Semigroups and the density matrix formulation of quantum mechanics

✍ Scribed by H. S. Green

• Publisher: John Wiley and Sons
• Year: 1980
• Tongue: English
• Weight: 716 KB
• Volume: 17
• Category: Article
• ISSN: 0020-7608
• DOI: 10.1002/qua.560170112

✦ Synopsis

Abstract

A formulation of quantum mechanics is presented based on the theory of semigroups and the associated enveloping algebras of functions defined on countable subsemigroups. The existence of a unique *‐involution is not assumed. The fundamental elements of a semigroup are identified with experimental precedures for the separation of subensembles from a given ensemble of experimental systems. Observables are represented as elements of enveloping algebras, and ensembles as density matrices within an enveloping algebra. The statistical properties of ensembles are expressed in terms of traces defined on the semigroup and its enveloping algebras. The elements and generators of the Poincaré group can be defined and interpreted in the usual way. A variety of applications is described, in which the theory of the density matrix plays an essential or effective role. Advantages associated with the resulting freedom from the limitations of Hilbert space are illustrated.