XXVI Workshop on Geometric Methods in Physics 1-7.07.2007

Denis Kochan

Quantization of dissipative systems

Geometrical formulation of classical mechanics with forces that are not necessarily potential-generated will be presented. Time evolution in that case is governed by certain canonical two-form $\Omega$, which is constructed purely from forces and the metric tensor entering the kinetic energy of the system. New type of variational principle employing differential two-form $\Omega$ and umbilical strings will be introduced. The Feynman path integral over histories of the system will be then rearranged to umbilical world-sheet functional integral in accordance with the proposed variational principle. In the case of potential-generated forces, world-sheet approach reduces to the standard quantum mechanics. As an example Quantum Mechanics with friction will be analyzed in detail.