Quantum Gravity School - March 2007

[francesca]

My notes are amazing... when I write in Italian!
while my comprehension of spoken English have to be improved
I hope that slides (if present) would be detailed enough,
maybe for comprehensive notes we have to wait for the school proceedings!

 

[Timbuqtu]

I will be going to Zakopane too! They have done a great job getting this group of lecturers together. Apart from the people addressed by others, I am pleased to see Jean-Marc Schlencker is going to tell something about 2+1 quantum gravity and related math, since I'm working in that area at the moment.

 

[francesca]

Hi Timothy,
plese tell something more about it!
I look at Schlencker's lecture topics, but it doesn't say so much to me...

​

 

[Timbuqtu]

Hi Francesca. I'll try to shortly summarize some (quantum) gravity in 2+1 dimensions:

The phase space of (classical) general relativity in 2+1 dimensions is much smaller than in 3+1 dimensions, because there are no local degrees of freedom (in absence of matter). The reason is that the Riemann tensor is fully determined by the Ricci tensor and the Einstein equations imply that the Ricci tensor vanishes in vacuum when cosmological constant \Lambda=0. Consequently spacetime is flat everywhere and looks like Minkowski space (or in case \Lambda \neq 0 it looks like deSitter or Anti-deSitter). All degrees of freedom reside in the ways of glueing patches of Minkowski space together to form your space time. So actually only topologically non-trivial spacetimes contain (physical) degrees of freedom.

Let's assume our spacetime to have the topology of [0,1]\times \Sigma where \Sigma is a closed two-dimensional surface. Now topologically such a surface is determined by only its genus g (number of tori attached to each other). Now the geometry of your spacetime is fully determined by the holonomies around a set of (2g) non-contractible loops in your spacetime. Now it turns out that this phase space (the set of holonomies) can be identified with the cotangent bundle of Teichmuller space of your surface \Sigma. Teichmuller space is the space of all inequivalent Riemann (i.e. complex one-dimensional) surfaces (of a specific genus), which is a widely studied topic in mathematics. Also the physical symplectic structure on your phase space can be canonically defined on your cotangent bundle. That's why a lot of math related to Riemann surface/hyperbolic geometry is studied in 2+1 GR.

Because our phase space is finite-dimensional in 2+1 dimensions (contrary to the 3+1 dimensional case) and the symplectic structure is pretty simple, we can (in principle) easily quantize such a system using normal quantum mechanics. This approach is called reduced phase space quantization because all constraints coming from equations of motion have already been applied and all gauge freedom has been removed before quantization. Of course there are several issues to solve, but despite that you can explicitly write down a quantum theory of your system (which in this case describes a spatially compact spacetime without matter or whatever).

I hope this clarifies the subject a little. Here is a good (introductive) review article on 2+1 dimensional quantum gravity if you want to read more: "Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe", S. Carlip, http://arxiv.org/abs/gr-qc/0409039.

-- Timothy

 

[francesca]

Definitely a clear introduction, thank you!