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Linear canonical transformation
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<p>In <a href="/facts/Hamiltonian_mechanics/2ReerlNA">Hamiltonian mechanics</a>, the linear canonical transformation (LCT) is a family of <a href="/facts/Integral_transform/AB2yjHJJ">integral transforms</a> that generalizes many classical transforms. It has 4 parameters and 1 constraint, so it is a 3-dimensional family, and can be visualized as the action of the <a href="/facts/Special_linear_group/rsahah9w">special linear group</a> <a href="/facts/M%C3%B6bius_transformation/YEkTTLfK">SL2(C)</a> on the <a href="/facts/Time%E2%80%93frequency_representation/b3BqBScJ">time–frequency plane</a> (domain). As this defines the original function up to a sign, this translates into an action of its <a href="/facts/Covering_space/bDRGasl7">double cover</a> on the original function space. </p><p>The LCT generalizes the <a href="/facts/Fourier_transform/XXhKJtc8">Fourier</a>, <a href="/facts/Fractional_Fourier_transform/kwH2qV0M">fractional Fourier</a>, <a href="/facts/Laplace_transform/0Ge4MIc9">Laplace</a>, <a href="/facts/Weierstrass_transform/1LNo4Bce">Gauss–Weierstrass</a>, <a href="/facts/Segal%E2%80%93Bargmann_space/wWQfnJQz">Bargmann</a> and the <a href="/facts/Fresnel_transform/lxyq42M4">Fresnel</a> transforms as particular cases. The name "linear canonical transformation" is from <a href="/facts/Canonical_transformation/J8vfkj37">canonical transformation</a>, a map that preserves the symplectic structure, as SL2(R) can also be interpreted as the <a href="/facts/Symplectic_group/G4CKBc7z">symplectic group</a> Sp2, and thus LCTs are the linear maps of the time–frequency domain which preserve the <a href="/facts/Symplectic_form/o95tWKBc">symplectic form</a>, and their action on the Hilbert space is given by the <a href="/facts/Metaplectic_group/xtmdUCdA">Metaplectic group</a>. </p><p>The basic properties of the transformations mentioned above, such as scaling, shift, coordinate multiplication are considered. Any linear canonical transformation is related to affine transformations in phase space, defined by time-frequency or position-momentum coordinates. </p>