Bohmian trajectories and the Path Integral Paradigm. Complexified Lagrangian Mechanics

TL;DR

This paper explores a complexified Lagrangian mechanics framework derived from Bohmian trajectories, unifying quantum and classical mechanics through modified Hamilton-Jacobi equations and path integrals.

Contribution

It introduces a novel complexified Hamilton-Jacobi equation incorporating quantum corrections and extends Feynman's path integral by including imaginary components.

Findings

Derived a complexified Hamilton-Jacobi equation with quantum corrections.

Unified quantum mechanics and entropy balance in a complex framework.

Expanded Feynman's path integral to include imaginary coordinate and momentum sectors.

Abstract

David Bohm shown that the Schr{\"o}dinger equation, that is a "visiting card" of quantum mechanics, can be decomposed onto two equations for real functions - action and probability density. The first equation is the Hamilton-Jacobi (HJ) equation, a "visiting card" of classical mechanics, to be modified by the Bohmian quantum potential. And the second is the continuity equation. The latter can be transformed to the entropy balance equation. The Bohmian quantum potential is transformed to two Bohmian quantum correctors. The first corrector modifies kinetic energy term of the HJ equation, and the second one modifies potential energy term. Unification of the quantum HJ equation and the entropy balance equation gives complexified HJ equation containing complex kinetic and potential terms. Imaginary parts of these terms have order of smallness about the Planck constant. The Bohmian quantum corrector is indispensable term modifying the Feynman's path integral by expanding coordinates and momenta to imaginary sector.