# Exploring potential hidden aspects of quantum field theory through numerical solution of the Klein–Gordon equation using the Yee algorithm

## TL;DR

This paper introduces a new computational method to solve the Klein–Gordon equation using a Maxwell-like approach, addressing issues like negative probability and revealing connections to Dirac spinors.

## Contribution

A novel reformulation of the Klein–Gordon equation using Maxwell–Heaviside-like equations and the Yee algorithm, enabling efficient numerical solutions and addressing negative probability.

## Key findings

- The reformulated Klein–Gordon equation introduces Conserved Maxwellian Fields (CMFs) with non-negative probability density.
- CMFs show structural similarities to Dirac spinors, especially in three spatial dimensions with monopole-like divergence.
- The method provides a computational framework for handling nonlinear and inhomogeneous systems efficiently.

## Abstract

This study presents a novel reformulation of the Klein–Gordon (KG) equation by embedding it within a system of first-order Maxwell–Heaviside (MH)-like equations, enabling its numerical solution using the finite-difference time-domain method based on the Yee algorithm. This approach extends the scalar KG field into a pair of fictitious Maxwellian vector fields. This reformulation not only provides an efficient computational framework, capable of handling nonlinearity and inhomogeneity, but also introduces a first-order structure with symmetric field dynamics. Plane-wave quantization of these fields reveals a conserved, non-negative quantity, forming what is termed Conserved Maxwellian Fields (CMFs), that addresses the longstanding issue of negative probability density in the conventional KG theory. Furthermore, the resulting CMFs exhibit deep structural analogies with Dirac spinors, particularly in three spatial dimensions, where only two CMF modes exist with monopole-like divergence. These findings bridge the gap between scalar field dynamics and electromagnetic field theory, offering both computational utility and potential insight into hidden structures in quantum field theory.