Something special about four dimensions?

[tom.stoer]

I know that string theory tells us that 10 space-time dimensions have a rather special property, namely that a certain central charge vanishes. This allows for an anomaly-free quantization of s.t.

Two questions:
- are there other statements (not necessarily from s.t.) that something special happens in 4 dimensions?
- are there s.t. based arguments that exactly 6 of 10 dimensions are compactified?

 

[marcus]

http://arXiv.org/abs/gr-qc/0404088
Quantum general relativity and the classification of smooth manifolds
Hendryk Pfeiffer
41 pages
(Submitted on 21 Apr 2004)
"The gauge symmetry of classical general relativity under space-time diffeomorphisms implies that any path integral quantization which can be interpreted as a sum over space-time geometries, gives rise to a formal invariant of smooth manifolds. This is an opportunity to review results on the classification of smooth, piecewise-linear and topological manifolds. It turns out that differential topology distinguishes the space-time dimension d=3+1 from any other lower or higher dimension and relates the sought-after path integral quantization of general relativity in d=3+1 with an open problem in topology, namely to construct non-trivial invariants of smooth manifolds using their piecewise-linear structure. In any dimension d<=5+1, the classification results provide us with triangulations of space-time which are not merely approximations nor introduce any physical cut-off, but which rather capture the full information about smooth manifolds up to diffeomorphism. Conditions on refinements of these triangulations reveal what replaces block-spin renormalization group transformations in theories with dynamical geometry. The classification results finally suggest that it is space-time dimension rather than absence of gravitons that renders pure gravity in d=2+1 a `topological' theory."

I've seen several papers bearing on the issue of why 4D is special. At the moment one comes to mind is Hendryk Pfeiffer's here. At the time he was at Cambridge DAMTP, then 2005-2007 at the Albert Einstein Institute (MPI Potsdam), and 2007-now has a research/teaching position at University of British Columbia:
http://www.math.ubc.ca/~pfeiffer/ [Broken]

 

[MTd2]

4D is so special because its is the only the only dimension that needs fractals to classify and describe its smooth structures. All the other dimensions are really dull in this aspect.

4D is also the threashold for the classifictions of regular polytopes: everything is boring after 5D and up, unless 5D is equivalent to a curved 4 manifold:

http://en.wikipedia.org/wiki/List_of_regular_polytopes

4D is the richest manifolds in these aspects, and I believe that this is way theories in 4D dimensions are so problematic, because you it is hard to find physicaly meaningful ways to measure fractals.

A personal rule of thumb to know that things stem from the crazyness of 4 manifolds, and so have a chance of being realistic, is to look for any theory with E8 group. I don't know any exception for these rule, but theories superstrings, LQG, Horava Lifgarbagez (I will explain later) fits here.

 

[tom.stoer]

Thanks.

I've already heard some basic facts about Donaldson's differential structures on 4-manifold, but I never saw any relation to physics. Are these topics still relevant if one replaces manifolds by something more fundamental, e.g. discrete structures?

 

[arivero]

Inverse square force law, the one that happens in 4D dimensions, is very peculiar both classically and quantum-mechanically.

 

[tom.stoer]

Yes, but that's derived from the 1/k² of massless propagators; therefore it's not fundamental.

 

[arivero]

Yes, but that's derived from the 1/k² of massless propagators; therefore it's not fundamental.

Hmm? It depends of the dimension; actually people is searching for extra dimensions by looking for deviations of the inverse square law.

 

[tom.stoer]

Yes, you are right.

What I am saying is that if you start with massless particles, then they will always have a 1/k² propagator in ANY dimension. If you calculate the potential (= the Fourier-Trf.) of 1/k² you get V(r)~1/r in 3 spatial dimensions, and of course other r-dependencies in other dimensions. If the "size" of a dimension is small (a circle) again you have a different potential V(r).

So the potential is an indicator of deviations from the usual 3 spatial dimensions. It's not the root - it's derived.

 

[tom.stoer]

Perhaps I should explain why I am interested in this question:

If you do ordinary physics, nothing is special about 4 dimensions. You can do classical physics, quantum mechanics and quantum field theory with an arbitrary number of spatial dimensions.

So you can ask if there's an underlying principle which explains why we are living in 4 dimension or why we are observing 4 dimensions. I do not know one!

If you do string theory again 4 is not special. 10 is special because only in 10 dimensions perturbative superstring theory is consistent. Again you can ask if there's an underlying principle which explains why only 3 out of 9 spatial dimensions are observed. I do not know one!

What I am looking / asking for is some hint or principle that makes 4 dimensions (3 spatial dimensions) special or somehow unique.



are there other statements (not necessarily from s.t.) that something special happens in 4 dimensions?
- are there s.t. based arguments that exactly 6 of 10 dimensions are compactified?

 

[Haelfix]

The 1/r^2 force law is a good example. It allows for stable orbits and things like that.

