June 25, 2018

Ontario Non-Commutative Geometry Seminar

February 11, 2003 - 3:00 pm
Hanfeng Li, University of Toronto
Strict Quantization of Almost Poisson Manifolds

In the passage from classical mechanics to quantum mechanics, smooth functions on symplectic manifolds (more generally, Poisson
manifolds) are replaced by operators on a Hilbert space and the Poisson bracket of smooth functions is replaced by the commutator of the operators. When one thinks of classical mechanics as a limit of quantum mechanics, Poisson brackets become limits of commutators.

The formal algebraic approach to such a process is called formal deformation quantization. Its existence for every symplectic manifold was proven first by De Wilde and Lecomte, while the general case of Poisson manifolds was proven by Kontsevich.

A stricter approach to such a process is strict (deformation) quantization, which is based on continuous fields of C*-algebras (hence
everything is really operators on Hilbert spaces, compared to the formal algebraic approach). However, the existence of strict quantization was known only for special cases.

In this talk, I will give a construction of (non-Hermitian) strict quantization for every almost Poisson manifold.

For more details on the thematic year, see Program Page