# Quantization of Forms on the Cotangent Bundle

@article{Voronov1999QuantizationOF, title={Quantization of Forms on the Cotangent Bundle}, author={Theodore Th. Voronov}, journal={Communications in Mathematical Physics}, year={1999}, volume={205}, pages={315-336} }

Abstract:We consider the following construction of quantization. For a Riemannian manifold $M$ the space of forms on T⋆M is made into a space of (full) symbols of operators acting on forms on M. This gives rise to the composition of symbols, which is a deformation of the (“super”)commutative multiplication of forms. The symbol calculus is exact for differential operators and the symbols that are polynomial in momenta. We calculate the symbols of natural Laplacians. (Some nice Weitzenböck like…

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## References

SHOWING 1-10 OF 25 REFERENCES

The normal symbol on Riemannian manifolds

- Mathematics
- 1996

For an arbitrary Riemannian manifold X and Hermitian vector bundles E and F over X we dene the notion of the normal symbol of a pseu- dodierential operator P from E to F. The normal symbol of P is a…

The Laplacian on a Riemannian Manifold: The Laplacian on a Riemannian Manifold

- Mathematics
- 1997

In this chapter we will generalize the Laplacian on Euclidean space to an operator on differential forms on a Riemannian manifold. By a Riemannian manifold, we roughly mean a manifold equipped with a…

Homogeneous Fedosov Star Products on Cotangent Bundles I: Weyl and Standard Ordering with Differential Operator Representation

- Mathematics
- 1998

Abstract:In this paper we construct homogeneous star products of Weyl type on every cotangent bundle T*Q by means of the Fedosov procedure using a symplectic torsion-free connection on T*Q…

An explicit *-product on the cotangent bundle of a Lie group

- Mathematics
- 1983

We give explicit formulas for a *-product on the cotangent bundle T*G of a Lie group G; these formulas involve on the one hand the multiplicative structure of the universal enveloping algebra U(G) of…

Quantization of supermanifolds and an analytic proof of the Atiyah-Singer index theorem

- Mathematics
- 1993

An analytic proof of the Atiyah-Singer index, theorem is given with the help of the tools of supermathematics. The index formula for the Dirac operator on a spinor manifold is obtained here by direct…

Introduction to the Theory of Supermanifolds

- Mathematics
- 1980

CONTENTSIntroduction Chapter I. Linear algebra in superspaces § 1. Linear superspaces § 2. Modules over superalgebras § 3. Matrix algebra § 4. Free modules § 5. Bilinear forms § 6. The supertrace §…

The Schrödinger Equation

- Mathematics
- 1991

1. General Concepts of Quantum Mechanics.- 1.1. Formulation of Basic Postulates.- 1.2. Some Corollaries of the Basic Postulates.- 1.3. Time Differentiation of Observables.- 1.4. Quantization.- 1.5.…

Pseudodifferential Operators and Linear Connections

- Mathematics
- 1997

The aim of the paper is to construct a calculus of pseudodifferential operators (PDOs) on a smooth manifold M without using local coordinate systems. Instead we deal with linear connections Γ of M.

Supersymmetry and the Atiyah-Singer index theorem

- Physics, Mathematics
- 1983

Using a recently introduced index for supersymmetric theories, we present a simple derivation of the Atiyah-Singer index theorem for classical complexes and itsG-index generalization using elementary…