Wednesday, August 24, 2016

What if the universe was like a pile of laundry?

What if the universe was like a pile of laundry?

Have one.

See this laundry pile? Looks just like our universe.

No?

Here, have another.

See it now? It’s got three dimensions and all.

But look again.

The shirts and towels, they’re really crinkled and interlocked two-dimensional surfaces.

Wait.

It’s one-dimensional yarn, knotted up tightly.

You ok?

Have another.

I see it clearly now. It’s everything at once, one-two-three dimensional. Just depends on how closely you look at it.

Amazing, don’t you think? What if our universe was just like that?

 Universal Laundry Pile.[Img Src: Clipartkid]

It doesn’t sound like a sober thought, but it’s got math behind it, so physicists think there might be something to it. Indeed the math piled up lately. They call it “dimensional reduction,” the idea that space on short distances has fewer than three dimensions – and it might help physicists to quantize gravity.

We’ve gotten used to space with additional dimensions, rolled up so small we can’t observe them. But how do you get rid of dimensions instead? To understand how it works we first have clarify what we mean by “dimension.”

We normally think about dimensions of space by picturing lines which spread from a point. How quickly the lines dilute with the distance from the point tells us the “Hausdorff dimension” of a space. The faster the lines diverge from each other with distance, the larger the Hausdorff dimension. If you speak through a pipe, for example, sound waves spread less and your voice carries farther. The pipe hence has a lower Hausdorff dimension than our normal 3-dimensional office cubicles. It’s the Hausdorff dimension that we colloquially refer to as just dimension.

For dimensional reduction, however, it is not the Hausdorff dimension which is relevant, but instead the “spectral dimension,” which is a slightly different concept. We can calculate it by first getting rid of the “time” in “space-time” and making it into space (period). We then place a random walker at one point and measure the probability that it returns to the same point during its walk. The smaller the average return probability, the higher the probability the walker gets lost, and the higher the number of spectral dimensions.

Normally, for a non-quantum space, both notions of dimension are identical. However, add quantum mechanics and the spectral dimension at short distances goes down from four to two. The return probability for short walks becomes larger than expected, and the walker is less likely to get lost – this is what physicists mean by “dimensional reduction.”

The spectral dimension is not necessarily an integer; it can take on any value. This value starts at 4 when quantum effects can be neglected, and decreases when the walker’s sensitivity to quantum effects at shortest distances increases. Physicists therefore also like to say that the spectral dimension “runs,” meaning its value depends on the resolution at which space-time is probed.

Dimensional reduction is an attractive idea because quantizing gravity is considerably easier in lower dimensions where the infinities that plague traditional attempts to quantize gravity go away. A theory with a reduced number of dimensions at shortest distances therefore has much higher chances to remain consistent and so to provide a meaningful theory for the quantum nature of space and time. Not so surprisingly thus, among physicists, dimensional reduction has received quite some attention lately.

This strange property of quantum-spaces was first found in Causal Dynamical Triangulation (hep-th/0505113), an approach to quantum gravity that relies on approximating curved spaces by triangular patches. In this work, the researchers did a numerical simulation of a random walk in such a triangulized quantum-space, and found that the spectral dimension goes down from four to two. Or actually to 1.80 ± 0.25 if you want to know precisely.

Instead of doing numerical simulations, it is also possible to study the spectral dimension mathematically, which has since been done in various other approaches. For this, physicists exploit that the behavior of the random walk is governed by a differential equation – the diffusion equation – which depends on the curvature of space. In quantum gravity, the curvature has quantum fluctuations, and then it’s instead its average value which enters the diffusion equation. From the diffusion equation one then calculates the return probability for the random walk.

This way, physicists have inferred the spectral dimension also in Asymptotically Safe Gravity (hep-th/0508202), an approach to quantum gravity which relies on the resolution-dependence (the “running”) of quantum field theories. And they found the same drop from four to two spectral dimensions.

Another indication comes from Loop Quantum Gravity, where the scaling of the area operator with length changes at short distances. In this case is somewhat questionable whether the notion of curvature makes sense at all on short distances. But ignoring this, one can construct the diffusion equation and finds that the spectral dimension drops from four to two (0812.2214).

