Klein–Gordon equation

The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second order in space and time and manifestly Lorentz covariant. It is a quantized version of the relativistic energy-momentum relation. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. Its theoretical relevance is similar to that of the Dirac equation. Electromagnetic interactions can be incorporated, forming the topic of scalar electrodynamics, but because common spinless particles like the pi-mesons are unstable and also experience the strong interaction (with unknown interaction term in the Hamiltonian), the practical utility is limited.

The equation can be put into the form of a Schrödinger equation. In this form it is two coupled differential equations, each of first order in time. The solutions have two components, reflecting the charge degree of freedom in relativity. It admits a conserved quantity, but this is not positive definite. The wave function cannot therefore be interpreted as a probability amplitude. The conserved quantity is instead interpreted as electric charge and the norm squared of the wave function is interpreted as a charge density. The equation describes all spinless particles with positive, negative as well as zero charge. Within the Feynman–Stueckelberg interpretation, particles and antiparticles are treated mathematically as if they propagate forward and backward in time respectively, the advanced propagator (as opposed to the retarded propagator) is employed for antiparticles. Physically, all particles move forward in time.

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