Friday, July 01, 2011

Why do we live in 3+1 dimensions? Another attempt.

It's been a while since we discussed the question why we experience no more and no less than 3 spatial dimensions. The last occasion was a paper by Karch and Randall who tried to shed some light on the issue, if not very convincingly. Now there's a new attempt on the arXiv:
    Spacetime Dimensionality from de Sitter Entropy
    By Arshad Momen and Rakibur Rahman
    arXiv: 1106.4548 [hep-th]

    We argue that the spontaneous creation of de Sitter universes favors three spatial dimensions. The conclusion relies on the causal-patch description of de Sitter space, where fiducial observers experience local thermal equilibrium up to a stretched horizon, on the holographic principle, and on some assumptions about the nature of gravity and the constituents of Hawking/Unruh radiation.

What they've done is to calculate the entropy and energy of the Unruh radiation in a causal patch of any one observer in a de Sitter spacetime with d spatial dimensions. Holding the energy fixed and making certain assumptions about the degrees of freedom of the particles in the radiation, the entropy has a local maxium at d= 2.97 spacelike dimensions, a minimum around 7 and goes to infinity for large d. Since the authors restrict themselves to d less or equal to 10, this seems to say for a given amount of energy the entropy is maximal for 3 spacelike dimensions. Assuming that the universe is created by quantum tunneling, the probability for creation is larger the larger the entropy, thus it would be likely then that we live in a space with 3 dimensions.

To calculate the entropy one needs a cutoff the value of which is fixed by matching it to the entropy associated with the de Sitter horizon, so that's where the holographic principle becomes important.

Not only is it crucial that they add an upper bound on the number of dimensions by some other argument, their counting also depends on the number of particles and the dimensions they can propagate into. They are assuming only massless particles contribute, and these are photons and gravitons. Massive particles even with small masses, the authors write, are "unacceptable" because then the cutoff could be sensitive to the Hubble parameter. By considering only photons and gravitons as massless particles they are assuming the standard model. So even in the best case one could say they have a correlation between the number of dimensions and the particle content. Also, in braneworld models the total number of spatial dimensions isn't necessarily the one determining degrees of freedom at low energy; a possibility the authors explicitly say they're not considering.

Thus, as much as I'd like to see a good answer to the question, I'm not very convinced by this one either.

38 comments:

Jérôme CHAUVET said...

Hi Bee,

Why are you're not convinced by this one either? Is it because of the fact there are too much restrictions put on the model?

Arun said...

So why don't we live in 2.97 dimensions? Is it because we all "know" that the number of dimensions has to be an integer?

Plato said...

Bee,

I guess, one can accept the framework with which to count?:)

String theorists describe the physics of black holes in five-dimensional spacetime. They found that these five-dimensional objects provide a good approximation of the quark-gluon plasma in one fewer dimension, a relationship similar to the one between a three-dimensional object and its two-dimensional shadow. Image: SLAC National Accelerator Laboratory

One millionth of a second after the Big Bang, the universe was a hot, dense sea of freely roaming particles called quarks and gluons. As the universe rapidly cooled, the particles joined together to form protons and neutrons, and the unique state of matter known as quark-gluon plasma disappeared.See: String theory may hold answers about quark-gluon plasma

Of course if you are a realist you will capture and hold onto words like "may, or could of," and say these things since they really don't "amount to much?" So one said, "I would feel more comfortable, until there is physical proof?"

Whose is actually closer to acceptance of the framework then those who speculate? The mathematics you choose?

So what is your stepping off point or question as to how to relate? You write papers.

Best,

Plato said...

Ummmmmm......nothings changed?

Best,

Plato said...

So what has changed experimentally? What conclusions had been drawn?

In summary, experiments at RHIC have shown that a very dense QCD medium is formed in high-energy heavy-ion collisions. Other measurements, namely elliptic flow and baryon-to-meson ratios, indicate that this medium is characterized by partonic degrees offreedom and that its expansion and cooling is well described by hydrodynamical models with high viscosity. Thus, this medium is more similar to a liquid than to a gas of gluons and quarks.Review on Heavy-Ion Physics

New Discovery about the Fabric of Space-Time


Quark Soup: Applied Superstring Theory

Uncle Al said...

