Spatial and phase space distributions of observable values in quantum mechanics and local quantization

Abstract

A fresh approach to phase space distributions in quantum mechanics is presented. Traditional theories in terms of quasi-probability distribution functions such as Wigner’s distribution function are shown to be incapable of producing a general and rigorous theory. The new approach introduces the idea of distributions of observable values rather than probabilities on the spatial space and generalized phase spaces. Consequently there is no need to resort to the notion of negative probability which plagues conventional theories. The new formulation is applicable to all quantum observables. The usual quantization rules associated with orthodox distribution functions such as Weyl’s rule are shown to be formal procedures which do not generally lead to self-adjoint operators. A local quantization scheme leading to well-defined local observables is proposed. This serves as further evidence in favour of the introduction of local observables in quantum mechanics.