✦ LIBER ✦

A generalization of the notion of polarization

✍ Scribed by V. Guillemin; S. Sternberg

Publisher: Springer
Year: 1986
Tongue: English
Weight: 763 KB
Volume: 4
Category: Article
ISSN: 0232-704X
DOI: 10.1007/bf00128051

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✦ Synopsis

Mi --N be a principal fiber bundle over N with structure group IR+. Given a symplectic form, co, on M, we will call the pair (M, A) a symplectic cone if, for every a£ER+, the diffeomorphism, r a , of M associated with a satisfies (4.4) Ta = a D . A problem which we have considered in several previous papers ([2], [31, [4], [51) is the "quantization problem" for symplectic cones. Roughly speaking this problem consists of associating with M an algebra of operators, A, and a symbolic calculus for which the symbols of the operators belonging to A are functions on M. For instance if M is the punctured cotangent bundle of a compact manifold B: M = TB def T*B -O the ring of pseudodifferential operators on B has such a symbolic calculus. Therefore, the quantization problem consists of constructing something like a ring of pseudodifferential operators for any symplectic cone, MI. One

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