Wigner–Weyl transform

In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for associating a "quantum mechanical" Hermitian operator with a "classical" distribution in phase space. The technique was first described by Hermann Weyl in 1927.

Example

The following demonstrates Weyl quantization on a simple, two-dimensional Euclidean phase space. Let the cordinates on phase space be $(q,p)$ and let f be a function defined everywhere on phase space. The Fourier transform of f is given by

${\widehat {f}}(a,b)={\frac {1}{(2\pi )^{2}}}\int \int f(q,p)e^{-i(aq+bp)}dq\,dp$

The associated Weyl operator is

$\Phi (f)=\int \int {\widehat {f}}(a,b)\Phi \left(e^{i(aQ+bP)}\right)da\,db$

Here, P and Q are taken to be self-adjoint (Hermitian) operators on some Hilbert space ${\mathcal {H}}$, such that the thier commutator is the identity on the Hilbert space:

$[P,Q]=PQ-QP=-i\hbar \,\operatorname {Id} _{\mathcal {H}}$

and $\hbar$ is the reduced Planck constant. These operators are thus seen to be the generators of a Lie algebra, the Heisenberg algebra. A general element of the algebra may thus be written as

$aQ+bP-i\hbar z\operatorname {Id} _{\mathcal {H}}$

The exponential map of an element of a Lie algebra is then an element of the corresponding Lie group. Thus,

$g=e^{(aQ+bP-i\hbar z)}$

is an element of the Heisenberg group. Given some particular group representation $\Phi$ of the Heisenberg group, the quantity

$\Phi \left(e^{(aQ+bP-i\hbar z)}\right)$

denotes the element of the representation corresponding to the group element g.

Properties

The above construction has a number of properties. If f is a real-valued function, then $\Phi (f)$ is self-adjoint. If f is an element of Schwartz space, then $\Phi (f)$ is trace-class.