[Paper Review] A new concept of deformation quantization, I. Normal order quantization on cotangent bundles

Markus J. Pflaum|arXiv (Cornell University)|Apr 23, 1996

Medical Imaging Techniques and Applications3 citations

TL;DR

This paper introduces a deformation quantization framework for classical observables on cotangent bundles using normal ordering, extending formal power series quantization to real values of Planck's constant ℏ. By employing a complete symbol calculus on manifolds, it constructs a rigorous, non-formal quantization procedure that preserves functoriality and applies to polynomial and symbol spaces, offering a new deformation-theoretic approach beyond formal expansions.

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.

Motivation & Objective

• To develop a deformation-theoretic framework for quantization that is not restricted to formal power series in ℏ.
• To define Dirac quantization of Poisson spaces of classical observables on cotangent bundles.
• To extend normal order quantization of polynomial observables to a full symbol calculus on manifolds.
• To ensure the quantization procedure is functorial under smooth maps between manifolds.
• To demonstrate that deformation quantization can be meaningful for all real values of ℏ, not just formal ones.

Proposed method

• Introduce deformation of noncommutative ringed spaces over commutative locally ringed spaces using algebraic geometry and complex analysis tools.
• Define Dirac quantization as a deformation of a commutative ringed space equipped with a Poisson structure.
• Construct normal order quantization for polynomial observables on cotangent bundles via a symbol calculus on manifolds.
• Use a complete symbol calculus to extend the quantization from polynomials to the full space of symbols on the cotangent bundle.
• Ensure the quantization respects smooth maps between manifolds, establishing functoriality.
• Treat ℏ as a real parameter, avoiding reliance on formal power series expansions.

Experimental results

Research questions

• RQ1 Can deformation quantization be formulated in a non-formal way, allowing physical values of ℏ?
• RQ2 How can normal ordering be systematically extended from polynomial observables to general symbols on cotangent bundles?
• RQ3 What is the role of symbol calculus in constructing a consistent quantization map on manifolds?
• RQ4 How does functoriality emerge in this deformation-theoretic quantization framework?
• RQ5 Can a Poisson space of classical observables be consistently quantized using normal ordering beyond formal power series?

Key findings

• The paper constructs a normal order quantization for polynomial observables on cotangent bundles using a complete symbol calculus.
• The quantization map is extended from polynomial observables to the full space of symbols on the cotangent bundle.
• The deformation parameter ℏ is allowed to take any real value, not just formal power series values.
• The quantization procedure is functorial, preserving structure under smooth maps between manifolds.
• A new deformation-theoretic framework for quantization is established, valid beyond formal expansions.
• The construction provides a rigorous realization of normal ordering in the context of deformation quantization on cotangent bundles.