Gravitons? All quantum gravitation theories are empirical disasters. Continuous geometries General Relativity and teleparallelism work. One sine wave cycle is also an odd-order polynomial... but drastically wrong outside the interval. This is elegant physical theory’s Achilles heel.

Genrich Altshuller, TRIZ, And Suddenly the Inventor Appeared. Orthodoxy excludes discovery. Heterodox observation falsifies assumed knowledge. Fundamentally mirror-symmetric massless physical theory is properly tested with opposite chirality self-similar atomic mass distributions. Do it the wrong way, seeking not validation but failure. Test it with opposite shoes,

http://www.mazepath.com/uncleal/erotor1.jpg
Two parity Eotvos experiments. The worst they can do is succeed.

Uncle Al said...

Gravitons? All quantum gravitation theories are empirical disasters. Continuous geometries General Relativity and teleparallelism work. One sine wave cycle is also an odd-order polynomial... but drastically wrong outside the interval. This is elegant physical theory’s Achilles heel.

Genrich Altshuller, TRIZ, And Suddenly the Inventor Appeared. Orthodoxy excludes discovery. Heterodox observation falsifies assumed knowledge. Fundamentally mirror-symmetric massless physical theory is properly tested with opposite chirality self-similar atomic mass distributions. Do it the wrong way, seeking not validation but failure. Test it with opposite shoes,

http://www.mazepath.com/uncleal/erotor1.jpg
Two parity Eotvos experiments. The worst they can do is succeed.

Bee said...

Hi Jérôme,

It's not so much restrictions as that I'm missing a coherent picture where all the made assumptions come naturally. At that point it's more like and then we assume this because somebody here has said that might be reasonable, and then we assume this because somebody has postulated this principle, and then we assume that because otherwise we've no paper to write. But why not bind the photons to a submanifold of dimension 5 and add some other U(1) symmetries and let's forget about the holographic principle because, well, it's called a principle because there's no prove for it. So then we're you left? On the other hand, it seems possible to me that the missing blanks can be filled in, so maybe I came across too pessimistic. Best,

B.

Bee said...

Dear Arun,

The paper doesn't say anything about it. Whether or not the number of dimensions has to be an integer depends on what fundamentally spacetime actually is. If it is a network for example, it's easy to imagine ways in which the dimensionality is not an integer. So, well, as I just said to Jérôme, I'm missing a complete picture. Without that, I don't even know where to begin trying to address your question. Best,

B.

Bee said...

Hi Plato,

Well, the spacetime is actually 10 dimensional, it's AdS_5 times S_5. The authors of the above paper want to do without string theory. I don't have problems with the AdS/CFT correspondence, except possibly that too many people seem to work on it. Best,

B.

Jérôme CHAUVET said...
This comment has been removed by the author.
Jérôme CHAUVET said...
This comment has been removed by the author.
Jérôme CHAUVET said...

Or the dimensionality of our great space perhaps depends on the observation viewpoint chosen, hence on the geometry needed to describe it... I mean, general relativity requires 3+1 dimensions in order to explain the dynamics of bodies due to gravitation. But now, if one oberves the organisation of stars and galaxies within a large part of the Universe, one will have probably better resort to fractal geometry and admit a non integer value for the correponding dimension.

My idea is that the number of dimensions may fit with what aspect of the universe one considers, and may thus be some subjective value tightly depending on the viewpoint.

Another thing could be that the movement of bodies in the universe, and its dynamics in general, would be equivalent to a big-big-big computation (cf. Sir Seth Loyd). In this case, 3+1 dimension would be simply the result of an optimisation process which the universe constantly undergoes. It is known that the more dimensions a space has, the more "degenerated" his algebra is . For example in the following, I write on the right what has been lost due to augmentation of dimension:

0) Real numbers

1) Complex numbers : - total order

2) Quaternions : - total order - commutativity

3) Octonions : total order - commutativity - associative property

4)Sédénions : total order - commutativity - associative property - alternativity

The bigger the dimension, the less easy-to-handle-algebraically it is for us. Sure black holes, stars and planets don't like it either :)

Regards,

Phil Warnell said...

Hi Bee,

Thanks for making note of this paper. Now all I have to do is try to understand it. Never the less I have my own take on why the universe is three dimensional. That is as QM dictates god must play dice, dice of less than 3 dimensions can’t roll and ones greater than 3 dimensions roll too well to ever be able stop rolling to make a determination :-)

Best,

Phil

Steven Colyer said...

