Systems with Many Particles

Abstract

We have learned to describe systems of finitely many particles with an algebra A of observables, and information about the systems with a state w on the algebra (cf. (III: 2.2.32)). As our main goal is the study of everyday matter, our framework will be that of nonrelativistic quantum theory. For the purposes of contrast, or of aiding intuition, we shall also have occasion to call upon classical mechanics, where states are measures on phase space, and extremal states are point measures. In either framework time-evolution can be represented as an automorphism a → a _t ,for a ∈ A in the Heisenberg picture. If desired, time-dependence can alternatively, in the Schrödinger picture, be put upon the state: w → w _t such that w _t (a) = w (a _t ). If the algebra is Abelian (classical mechanics), then the point of an extremal state moves along a classical trajectory in phase-space.

Macroscopic bodies act in an irreversible and deterministic manner in contrast with the reversible and indeterministic character of the underlying laws of quantum physics. How can the apparent contradiction be understood?