Representation theory for classical mechanics

Abstract

A representation theory for classical mechanics is proposed. It deals with the unitary group induced in the “classical” Hilbert space of Koopman and Von Neumann by the canonical realizations of a Lie group g of transformations. The concept of “classical” Lie algebra of g is defined by supplementing the usual commutation relations of its ordinary Lie algebra, with a set of commutation relations playing the role of supplementary conditions. Such conditions imply that the same infinitesimal generators of the canonical realizations of g are defined as additional operators in the “classical” Hilbert space and make possible the introduction of classical observables into the theory. The irreducible “classical” representations, i.e., the irreducible representations in the usual sense of the “classical” algebra, are classified. A selection of such “classical” representations leads to the characterization of all the irreducible representations which actually are the unitary counterpart of the canonical realizations of the group g . They are called the “canonical” representations. The theory is applied to work out the “canonical” representations of the rotation group in three dimensions and of the Euclidean group of the plane.