A new concept of deformation quantization, I. Normal order quantization on cotangent bundles

TL;DR

This paper introduces a deformation theoretical framework for quantization, specifically constructing a normal order quantization on cotangent bundles that extends to a broad class of symbols and allows real values of the deformation parameter.

Contribution

It develops a deformation approach to quantization using algebraic geometry tools, extending normal order quantization to symbols on cotangent bundles with functorial properties.

Findings

Constructed a normal order quantization for polynomial observables on cotangent bundles.

Extended quantization to a Poisson space of symbols using symbol calculus.

Demonstrated that the deformation parameter can be any real number.

Abstract

In this work we give a deformation theoretical approach to the problem of quantization. First the notion of a deformation of a noncommutative ringed space over a commutative locally ringed space is introduced within a language coming from Algebraic Geometry and Complex Analysis. Then we define what a Dirac quantization of a commutative ringed space with a Poisson structure, the space of classical observables, is. Afterwards the normal order quantization of the Poisson space of classical polynomial observables on a cotangent bundle is constructed. By using a complete symbol calculus on manifolds we succeed in extending the normal order quantization of polynomial observables to a quantization of a Poisson space of symbols on a cotangent bundle. Furthermore we consider functorial properties of these quantizations. Altogether it is shown that a deformation theoretical approach to quantization is possible not only in a formal sense but also such that the deformation parameter $\hbar$ can attain any real value.