Hi Bee, thanks for this, You know what an Engineer would say about this, right? He (or she) would say, hmm, 2.97 D? Let's restrict ourselves to 2 significant digits, and round that up to 3.0. :-)

Is there anyway possible to test this? Something out there has to explain the cosmological constant and why it's so small. For a while I had an idea that there is a 4th spacial dimension, but it's very small, on the order of a total 3.00000001 dS (dimensions of Space)or something.

This paper reminds me of a backup thought I had (because I do believe the work of Simon Donaldson blows 4 spacial dimensions out of the water), which is as this paper implies 2.97dS, or something close. That may or may not explain Dark Energy, which I have the niggling feeling is a geometrical consequence of ... whatEVer our Universe is, and thus perhaps the CC.

I do not believe, nor do I know, if Renate Loll et. al.'s "Casual Dynamical Triangulations" (CDT) have been disproved, however the inkblot tabletop (!) experiment where a 1dS+1dT "thing" can naturally expand to almost, but not quite, 3 spacial dimensions.

Interesting stuff in any event, Bee, thanks for posting this.

Neil Bates said...

I considered how electromagnetic interactions would go if space had various dimensions (making logical extrapolations, simplest for example: E = qr^(1-N) where N is number of space dimensions) and found inconsistencies with some complex situations if N was other than three. Hard to get up to pro level, had it wrangling around AJP but they didn't take it - said couldn't follow some of the stuff (it can get complicated.) More at blog.

Bee said...

Hi Steven,

For what I understand the CDT folks have made a lot of progress in the last years. There's a recent review paper by Ambjorn on the arXiv which is quite interesting (will see if I find the reference). However, there has so far no phenomenology come out for all I know, but I don't think it will take that much longer. Best,

B.

Bee said...

PS: I meant this paper. It's very readable.

Giotis said...

I don't quite understand. For a tunneling from nothing the inflationary universe demands initialy large CC and very low entropy (S=3πG/Λ) not the other way around.

Bee said...

Hi Giotis,

The entropy they consider is the entropy of the Unruh radiation in a dS space with some CC, and they construct it by introducing local temperatures T(r) for local observers. Similarly, the energy they consider is the energy associated to the energy density of the radiation. The whole reason why they get a nontrivial factor is that the entropy density and the energy density are different powers of the temperature which is now position dependent. If you integrate both over the local patch, you end up with the ratio of the integrals, each of which spits out some hypergeometric functions. As you probably know, these functions are not monotonic, thus the ratio has a non-trivial dependence on the exponent of T(r). Best,

B.

Neil Bates said...

@Giotis "tunneling from nothing ..." This careless phrase gets around and it is logically wrong A space with rules and statistics (requiring some kind of time standard) of occurrence of anything, including "universes", is not "nothing." The baseline states are called vacua - but anything with any properties at all such as what would come from it (instead of say, something of different features), any number of dimensions at all, etc. is a "something." Imagine it as you will, it is inappropriate to call such things nothing.

The findings about virtual particles, Casimir effect etc. did not prove that "nothingness" had such properties and potential,instead they showed that space really wasn't a "nothing" after all. Still true apart from existence or not of an aether. (Usage note: many think that spelling good to specify a medium in physics, not the chemical type. Helps with search, right?)

Heh, "flings" word verify. Are universes "flings" from the so-called vacuum (call it a plenum ...)

Giotis said...

Hi Bee,

Ok I didn't read the details but this is still Gibbons-hawking entropy just in another formulation, right? But anyway I don't see how this ansewrs my question? I gave it a quick look and they say somewhere: "we will argue that an infationary cosmic origin implies three large dimensions of space" and later "we argued that the spontaneous creation of infationary universes favors three dimensions of space", but then they use the Hartle-Hawking wave function to argue that tunelling favors large intitial entropy. My understanding is that these two points are in contradiction with each other since HH wave function is in contradiction with inflation which requires large initial CC and low entropy. HH prefers large entropy and zero CC and thus forbits inflation. It is known that HH wave-function has serious problems and cannot be trusted because it contrdicts obervation and inflation especially.

So i think that this can't be consistend; you can't have it both ways.

Plato said...

