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

• 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
• ISSN: 0377-9017; 1573-0530
• DOI: 10.1007/s11005-008-0280-5

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 .