zbMATH — the first resource for mathematics

Applications of the deformations of the algebraic structures to geometry and mathematical physics. (English) Zbl 0674.58021
Deformation theory of algebras and structures and applications, Nato Adv. Study Inst., Castelvecchio-Pascoli/Italy 1986, Nato ASI Ser., Ser. C 247, 855-896 (1988).
[For the entire collection see Zbl 0654.00006.]
Some properties and applications of the deformations of an associative algebra and of the Poisson Lie algebra of a symplectic manifold are given. It is shown that such deformations give a new invariant approach for Quantum Mechanics.
The symplectic geometry and its generalizations are introduced and then the deformations of the algebraic structures defined on a symplectic manifold are described. The star-product and Vey star-product on a symplectic manifold are defined and the existence conditions for such products are established. The quantization is introduced directly in terms of ordinary functions (or distributions) and star products without reference to operators. Some applications to statistical mechanics are given.
It is shown that the star-product leads to a natural and simple expression for the condition characterizing the Gibbs states in statistical mechanics (both classical and quantum) in the context of the conformal symplectic geometry. One considers only dynamical systems with a finite number of degrees of freedom, but it is claimed that the approach and a significant part of the results can be extended to physical fields.
Reviewer: G.Zet

53D50 Geometric quantization
82B05 Classical equilibrium statistical mechanics (general)
58H15 Deformations of general structures on manifolds
16S80 Deformations of associative rings