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

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

1996-04-23

arxiv.org/abs/hep-th/9604144v1

Physics

High Energy Physics

High Energy Physics - Theory

postscript-file, 70 pages, also available at ftp://ftp.math.tu-berlin.de/pub/Preprints/sfb288/abstract186.html

Scientific paper

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.

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