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  "body": "\n## Definition\n\nWrite points of $\\R^4$ as $x = (x^0, x^1, x^2, x^3)$ with $x^0 = t$, and let $\\eta = \\mathrm{diag}(1, -1, -1, -1)$.\nLet $\\gamma^0, \\gamma^1, \\gamma^2, \\gamma^3$ be complex $4 \\times 4$ matrices with\n$$\\gamma^\\mu \\gamma^\\nu + \\gamma^\\nu \\gamma^\\mu = 2 \\eta^{\\mu\\nu} I \\quad \\text{for all } \\mu, \\nu \\in \\set{0, 1, 2, 3}.$$\nLet $m \\ge 0$ be a constant, the mass.\nThe *Dirac equation* is the [[partial differential equation|GC5BDZ6QBC]]\n$$\\Big( i \\sum_{\\mu=0}^{3} \\gamma^\\mu \\partial_\\mu - m \\Big) \\psi = 0$$\nfor an unknown function $\\psi \\colon \\R^4 \\to \\C^4$, where $\\partial_\\mu$ is the [[partial derivative|09EGC6F8PT]] in $x^\\mu$.\nUnits are chosen with $\\hbar = c = 1$.\n\n## Remarks\n\nThe matrices $\\gamma^\\mu$ generate a representation of the [[Clifford algebra|A38B1QFK20]] of $(\\R^4, \\eta)$.\nA standard choice is the Dirac representation, in which $\\gamma^0 = \\mathrm{diag}(1, 1, -1, -1)$.\n\nExample: in the Dirac representation, $\\psi(x) = e^{-imt} (1, 0, 0, 0)$ is a solution.\nIndeed $\\psi$ depends only on $t$, and $i \\gamma^0 \\partial_t \\psi = i \\gamma^0 (-im) \\psi = m \\gamma^0 \\psi = m \\psi$, since $(1, 0, 0, 0)$ is an eigenvector of $\\gamma^0$ for the eigenvalue $1$.\n\nNon-example: for $m \u003e 0$, $\\psi(x) = e^{imt} (1, 0, 0, 0)$ is not a solution, because $i \\gamma^0 \\partial_t \\psi - m \\psi = -2m \\psi \\ne 0$.\n\nApplying the operator $i \\gamma^\\nu \\partial_\\nu + m$ to the equation and using the anticommutation relations gives $(\\partial_t^2 - \\Delta + m^2) \\psi = 0$, the Klein–Gordon equation for each component.\n",
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