Quantum kinematics and projective unitary representations of abelian groups

Abstract

The kinematics of quantum systems, as is well known, are generally governed by the Heisenberg commutation rules. It was Hermann Weyl who suggested the consideration of the global versions of these rules which define a projective unitary representation of the additive group of the classical phase space. This prompted Weyl to formulate his principle that quantum kinematics is always governed by a projective unitary representation of some abelian group. The ideas of Weyl, and their natural development by his successors, notably Stone, Von Neumann, Mackey, Segal, Shale, Weil, and many others have had and still continue to have a great impact on many areas of mathematics and physics. This paper is an exposition of some of the themes that flow from these ideas.