Wigner–Weyl transform

In quantum mechanics, the Wigner–Weyl transform or Weyl–Wigner transform is the invertible mapping between functions in the quantum phase space formulation and Hilbert space operators in the Schrödinger picture.

Often the mapping from functions on phase space to operators is called the Weyl transform, 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 want for quantization and therefore sometimes yields unphysical answers.

Nevertheless, in a mathematical sense, 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 Wigner quasi-probability distribution is the Wigner transform of the quantum density matrix, 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 Planck's constant ħ, as it does in some familiar cases involving angular momentum. This invertible representation change then allows one to express quantum mechanics in phase space, as was appreciated in the 1940s by Groenewold and Moyal.

The following illustrates the Weyl transformation on the simplest, two-dimensional Euclidean phase space. Let the coordinates on phase space be (q,p), and let f be a function defined everywhere on phase space.

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