Weyl Quantization

Weyl Quantization

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 phase space to operators is called the Weyl transform whereas the mapping from operators to 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 or phase-space quantization. It is now understood that Weyl quantization is not always well defined and sometimes gives unphysical answers.

Nevertheless, the mapping within quantum mechanics between the phase space and operator representations is well defined and is given by the Wigner–Weyl transform. Most importantly, the Wigner quasi-probability distribution is the Wigner transform of the quantum density matrix, and the density matrix is the Weyl transform of the Wigner function. In some contrast to Weyl's original intentions in seeking a consistent quantization scheme, this map merely amounts to a change of representation. It need not connect "classical" with "quantum" quantities: the starting phase-space function may well depend on Planck's constant ħ. Indeed, in some familiar cases involving angular momentum, it does. This invertible representation change then allows expressing quantum mechanics in phase space, as was appreciated in the 1940s by Groenewold and Moyal.