Nov 9, 2005

Quantum Gravity: CDT

Causal Dynamical Triangulations (CDT) is a very recent approach to quantum gravity. Like LQG it didn't use any new principles or symmetries, but tries to quantize gravity using the Feynman path-integral approach to quantum mechanics.

The Feynman path-integral in quantum mechanics is a formalism that allows us to calculate the probability of scattering processes in quantum field theory (QFT) in a simple way that can be associated with graphs to easier visualization and calculation. The idea is that there is a quantity named the propagator that can be calculated as the weighted sum of the possible ways that a particle has to going from a point to the other of spacetime. In classical physics, there are some paths that are forbidden, the ones where the particle needs to travel faster than light, but the idea of Feynman was that in quantum mechanics all paths are allowed, but the resulting path is calculated by interference of all the paths. All the scattering probabilities can be calculated using the propagator and this approach allowed the quantization of electromagnetic theory (QED). But gravity is a very trick force and resisted to first attempts to be quantized in this way.

In gravity, the analogous quantity to a path in spacetime is a path in the space of all possible geometries of the universe. But this summation is divergent, i.e., the result is infinite and a lot of work has been done to try to find a way to make this sum convergent. The CDT approach consists of approximating spacetime by a mesh of triangles (in this case, 4-dimensional triangles), make the summation and then calculate it in the limit where the size of the triangles goes to zero. In this continuous limit, it's expected that the resulting theory is well behaved. The first results show that it could be.

CDT was developed by Renate Loll and Jan Ambjorn and has the advantage that a lot of simulations can be done and some interesting results appeared. One of the most interesting results is the possibility of spacetime to have different dimensionalities in different scales. This appears to be a little strange, but technically what happens is that if you put a particle moving at random (technically, executing a random walk) in space, its behavior is similar to a particle walking in a 2-dimensional space for short times (what is interpreted as for short scales) and similar to a particle walking in a 4-dimensional space for longer times. Mathematically what happens is that the particle is performing a diffusion in a fractal space and we have a formula for this that gives the so-called spectral dimension for the diffusion. The spectral dimension can be calculated by fitting a curve to the graphic of how much the particle walked versus the time spent and inserting the result in the formula. That is what they did and found these results.

As I said, CDT is new and there are not so many people working on it in the world, but if you are interested, search for it and for papers published by Loll and Ambjorn in the arXiv. They always put a preprint of their work there. And take a look at the discussions in the "Strings, Branes and LGQ" section of Physics Forums. You can always learn a lot of things there.

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