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.[1] 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 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. 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 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]
The following explains the Weyl transformation on the simplest, two-dimensional Euclidean phase space. Let the coordinates on phase space be, and let be a function defined everywhere on 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. We assume that the exponentiated operators
eiaQ
eibP
The Weyl transform (or Weyl quantization) of the function is given by the following operator in Hilbert space,[5] [6] Throughout, ħ is the reduced Planck constant.
It is instructive to perform the and integrals in the above formula first, which has the effect of computing the ordinary Fourier transform
\tilde{f}
ei(aQ+bP)
\Phi[f]=
| 1 | |
| (2\pi)2 |
\iint\tilde{f}(a,b)eiaQ+ibPdadb
We may therefore think of the Weyl map as follows: We take the ordinary Fourier transform of the function
f(p,q)
P
Q
A less symmetric form, but handy for applications, is the following,
\Phi[f]=
| 2 | |
| (2\pi\hbar)3/2 |
\iint\iintdqdpd\tilde{x}d\tilde{p}
| ||||||
| e |
\tilde{p}-2(\tilde{p}-p)(\tilde{x}-q))}~f(q,p)~|\tilde{x}\rangle\langle\tilde{p}|.
The Weyl map may then also be expressed in terms of the integral kernel matrix elements of this operator,[8]
\langlex|\Phi[f]|y\rangle=
| infty | |
| \int | |
| -infty |
{dp\overh}~eip(x-y)/\hbar~f\left({x+y\over2},p\right).
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 back to the original phase-space kernel function,
For example, the Wigner map of the oscillator thermal distribution operator
\exp(-\beta(P2+Q2)/2)
\exp\star\left(-\beta(p2+q2)/2\right)=\left(\cosh\left(
| \beta\hbar | |
| 2 |
\right)\right)-1\exp\left(
| -2 | \tanh\left( | |
| \hbar |
| \beta\hbar | |
| 2 |
\right)(p2+q2)/2\right).
If one replaces
\Phi[f]
\Phi[f]=h\iintdadb~eiaQ+ibP\operatorname{Tr}(e-iaQ-ibP\Phi).
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
q
p
Q
P
The action of the Weyl quantization on polynomial functions of
q
p
(aq+bp)n\longmapsto(aQ+bP)n
a
b
qkpl
k
Q
l
P
6p2q2~~\longmapsto~~P2Q2+Q2P2+PQPQ+PQ2P+QPQP+QP2Q.
While this result is conceptually natural, it is not convenient for computations when
k
l
pmqn~~\longmapsto~~{1\over2n}
| n | |
| \sum | |
| r=0 |
{n\chooser}QrPmQn-r={1\over
| m | |
| 2 | |
| s=0 |
{m\chooses}PsQnPm-s.
p2q2
p2q2
P2Q2
QP2Q
Q2P2
m=n=2
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.) For example, Moyal showed the
Theorem: If
f(q,p)
g(q,p)
| \Phi(\{f,g\})= | 1 |
| i\hbar |
[\Phi(f),\Phi(g)]
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.