Bee:I don't have problems with the AdS/CFT correspondence, except possibly that too many people seem to work on it.

Yes, here from coment disregarding link to your article. How they approach it..as too, what currently exists in formulations and foundation.

I think on the point of phenomenology we both agree on working principals, tied to mathematical forays into the theoretical, must have substance or approach?

While Neil points out vacuum base perception these are ideals from which asymmetrical ideas emerge and point toward foundational 3+1. Standing pencils, or, stones rolling down hills "into valleys."

It's recognizing that while you work from euclidean there is a move to non-euclidean understandings. Qui /non, or, how could you worked dynamical relation to that which exists > or < then 180 degrees? It's a geometrical projection of geometry?

So they use a conformal field theory approach to be representational? Do you agree? It is mathematical pleasing, Non?

From a reductionist perspective the 3+1 then, is a result of.

The point is, how can you contend from that which is greater then 3+1, and your asking dimensional possibility? Yes?

A Platonic viewing at the 3+1 "is as if" you are moving away "from 3+1" how is that possible?

It means the platonic view is capable of seeing the result?;)

He may look at collision processes and the scope through which the energy used identifies a greater complexity of that moment.

QGP, recognizes that moment.

Recognizes a relation to the nature of the cosmos and the timing from minutes to microseconds? So you see, where is perspective pushed.

It sees not from outside some boundary but very well within the confines of this universe:)

Best,

Plato said...

For those like Phil who cannot see probabilist outcome as being greater then the dice through which one throws, rest easy then, and visualize sound "as a connecting principal" that is greater then the billiard balls that connect through trajectory alone.

So realists then, as to the substance of outcome "only see a billiard table," or, dice thrown?:)Is this not logically correct?:)

I like Jerome's examples as he understands the geometry of?:)Digitally and fractionally pleasing, it all gets smeared out yet, is representational of outcome "as it is written" as one peers to patterns that repeat them self? Through the wave transformation one is very capable of pinning down location?

Yet, one will progress no further, and then again see the result of a greater potential as those same patterns? Are they representational in the Platonic view as seeing the Platonic solids as a foundational approach? Yes, but there is recognition now as too "a greater potential" then for those who proclaim them self realists:)Oui/Non?

It's best one realizes how string theorists work then the continued bickering from the realists and from Uncle?:)LOL

Conformal Field theory is representational?

Best,

Kea said...

Because the dimension raising Gray tensor product generates dynamics from the tricategorical realm of braids and bioctonions for quantum logic, where the three valued quantum number "rest mass" is captured in the three strands needed to describe the color kinematic duality of modern perturbative supergravity. Duh.

Bee said...

Hi Giotis,

What you say is correct, but note that they do not ask what CC maximizes S, but what d maximizes S for fixed E. Or, in other words, they don't say anything about the value of the CC. Look at Eq (19) in the paper, the ratio is (for small epsilon) independent of \Lambda. Best,

B.

Steven Colyer said...

Hi Bee,

Thank you very much for that readable paper on CDT by Loll et. al., the very three founders of Causal Dynamical Triangulations.

I would also like to back up everything Kea said, and I will as soon as I understand what the hell she said. You're amazing Kea. ;-) Are you on Facebook?

Speaking of dimensions, I have to buy a new computer, and hell no not a laptop those damn things break too much.

What are the current "dimensions" of a computer these days? I only buy a new one every five years, using spit and rubber bands to keep them together toward the bitter end.

So far I have: Price, RAM, ROM, graphics card,Micropocessor ... what else?

The reason i ask is I think i'm going to try and teach Calculus and physics and maybe Algebra at www.tutor.com, an APS approved website. And hoo boy do they have specific requirements, and i'm pretty sure this POS 'puter I have, when it's running, doesn't meet them. Here are the requirements:

Windows XP (w/Service Pack 3), Vista (w/Service Pack 2), or Windows 7 (w/Service pack 1)
2.0 GB of system memory (RAM)
Single Core Processor at 2.4 GHz or Dual Core Processor at 1.8 GHz (manufactured on or after 2004)
High-speed internet access
Microsoft .NET 3.5 SP1
Microsoft Internet Explorer 7.0 or higher

Bee said...

