Closedness of Star Products and Cohomologies

Flato, M.; Sternheimer, Daniel

doi:10.1007/978-1-4612-0261-5_8

Cited by 8 publications

(2 citation statements)

References 40 publications

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“…For completeness we give here a brief review on deformation quantization and star-products; a full treatment can be found in [2], [3] and a recent review in [10]. Let M be a Poisson manifold.…”

Section: Deformation Quantizationmentioning

confidence: 99%

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Deformation quantization and Nambu Mechanics

et al. 1997

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Starting from deformation quantization (star-products), the quantization problem of Nambu Mechanics is investigated. After considering some impossibilities and pushing some analogies with field quantization, a solution to the quantization problem is presented in the novel approach of Zariski quantization of fields (observables, functions, in this case polynomials). This quantization is based on the factorization over R of polynomials in several real variables. We quantize the infinite-dimensional algebra of fields generated by the polynomials by defining a deformation of this algebra which is Abelian, associative and distributive. This procedure is then adapted to derivatives (needed for the Nambu brackets), which ensures the validity of the Fundamental Identity of Nambu Mechanics also at the quantum level. Our construction is in fact more general than the particular case considered here: it can be utilized for quite general defining identities and for much more general star-products.

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Section: Deformation Quantizationmentioning

confidence: 99%

“…Our construction can be generalized to any orbit of the coadjoint action of a Lie algebra on its dual (the case of R 3 corresponds to su(2) ⋆ ). In that case, instead of the Moyal product appearing in the evaluation map, one can use a covariant star-product on the orbit [10].…”

Section: Zariski Productmentioning

confidence: 99%

Deformation quantization and Nambu Mechanics

et al. 1997

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The Differential Geometry of Fedosov’s Quantization

Emmrich

Weinstein

1994

Lie Theory and Geometry

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B. Fedosov has given a simple and very natural construction of a deformation quantization for any symplectic manifold, using a flat connection on the bundle of formal Weyl algebras associated to the tangent bundle of a symplectic manifold. The connection is obtained by affinizing, nonlinearizing, and iteratively flattening a given torsion free symplectic connection. In this paper, a classical analog of Fedosov's operations on connections is analyzed and shown to produce the usual exponential mapping of a linear connection on an ordinary manifold. A symplectic version is also analyzed. Finally, some remarks are made on the implications for deformation quantization of Fedosov's index theorem on general symplectic manifolds.

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Star Products: Their Ubiquity and Unicity

Sternheimer

1995

Modern Group Theoretical Methods in Physics

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Closedness of Star Products and Cohomologies

Cited by 8 publications

References 40 publications

Deformation quantization and Nambu Mechanics

Deformation quantization and Nambu Mechanics

The Differential Geometry of Fedosov’s Quantization

Star Products: Their Ubiquity and Unicity

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