And then there is Horava-Lifshitz gravity, yet another modification of gravity which some believe helps with quantizing it. Here too, dimensional reduction has been found (0902.3657).

It is difficult to visualize what is happening with the dimensionality of space if it goes down continuously, rather than in discrete steps as in the example with the laundry pile. Maybe a good way to picture it, as Calcagni, Eichhorn and Saueressig suggest, is to think of the quantum fluctuations of space-time hindering a particle’s random walk, thereby slowing it down. It wouldn’t have to be that way. Quantum fluctuations could also kick the particle around wildly, thereby increasing the spectral dimension rather than decreasing it. But that’s not what the math tells us.

One shouldn’t take this picture too seriously though, because we’re talking about a random walk in space, not space-time, and so it’s not a real physical process. Turning time into space might seem strange, but it is a common mathematical simplification which is often used for calculations in quantum theory. Still, it makes it difficult to interpret what is happening physically.

I find it intriguing that several different approaches to quantum gravity share a behavior like this. Maybe it is a general property of quantum space-time. But then, there are many different types of random walks, and while these different approaches to quantum gravity share a similar scaling behavior for the spectral dimension, they differ in the type of random walk that produces this scaling (1304.7247). So maybe the similarities are only superficial.

And of course this idea has no observational evidence speaking for it. Maybe never will. But one day, I’m sure, all the math will click into place and everything will make perfect sense. Meanwhile, have another.

[This article first appeared on Starts With A Bang under the title Dimensional Reduction: The Key To Physics' Greatest Mystery?]

1. A good argument against trusting observation in the absence of a mathematical theory.

2. Sabine,

The link to "have another" is broken.

3. The link at "have another" did not work. Did you mean Spontaneous Dimensional Reduction in Quantum Gravity ?

4. t h ray,

Or against trusting a mathematical theory in the absence of observation...

5. Big Henry,

Thanks for letting me know, I've fixed this!

6. Glenn,

Yes, that's where the link should have pointed! I've fixed this now, thanks for letting me know.

7. "What if the universe was like a pile of laundry?" The subjunctive case of "to be," "were like," is better.

Running dimensionality and the Casimir effect suggest atomic force microscopy for measuring gravitation/distance. EM relative amplitude suggests otherwise. If gravitation is fractal at small separations, magnification or tessellation won't simplify it. If black hole event horizons are curved 2-spheres (simply connected 2-dimensional manifolds of constant positive curvature) without enclosed volume or a "central" singularity, analysis-resisting QM to information theory disappear. Or, explain observed LIGO event GW150914's last 0.2 seconds (ringdown!) versus hectares of theory that demand otherwise.

8. "The shirts and towels, they’re really crinkled and interlocked two-dimensional surfaces." That part sounds a bit misleading because the pile is physically still 3D and the 2D surface is mathematical. I only mention this because you mention near the end, "walk in space, not space-time, and so it’s not a real physical process"; some forget to consider what can only be a mathematical existence vs what is a physical one.

9. Do either of these definitions of dimension coincide with fractal dimension?

10. Bee,

"Or against trusting a mathematical theory in the absence of observation.."

Disagree. A mathematical theory can exist independent of observation, while an observation without theory-dependence is meaningless. Think: Penzias & Wilson and CMB. If the big bang theory of cosmology had not preceded its discovery, what would one have called the radiation but "static of unknown origin"?

It's those " ... free inventions of the human mind ... " as Einstein put it, that given meaning to noise.

Best,
Tom

11. No the insight is very good. It's obvious, but not pejoratively for the individual to speak first.
But even good insights say little about the future direction of research science. What must be true is the one that speaks there. This excerpt says the most:
"I find it intriguing that several different approaches to quantum gravity share a behavior like this."

12. Interesting piece, as always. Re taking the time out of spacetime are you suggesting that this step toward quantum gravity may in any way require repudiation of the spacetime notion and thus GR?

13. There also doesn't seem to be an active hyperlink with the reference, 1304.7247, concerning the differing types of random walk, which seems to be here:

http://arxiv.org/pdf/1304.7247.pdf

14. andrew,

To my knowledge, no.

