Chapter 10: The Quantum State as Spatial Displacement

We give a simple demonstration that the Schrödinger equation of wave mechanics may be recast as a self-contained second-order Newtonian law for a congruence of spacetime trajectories. This implies that a pictorial representation of the quantum state as a displacement function is implicit in the quantum description, complementary to and independent of the wave function. Quantum evolution is described by 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 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.