Does chaos – namely extreme sensitivity to initial condition in a complex system – exist in quantum systems, such as single atoms? Extensive research by physicists yielded a disappointing answer: probably not. Nevertheless, along the way it was found that chaos (or the lack of it) is reflected in quantum systems in other ways. For instance, the energy levels of chaotic quantum systems were found to display typical statistical behavior, which seems to be described by a new field of mathematics called Random Matrix Theory.
Surprisingly enough, some of the mathematical questions that arose in connection to Quantum Chaos had a lot to do with Number Theory, an ancient discipline, which deals with the structure of integer arithmetic. Moreover, insights from Quantum Chaos have been employed in order to find fresh approaches to one of the most important unsolved problems in mathematics, the Riemann Hypothesis.
In the past few years, Prof. Zeev Rudnick from the School of Mathematical Sciences at Tel Aviv University, has been exploring different aspects of Quantum Chaos and Number Theory. He has contributed to one of the most surprising discoveries concerning the Riemann Hypothesis, namely, that the Riemann zeros appear to display the same statistics of those which are believed to be present in energy levels of quantum chaotic systems.
In 2012, Prof. Rudnick’s research project, Arithmetic and Quantum Chaos, was awarded an Advanced Grant by the European Research Council (ERC). As part of the project, Rudnick and his associates will continue on in examining new approaches to interconnections between Number Theory and Quantum Chaos.
Further reading: Zeev Rudnick, “What is Quantum Chaos?“, Notices of the AMS.