belief system. Can the holographic universe encounter an irreducible case?

6. It is strange to notice that Kaluza-Klein or string/M theorie(s) seem to require more dimensions (10 or 11?) while some alternative (loop)quantum gravity theories ask purpotedly for less dimensions (2?)... This would make a nice sequel for the "Alice and Bob in wonderland" cartoon series! The title could be : "How sure are we that space-time is 4D?". Alice, the string expert, would start, talking about how one goes from 4 to 4+6 (Hausdorff?) dimensions, Bob would be the loop expert, talking about the opposit 4 from 2 (spectral) dimensions hypothesis revealing the puzzle. Then Alice would say "We need more powerful ideas ... new quantum(noncommutative?) ideas(geometries?)!
Alternative ending : Alice could give a hypothetical solution 4+6 = 2 (modulo 8). Her last sentence could be : I wonder if that was Connes and Chamseddines idea ;-)(http://arxiv.org/pdf/1304.8050.pdf and http://golem.ph.utexas.edu/category/2006/09/connes_on_spectral_geometry_of_1.html)

7. /* Dimensional reduction means that on short distances the dimension of space-time decreases. */

The same effects manifest itself at large scales too. In water surface analogy of space-time it's the result of the dispersive character of surface wave spreading at short (Brownian noise) or large (gravity waves) dimensional/mass-energy density scales. Unfortunately at even shorter/larger scales the same scattering violates the dimensional geometry itself, so that the validity scope of spectral/hyperdimensional models remains limited.

8. There may be already an experimental evidence for the short-distance dimensional reduction: planar alignment of events in cosmic-ray experiments (see arxiv:1304.6444 by D. Stojkovic).

9. Mikovic:

We previously discussed the model by Stojkovic et al here. I don't see that there is a direct relation to the dimensional reduction I discussed here. The model by Stojkovic has issues with Lorentz invariance violation and (or maybe: because) for all I can tell they are talking about the Hausdorff dimension, not the spectral dimension. Best,

B.

10. ""The whole of organic chemistry reduces to flat non-crossing 2-D Schlegel diagrams. ""

Uncle Al,
that used to be the case up to the 30ties, predominantly a question of printing technology (= cost of "pictures")
Later more and more sophisticated drawings were printed and helped to transmit/teach stereochemical information.
That "spontaneous" trefoil is nothing to write home about, because You get - as always - left and right-handed trefoils in the same amount.
Georg

11. They should find another name for it. Dimensional reduction is a well know process in higher dimensional theories like String theory where you go compactify a dimension on a circle and then you keep only the zero modes in the Fourier expansion of the various fields.

12. @Georg:
1) Sterochemistry is irrelevant to Schlegel diagram non-crossing planarity. All [m.n]chiralanes (atom at center) are chiral K_5 molecules. Chiral hollow [m]chirolanes; point groups T, O, and I chiral fullerenes are not K_5 molecules. The latter are all perfectly mathematically chiral, Petitjean's CHI = 1 exactly. K_5 molecules are published that are not chiral.

Don't take my word for it. At www.mazepath.com/uncleal/, six examples of those fullerenes' atom coordinates: c44.xyz, c52.xyz, c92.xyz, c100.xyz, c140.xyz, c260.xyz.

2) The trefoil knots are strictly homochiral product. The knots only form when all four stereogenic centers in the planar precursor, two cysteines and two beta-amino alanines, are geometrically (but not CIP!) homochiral. Mixed intra-molecule chirality gives circular oligomers.

3) "The whole of organic chemistry reduces to flat non-crossing 2-D Schlegel diagrams." Except for about a dozen K_5 molecules of some 60 million CAS organic entries, that is absolutely true.

ORTEP display changes nothing in the geometry. Gauge transformations are labels. Define anything you like. Proper orthochronous Lorentz symmetries and discrete symmetries are fundamental. Parity is outside Noether's theorems (re Lie groups) for being absolutely abruptly discontinuous. It is the key to finding incomplete founding postulates in physics. Euclid assuming all triangles' interior angle sum to 180 degrees was Bolyai's pry bar.

13. Hi Giotis,

Yes, it would be better to use another name. "Running spectral dimensions" eg would work well. But you know how it is with names, which one sticks is an emergent process in the community and difficult to influence. Best,

B.

14. Hi Sabine,
I was not referring to the Stojkovic model, but I was referring to the cosmic ray experiments he was citing, where it seems that the effective spacetime dimension is reduced to 2 at high energies. I wanted to hear what is your opinion about those experiments.

15. An alignment of outcomes of a scattering event does not imply "that the effective spacetime dimension is reduced to 2 at high energies". Besides this, I think it's not statistically very significant and one would like to see better data before claiming the SM needs scale dependent dimensions of one type or the other. Don't misunderstand me: This is not to say that I think it's uninteresting, just to say that at this point it seems premature to get excited.