A Brief Introduction to Quantum Geometry

Deformation Quantization of Symplectic Manifolds

A very interesting class of examples of noncommutative *-algebras can be obtained by deforming the algebras of smooth functions over symplectic manifolds M, so that a quantum correspondence principle holds. This requirement is actually a central problem in the deformation quantization of symplectic manifolds. More precisely, let be the (commutative) algebra of smooth functions on a symplectic manifold M. Let be the associated algebra of formal power series with a formal parameter . We say that a new associative product introduced in the space satisfies the correspondence principle, if

where {,} are Poisson brackets associated to the symplectic manifold M. The motivating idea behind this definition is that M plays the role of the phase space of a classical mechanical system. We assume that this classical system has a quantum couterpart, described by a noncommutative algebra, which is in fact equipped with the new product . The parameter plays the role of the Planck constant. The correspondence principle tells us that quantum commutator coincides with the classical Poisson bracket, modulo terms of the order of .

There exists an intrinsically geometrical construction [F] of a noncommutative product of the described type, for every symplectic manifold M. The following is a very brief sketch of the construction. We start from the formal Weyl algebra bundle W[M] associated to , where is the initial symplectic form on M. In other words, the fibers of W[M] are the Weyl algebras associated to the tangent spaces of M, equipped with the symplectic scalar product induced by (a formal symplectic product). Let be the algebra of formal power series with coefficients in the smooth sections of W[M]. It turns out that every symplectic torsion-free connection on M induces an injective map with the following properties:

[i] The image of is a subalgebra of . In fact this image coincides with the kernel in of a naturally associated differential D, acting in the algebra consisting of W[M]-valued differential forms on M.

[ii] A new product in defined by

satisfies the above mentioned correspondence principle.

The final and crucial (from the physical viewpoint) step in the quantization of the considered system is to incorporate the construction in the conceptual framework of C*-algebraic physics [BR]. This is done by constructing a C*-algebra , by completing the appropriate *-subalgebra of , and considering irreducible representations and superselection sectors of the completed algebra .