## Monday, May 08, 2006

### Why do we live in 3+1 dimensions?

People warned me that writing a blog would take a lot of time. Hopelessly naive as I am, I thought, well, I would just not post anything if I am too busy. It seems, I underestimated the persistent interest of my fellow readers.

So, I wrote last week that I volunteered to bring an Honest Question to our gravity lunch. This meeting takes place here at the Department pf Physics at UCSB every Friday at noon. Lately, the discussion has been mostly about black thingies. I tried to come up with a question that would roughly fit into the string dominated atmosphere and the time constraint, and eventually settled on "Why do we live in 3+1 dimensions?"

The last time I wrote this question down, someone was so nice to tell me that 3+1=4. Therefore, let me point out that the question actually consists of two parts: a) why 3 spacelike dimensions and b) why Lorentzian signature -- I will only discuss a) in the following.

But of course I had to make the question more complicated to be appropriate for the gravity lunch. To do so, I picked the paper

Relaxing to Three Dimensions
Authors: Andreas Karch, Lisa Randall
Phys.Rev.Lett. 95 (2005) 161601 (hep-th/0506053)

Last time I was at PI, I happened to hear a seminar by Lisa Randall about the paper, you can find it online at the streaming seminars (click on "seminar series", then "filter by presenter" Lisa Randall - comes under L not R, and hit "search" -- they promised they are working on an improvement...).

While writing this post, I also found an article about the paper from economist.com

A braney theory: An explanation for the anthropic principle comes a little closer

Here is a quick summary: The idea is to take a 9+1 dimensional non-compactified spacetime. Fill it with gases of d-branes, each with an energy density and pressure. And let it expand with a Friedmann-Robertson-Walker (FRW) ansatz, i.e. homogeneous and isotropic. The paper is quite impressing, as it only contains 4 equations, which are the FRW equations in higher dimensions.

Now the question is what happens to the gases of the d-Branes.

1. When the branes don't interact, the energy density will dilute slower the larger the dimension of the brane - because it can dilute only into the dimensions it does not occupy. In terms of the FRW scale parameter a, the density goes with a power -9+d.

2. But they can interact, i.e. branes can meet anti-branes and decay. This decay goes slower the larger the dimensionality of the brane - because there is less space to decay into. In terms of the time t, the density goes with -9+d.

3. Then the question remains whether d-branes do find each other to interact. It turns out from dimensional arguments that they will generically find each other and attempt to decay when 2*(d+1) is larger or equal 9.

From 3. it follows that 3-branes are those with the largest dimensionality that will not interact. From those that will not interact, they are also those whose energy density will dilute the least. For d larger than 3, the 9-branes do always overlap and therefore are gone. The 8-branes are apparently more complicated, but can be argued away. The only argument for the latter that I understood was that in some scenarios there just are no 8-branes. Let's assume that works.

Then, in terms of energy densities, 3-branes and 7-branes will dominate. Such the conclusion of the paper. I understand why one would like to have 3-branes. As to the 7-branes, the paper states

A configuration that is a natural candidate for four-dimensional gravity is the intersection of three 7-branes where the intersection has spacetime dimension four.

Among others, a question I raised on Friday was why this is natural. I learned that the physics on such an intersection allows chiral fermions. That explains why they write it is natural to live at this intersection. But not why it is natural.

More importantly, I fail to see why the densities of the gases are relevant for the question why we experience three dimensions. Even if the higher dimensional thingies decay, the lower dimensional ones are still around, no matter what their density is. Why is the energy density the selection criterion?

And another point that I still don't understand is how it is possible that the ongoing time-dependence in the bulk does not influence the physics (locally localized gravity) on the brane or brane/intersection. I mean, one has to make sure that things we call constant actually are constant (restrictions apply).

Bottom-line: I like the paper, I like the idea and the minimalistic setup. Unfortunately, it seems to me some of the arguments are more wishful thinking than strict conclusions.

I have certainly heard weirder things. I once sat through a seminar while the speaker explained that our universe has 10 dimensions because Pi^2 is approximately 10, and we live on a 3-dimensional submanifold because Pi is close by 3. No, I can't recall the name of the speaker, and I never heard of him again.

On the other hand, it is indeed puzzling that dimensional regularization works only in 4 dimensions, isn't it?

For those of you who are interested, here are some further references

Why do we live in 3+1 dimensions?
Ruth Durrer, Martin Kunz, Mairi Sakellariadou

Brane Gases in the Early Universe
S. Alexander, R.Brandenberger, D.Easson

On the dimensionality of spacetime
Max Tegmark

(thanks to Garrett for the reference)

Why is Space Three Dimensional
Ingemar Bengtsson

Oh, and due to a certain lack of volunteers, I agreed to bring another question for the next gravity lunch. Any suggestions?

1. Oh no...this is all wrong...

The three comes from the fact that "Da Vinci Code" has three words, and the seven comes from the fact that "opus dei" has seven letters.

