# 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.

## Key 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.

---
Source: https://tomesphere.com/paper/1905.13057