CONVERGENCE OF STAR PRODUCTS: OLD AND NEW

In this talk, I will explain the issue of convergence for formal star products. While the theory of formal deformation quantizations of general Poisson manifolds is well-understood and explored by many people, for physical applications, formal power series in h are not yet enough to have physically meaningful models of quantum theories. Thus, the formal character of the deformation parameter h has to be overcomed. I will discuss several competing options of how this can be achieved. Unlike the general formal deformation quantization, no general existence or classification results are available. This forces one to investigate (classes of) examples to study the phenomena first, hopefully leading to more general scenarios where more general theorems can be shown. In this talk, I will discuss several of these examples by constructing rather explicit locally convex algebras out of the formal star product algebras.