View source for Massless Klein-Gordon equation

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The [[Klein–Gordon equation|Klein–Gordon equation]] [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]]

\begin{equation*} 0 = \left[ - \left( \frac { \partial } { \partial t } - i \frac { q e } { \hbar } \phi \right) ^ { 2 } + \right. \end{equation*}

\begin{equation*} \left.+ c ^ { 2 } \left( \nabla - i \frac { q e } { \hbar c } A \right) ^ { 2 } + \frac { c ^ { 4 } m ^ { 2 } } { \hbar ^ { 2 } } \right] \psi ( t , \mathbf{x} ) \end{equation*}

for the case where the mass parameter $m$ is equal to zero. The constant $c$ stands for the speed of light, $e$ is the charge of the positron, $\hbar = h / 2 \pi$ where $h$ is the [[Planck constant|Planck constant]], $( t , \mathbf{x} )$ are the time, respectively space, variables, and $i$ is the imaginary unit. The (complex-valued) solution $\psi$ describes the wave function of a relativistic spinless and massless particle with charge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130090/m13009011.png"/> in the exterior electro-magnetic field $( \phi , \mathbf{A} )$. It is a second-order, [[Hyperbolic partial differential equation|hyperbolic partial differential equation]]. Solutions are being studied in, e.g., [[#References|[a4]]], [[#References|[a5]]].

If the outer field is zero, $( \phi , \mathbf{A} ) = 0$, or the coupling of the spin to the magnetic potential $\mathbf{A}$ 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|wave equation]] with wave speed $c$ and is independent of the magnitude of Planck's constant $h$. 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|Massless field]]; [[Massive field|Massive field]]).

The interpretation of the wave function $\psi$ as a quantum mechanical  "probability amplitude"  (similarly as in the case of the [[Schrödinger equation|Schrödinger equation]]), however, is not consistent, since the quantity $\int _ { \mathbf{R} ^ { 3 } } | \psi ( t ,  \mathbf{x} ) | ^ { 2 } d  \mathbf{x}$ in general depends on the time parameter $t$. 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 $\psi ( t , \mathbf{x} )$ as a quantum field (cf. [[Quantum field theory|Quantum field theory]]), see e.g. [[#References|[a6]]], [[#References|[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. [[#References|[a8]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  O. Klein,   "Quantentheorie und fünfdimensionale Relativitätstheorie"  ''Z. f. Phys.'' , '''37'''  (1926)  pp. 895</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  O. Gordon,   "Der Comptoneffekt nach der Schrödingerschen Theorie"  ''Z. f. Phys.'' , '''40'''  (1926)  pp. 117</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  E. Schrödinger,   "Quantisierung als Eigenwertproblem IV"  ''Ann. Phys.'' , '''81'''  (1926)  pp. 109</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  L. Gross,   "Norm invariance of mass zero equations under the conformal group"  ''J. Math. Phys.'' , '''5'''  (1964)  pp. 687–695</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  E.M. de Jager,   "The Lorentz-invariant solutions of the Klein–Gordon equation I-II"  ''Indag. Math.'' , '''25'''  (1963)  pp. 515–531; 546–558</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  R. Jost,   "The general theory of quantised fields" , Amer. Math. Soc.  (1965)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  S. Weinberg,   "The quantum theory of fields" , '''I''' , Cambridge Univ. Press  (1995)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  S.A. Fulling,   "Aspects of quantum field theory in curved space-time" , Cambridge Univ. Press  (1989)</td></tr></table>