Originally posted by sonhouse
The Hausdorf dimension of 2D Quantum Gravity is 4 (this is a measure of how the volume of a disc changes with it's radius). However the spectral dimension is 2 (This is another way of probing dimension by looking at diffusion rates). There's a fairly obvious similarity with string theory - the world sheet of a string is 2 dimensional. But in a string theory the space-time dimensions, including any unseen ones, are described as fields on the surface of the string.
I'd need to read the reference, but they've got extra degrees of freedom appearing in the theory in a way that sounds like it's connected with Hausdorf dimension. I'd describe myself as skeptical on theoretical grounds for the following reasons:
The universe acquires extra dimensionality by making it's one space like and one time like dimension do more work. This means that two points separated by a small distance in 3 dimensional space could be separated by a very large distance along the underlying world sheet. In fact you could have two points which should be causally connected being disconnected on the world sheet. I think this could cause fine-tuning difficulties.
Although the underlieing theory is 2 dimensional the is only way of getting the Hausdorf dimension to 2 renders the theory inconsistent, There is no natural way of getting the Hausdorf dimension to come out at 3 (you can probably force it somehow, but then you have to a fine tune your theory to infinite precision).
As a matter of experimental observation it is believed that the speed of light is the same in all directions. In this theory there is considerable potential for the speed of light to be faster in one direction than another depending on how the 1+1 space-time is folded up.
They have to introduce an additional mechanism to make the spectral dimension come out the same as the Hausdorf dimension, and explain how we have such a flat geometry in 3+1 dimensions. 2D quantum gravity does not produce smooth universes. What's more it is only renormilisable if there is less than one type of particle (*). Worse still it doesn't generate a length scale properly (there is no mass gap), This means that it is not a consistent theory (**), certainly not a T.o.E. candidate - incidentally string theory is immune to this because of technical differences in the construction of the theories.
The evidence from cosmic rays isn't stunningly convincing as they haven't ruled out other possibilities. For example, you could form a Brane World inspired hypothesis that really it's evidence of TeV scale micro black holes, and the reason for the apparently 2 dimensional scattering has some connection with the axial symmetry of the micro-black hole undergoing quantum evaporation. You then have to show that an evaporating black hole would emit on the ecliptic plane of the hole. What I'm trying to illustrate is that the cosmic ray evidence doesn't prove much unless it rules out other candidates.
I'm not sure that reducing the number of dimensions of the universe in this way is really helpful. Since LHC probes TeV scale physics it should rule out this theory quickly. Interesting that there's a directional bias to scattering from very high energy cosmic rays.
(*) The technical requirement is that the total conformal charge of all matter fields coupled to the 1+1 (or 2+0) universe has to be less than 1 for the theory to be renormilizable.
(**) A mass gap is required for a Quantum field theory to be consistent. This is why the Clay institute is offering $1,000,000 for a proof. Quantum electro-dynamics is an example of a field theory that has no mass gap and the theory breaks down at ultra-high energies at something called the Landau pole, where the running fine-structure constant (and hence the electric charge) becomes infinite. This is no great problem for Q,E,D, as we expect it to merge with other forces at high energy in a way we don't yet understand, but long before the Landau pole is reached.