(sorry, I couldn't help myself)

2. Dear Bee,

what is this point about the chiral fermions? What restrictions do they imply, and why? And what happens if there aren't any chiral fermions in a therory?

Best, Stefan

3. Stefan said: what is this point about the chiral fermions? What restrictions do they imply, and why? And what happens if there aren't any chiral fermions in a therory?

Well, if there are no chiral fermions, than the theory is not the standard model. That's the point.

a said: The three comes from the fact that "Da Vinci Code" has three words

Interesting point. One should not underestimate the power of trinity.

Besides the holy one, we have 3 colors, 3 flavors, 3 generations, 3 volumes Lord of the Rings, 3 parts of Matrix, 3 genders (male, female, do not wish to specify), we have 3 Doors Down, the first three minutes, and the last three minutes, and recently we also have triply special relativity. It's completely obvious that we MUST live in 3 dimensions.

4. We may not live in 3+1 dimensions, but photons sure seem to.

5. We may not live in 3+1 dimensions, but photons sure seem to.

We = quarks,gluons,electrons?

6. Okay, so that was a bit obtuse on my part. My point was that unless someone has built a direct strong or weak force detector our observations of the universe are mediated by electromagnetic interactions, correct?

Unless there is something horribly wrong with my knowledge of particle experiments, we infere the other forces by looking at the behavior of particles in electromagnetic fields. I mean heck, even gravity is infered in a sort of way. If our bones, cells, etc... didn't happen to be held together by chemical bonds would we sense gravity (yes I see the tautological joke in that statement ;o). When you hold a weight in your hand, the pull you feel may originate in gravity, but its mechanical translation through your musco-skeletal is all EM mediated. Furthermore we infere the presence of gravity on large scales by observations of paths, again observations mediated by EM.

Okay lets try another tactic. Why do we humans experience 3+1 dimensions, well its because we experience the universe through wonderfull little massless, chargless, bosons that happen to have 3+1 dimensions to move in. So this brings up the question, why do photons move in 3+1, why not something else?

So the challenge is using only a Hilbert space, U(1) symmetry on that space, c, h, and Lorentz symmetry (perserves the matrix n=2aa^t-1, a:vector, a^t:transpose) can you come up with a really solid 3+1 argument, if not what has to be added to do so? (sorry anthropic agruments using Green's function in higher or lower dimensions will not be graded ;o)

And yes I would really like to discard the idea that other fundamental particles move through higher dimensions, but first one has to figure out what the predications of such a hypothesis would be. Once we've done that then its a matter of reviewing past physics experiments for data that contradicts the hypothesis. For example, can we eliminate the hypothesis that rest mass is the movement of massless particle in higher dimensions?

Now it may be that we can't eliminate higher dimensional hypothesises, mostly because you can always muck about with a theory to make it fit. So the question becomes one of comparison: which theory requires a greater degree of theoretical controtionism? 3+1 or other higher dimensional theories? If higher dimensional theories turn out to always be more complicated then we have come full circle and via Occam's Razor must accept 3+1 as an empirical fact that cannot be infered mathematically. Or in other words, we may in fact live in a higher dimensional universe, but the most intellectually concise theory that explains our observations is 3+1.

Once again sorry for being obtuse. Hope that makes me sound a little less like a cracked-pot.

7. insightaction said: Once again sorry for being obtuse. Hope that makes me sound a little less like a cracked-pot.

Well, it's my blog and I allow crack-pots :-) Anyway, I didn't mean to make fun out of your comment.

insightaction said: And yes I would really like to discard the idea that other fundamental particles move through higher dimensions, but first one has to figure out what the predications of such a hypothesis would be. Once we've done that then its a matter of reviewing past physics experiments for data that contradicts the hypothesis. For example, can we eliminate the hypothesis that rest mass is the movement of massless particle in higher dimensions?

You are confusing me. This hypothesis has been investigated en detail in the last years.

Suppose you have additional dimensions. You need to compactify them (at least you need a finite volume a la Randall Sundrum). The reason is essentially that you otherwise can't confine gravity and can't reproduce the 3+1 dimensional Newton's potential (I think anti-gravitation provides another way to confine gravity, but thats a different issue).

When particles of the standard model are allowed into the extra dimensions and these are compactified (comes under the name 'universal extra dimensions'), then these particles can have KK-excitations that come in discrete steps. That's a prediction, and it can be tested at colliders. No such excitations have been seen so far. That sets an upper bound on the radius of these dimensions.

Suppose the rest-mass of the electron was due to it's movement into a additional dimensions. Then it should have excitations with 2,3,4,5,6... times the electron mass - we would already have seen that.

Can give you plenty more reference if interest. A good starting point is actually the review in the particle data booklet, which comes with plenty of further references.

Best,

B.

8. The confusion was because I was rambling.

Shows the danger of making off the cuff remarks about the nature of observation.

9. insigtaction said: The confusion was because I was rambling.