Steven: I'm totally not the person to ask about computers n stuff, also off-topic. My sense is however that whatever PC you buy today will fulfill the requirements. Also, I've exclusively had laptops since '03 and none of them ever 'broke.' If I consider how often I've dropped them, it's quite amazing actually. Best,

B.

Plato said...

Oh K...K..Kea?;)

A summary for sure then.

While I may have been asleep for last 15 years and "a Smoit," it was just a laymen trying to stay abreast of the knowledge mathematical convergence.

It's real Beee.....:)

Best,

Phil Warnell said...
This comment has been removed by the author.
Steven Colyer said...

Hi Bee,

It wasn't about computers, it was about dimensions. As you know being both Mathematician and Physicist, the 2 fields have slightly different definitions of what constitutes: "dimension."

Here's another example:

You have to drive from point A to point B, say, across Arizona, or whatever. How many different dimensions are involved?

Well, most would say 2, or 3, if they remember time is a factor. But there's a 4th: cost, based on whether you're driving a Volkswagen Beetle, or a Supersized Sean Hannity-approved gas-guzzling Ford Expedition. Makes a difference, and has nothing to do with length or time. Although mileage does connect with length.

Bee said...

Hi Stephen,

Well, as I already said above, the dimensions the authors are concerned with are the dimensions of the background manifold. Best,

B.

Plato said...

Hi Bee,

So many Stefan's, Stevens, or Stephens, it's starting to get confusing:)

Any theoretic development should be phenomenology applied....if you don't...it's mere speculation?:)

I was looking for "holes" that may account for energy leakage ->toward extra-> dimensions....but hey, that's just me speculating:)Not all the energy has been accounted for?

Best,

Plato said...

Bee,

Even the famous helium-3, which can flow out of a container via capillary forces, does not count as a perfect fluid.What black holes teach about strongly coupled particles by Clifford V. Johnson and Peter Steinberg....May of Last Year.

If helium-3 is used in cooling energy containment and was to be considered within LHC, wouldn't such example be applicable as to thinking about capillary routes as holes? Energy loss attributed too?

Layman wondering.

Best,

Zephir said...

For me this attempt appears pretty good (I'd expect slightly more complex curve at the higher-dimensional end, though). The N = 2.97 result could correspond the fact, we are living in hyperbolic space-time geometry due the CMBR. It's probably the second most important postdiction of holographic model after the Verlinde's entropic gravity model.

Bee said...

Hi Plato,

If you wanted to have 'capillaries' on the boundary, it would no longer be homogeneous. I don't know how one would do that, and I'm not aware of any literature on it. Also, it is somewhat misleading to think of a black hole as an actual 'hole.' There is nothing 'leaking' there to anywhere, it just doesn't come out again. I know it's all over scifi, but you can't pass through a black hole and get out elsewhere. Best,

B.

Plato said...

Bee:Also, it is somewhat misleading to think of a black hole as an actual 'hole.' There is nothing 'leaking' there to anywhere,

It is a relationship between what is understood in relation to cosmic particle collisions(particle decay accounted for as energy expressions)and the relationship of what is accounted for in LHC collision processes as an area caloric illustrated.

This "area" is identified as to being in both location, is to show "beyond backdrop measure" to illustrate faster than light entities traveling through earth's medium of expressions as to back further backdrops like ICECUBE, Pierre Auger experiments...our previous talks on energy allocation with regard to Fly's eye energy determination and cutoffs etc.

Of course we are trying to determine total energy calculation and accountability as to presence of particle determinations as a total energy "before" energy collision. See?

Best,

Plato said...

Bee:I don't know how one would do that, and I'm not aware of any literature on it

Again I am pointing toward "states of existence" with regard to similarities of expression generalized in layman view as too, characteristics of QGP and helium-3 containment.

If you can identify similar expressions as to relation "of an area of leakage and align them" then this leakage 'is the pathway for energy expressions and accounting' that leads to total energy?

This imbalance is thought to arise because one of the jets traversed the hot and dense matter, transferring a substantial fraction of its energy to the medium in a way that is not recovered by the reconstruction of the jets.See: ALICE enters new territory in heavy-ion collisions

Jet quenching? Again what do we see by using measures like Gran Sasso after collision process has taken place? All measures around the Ring yet this backdrop measure is distant through earth to help us measure the nature of earth from one point to another. Volcanoes.

The relativistic nature of muon detection process.

Best,