"Deformation theory: A powerful tool in physics modeling"

Abstract

Deformation theory has proved to be a powerful tool -- so far, a posteriori -- in modeling physical reality. We start with a short historical and philosophical review of the context and concentrate this presentation on three interrelated directions where deformation theory is essential in bringing a new framework -- which has then to be developed using adapted tools, some of which come from the deformation aspect.

Minkowskian space-time can be deformed into Anti de Sitter, where massless particles become composite (also dynamically): this opens new perspectives in particle physics, at least at the electroweak level, including prediction of new mesons. Nonlinear group representations and covariant field equations, coming from interactions, can be viewed as some deformation of their linear (free) part: recognizing this fact can provide a good framework for treating problems in this area. Last but not least, (algebras associated with) classical mechanics (and field theory) on a Poisson phase space can be deformed to (algebras associated with) quantum mechanics (and quantum field theory). That is now a frontier domain in mathematics and theoretical physics called deformation quantization, with multiple ramifications, avatars and connections. These include representation theory, quantum groups (when considering Hopf algebras instead of associative or Lie algebras), noncommutative geometry and manifolds, algebraic geometry (even algebraic curves `a la Zagier), number theory, and of course what is regrouped under the name of M-theory.

In these lectures we shall look at these from the unifying point of view of deformation theory and refer to a limited number of papers as a starting point for further study.