Shows the danger of making off the cuff remarks about the nature of observation.

thanks for the comment anyway:
I LOVE questions that I can answer :-)

10. there were no references to a FRW model incorporating higher dimensions (curse you google), but quite a grade A blog.

11. Hello. It's great to find a quality blog about frontier issues like extra dimensions. I've posted about why space must have three large dimensions on my own blog, tyrannogenius. I posted the same to another thread, but this is so relevant here (since it is presumed, falsely IMHO, that there isn't a good reason for the specification 3 + 1 dimensions):

In April, I listened to an interview on Public Radio with physicist Lisa Randall. She is a top theorist on foundational theory of why the universe is the way it is. That means string, branes, and such. One of the venerable questions is: why is space three-dimensional? It may seem natural to have three dimensions of space and one of time, but mathematically there can be any number of dimensions (think of specifying points using 4, 10, etc. variables.) Physicists, including Lisa, say they can't see why space *had* to have three dimensions. Check out this.

However, they have come up with reasons why three large-scale dimensions would be more likely to expand out of a larger set (usually thought of as 10 or 11) of original, perhaps tiny dimensions. Below follows a statement adapted from my post to radio open source in response to the interview, and outlining my own efforts to answer this question.

I have been working myself on the question, why are there three *large* dimensions of space? (There are probably more, like a total of 10 or 11 space dimensions, but the rest are curled up very small or otherwise inaccessible.) After extrapolating electromagnetic interactions to spaces of other dimensions, I found at least two arguments:

1. In spaces with other than one or three dimensions, an oscillating charge does not project the same *average* field along the axis of oscillation as the rest value. That is due to two things: the combination of "projection" of its retarded distance - where it would be had it continued at the velocity it had when light left it - and the distortion of the field due to Lorentz contraction, which weakens it to gamma^(1-N) the value it has at rest. N is the number of large space dimensions. (We also must take into account the Doppler shift of projection intervals. Heh, it’s not quite as complicated as it sounds.) Remember that the Coulombic electric field intensity is given as E = qr^(1-N) due to field spreading. This amplifies the effect of the oscillating charge’s apparent position being close (projected from approaching cycle) to a second “target” charge at rest. It increasingly swamps the weakening effect of the gamma factor as N goes above three and is incorrect when N = 2. That would impose a net force on a second "target" charge unequal to that on the oscillating charge, and violate conservation of momentum and energy. The one-dimensional case is ruled out due to infinite potential energy as is the 2-D case (why didn’t A. K. Dewdney realize that about the 2-D Planiverse?)

2. Let two charges be connected by a reasonably rigid rod. Then, accelerate the rod along its length. The combined force between the charges will be derived from the sort of considerations given in (1.), as the projected field of each charge catches up to the other charge. Then we must take into account the extra force created by the action of acceleration on the relativistic stress-correction to the momentum and energy of the rod. Only in three dimensions of space does that equal in net the effective inertia the charges should have given their potential energy. (In higher dimensions, taking the integral of f = q1q2/r^(N-1), that potential w.r.t. infinity is: -q1q2r^(2-N)/(2-N).)

I hope I can publish the full development of this before long. I don’t think anyone else has an explicit proof that N *must* equal three, only reasons it was more likely to form, or oddities like being unfriendly to life, distorted wave propagation (see Barrow and Tipler’s _The Anthropic Cosmological Principle_ for great discussion of this.)

Neil Bates

12. By AWT the reality is formed by density fluctuations of dense particle environement and we are living in 3D space because just the 3D hyperspheres exhibits the most dense packing (compare the Kepler conjecture for kissing hyperspheres packing).

13. This thread surprises me, given the following references:

Tegmark, Max (1998) “Is the ‘theory of everything’ merely the ultimate ensemble theory?” Annals of Physics 270: 1-29. §IV.B. http://arxiv.org/abs/gr-qc/9704009v2.

Tegmark, Max (1997) “On the dimensionality of spacetime,” Classic and Quantum Gravity 14: L69-L75, p. L69.
http://www.iop.org/EJ/abstract/0264-9381/14/4/002 .

Barrow & Tipler (1986: §4.8) claim that this sort of reasoning goes back to Ehrenfest (1917), and was prominent in the 1950s writings of Gerald Whitrow.

The upshot: we do not truly know "why" reality is embedded in 3+1 Minkowski spacetime. But we do know the 3+1 is "well-behaved" and maximally interesting (kudos to you, Leibniz!). In particular, 3+1 is mathematically most rich. (Perelman earned his fame by proving the case D=3.)

14. it would seem relevant that
if you complexify octonion algebra
you get -+++ and +---, unlike
Clifford algebra where those two
are not algebraically isomorphic
over R.

15. CONJECTURE: it will take understanding what octonions are about to make sense of why space is three dimensional.

Also - Ingemar's paper mentions the garden. We would not think much of an explanation if the 3 space had only photons or electrons and positrons - we really need to expect it to make sense with space + hydrogen atoms.

16. Thanks for a very useful blog post, Bee.

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