The geometrization of quantum mechanics, the nonlinear Klein–Gordon equation, Finsler gravity and phase spaces

Abstract

The Geometrization of Quantum Mechanics proposed in this work is based on the postulate that the quantum probability density can curve the classical spacetime. It is shown that the gravitational field produced by smearing a point-mass Mo at r=0 throughout all of space (in a spherically symmetric fashion) can be interpreted as the gravitational field generated by a self-gravitating anisotropic fluid droplet of mass density 4πMor2φ∗(r)φ(r) and which is sourced by the probabilitycloud (associated with a spinless point-particle of mass Mo) permeating a 3-spatial domain region D3=∫4πr2dr at any time t. Classically one may smear the point mass in any way we wish leading to arbitrary density configurations ρ(r). However, Quantum Mechanically this is not the case because the radial mass configuration M(r) must obey a key third order nonlinear differential equation (nonlinear extension of the Klein–Gordon equation) displayed in this work and which is the static spherically symmetric relativistic analog of the Newton–Schrödinger equation. We conclude by extending our proposal to the Lagrange–Finsler and Hamilton–Cartan geometry of (co) tangent spaces and involving the relativistic version of Bohm’s Quantum Potential. By further postulating that the quasi-probability Wigner distribution W(x,p)curves phase spaces, and by encompassing the Finsler-like geometry of the cotangent-bundle with phase space quantum mechanics, one can naturally incorporate the noncommutative and non-local Moyal star product (there are also non-associative star products as well). To conclude, Phase Space is the arena where to implement the space–time–matter unification program. It is our belief this is the right platform where the quantization of spacetime and the quantization in spacetime will coalesce.