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  "body": "\n## Summary\n\nA Fourier integral operator is an operator of the form $u \\mapsto \\int e^{i \\varphi(x, \\xi)} a(x, \\xi) \\hat{u}(\\xi) \\, d\\xi / (2\\pi)^n$.\nHere the phase $\\varphi$ is homogeneous of degree one in $\\xi$ and $a$ is a symbol.\nThe phase $x \\cdot \\xi$ gives the [[pseudo-differential operators|FF27N0BWTP]], and other phases move singularities along a canonical transformation of the cotangent bundle.\nSolution operators of strictly hyperbolic equations, such as the wave equation, are Fourier integral operators.\nThe general theory was developed by Lars Hörmander.\n\n## Explanations\n\n### L1 human\n\nA pseudo-differential operator can smooth or sharpen a function, but it does not move its singularities, such as jumps or kinks.\nThe solution of the wave equation does move them: a sharp wave front travels at the speed of light.\nFourier integral operators are the class of operators that can move singularities in a controlled way.\nFor example, the operator $e^{it\\sqrt{-\\Delta}}$, which multiplies the frequency $\\xi$ by $e^{it\\abs{\\xi}}$, moves each singularity a distance $t$ in a direction set by its frequency.\n",
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