Chapter 19 Dynamical Deformation of Algebras of Observables

Abstract

Publisher Summary This chapter describes the dynamical deformation of the algebras of observables. The dynamical structure of Hamiltonian systems is given by one-parameter groups of automorphisms of the underlying kinematical structure, which represent the motion of the systems with time. In classical Hamiltonian mechanics, a group of diffeomorphisms is present on a smooth manifold. In quantum dynamics of Hamiltonian systems, a group of unitary operators is used on a Hilbert space. The time development can be given by a strongly continuous one-parameter group of automorphisms of a C ∗ -algebra. It is conventional to describe the symmetries of systems by a group of automorphism of the basic kinematical structures. In non-Hamiltonian quantum mechanics, automorphisms and endomorphisms cannot be used to describe dynamics. The definitions and certain theorems of a deformation theory for operator algebras are discussed in the chapter. The second cohomology group H 2 (M,M) of an algebra M is interpreted as the group of infinitesimal deformations of M in the same way that the first cohomology group H 1 (M,M) is interpreted as the group of infinitesimal automorphisms.