Quantization as a dimensional reduction phenomenon

Abstract

Classical mechanics, in the operatorial formulation of Koopman and von Neumann, can be written also in a functional form. In this form two Grassmann partners of time make their natural appearance extending in this manner time to a three dimensional supermanifold. Quantization is then achieved by a process of dimensional reduction of this supermanifold. We prove that this procedure is equivalent to the well-known method of geometric quantization.Comment: 19 pages, Talk given by EG at the conference "On the Present Status of Quantum Mechanics", Mali Losinj, Croatia, September 2005. New results are contained in the last part of the pape