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  "body": "\n## Definition\n\nLet $n \\ge 1$ and $m \\ge 0$.\nThe *Klein–Gordon equation* with mass $m$ is the [[hyperbolic equation|92C3S03AQZ]] for $u \\colon \\R^n \\times \\R \\to \\C$:\n$$\\frac{\\partial^2 u}{\\partial t^2} - \\Delta u + m^2 u = 0,$$\nwhere $\\Delta$ is the [[Laplacian|0E0B6RA8K0]] in $x$.\nWith the speed of light $c$ and the reduced Planck constant $\\hbar$ restored, it reads $\\frac{1}{c^2} u_{tt} - \\Delta u + \\frac{m^2 c^2}{\\hbar^2} u = 0$.\n\n## Remarks\n\nExample: for $k \\in \\R^n$ and $\\omega = \\sqrt{\\abs{k}^2 + m^2}$, the plane wave $u(x, t) = e^{i(k \\cdot x - \\omega t)}$ is a solution, since $u_{tt} = -\\omega^2 u$ and $\\Delta u = -\\abs{k}^2 u$, so $u_{tt} - \\Delta u + m^2 u = (-\\omega^2 + \\abs{k}^2 + m^2) u = 0$.\nNon-example: the plane wave $e^{i(k \\cdot x - \\abs{k} t)}$ solves the [[wave equation|5EHD2XBQDG]] but not the Klein–Gordon equation with $m \u003e 0$, since then the left side is $m^2 u \\ne 0$.\nFor $m = 0$ the equation is the wave equation with speed $1$.\nThe conserved energy is $E(t) = \\frac{1}{2}\\int_{\\R^n} \\big(\\abs{u_t}^2 + \\abs{\\nabla u}^2 + m^2 \\abs{u}^2\\big)\\, dx$.\n",
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