Narrow Results

A novel method for quantization of non-Lagrangian (open) systems is proposed. It is argued that the essential object, which provides both classical and quantum evolution, is a certain canonical two-form defined in extended velocity space. In this setting classical dynamics is recovered from the stringy-type variational principle, which employs umbilical surfaces instead of histories of the system. Quantization is then accomplished in accordance with the introduced variational principle. The path integral for the transition probability amplitude (propagator) is rearranged to a surface functional integral. In the standard case of closed (Lagrangian) systems the presented method reduces to the standard Feynman's approach. The inverse problem of the calculus of variation, the problem of quantization ambiguity and the quantum mechanics in the presence of friction are analyzed in detail.

New method of quantization is presented. It is based on classical Newton–Lagrange equations of motion (representing the fundamental physical law of mechanics) rather than on their traditional Lagrangian and/or Hamiltonian precursors. It is shown that classical dynamics is governed by canonical two-form Ω, which embodies kinetic energy and forces acting within the system. New type of variational principle employing differential two-form Ω and "umbilical strings" is introduced. The Feynman path integral over histories of the system is 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 is analyzed in detail.