Email: p.forrester@ms.unimelb.edu.au
Phone: +61 (0)3 9344 9386
Department of Mathematics and Statistics, The University of Melbourne, Parkville, Vic 3052, Australia.
Quantum systems which are classically chaotic display universal features in the statistical properties of their spectrum and wavefunction which are correctly described by random matrices and random polynomials respectively. In fact the same universal features have been observed in systems as seemingly diverse as the sound spectrum of vibrating pieces of metal to the large zeros of the Riemann zeta function on the critical line. I aim to calculate the analytic form of the universal features from the mathematical desciption in terms of random matrices and random polynomials, a task which requires exploiting the expansive structures inherent therein. The agreement between the analytic form and empirical data can be quite spectacular: in collaborative work with A.M.~Odlyzko I compared a newly determined statistic expressed in terms of a Painlev\'e transcendent with empirical data for the large Riemann zeta function zeros and found 4 decimal agreement.
Integrable quantum many body problems.
In general the two-body problem is integrable while the three-body problem is not. However there are a small number of very special systems for which the full $N$-body problem is integrable. One such example is the quantum many body problem on a line or circle with $1/r^2$ pairwise interactions. This problem has a deep algebraic structure in terms of double affine Hecke algebras. Furthermore, it is possible to develop the properties of the eigenfunctions to the extent that ground state correlation functions can be computed. I was the first person to calculate such correlations, which as well as having the mathematical interest of being intimately related to higher dimensional generalized hypergeometric functions, explicitly exhibit the fractional statistical charge carried by the elementary excitations. This latter notion is the cornerstone of Laughlin's theory of the fractional quantum Hall effect, which earned him the 1998 Nobel prize for physics.
Statistical mechanics of log-potential Coulomb systems.
The logarithmic potential is intimately related to topological charge -- for example vortices in a fluid carry a topological charge determined by the circulation, and the energy between two vortices is proportional to the logarithm of the separation. The logarithmic potential is also the potential between two-dimensional electric charges, so properties of the two-dimensional Coulomb gas can be directly related to properties of systems with topological charges. In a celebrated analysis, Kosterlitz and Thouless identified a pairing phase transition in the two-dimensional Coulomb gas. They immediately realized that this mechanism, with the vortices playing the role of the charges, was responsible for the superfluid--normal fluid transition in liquid Helium films. In my studies of the two-dimensional Coulomb gas I have exploited the fact that at a special value of the coupling the system is equivalent to the Dirac field and so is exactly solvable. This has provided an analytic laboratory on which to test approximate physical theories, and has also led to the discovery of new universal features of Coulomb systems in their conductive phase.
My most recent (as of February 2000) preprints are 'Analytic properties of the structure function for the one-dimensional one-component log-gas' (with B. Jancovici and D.S. McAnally; Jan. '00 cond-mat archive) and `Exact Wigner surmise type evaluation of the spacing distribution in the bulk of the scaled random matrix ensembles' (with N.S. Witte; Feb. '00).
Where: Richard Berry Building, Corner of Swanston St and Monash Rd (extension of Faraday St), Gate 4, Melbourne University)
When: Thurs. 3rd, Friday 4th of February 2000, 2-4 p.m. (4 hrs. total)
Cost: $25, Concession $15
Booking: Ask for a registration form to be sent out.