Exploring the propagation of relativistic quantum wavepackets in the trajectory-based formulation

TL;DR

This paper introduces a trajectory-based formulation of relativistic quantum mechanics that produces well-behaved Gaussian wavepackets, avoiding the counterintuitive behaviors seen in traditional wavefunction approaches.

Contribution

It extends a nonrelativistic trajectory-based quantum framework to the relativistic domain, resulting in physically consistent wavepacket solutions.

Findings

Probability density remains positive and localized.

Wavepackets propagate along straight paths in spacetime.

Probability is conserved in any inertial frame.

Abstract

In the context of nonrelativistic quantum mechanics, Gaussian wavepacket solutions of the time-dependent Schr\"odinger equation provide useful physical insight. This is not the case for relativistic quantum mechanics, however, for which both the Klein-Gordon and Dirac wave equations result in strange and counterintuitive wavepacket behaviors, even for free-particle Gaussians. These behaviors include zitterbewegung and other interference effects. As a potential remedy, this paper explores a new trajectory-based formulation of quantum mechanics, in which the wavefunction plays no role [Phys. Rev. X, 4, 040002 (2014)]. Quantum states are represented as ensembles of trajectories, whose mutual interaction is the source of all quantum effects observed in nature---suggesting a "many interacting worlds" interpretation. It is shown that the relativistic generalization of the trajectory-based formulation results in well-behaved free-particle Gaussian wavepacket solutions. In particular, probability density is positive and well-localized everywhere, and its spatial integral is conserved over time---in any inertial frame. Finally, the ensemble-averaged wavepacket motion is along a straight line path through spacetime. In this manner, the pathologies of the wave-based relativistic quantum theory, as applied to wavepacket propagation, are avoided.