What is quantum chaos?

In January, the Notices of the American Mathematical Society published a short article in their 'What is...' series called What is... Quantum Chaos. It was written by Ze'ev Rudnick, and if you look very closely, you'll find that I am mentioned in the article. (Alex Barnett, another quantum chaotician from my generation, is also mentioned. Lots of old people are mentioned too.)

The article does a very nice job of describing the gist of quantum chaos, so I wanted to post a link to it here (given that this field forms the namesake for my blog, and also given that I wrote my dissertation on this topic). Here is the wikipedia page for quantum chaos. (By the way, if you google 'wikipedia quantum chaos', my research synopsis web page comes up third. This was a happy byproduct of linking to the page in order to help prospective students and postdocs know what the subject is in case they want to work with me on it.) A 1992 article that Martin Gutzwiller wrote in Scientific American after encouragement by Predrag Cvitanovic provides an excellent introduction to the subject for intelligent people who aren't physicists or mathematicians.

To explain very briefly, quantum systems can't actually exhibit a rigorous form of sensitive dependence on initial conditions (the butterfly effect; small differences in initial conditions leading to exponentially large divergence of trajectories) the way that classical systems can. (For one thing, there is the issue of defining something that corresponds to a trajectory in quantum mechanics, though the headaches don't end there.) However, if I hold a classical chaotic system in one hand (for simplicity, say that it's fully chaotic rather than mixed) and a classically non-chaotic (regular, integrable) system in the other and I quantize them both, I can tell which one was which based on certain properties (like the distributions of the spectra, the scarring of classical periodic orbits, and so on) of the quantum systems even though I can't define chaos rigorously in those systems. In the case of mixed systems with well-separated regular and chaotic regions, one can see the signatures of the different regions. (For example, see the paper that my student, Tom Mainiero, and I published in Chaos in December 2007. Also see recent work by Alex Barnett on quantum mushroom billiards.) If the different types of regions are not well-separated, then it can get pretty hard and lots of subtleties ensue. (Actually, lots of subtleties ensue even before you start dealing with the quantization of systems with poorly-separated mixtures of regular and chaotic regions. It's just that there are even less tractable subtleties that arise when the regions are completely interspersed with each other.)

This entry was longer than I intended, but I hope it can give normal people some idea of what quantum chaos is.