Geometrization Of Quantum Mechanics Research Articles

The geometrization of quantum mechanics, frozen stars, the Bohm–Poisson and nonlinear Klein–Gordon equations

We revisit the nonlinear Klein–Gordon-like equation that was proposed by us which captures how quantum mechanical probability densities curve spacetime, and find an exact solution that may appear to be “trivial” but with important physical implications related to the physics of frozen stars and with Mach’s principle. The nonlinear Klein–Gordon-like equation is essentially the static spherically symmetric relativistic analog of the Newton–Schrödinger equation. We finalize by studying the higher-dimensional generalizations of the nonlinear Klein–Gordon-like equation and examine the relativistic Bohm–Poisson equation as yet another equation capturing the interplay between quantum mechanical probability densities and gravity.

Relativistic canonical commutation relations and the geometrization of quantum mechanics

It is shown how the relativistic canonical communication relations (RCCRs) for a massive particle with spin are essentially uniquely determined by the requirements that the set of Poincare group generators form a covariant tensor operator, and that a certain covariance condition be satisfied. An outline of the connection between stochastic-phase-space representations of the Poincare group and realizations of the RCCRs is presented, and the possibility of generalization to curved space-times with internal gauge symmetry is considered.

Geometrization of Quantum Mechanics and the New Interpretation of the Scalar Product in Hilbert Space

It is shown that the new interpretation of the scalar product in Hilbert space recently proposed by Aharanov, Albert, and Au is in fact the one underlying the stochastic phase-space formulation of nonrelativistic quantum mechanics. The extrapolation of this interpretation to the relativistic domain and its relationship to the program of geometrizing quantum theory and quantizing space-time are discussed.