Number Theory Related to Quantum Chaos

University Of Georgia Research Foundation Inc, Athens GA

Investigators

Abstract

Par Kurlberg ABSTRACT: In this project, the Co-Principal Investigator will study some number theoretical questions related to mathematical physics. Quantum Chaos is concerned with how chaos in classical systems manifests itself in terms of quantum mechanics. For instance, can chaos, or lack thereof, be detected in the statistical behavior of the energy levels when a classical system is quantized? Pursuing this question for the ``boxed harmonic oscillator'', a simple system whose energy levels show interesting spacing behavior, led the co-P.I. to consider spacings between squares modulo highly composite integers. The first goal of this project is to generalize the results on spacings of squares to spacings between values of more general polynomials, both modulo highly composite integers, as well as modulo primes. In this context it is also interesting to study the distribution of the cardinality of the images of these polynomial maps. Another aspect of Quantum Chaos is the quantum mechanical analogue of classical ergodicity. The ``cat map'' is a simple model of a chaotic system, and its quantization has strong arithmetical features. Using these special features, it is possible to express Quantum Ergodicity for the cat map in terms of exponential sums. A second goal of this proposal is to prove quantum unique ergodicity for the quantized cat map using equidistribution results for exponential sums. The main objective of this project is to explore the connection between Quantum Chaos and Number Theory. Quantum Chaos tries to answer important questions in Physics regarding quantum mechanical analogues of classical chaos. The subject has applications to microelectronics, and is hence of interest to the computer and communications industry. Number Theory is concerned with properties of whole numbers and is one of the oldest branches of mathematics. For a long time it was studied mostly because of aesthetic reasons, but in the last few decades the field has been of fundamental importance to cryptography