Abstract
It is possible to give a complete description of Classical Mechanics in terms of symplectic geometry and Poisson brackets. It is the essential of the hamiltonian formalism. In a common program with Flato, D. Sternheimer and J. Vey and other scientists (Fronsdal, Arnal, M. Cahen, S. Gutt, M. de Wilde) we have studied properties and applications of the deformations of the trivial associative algebra and of the Poisson Lie algebra associated with a symplectic manifold. Such deformations give a new approach for Quantum Mechanics; this approach has been developped in other papers ((1),(2)). In this lecture, I will give recent results concerning the existence and the equivalence of associative deformations (or *υ-products).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Flato, A. Lichnerowicz, D. Sternheimer. C.R. Acad. Sci. Paris, 283 A, (1976), pp. 19–24.
F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer. Lett. in Math. Phys. 1, (1977), pp. 521–530;
F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer. Ann.of Physics 111, (1978), PP. 61–132.
J. Vey. Comm. Math. Helv. 50, (1975), pp. 421–454.
A. Avez and A. Lichnerowicz. C.R. Acad. Sci. Paris, 275 A, (1972) pp. 11–24.
M. Gerstenhaber. Ann. of Math. 79, (1964), pp. 59–103.
A. Lichnerowicz. C.R. Ac. Sci. Paris, 286 A, (1978), pp. 49–53;
A. Lichnerowicz. Ann. di Matematica t 123, (1980), pp. 287–330.
M. Neroslavsky and A. Vlasov. C.R. Acad. Sci. Paris I, t 292, (1981), pp. 71–75.
S. Gutt. Lett. in Math. Phys. 3(1979), pp. 297–310; Ann. Inst. H. Poincaré 27, (1980).
A. Lichnerowicz. C.R. Acad. Sci. Paris 287 A, (1978) pp. 1029–1033;
A. Lichnerowicz. Lett. in Math. Phys. 3, (1979), pp. 495–502.
M. Cahen, S. Gutt and M. De Wilde. Local cohomology of the algebra of C¥ -functions on a connected manifold; Letters in Math. Phys. 4(1980), pp. 157–167.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1983 D. Reidel Publishing Company
About this chapter
Cite this chapter
Lichnerowicz, A. (1983). Deformations of Algebras Associated with a Symplectic Manifold. In: Cahen, M., De Wilde, M., Lemaire, L., Vanhecke, L. (eds) Differential Geometry and Mathematical Physics. Mathematical Physics Studies, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-7022-9_6
Download citation
DOI: https://doi.org/10.1007/978-94-009-7022-9_6
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-277-1508-1
Online ISBN: 978-94-009-7022-9
eBook Packages: Springer Book Archive