Klein–Gordon equation

Definition

The Klein–Gordon equation for a scalar field $\phi$ of mass $m\geq0$ on a Lorentzian manifold is

(\Box_g+m^2)\phi=0

in the $(-+\cdots+)$ metric and wave-operator convention of this collection. Equivalently, $\phi$ lies in the kernel of the Klein–Gordon operator.

Massive and massless cases

When $m>0$ , flat-spacetime plane waves satisfy $\omega^2=|k|^2+m^2$ . When $m=0$ , the equation reduces to the wave equation. A curvature-coupled scalar field instead satisfies

(\Box_g+m^2+\xi\operatorname{Scal}_g)\phi=0.

Here $\operatorname{Scal}_g$ is the scalar curvature. Minimal coupling means $\xi=0$ ; conformal coupling uses the dimension-dependent coefficient that makes the massless equation conformally covariant.

Initial data and propagation

Because the Klein–Gordon operator is normally hyperbolic, its characteristics are null even when $m>0$ . Its global existence, uniqueness, and finite-propagation properties on globally hyperbolic spacetimes follow from the Cauchy theorem for normally hyperbolic operators.

Relation to the Dirac equation

In flat spacetime the free Dirac equation factors a Klein–Gordon operator: each component of a free Dirac spinor satisfies the Klein–Gordon equation. Curvature and gauge coupling add lower-order terms to the corresponding squared Dirac operator, so the flat factorization should not be transferred unchanged to general backgrounds.

Terminology

The equation is named for Oskar Klein and Walter Gordon. “Klein–Gordan” is a common misspelling, but Klein–Gordon is the standard spelling.

References

Christian Bär, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, European Mathematical Society, 2007. Publisher record. Relevant: Chapters 3–4.
Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995. Publisher record. Relevant: Chapter 2.