KleinGordon

The Klein-Gordon equation is a relativistic wave equation for a spin-0 (scalar) field. It provides a relativistic generalization of the Schrödinger equation for a free particle. In flat spacetime with a metric signature, the equation is (∂^μ ∂_μ + m^2) φ = 0, or equivalently □ φ + m^2 φ = 0, where □ is the d'Alembert operator. In natural units (c = ħ = 1).

Solutions to the equation include plane waves φ ∝ e^{-i p·x} with the four-momentum satisfying p^μ p_μ = m^2. The theory supports real and complex scalar fields; upon quantization it becomes a quantum field theory of spin-0 particles. For a real field, there is no conserved charge, while a complex field carries a conserved U(1) charge.

Historically, the equation was proposed in the 1920s by Oskar Klein and Walter Gordon as an attempt to formulate a relativistic quantum equation for a single particle. While it correctly encodes the relativistic energy–momentum relation, its probabilistic interpretation as a single-particle wavefunction is problematic, since the associated probability density is not positive definite. Today the Klein-Gordon equation is viewed as the equation of motion for a free scalar quantum field, and it underpins scalar field theory.

Extensions and applications include its covariant generalization to curved spacetime, (□_g + m^2 + ξR) φ = 0, with possible curvature coupling. In particle physics, a complex Klein-Gordon field describes charged scalar particles, and the Higgs field in the Standard Model is a fundamental example of a scalar field described by related equations. Scalar fields also appear in cosmology and inflationary theories.