Klein-Gordon equation

The Klein–Gordon equation is the simplest relativistic wave equation, describing a spinless particle in a way consistent with special relativity. It is obtained by taking Einstein's relation between energy, momentum, and mass and translating it directly into a quantum wave equation. Because it treats time and space symmetrically, it succeeds where the Schrödinger equation, built only for low speeds, breaks down.

Historically it caused as much trouble as triumph. Read as an equation for a single particle's wavefunction, it yields negative energies and a probability density that can go negative — both nonsensical for a literal probability. These flaws were why Dirac sought his own equation for the electron, and why the Klein–Gordon equation was for a time set aside as a failed attempt at a one-particle theory.

Its true meaning appeared once it was reinterpreted not as a wavefunction equation but as a field equation. As the equation of motion for a quantum field, it describes spinless particles such as the pion or the Higgs boson, with the troublesome negative-energy solutions becoming antiparticles and the probability worry dissolving entirely. It thus stands as the gentlest entry point into relativistic quantum field theory.