Still over the years one learns a number of peculiarities of 4dimensions that are in some sense borderline cases between trivial and/or impossible to solve. Clifford algebras, spin structures, various geometric factoids etc etc

Of course, there's a natural numerology/coincidence bias going on as well, since we are accustomed to primarily dealing with 4d

 

[ConradDJ]

The 1/r^2 force law is a good example. It allows for stable orbits and things like that.

"Stable orbits" prompts me to ask -- is the stability of atomic structure in some way dependent on 4 dimensions?

Atoms may not seem "fundamental" -- but without atoms, I doubt there's any way to physically measure space or time. And at least so far as QM is concerned, it seems to be a significant feature of our universe that there are physical means of measuring its parameters.

 

[tom.stoer]

If you increase the number of dimensions, you get 1/r,1/r², 1/r³ potentials etc. I don't think that the Hamiltonian is well-behaved, perhaps not even bounded from below.

But again, that's a bit like the anthropic principle. "the world is four-dim. because we are here ..." I don't like this principle.

 

[arivero]

are there other statements (not necessarily from s.t.) that something special happens in 4 dimensions?
- are there s.t. based arguments that exactly 6 of 10 dimensions are compactified?

Perhaps it could be argued in reverse way: that 6 dimensions is the most we can compactify.

For instance, 11 dimensions is the max for supergravity and seven dimensions is the minimum for an action of SU(3)xSU(2)xU(1). So if both things were fundamentals, 4 follows from 11-7.

 

[arivero]

http://arxiv.org/abs/0808.3667 tells about 24 and 96
http://arxiv.org/abs/gr-qc/0404086 (and some of http://arxiv.org/abs/gr-qc/0603123 ) puts attention to the question of the relationship between potential, force law, and fundamental constants: only for D=4 Newton constant can be removed from some equations.

 

[arivero]

Ah, What about the degrees of freedom of the massless graviton D(D-3)/2? Taken at face value, should it imply that for D < 4 there is no such graviton?

D    dof
==   ==
11   44  
10   35
9    27
8    20
7    14
6    9
5    5
4    2
3    0

Off topic: what does it happen with the extra degrees of freedom as you compactify or decompactify? I read time ago the generic answer that they are swept under the carpet of the Kaluza Klein tower of states.

For instance, to build a U(1) theory you go to dimension 5, the uncompactified theory has 5 degrees of freedom, the Kaluza Klein theory has 2 from the graviton and 2 from the photon. Similarly, for 4gravityxSU(3)xU(1) you have 2+16+2 = 20, while the decompacfied theory has, minimum, D=9 and thus 27 degrees of freedom.

 

[tom.stoer]

... 11 dimensions is the max for supergravity and seven dimensions is the minimum for an action of SU(3)xSU(2)xU(1). So if both things were fundamentals, 4 follows from 11-7.

OK, I "understand" the 11 dimensions: the reason is that for D>11 one would have particles with spin>2 and nobody is able to quantize them consisently. I don't know why quantization fails, I only heard that it fails.

What do you mean by "7 is the min. for the standard model". Do you mean that if you have SUGRA in 11 dim. and you compactify less than 7, the resulting theory cannot have this special gauge group? E.g. only a smaller one? Again this would be somehow anthropic reasoning.

In the very end compactifying less than 7 or more than 7 seems to be consistent. It does not fit to the world, but there's no (mathematical) principle that excludes those possibilities, right?

Regarding D<4: it is said that in D=3 gravity is a topological field theory; that would mean that there are no dynamical degrees of freedeom.

 

[arivero]

What do you mean by "7 is the min. for the standard model". Do you mean that if you have SUGRA in 11 dim. and you compactify less than 7, the resulting theory cannot have this special gauge group? E.g. only a smaller one?

Exactly. And here I agree that it is "anthropic" or, to use a more classical word, "empiricist". But if you were able to justify on first principles the standard model, then the input should not be empirical any more, so it was worth to mention it. Note also that the respective minimal dimensions of the compactified spaces for SU(3), SU(2) and U(1) are 4, 2, 1. You could relate it to associative division algebras, spinor representations, or some other exotic math, who knows.

Regarding D<4: it is said that in D=3 gravity is a topological field theory; that would mean that there are no dynamical degrees of freedeom.

So if we ask the compactification to be the maximum you can do while still keeping dynamical degrees of freedom for gravitation, we are again back to D=4. My opinion, I would not be surprised if this argument were proved to be related to the ones on stability etc and not an independent one.

 

[tom.stoer]

Fine, thanks! I got the point - but I am still not satisified :-)

 

[arivero]

http://www-spires.fnal.gov/spires/find/hep/www?j=PHLTA,B97,233

Abstract

In d-dimensional unified theories that, along with gravity, contain an antisymmetric tensor field of rank s-1, preferential compactification of d-s or of s space-like dimensions is found to occur. This is the case in 11-dimensional supergravity where s = 4.