15. No mention of Kaluza-Klein as an example of dimensional reduction? Maybe I'm just not getting it. Anyway, love the blog and the ideas you're proposing.

16. "What if the universe was like a pile of laundry?"

You would still be missing 1 sock.

17. Mathematically, the progression is moving from ordinary smooth manifolds with commutative rings of functions acting locally on their spectra to noncommutative algebras acting on noncommutative spaces (that replace the familiar open sets isomorphic to R^n). This is the essence of noncommutative geometry, where Connes takes the noncommutative algebra to have the full structure of a C*-algebra. Hence, in this formalism, a priori our "noncommutative open sets" are fuzzy and only "crystallize" once one spectrally resolves an operator of the local C*-algebra. This is why matrix models have had great success in quantum gravity, especially in recovering the mathematical structure of branes in string theory. However, one can go further than the C*-algebra framework into structures that posess quasiconformal symmetry, as in the case of E8 acting non-linearly on its 57. This is where the fun begins.

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19. Bill,

In Kaluza-Klein one *adds* dimensions. You can then make them unnoticeable by rolling them up ("compactifying") them to small radius. In this case, there will be more dimensions that become accessible at higher energy (short distances), not less.

20. Steven,

The paths that you talk about have dimension two because they're paths of (classical) particles. If you want to know quantum properties of other things, this generally isn't the case. Eg, Causal Dynamical Triangulation uses the Feynman path integral approach, but the "paths" in this case are space-times each. Best,

B.

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22. Steven,

Well, they don't (lie in that subspace). Really I think you should read the CDT papers I've linked to, I don't get the impression you know what you're talking about.

23. The problem with such speculative ideas is that they are not based on an experimental fact. This leads to inconsistencies as the theory is developed which unfortunately will be explained away by other speculative propositions feeding an infinite loop and diverting human and financial resources from the actual problem. This problem is becoming pandemic in fundamental physics. The firewall problem, SUSY (which is undergoing postmortem as we speak ) unfalsifiable quantum gravity theories etc.are just but a few examples. The LIGO results are a good starting point for BSM physics. Why? As Uncle points out there is no evidence for infinite redshifts and time dilation. This could be the first signs that GR is collapsing and quantum gravity effects are taking over.

24. @Meta Tron Given a non-commutative construct, reverse the signs of one coordinate (mirror image, S_1 improper rotation axis) and all coordinates (parity inversion, S_2 improper rotation axis). Are sign-reversed constructs exactly superposable upon the originals? If not, non-commutative constructs are chiral.

Chiral entities cannot possess S_n symmetries. Baryogenesis' Sakharov conditions require chiral asymmetry. Noetherian coupling of exact spatial isotropy with angular momentum conservation, given fundamental chiral anisotropy, leaks Milgrom acceleration. The Tully-Fisher relation is universal; no dark matter.

http://classes.uleth.ca/200303/chem3810a/Part1A.pdf
Viam sapientiae mundi, per quam pervenitur.

25. In a perfect rotationally symmetric washingmachine universe, find a breaking symmetry mechanism that produce
1) a chromatic number violation for pants
2) a parity violation for black socks

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27. Alexey,

Excuse me, I've never called anyone in my life a "crook" - it's a word that isn't even in my vocabulary, so please stop accusing me of things I've never done.

I think you didn't get the point this post. It wasn't to say that a pile of laundry is a good metaphor, it was to say we all take our inspirations from simple ideas, but in the end it needs to be converted into solid math. There are lots of references in that post which you can click on and get to the relevant papers. Best,

B.

28. "The mass of the Higgs is much smaller than the Planck mass. So what? The spatial curvature of the universe is almost zero, the cosmological constant tiny, and the electric dipole moment of the neutron is for all we know absent. Why should that bother me?"

Whether or not it bothers you depends on your goal. One sense those questions are important is that they help delineate the limiting bounds of current explanation. The significance of the bounds is obviously that breakthrough theories tend to encapsulate the best of scientific theory to date. Best theories don't get proved wrong, they just get encapsulated within something that reduces back to that earlier theory in some ground state.