Quantization of dissipative systems—some irresponsible speculations

Abstract

The Newton‐Lagrange equations of motion represent the fundamental law of mechanics. Their traditional Lagrangian and/or Hamiltonian precursors when available are essential in the context of quantization. However, there are situations that lack Lagrangian and/or Hamiltonian settings. This paper discusses classical and quantal dynamics of such systems and presents some irresponsible speculations by introducing a certain canonical two‐form Ω. By its construction Ω embodies kinetic energy and forces acting within the system (not their potential). A new type of variational principle is introduced, where variation is performed over a set of “umbilical surfaces” instead of system histories. It provides correct Newton‐Lagrange equations of motion and something more. The quantization is inspired by the Feynman functional integral approach. The quintessence is to rearrange path integral into an “umbilical world‐sheet” integral in accordance with the proposed variational principle. In the case of potential‐generated...