Massless Klein-Gordon equation

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The Klein–Gordon equation [a1], [a2], [a3]

for the case where the mass parameter is equal to zero. The constant stands for the speed of light, is the charge of the positron, where is the Planck constant, are the time, respectively space, variables, and is the imaginary unit. The (complex-valued) solution describes the wave function of a relativistic spinless and massless particle with charge in the exterior electro-magnetic field . It is a second-order, hyperbolic partial differential equation. Solutions are being studied in, e.g., [a4], [a5].

If the outer field is zero, , or the coupling of the spin to the magnetic potential can be neglected, the massless Klein–Gordon equation also can be used for the description of massless spin-carrying particles, such as e.g. photons. In the case without outer fields the massless Klein–Gordon equation becomes equivalent to the wave equation with wave speed and is independent of the magnitude of Planck's constant . This explains, why the wave nature of massless particles, such as e.g. photons ( "light" ), can also be observed on a macroscopic scale — in contrast with the wave nature of massive particles (cf.also Massless field; Massive field).

The interpretation of the wave function as a quantum mechanical "probability amplitude" (similarly as in the case of the Schrödinger equation), however, is not consistent, since the quantity in general depends on the time parameter . Furthermore, the existence of negative frequency solutions is in contrast with the required lower boundedness of the energy ( "stability of matter" ). These problems are resolved through a re-interpretation of as a quantum field (cf. Quantum field theory), see e.g. [a6], [a7].

In recent time (as of 2000) solutions of the Klein–Gordon equation on Lorentzian manifolds have attracted increasing attention in connection with the theory of quantized fields on curved space-time, cf. e.g. [a8].


[a1] O. Klein, "Quantentheorie und fünfdimensionale Relativitätstheorie" Z. f. Phys. , 37 (1926) pp. 895
[a2] O. Gordon, "Der Comptoneffekt nach der Schrödingerschen Theorie" Z. f. Phys. , 40 (1926) pp. 117
[a3] E. Schrödinger, "Quantisierung als Eigenwertproblem IV" Ann. Phys. , 81 (1926) pp. 109
[a4] L. Gross, "Norm invariance of mass zero equations under the conformal group" J. Math. Phys. , 5 (1964) pp. 687–695
[a5] E.M. de Jager, "The Lorentz-invariant solutions of the Klein–Gordon equation I-II" Indag. Math. , 25 (1963) pp. 515–531; 546–558
[a6] R. Jost, "The general theory of quantised fields" , Amer. Math. Soc. (1965)
[a7] S. Weinberg, "The quantum theory of fields" , I , Cambridge Univ. Press (1995)
[a8] S.A. Fulling, "Aspects of quantum field theory in curved space-time" , Cambridge Univ. Press (1989)
How to Cite This Entry:
Massless Klein-Gordon equation. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by S. AlbeverioH. Gottschalk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article