X iv : m at h - ph / 9 90 10 05 v 1 1 4 Ja n 19 99 Classical Mechanics and Geometric Quantization on an Infinite Dimensional Disc and Grassmannian O . T . Turgut Institut

We discuss the classical mechanics on the Grassmannian and the Disc modeled on the ideal L (2,∞). We apply methods of geometric quantization to these systems. Their relation to a flat symplectic space is also discussed. 1 Introduction We will analyze geometric quantization of a classical system which has as its phase space the infinite dimensional Grassmannian or the Disc modeled on the ideal L (2,∞) (H + , H −). There are two motivations for our work. The classical dynamics studied should correspond to the large-N c limit of a quantum system which requires a logarithmic renormalization. Its quantization should give us an understanding of this system in the Schrödinger picture. This picture has some advantages over the scattering matrix, as well-known in the physics literature. The second is to study and understand infinite dimensional systems, their quantization should lead to some interesting mathematical questions. A good example is typical two dimensional field theory models, which do not require a renormalization but only a normal ordering [20, 21]. It will be interesting to develop the necessary tools for more complicated systems, and perhaps give a more precise meaning to renormalized field theories. We should