If life grows over your head, your closest pop psy magazine recommends dividing it up into small, manageable chunks. Physicists too apply this method in difficult situations. Discrete approximations – taking a system apart into chunks – are enormously useful to understand emergent properties and to control misbehavior, such as divergences. Discretization is the basis of numerical simulations, but can also be used in an analytic approach, when the size of chunks is eventually taken towards zero.Understanding space and time in the early universe is such a difficult situation where gravity is misbehaved and quantum effects of gravity should become important, yet we don’t know how to deal with them. Discretizing the system and treating it similar to other quantum systems is the maybe most conservative approach one can think of, yet it is challenging. Normally, discretization is used for a system within space and time. Now it is space and time themselves that are being discretized. There is no underlying geometry as reference on which to discretize.
Causal Dynamical Triangulations (CDT), pioneered by Loll, Ambjørn and Jurkiewicz, realizes this most conservative approach towards quantum gravity. Geometry is decomposed into triangular chunks (or their higher-dimensional versions respectively) and all possible geometries are summed over in a path integral (after Wick-rotation) with the weight given by the discretized curvature. The curvature is encoded in the way the chunks are connected to each other. The term ‘causal’ refers to a selection principle for geometries that are being summed over. In the end, the continuum limit can be taken, so this approach in an by itself doesn’t mean that spacetime fundamentally is discrete, just that it can be approximated by a discretization procedure.
The path integral that plays the central role here is Feynman’s famous brain child in which a quantum system takes all possible paths, and observables are computed by suitably summing up all possible contributions. It is the mathematical formulation of the statement that the electron goes through both slits. In CDT it’s space-time that goes through all allowed chunk configurations.
Evaluating the path integral of the triangulations is computationally highly intensive, but simple universes can now be simulated numerically. The results that have been found during the last years are promising: The approach produces a smooth extended geometry that appears well-behaved. This doesn’t sound like much, but keep in mind that they didn’t start with anything resembling geometry! It’s discrete things glued together, but it reproduces a universe with a well-behaved geometry like the one we see around.
Or does it?
The path integral of CDT contains free parameters, and most recently the simulations found that the properties of the universe it describes depend on the value of the parameters. I find this very intriguing because it means that, if space-time's quantum properties are captured by CDT, then space-time has various different phases, much like water has different phases.
In the image below you see the phase-diagram of space-time in CDT.
|CDT phase diagram of space-time. |
After Fig 5 in 1302.2173
The parameter κ is proportional to the inverse of Newton’s constant, and the parameter Δ quantifies the (difference in the) abundance of two different types of chunks that space-time is built up of. The phase marked C in the upper left, with the Hubble image, is where one finds a geometry resembling our universe. In the phase marked A to the right space-time falls apart into causally disconnected pieces. In the phase marked B at the bottom, space-time clumps together into a highly connected graph with a small diameter that doesn’t resemble any geometry. The numerical simulations indicate that the transition between the phases C and A is first order, and between C and B it’s second order.
In summary, in phase A everything is disconnected. In phase B everything is connected. In phase C you can share images of your lunch with people you don’t know on facebook.
Now you might say, well, but the parameters are what they are and facebook is what it is. But in quantum theory, parameters tend to depend on the scale, that is the distance or energies by which a system is probed. Physicists say “constant’s run”, which just rephrases the somewhat embarrassing statement that a constant is not constant. Since our universe is not in thermal equilibrium and has cooled down from a state of high temperature, constants have been running, and our universe can thus have passed through various phases in parameter space.
Of course it might be that CDT in the end is not the right way to describe the quantum properties of gravity. But I find this a very interesting development, because such a geometric phase transition might have left observable traces and brings us one step closer to experimental evidence for quantum gravity.
You can find a very good brief summary of CDT here, and the details eg in this paper.
Images used in the background of the phase-diagram, are from here, here and here.