Trajectory-state theory of the Klein-Gordon field

TL;DR

This paper introduces a trajectory-based framework for the Klein-Gordon field, representing quantum states as congruences of trajectories with internal time, maintaining Lorentz covariance and extending classical wave solutions.

Contribution

It develops a novel trajectory construction for the Klein-Gordon field that captures quantum states as congruences of trajectories with an internal scalar time, preserving Lorentz invariance.

Findings

Trajectory construction for massless wave equation in higher dimensions.

Representation of quantum states via 3-trajectories and internal time.

Lorentz covariance of the trajectory model established.

Abstract

We develop a trajectory construction of solutions to the massless wave equation in n+1 dimensions and hence show that the quantum state of a massive relativistic system in 3+1 dimensions may be represented by a stand-alone four-dimensional congruence comprising a continuum of 3-trajectories coupled to an internal scalar time coordinate. A real Klein-Gordon amplitude is the current density generated by the temporal gradient of the internal time. Complex amplitudes are generated by a two-phase flow. The Lorentz covariance of the trajectory model is established.