The Lie–Rinehart Universal Poisson Algebra of Classical and Quantum Mechanics

The Lie–Rinehart algebra of a (connected) manifold , defined by the Lie structure of the vector fields, their action and their module structure over , is a common, diffeomorphism invariant, algebra for both classical and quantum mechanics. Its (noncommutative) Poisson universal enveloping algebra ,...

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Bibliographic Details
Published in:Letters in mathematical physics 2008-12, Vol.86 (2-3), p.135-150
Main Authors: Morchio, Giovanni, Strocchi, Franco
Format: Article
Language:English
Subjects:
Complex Systems
Geometry
Group Theory and Generalizations
Mathematical and Computational Physics
Physics
Physics and Astronomy
Theoretical
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Summary:The Lie–Rinehart algebra of a (connected) manifold , defined by the Lie structure of the vector fields, their action and their module structure over , is a common, diffeomorphism invariant, algebra for both classical and quantum mechanics. Its (noncommutative) Poisson universal enveloping algebra , with the Lie–Rinehart product identified with the symmetric product, contains a central variable (a central sequence for non-compact ) which relates the commutators to the Lie products. Classical and quantum mechanics are its only factorial realizations, corresponding to Z  =  i z , z  =  0 and , respectively; canonical quantization uniquely follows from such a general geometrical structure. For , the regular factorial Hilbert space representations of describe quantum mechanics on . For z  =  0, if Diff( ) is unitarily implemented, they are unitarily equivalent, up to multiplicity, to the representation defined by classical mechanics on .
ISSN:0377-9017
1573-0530
DOI:10.1007/s11005-008-0280-5