Direct quantization of equations of motion: from classical dynamics to transition amplitudes via strings

TL;DR

This paper introduces a novel quantization method based on classical equations of motion and a two-form variational principle, leading to a path integral over world-sheets that generalizes standard quantum mechanics.

Contribution

It proposes a new quantization framework using classical Newton-Lagrange equations and differential two-forms, extending quantum mechanics to systems with friction.

Findings

Reformulation of quantum mechanics via umbilical world-sheet integrals.

Reduction to standard quantum mechanics for potential forces.

Application to quantum systems with friction.

Abstract

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 $\Omega$, which embodies kinetic energy and forces acting within the system. New type of variational principle employing differential two-form $\Omega$ 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.