# Infinite dimensions?

1. Oct 23, 2010

### Nikola Mitic

Is there a possibility that there are infinite dimensions in our universe.M theory predicts that there are 11 dimensions,but why not more?

2. Oct 24, 2010

### tom.stoer

In bosonic string theory it's rather easy. One starts with N-dimensional flat space in which bononic strings move;one finds that the requirement that a certain anomaly cancles and that the quantized theory has a chance to be consistent uniquely fixes the spacetime dimension N=26.

In superstring theory the calculation is more involved but one ends up via similar reasoning with N=10.

Up to now I talked about free or perturbative string theory.

No it becomes interesting. Already in the heterotic string there is an interesting split between internal degrees of freedom sitting in an 16-dim. "quotient space" or "lattice" and 10-dim. spacetime. These 10+16=26 give us something like the original 26 dimensions of the bosonic string back. The degrees of freedom from these 16 dimensions are more comparable to gauge degrees of freedom than to spacetime degrees of freedom.

What I want to say is that the difference between spacetime dimensions and other "dimensions" becomes foggy in string theory. In M-theory the 11th dimension is not introduced by hand, but it emerges from a certain limt of 10-dim. superstring theory where the 1-dim. strings start to stretch into a new dimension and become effectively 2 dimensional. Studying the corresponding SUGRA theory everything looks like ordinary 11 dim. spacetime, but again this is only a certain limit.

What I want to say is that I am pretty sure that we haven't fully understood the concept of dimensionality at all. I think there is an underlying mathematical structure which "breaks" in some sense and leaves us with 4 "parameters" which look like dimensions. But I don't think that this bottom-up approach using the ordinary language of dimensions will be successful; I think that the whole concept of dimensions and manifolds is emergent and may apply only in certain limits.

A remark regarding other QG theories: In LQG one starts with a graph of spin networks which has no dimension at all. Embedding an arbitrary graph into a manifold such that the graph is dual to a triangulation of thast manifold could require arbitrary high dimensions! The way the theory has been constructed historically could cause the impression that it requires a three-dim. manifold, but this is not correct. It is exactly the other way round: it requires a graph from which a manifold emerges in some appropriate limit.

It is interesting to note that LQG and some other theories seem to propose that the low-energy limit looks like a 4-dim. manifold, whereas near the Planck-scale it looks like a 2-dim. manifold! Let's see how the dimension is determined: one looks at a diffusion process and compares a rather general mathematical object (which does not even need a dimension or a manifold for its definition) with some well-known results on manifolds. Then one calculates the "dimension" w/o defining a manifold and finds that the dimension runs continuously between 2 and 4 depending on the energy. So the expectation is that at low energies one should get 4-dim. spacetime back whereas at high energies = near the Planck scale everybody expects that there is no manifold at all!

Back to string theory: I do not say that these alternative approaches are better than string theory, but I think that there is one important lesson even for string theory: the theory is still waiting to be reformulated in a way that is more or less independent from the concept of an underlying manifold (with well-defined dimensionality) on which strings move. After this reformulation manifolds and dimensions will disappear completely and survive only as emergent concepts

Look at temperature: it can be defined only in special scenarios; one single particle does not have a temperature, a laser beam doen not have a temperature.

Look at water: the 2-dim. surface of a lake and waves on that surface are apropriate only in a certain. Looking through a magnifying glass the surface (and its dimension) disappears, instead one finds individual molecules w/o having a ceratin dimension. In addition there is no water between the molecules.

I am pretty sure that something like this will happen with spacetime, too.

Last edited: Oct 24, 2010
3. Oct 24, 2010

### Nikola Mitic

Thanks much for the explanation.

4. Oct 24, 2010

### arivero

An additional problem is that string theoretists have "decaffeinated" the whole concept because they need orbifolds and very exotic structures. Before this apotheosis, or perhaps ragnarok, of the string, it was very well understood that 11=7+4. This is because in 11 dimensions they are two natural bosonic objects, one with two indexes, the graviton, and another one with three indexes (the putative source of the membrane solutions). Due to the three indexes of this field, the compactifications of 11D space always divide in 7+4, the only doubt being which of the two terms is perceived. Mathematically, a space with D=7 was as likely as an space with D=4, from the point of view of compactified supergravity. Only by incorporating cosmology or thermodinamics, the D=4 solution was singled out.

5. Oct 24, 2010

### atyy

I had to look it up!

Ragnarök, n. The destruction or ‘twilight’ of the gods; spec. the last battle of this world, in which gods and men will be defeated by monsters and the sun will grow dark.

Well, isn't string theory supposed to predict our universe is unstable?

6. Oct 24, 2010

### arivero

Look also "apo*theo*sis"