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  "body": "\n## Definition\n\nLet $n \\ge 1$ and $m \\in \\R$.\nLet $\\varphi \\colon \\R^n \\times (\\R^n \\setminus \\set{0}) \\to \\R$ be a smooth function such that:\n\n1. $\\varphi(x, t\\xi) = t \\, \\varphi(x, \\xi)$ for all $t \u003e 0$, $x \\in \\R^n$ and $\\xi \\ne 0$.\n2. $\\det \\big( \\partial^2 \\varphi / \\partial x_j \\partial \\xi_k \\big)_{j,k} \\ne 0$ everywhere.\n\nLet $a \\in S^m(\\R^n \\times \\R^n)$ be a symbol in the [[symbol class|AM4RNWVV4Q]] $S^m = S^m_{1,0}$ with $a(x, \\xi) = 0$ for $\\abs{\\xi} \\le 1$.\nFor $u$ in the [[Schwartz space|CG9JEATW3Q]] $\\mathcal{S}(\\R^n)$ with [[Fourier transform|DPVM34ADZV]] $\\hat{u}$, put\n$$T u(x) = \\frac{1}{(2\\pi)^n} \\int_{\\R^n} e^{i \\varphi(x, \\xi)} a(x, \\xi) \\, \\hat{u}(\\xi) \\, d\\xi.$$\nThe operator $T$ is a *Fourier integral operator* of order $m$ with *phase function* $\\varphi$ and *amplitude* $a$.\n\n## Remarks\n\nExample: the phase $\\varphi(x, \\xi) = x \\cdot \\xi$ satisfies both conditions, with mixed Hessian the identity matrix, and gives the [[pseudo-differential operator|FF27N0BWTP]] $\\mathrm{Op}(a)$.\nFor fixed $t \\in \\R$, the phase $\\varphi(x, \\xi) = x \\cdot \\xi + t \\abs{\\xi}$ also satisfies both conditions, and with a symbol $a$ equal to $1$ for $\\abs{\\xi} \\ge 2$ it gives $e^{i t \\sqrt{-\\Delta}}$ up to a smoothing operator.\n\nNon-example: $\\varphi(x, \\xi) = x \\cdot \\xi + \\abs{\\xi}^2$ fails the first condition, since it is not homogeneous of degree one in $\\xi$; it is the phase of a Schrödinger evolution.\n\nThe general definition, due to Hörmander, allows phases $\\varphi(x, y, \\theta)$ with extra variables and works on manifolds.\nThere a Fourier integral operator is associated with a canonical relation, a [[Lagrangian submanifold|H8ACPTMX08]] of the product of the cotangent bundles, and it moves wave front sets along that relation.\n",
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