Observables, maximal symmetric operators, POV measures and their applications in quantum mechanics

Abstract

Approximate observables, their description in terms of POV measures and the subsequent extension of the orthodox notion of observable are examined. The sense in which the approximate observables are non-ideal observables is considered and a generalised concept of ideal observable is proposed. The conventional requirement of self-adjointness is relaxed to one of maximal symmetry and the PV measures are replaced by the larger set of generalised spectral measures of maximal symmetric operators. The implications for the quantisation problem are discussed and we give some new examples of observables which have no orthodox representation, i.e. observables which can be identified with non-self-adjoint maximal symmetric operators. Particular attention is paid to those aspects of the orthodox theory which rely on specific properties of self-adjoint operators, such as their association with the infinitesimal generators of groups of unitary operators, and we look at the analogous properties in our generalised theory.