In quantum mechanics, the Wigner–Weyl transform or Weyl–Wigner transform (after Hermann Weyl and Eugene Wigner) 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 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.
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 normal coordinates of such maps.
(Groenewold's theorem 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.
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 express quantum mechanics in phase space, as was appreciated in the 1940s by Hilbrand J. Groenewold[2] and José Enrique Moyal.
[3][4] In more generality, Weyl quantization is studied in cases where the phase space is a symplectic manifold, or possibly a Poisson manifold.
Related structures include the Poisson–Lie groups and Kac–Moody algebras.
The following explains the Weyl transformation on the simplest, two-dimensional Euclidean phase space.
In what follows, we fix operators P and Q satisfying the canonical commutation relations, such as the usual position and momentum operators in the Schrödinger representation.
constitute an irreducible representation of the Weyl relations, so that the Stone–von Neumann theorem (guaranteeing uniqueness of the canonical commutation relations) holds.
It is instructive to perform the p and q integrals in the above formula first, which has the effect of computing the ordinary Fourier transform
, but then when applying the Fourier inversion formula, we substitute the quantum operators
for the original classical variables p and q, thus obtaining a "quantum version of f." A less symmetric form, but handy for applications, is the following, The Weyl map may then also be expressed in terms of the integral kernel matrix elements of this operator,[8] The inverse of the above Weyl map is the Wigner map (or Wigner transform), which was introduced by Eugene Wigner,[9] which takes the operator Φ back to the original phase-space kernel function f,
For example, the Wigner map of the oscillator thermal distribution operator
in the above expression with an arbitrary operator, the resulting function f may depend on the reduced Planck constant ħ, and may well describe quantum-mechanical processes, provided it is properly composed through the star product, below.
[10] In turn, the Weyl map of the Wigner map is summarized by Groenewold's formula,[6] While the above formulas give a nice understanding of the Weyl quantization of a very general observable on phase space, they are not very convenient for computing on simple observables, such as those that are polynomials in
In later sections, we will see that on such polynomials, the Weyl quantization represents the totally symmetric ordering of the noncommuting operators
For example, the Wigner map of the quantum angular-momentum-squared operator L2 is not just the classical angular momentum squared, but it further contains an offset term −3ħ2/2, which accounts for the nonvanishing angular momentum of the ground-state Bohr orbit.
The action of the Weyl quantization on polynomial functions of
is completely determined by the following symmetric formula:[11] for all complex numbers
From this formula, it is not hard to show that the Weyl quantization on a function of the form
For example, we have While this result is conceptually natural, it is not convenient for computations when
There is no contradiction, however, since the canonical commutation relations allow for more than one expression for the same operator.
(The reader may find it instructive to use the commutation relations to rewrite the totally symmetric formula for the case of
It is widely thought that the Weyl quantization, among all quantization schemes, comes as close as possible to mapping the Poisson bracket on the classical side to the commutator on the quantum side.
(An exact correspondence is impossible, in light of Groenewold's theorem.)