Tuesday, January 27, 2009

Hořava, Lifshitz, Cotton, and UV general relativity

Let me start with some fun:

Click the picture of April Motl, my very distant relative who is "getting to the heart of the matter", too.
;-) Amusingly enough, in 1998, I was using pen name April Lumo for a while.

Petr Hořava wrote an interesting preprint:
Quantum gravity at a Lifshitz point
(see also: November 2007 talk in Santa Barbara)
He wants to find a "smaller" theory of quantum gravity than string theory, so he looks at the hypothetical UV fixed point (a theory without a preferred scale) that could flow to Einstein's equations at long distances. Fixed points are an intellectual value that the CMT and HEP cultures share.
See also NYU about Hořava-Lifshitz gravity for more comments about the paper and the sociology surrounding it...
This research program has been unsuccessfully tried many times in the past. The new twist is that his proposed fixed point is non-relativistic. Normal scale-invariant relativistic theories have a scaling symmetry that affects space and time equally. Dispersion relations tell us that "E=p" and we say that the exponent "z=1". Ordinary non-relativistic mechanics scales them differently and "E=p^2/2m", giving "z=2". His starting point is even more non-relativistic, with "z=3". But he wants to get to "z=1" at long distances.

That means that his short distance action for the metric is
S = two_time_derivatives (metric) + six_space_derivatives (metric)
Note that "z=6/2=3". This theory is claimed to be "renormalizable", non-relativistic theory and governs short distance phenomena. He perturbs it by a deformation, especially by the spatial part of the Einstein-Hilbert action, and claims that it flows to ordinary general relativity at long distances as the full Ricci scalar gets completed, giving you general relativity from a seemingly well-behaved (but non-relativistic) short-distance starting point.

Note that the action (with the "dt dx dy dz" measure) is dimensionless (and classically scale-invariant) if you assign your time with the dimension of "1/E", like always, spatial coordinates with the dimension "1/E^{1/3}" (because "z=3"), and if the metric tensor is dimensionless. In some sense, all the three spatial dimensions (or their corresponding derivatives) contribute in the same way as one spatial coordinate in two-dimensional CFTs: in this sense, the scaling of his theory is analogous to two-dimensional CFTs.

If you care, the "two_time_derivative" term includes the extrinsic curvature tensor "K_{ij}" at a spatial slice, constructed out of the metric, while the "six_spatial_derivative" term includes the squared derivatives of the (spatial part of the) Riemann curvature, in specific tensor structure combinations linked to the Cotton tensor (essentially the curl of the traceless part of the Ricci tensor).

Of course, because of the Lorentz violation in the UV, the theory is incompatible with string theory. I have several additional problems with the theory, especially the following four:
  • the 4D diffeomorphisms are broken (down to the foliation preserving ones) so that you can't really get rid of all the unphysical degrees of the freedom in the metric (ghostly negative-normed gravitational waves); see also Charmousis et al. and the discussion about it on TRF
  • the fundamental Lorentz violation of the theory leads to apparently serious inconsistencies, including perpetuum mobile machines constructed out of black holes
  • the Lorentz violation in the infrared requires a lot of fine-tuning, especially once the matter fields are added; on the other hand, even the very existence of the flow from "z=3" to "z=1" seems to be speculative in character
  • I don't quite see that the UV theory is scale-invariant and consistent even at the interacting, quantum level: the Lifshitz toy model from condensed-matter physics could be too simple an example to learn from because it's just a free scalar with an adjustable "z" and therefore offers no loop corrections
Concerning the last point, I have no experience with the power laws and physical impact of "beta functions" in non-relativistic theories (can they cause dimensional transmutation for "z=3"?), so it is plausible that Petr knows an optimistic answer to this one but I am afraid he would agree with the previous two problems.

Well, I also don't know how to reconcile the theory with the existence of other forces and matter, but one shouldn't repeat these standard arguments against such a "small" theory. It's an interesting paper, at least as a provocation.

Google Talk: Edward Farhi defends the LHC (instead of quantum computers). Via Dmitry.

Three generations are natural in M- and F-theory

The following paper, by Jacob Bourjaily, is probably more realistic. He describes the model building in F-theory (Vafa et al.) and M-theory in a way that links them. For example, the interactions due to some triple intersections in F-theory are related to some three-cycles in M-theory. The E8 ALE singularity that enters the F-theory model building (on fibrations of the partly resolved E8 singularity) is exactly large enough to accommodate some of the most realistic stringy vacua known to date.

Moreover, in the F-theory picture, he argues that three generations are natural. By decomposing the Killing-Cartan-Surfer-Dude group :-), E8, into the E6 x U(1) x U(1) subgroup, we get triples of copies of "27" representations of E6. Well, that is - using my words - because U(1) x U(1) can be thought of as the Cartan algebra of the E6 centralizer SU(3) inside E8 which has the "Z3 symmetry" inside SU(3). This fact gets reflected to the multiplicity of "27" that one gets for some sensible resolutions.

This decomposition of "248" of E8 as
  • 248 = 78 x 1 + 1 x 8 + 27 x 3 + 27' x 3'
has always been suggestive of three generations in "27". However, one could never make this argument directly because the SU(3) was normally broken and there was no reason to get the whole triples. If the E8 ALE singularity is a fiber, the situation is revived a bit because the U(1) x U(1) Cartan subalgebra becomes "more real" and "more unbroken" than the SU(3) was previously.

P.S.: I haven't read this paper or his previous papers in detail, I am of course aware that they're getting no positive feedback from the experts in these compactifications, but at the same moment, with the level of reading I've dedicated to these papers, I am not able to see their lethal flaws so I consider it fair that it is being mentioned here, especially because many much more obviously wrong papers are being promoted on the Internet.

1 comment:

  1. Haha ! I send you for fun an epic drawing about the Theory of Relativity of Einstein i did;p Hope you will enjoy it !