Wigner–Weyl transform

In <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, the Wigner–Weyl transform or Weyl–Wigner transform (after <a href="/facts/Hermann_Weyl/4C5HhyGU">Hermann Weyl</a> and <a href="/facts/Eugene_Wigner/HvWgw4KX">Eugene Wigner</a>) is the invertible mapping between functions in the quantum <a href="/facts/Phase_space_formulation/ogvtdMNs">phase space formulation</a> and <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> <a href="/facts/Operator_(mathematics)/1VWgEdjS">operators</a> in the <a href="/facts/Schr%25C3%25B6dinger_picture/4VK89MbE">Schrödinger picture</a>.
Often the mapping from functions on phase space to operators is called the Weyl transform or Weyl quantization, whereas the inverse mapping, from operators to functions on phase space, is called the Wigner transform. This mapping was originally devised by Hermann Weyl in 1927 in an attempt to map symmetrized classical phase space functions to operators, a procedure known as Weyl quantization. It is now understood that Weyl quantization does not satisfy all the properties one would require for consistent quantization and therefore sometimes yields unphysical answers. On the other hand, some of the nice properties described below suggest that if one seeks a single consistent procedure mapping functions on the classical phase space to operators, the Weyl quantization is the best option: a sort of <a href="/facts/Normal_coordinates/yEPcdFUd">normal coordinates</a> of such maps. (<a href="/facts/Groenewold%2527s_theorem/AKHX2JQn">Groenewold's theorem</a> asserts that no such map can have all the ideal properties one would desire.)
Regardless, the Weyl–Wigner transform is a well-defined integral transform between the phase-space and operator representations, and yields insight into the workings of quantum mechanics. Most importantly, the <a href="/facts/Wigner_quasi-probability_distribution/5tyVfdFF">Wigner quasi-probability distribution</a> is the Wigner transform of the quantum <a href="/facts/Density_matrix/A1XdylJa">density matrix</a>, and, conversely, the density matrix is the Weyl transform of the Wigner function. 
In contrast to Weyl's original intentions in seeking a consistent quantization scheme, this map merely amounts to a change of representation within quantum mechanics; it need not connect "classical" with "quantum" quantities. For example, the phase-space function may depend explicitly on the reduced Planck constant ħ, as it does in some familiar cases involving angular momentum. This invertible representation change then allows one to <a href="/facts/Phase_space_formulation/ogvtdMNs">express quantum mechanics in phase space</a>, as was appreciated in the 1940s by <a href="/facts/Hilbrand_J._Groenewold/DSVTxPdY">Hilbrand J. Groenewold</a> and <a href="/facts/Jos%25C3%25A9_Enrique_Moyal/eHbcpprk">José Enrique Moyal</a>.
In more generality, Weyl quantization is studied in cases where the phase space is a <a href="/facts/Symplectic_manifold/JvNrJkC7">symplectic manifold</a>, or possibly a <a href="/facts/Poisson_manifold/LvAmEzul">Poisson manifold</a>. Related structures include the <a href="/facts/Poisson%25E2%2580%2593Lie_group/zxJ6JMrI">Poisson–Lie groups</a> and <a href="/facts/Kac%25E2%2580%2593Moody_algebra/N8caXyvw">Kac–Moody algebras</a>.

Wigner–Weyl transform open-in-new

Wigner–Weyl transform