Groups, Semigroups, and Generators

Physical theories consist essentially of two elements, a kinematical structure describing the instantaneous states and observables of the system, and a dynamical rule describing the change of these states and observables with time. In the classical mechanics of point particles a state is represented by a point in a differentiable manifold and the observables by functions over the manifold. In the quantum mechanics of systems with a finite number of degrees of freedom the states are given by rays in a Hilbert space and the observables by operators acting on the space. For particle systems with an infinite number of degrees of freedom we intend to identify the states with states over appropriate algebras of fields, or operators. In each of these examples the dynamical description of the system is given by a flow, a one-parameter group of automorphisms of the underlying kinematical structure, which represents the motion of the system with time. In classical mechanics one has a group of diffeomorphisms, in quantum mechanics a group of unitary operators on the Hilbert space, and for systems with an infinite number of degrees of freedom a group of automorphisms of the algebra of observables. It is also conventional to describe other symmetries of physical systems by groups of automorphisms of the basic kinematic structure and in this chapter, and Chapter 4, we study various aspects of this group-theoretic description.