Member of the Theoretical Physics Group

 

 

Research

My research concerns supermanifolds.  A supermanifold is a space with some commuting and some anticommuting coordinates.

An account of the theory of supermanifolds, together with applications of these spaces, is given in the book 

Supermanifolds: theory and applications

 

During the last few years I have developed a theory of path integration on supermanifolds, building on the standard approach to path integration on Riemann manifolds. The motivation for this work is to bring mathematical rigour in a direct way into the heuristically powerful physicists' path integral approach. These methods give information about the evolution operator in supersymmetric quantum mechanics, and have many geometric applications, for instance to the Atiyah-Singer index theorem.

Recently I have been working on the canonical quantization of theories with symmetry, particularly considering the role of superspace path integrals for the BRST Hamiltonian. Superspace is involved because the extra 'ghost' degrees of freedom are represented using anticommuting variables. Some papers on this work are
 

"Gauge fixing and BFV quantization". Classical and Quantum Gravity, 17:389-397, 2000

  which uses supertrace arguments to show how the BRST commutator term in the Hamiltonian regularises the path integral, and

          "The topological particle and Morse theory". Classical and Quantum Gravity, 17:3703-3714, 2000

  in which it is shown that the supersymmetric model used by Witten in his celebrated paper on Morse theory can be obtained by BRST quantisation of the topological particle, and the tunelling calculations for the model are carried out rigorously using a new stochastic calculus technique, which incidentally show why the WKB method for quantum tunelling has improved accuracy in supersymmetric theories.

 

A paper concerning canonical quantization of topological quantum theories which shows how different choices of gauge fixing can lead to different quantum theories:

                        ‘Gauge fixing and equivariant cohomology’ Classical and Quantum Gravity, 22 :4083-4094 (2005)

A survey article on my work on path integration in superspace:

                        Supersymmetry and Brownian motion on supermanifolds

A joint paper with Rémi Léandre constructing equivariant cohomology on loop groups:

                        Equivariant Cohomology, Fock Space and Loop Groups

A paper which extends the standard BRST quantization of Hamiltonian systems with symmetry to systems with reducible symmetry:

                               Equivariant BRST quantization and reducible symmetries      J. Phys. A: Math. Theor. 40 (2007) 4649-4663

 

                               BRST quantization in the canonical setting

lecture given at XXVII Workshop on Geometric Methods in Physics, Bialowieza, Poland,  June 29 - July 5 2007

Copyright (2007) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.  The article appeared in AIP Conference Proceedings vol 956,

eds: Piotr Kielanowski, Anatol Odzijewicz, Martin Schlichenmaier and Theodore Voronov, pages 15-26

 

and may be found at http://link.aip.org/link/?APCPCS/956/15/1

 

   I am currently a member of ACME  

 

  Some uses of mathematics