The quantum state as spatial displacement

TL;DR

This paper demonstrates that the Schrödinger equation can be reformulated as a Newtonian-like law for trajectories, offering a new geometric perspective on quantum states and their evolution.

Contribution

It introduces a trajectory-based reformulation of quantum mechanics, connecting wave and particle pictures via gauge potentials and canonical transformations.

Findings

Quantum evolution as a deterministic coordinate transformation

Wave-mechanical and trajectory pictures linked by canonical transformation

Provides an alternative basis for quantum operator calculus

Abstract

We give a simple demonstration that the Schr\"odinger equation may be recast as a self-contained second-order Newtonian law for a congruence of spacetime trajectories. This provides a pictorial representation of the quantum state as the displacement function of the collective whereby quantum evolution is represented as the deterministic unfolding of a continuous coordinate transformation. Introducing gauge potentials for the density and current density it is shown that the wave-mechanical and trajectory pictures are connected by a canonical transformation. The canonical trajectory theory is shown to provide an alternative basis for the quantum operator calculus and the issue of the observability of the quantum state is examined within this context. The construction illuminates some of the problems involved in connecting the quantum and classical descriptions.