How to Quantize Forces(?): An Academic Essay How the Strings Could Enter Classical Mechanics

TL;DR

This paper introduces a geometric framework for classical mechanics with non-potential forces, proposing a dissipative quantization method using a two-form and a string-based path integral approach, extending quantum mechanics to dissipative systems.

Contribution

It develops a geometric formulation of classical mechanics for non-potential forces and introduces a novel dissipative quantization method using string functional integrals.

Findings

Reformulation of classical mechanics using an odd-dimensional contact bundle.

Proposal of a dissipative quantization approach via a two-form $\Omega$.

Reduction to standard quantum mechanics in the potential force case.

Abstract

Geometrical formulation of classical mechanics with forces that are not necessarily potential-generated is presented. It is shown that a natural geometrical "playground" for a mechanical system of point particles lacking Lagrangian and/or Hamiltonian description is an odd dimensional line element contact bundle. Time evolution is governed by certain canonical two-form $\Omega$ (an analog of $dp/\dq-dH/\dt$), which is constructed purely from forces and the metric tensor entering the kinetic energy of the system. Attempt to "dissipative quantization" in terms of the two-form $\Omega$ is proposed. The Feynman's path integral over histories of the system is rearranged to a "world-sheet" functional integral. The "umbilical string" surfaces entering the theory connect the classical trajectory of the system and the given Feynman history. In the special case of potential-generated forces, "world-sheet" approach precisely reduces to the standard quantum mechanics. However, a transition probability amplitude expressed in terms of "string functional integral" is applicable (at least academically) when a general dissipative environment is discussed.