University of Lethbridge

Biography

Mark is a professor in the Department of Physics and Astronomy. He has been a faculty member since January 1991. Prof. Walton is a theoretical and mathematical physicist. His B.Sc. (Honours) is from Dalhousie University in Halifax, and his M.Sc and Ph.D. (1987) were both completed at McGill University in Montreal. Mark's M.Sc. research was in elementary particle physics (or high energy physics) and his Ph.D. research was in string theory. During an NSERC Postdoctoral Fellowship at the Stanford Linear Accelerator Center he switched research fields to conformal field theory and related topics (a fairly non-technical introduction to conformal field theory is: J. Cardy, Physics World, June, 1993, pg 29). His 2nd postdoctoral position, at Université Laval, was shortened when he was offered the faculty position here at the U of L. In recent years, Prof. Walton has been studying quantum mechanics, and in particular, its phase-space formulation. Mark is interested in how classical mechanics emerges from quantum theory, and in the foundations of quantum mechanics. Throughout his research career, Prof. Walton has applied the math of Lie groups and algebras to various physical systems.

Selected Publications

B. Belchev*, S.G. Neale*, M.A. Walton, Flow of S-matrix Poles for Elementary Potentials, accepted 8/11 by the Canadian Journal of Physics

B. Belchev*, M.A. Walton, Solving for the Wigner Functions of the Morse Potential, J. Phys. A: Math.Theor. 43 (2010) 225206

B. Belchev*, M.A. Walton, On Robin Boundary Conditions and the Morse Potential in Quantum Mechanics, J. Phys. A: Math.Theor. 43 (2010) 085301

B. Belchev*, M.A. Walton, On Wigner Functions and a Damped Star Product in Dissipative Phase-space Quantum Mechanics, Ann. Phys. 324 (2009) 670

N. Okeke*, M.A. Walton, On Character Generators for Simple Lie Algebras, J. Phys. A: Math.Theor. 40 (2007) 8873

*students

Research Interests

Conformal Field Theory & Phase-Space Quantum Mechanics

Fields describe an enormous range of phenomena in physics. The predictions of a field theory are often difficult to work out, however, unless a system can be described as a small perturbation of another well-understood one. Some field theories do exist, however, that can be solved non-perturbatively. Conformal (quantum) field theories (CFTs) are examples. I study them because I believe they will teach us something to help us solve other physical theories. Happily, CFTs also find many uses, in statistical physics and condensed matter physics, for examples. My research focuses on improving the understanding of their mathematical properties, so that their predictions can be worked out, and new applications can be found.

Quantum field theory (QFT) is consistent with all experiments performed to date. It treats the elementary particles as structureless points. In spite of enormous effort, however, a QFT of gravity has not been constructed. String theory posits that the elementary particles are not point-like, but tiny strings, so small that their structure has not yet been detected. The payoff is that string theory can describe a unified theory of forces, including quantum gravity.

CFT also applies to string theory. Strings sweep out a two-dimensional world sheet in spacetime, and CFT describes the physics on the string world sheet. Part of my research focuses on the application of CFT to the study of string theory. I am also interested in the mathematics of CFT, for example, the infinite-dimensional affine Kac-Moody algebras and the semi-simple Lie algebras they are based on that are important in CFT.

A second important line of research deals with a different way of doing quantum mechanics, known as phase-space quantum mechanics (or deformation quantization, or Wigner-Weyl quantization, among other names). In it, observables are not represented by operators, but rather as ordinary functions on phase space. That way, connections with classical mechanics are often most transparent. For several years now, I have studied the phase-space quantization of simple systems that have unusual or difficult quantum aspects. They include the damped harmonic oscillator, boundaries in phase space, and point interactions. The goal is to deepen our understanding of these systems by looking at them from this new, different point of view.

Current Research and Creative Activity

Title	Location	Grant Information	Principal Investigator	Co Researchers
Conformal field theory, star quantization, and matrix models, for physical applications		Natural Sciences and Engineering Research Council (NSERC), $40,000/year for 5 years, 2006-11.	M.A. Walton, University of Lethbridge

Previous Research

Title	Grant Agency	Completion Date
Conformal field theory with applications to string theory	Natural Sciences and Engineering Research Council (NSERC)	2006
Conformal field theory	Natural Sciences and Engineering Research Council (NSERC)	2002
Conformal field theory and related systems	Natural Sciences and Engineering Research Council (NSERC)	1999