Friday, September 9, 2016

What's missing in Quantum Gravity I: Locality

I'm planning to put up some pretty nice renders of characters imported from DAZ studio into Blender, but before doing that I will write about something that I have thought about for a long time and think that I have some rather unique insights about (and something that motivates the name of this blog): how to construct a quantum theory of gravity. For a more exhaustive discussion of the topic in this and the next post, see arXiv:1407.6378 [gr-qc].

In the beginning of the previous century, two great theories were found that describe all of physics: Einstein's General Relativity (GR), which describes gravity, and Quantum Theory (including Quantum Mechanics and Quantum Field Theory (QFT)), which describes everything else. The problem is that these theories seem to be mutually incompatible. But we know that Nature exists and hence cannot be inconsistent, so a consistent theory of Quantum Gravity (QG) has to exist, and it must reduce to GR and QFT in appropriate limits.

For almost a century many great minds have tried to solve this conundrum. The most popular approaches in recent decades have been string theory and, to a lesser extent, loop quantum gravity. However, these and other approaches all have problems of their own, especially since the LHC recently has ruled out supersymmetry beyond reasonable doubt (which, incidentally, is not the same thing as beyond every doubt).

My own proposal is to go back to basics, and simple postulate that QG combines the main properties of GR and QFT: gravity, quantum theory, and locality.

Now, this may seem as a very natural and innocent assumption, but it is in fact very controversial. The reason is that there is a well-known theorem stating that there are no local observables at all in QG, unless classical and quantum gravity have different sets of gauge symmetries. Actually, if you have seen this statement before, it was certainly without the caveat, but it must be mentioned because it is the weak spot of the no-go theorem. In order to prove this, one assumes that the group of all spacetime diffeomorphisms, which is the gauge symmetry of classical GR, remains a gauge symmetry after quantization.

So, we need something that converts spacetime diffeomorphisms from a classical gauge symmetry to an ordinary quantum symmetry, which acts on the Hilbert space rather than reducing it. This something is called an extension. It is well known that the diffeomorphism algebra (the infinitesimal version of the diffeomorphism group) on the circle admits a central extension called the Virasoro algebra. This is a celebrated part of modern theoretical physics. It first appeared in string theory, but later found an experimentally successful application in condensed matter, in the theory of two-dimensional phase transition.

So the diffeomorphism algebra in one dimension has an extension, but we know that spacetime has four dimensions (at least). So we need a multi-dimensional Virasoro algebra, which is a nontrivial extension of the diffeomorphism algebra in higher (and in particular four) dimensions. There are several arguments why such an extension cannot exist, and I will address those in a later post, but exists it does. In fact, there are even two of them, that were discovered 25 years ago by Rao and Moody and myself, respectively.