How to quantize forces (?): An academic essay on how the strings could enter classical mechanics

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 Ω (an analog of d p ∧ d q − d H ∧ d t ), 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 Ω is proposed. The Feynman 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.