Conformal Structure of Quantum Wave Mechanics

TL;DR

This paper proposes a geometric interpretation of quantum mechanics by embedding quantum terms into a conformally modified spacetime metric, linking quantum fields with spacetime geometry without addressing measurement.

Contribution

It introduces a conformal metric framework where quantum effects are represented by a conformal factor, unifying quantum theory and spacetime geometry in a novel geometric structure.

Findings

Quantum terms are represented as scalar curvature of a conformal dilation covector.

A modified metric gc incorporates quantum effects while preserving classical lengths and angles.

The approach extends to all quantum fields satisfying the Klein Gordon equation.

Abstract

This work interprets the quantum terms in a Lagrangian, and consequently of the wave equation and momentum tensor, in terms of a modified spacetime metric. Part I interprets the quantum terms in the Lagrangian of a Klein Gordon field as scalar curvature of conformal dilation covector nm that is proportional to hbar times the gradient of wave amplitude R. Part II replaces conformal dilation with a conformal factor rho that defines a modified spacetime metric gc = exp(rho) g, where g is the gravitational metric. Quantum terms appear only in metric gc and its metric connection coefficients. Metric gc preserves lengths and angles in classical physics and in the domain of the quantum field itself. gc combines concepts of quantum theory and spacetime geometry in one structure. The conformal factor can be interpreted as the limit of a distribution of inclusions and voids in a lattice that cause the metric to bulge or contract. All components of all free quantum fields satisfy the Klein Gordon equation, so this interpretation extends to all quantum fields. Measurement operations, and elements of quantum field theory are not considered.