View source for Wigner-Weyl transform

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''Weyl–Wigner transform''

Let $\psi ( t )$, $t \in \mathbf{R} ^ { + }$, be a ray in $\mathcal{H} = L ^ { 2 } ( \mathbf{R} ^ { 3 N } )$. Then, for each $t$, the Wigner transform of $\psi$ is

\begin{equation*} \psi _ { \operatorname{w} } ( x , p , t ) = \int _ { \mathbf{R} ^ { 3 N } } e ^ { i p z / \hbar } \overline { \psi } \left( x + \frac { z } { 2 } , t \right) \psi \left( x - \frac { z } { 2 } , t \right) d z, \end{equation*}

where $\hbar$ is Planck's constant. The quantity $f _ { \mathbf{W} } = ( 2 \pi \hbar ) ^ { - 3 N } \psi _ { \mathbf{W} }$ is called the Wigner function. It was introduced by E.P. Wigner in 1932, [[#References|[a1]]], who interpreted $f _ { \text{w} }$ as a quasi-probability density in the phase space $\mathbf{R} _ { p } ^ { 3 N } \times \mathbf{R} _ { x } ^ { 3 N }$ and showed that it obeyed a kinetic pseudo-differential equation (the Wigner equation) of the form $\dot{f} _ { \text{W} } + p \cdot \nabla f _ { \text{W} } = P f _ { \text{W} }$, where $P$ is a [[Pseudo-differential operator|pseudo-differential operator]] with symbol defined by the potential energy of the system. Wigner went on to discuss how $f _ { \text{w} }$ might be used to calculate quantities of physical interest. In particular, the density is $n ( x , t ) = \int _ { \mathbf{R} ^ { 3 N } } f _ { \text{w} } d p$. Since, in general, the potential energy depends on the density, the Wigner equation is non-linear.

Generalizing to a mixed state, described not by a wave function but by a von Neumann density matrix [[#References|[a2]]]

\begin{equation*} \rho = \sum \lambda _ { i } P _ { i } , \quad 0 \leq \lambda _ { i } \leq 1 , \sum \lambda _ { i } = 1 \end{equation*}

($P_ i$ is the projection onto the vector $\psi _ { i }$):

\begin{equation*} \psi _ { \text{w} } = \sum \lambda _ { i } \int _ { \mathbf{R} ^ { 3 N } } e ^ { i p z / \hbar } \overline { \psi } _ { i } \left( x + \frac { z } { 2 } \right) \psi  _ { i }   \left( x - \frac { z } { 2 } \right) d z. \end{equation*}

Generalizing further, let $A$ be a (bounded) operator on $\mathcal{H}$. Let $\{ u _ { i } \}$ be a basis for $\mathcal{H}$ and write $A _ { k l }$ for $( u _ { k } , A u _ { l } )$, where $( \, . \, , \, . \, )$ is the inner product in $\mathcal{H}$. Then the Wigner transform $A _ { \text{W} }$ of the operator $A$ is

\begin{equation*} A _ { \text{w} } ( x , p ) = \end{equation*}

\begin{equation*} = \sum _ { k , l } A _ { k l } \int _ { \mathbf{R} ^ { 3 N } } e ^ { i p z / \hbar } u _ {  k } \left( x - \frac { z } { 2 } \right) \overline { u_l }  \left( x + \frac { z } { 2 } \right) d z. \end{equation*}

In particular, if $B$ is a trace-class operator on $\mathcal{H}$ and $A$ is bounded as above,

\begin{equation*} \operatorname { Tr } A B = \int _ { \mathbf{R} ^ { 3 N } \times \mathbf{R} ^ { 3 N } } A _ { \mathbf{w} } B _ { \mathbf{w} } d x d p. \end{equation*}

The Wigner transform of an operator is related to the Weyl transform [[#References|[a3]]] of a phase-space function, introduced by H. Weyl in 1950 in an attempt to relate classical and quantum mechanics. Indeed, let $f ( x , p )$ be an appropriate function in ${\bf R} _ { x } ^ { 3 N } \times {\bf R} _ { p } ^ { 3 N }$ (see [[#References|[a4]]] for a definition of  "appropriate" ). Then the Weyl transform of $f$, $\Omega f$, is defined in terms of the [[Fourier transform|Fourier transform]] $\phi$ of $f$ as [[#References|[a5]]]

\begin{equation*} \phi ( \sigma , \tau ) = \int _ { \mathbf{R} ^ { 3 N } \times \mathbf{R} ^ { 3 N } } e ^ { i ( \sigma x + r \cdot p ) / \hbar } f ( x , p ) d x d p. \end{equation*}

Here, $\Omega f = F$ is the operator

\begin{equation*} F = ( 2 \pi \hbar ) ^ { - 6 N } \int _ { \mathbf R ^ { 3 N } \times \mathbf R ^ { 3 N } } e ^ { i ( \sigma .X + r. P ) / \hbar } \phi ( \sigma , \tau ) d \sigma d \tau \end{equation*}

and $X$ is the multiplication operator on $L^{2}$ defined by $( X \psi ) ( x ) = x \psi ( x )$ and $P = - i \hbar \nabla _ { x }$. These are the usual position and momentum operators of quantum mechanics [[#References|[a2]]]. The Weyl and Wigner transforms are mutual inverses: $( \Omega f ) _ {  \operatorname{w} } = f$ and $\Omega A _ { W } = A$ [[#References|[a5]]].

Serious mathematical interest in the Wigner transform revived in 1985, when H. Neunzert published [[#References|[a6]]]. Since then, most mathematical attention has been paid to existence-uniqueness theory for the Wigner equation in $\mathbf{R} ^ { 3 }$ and, more recently, in a closed proper subset of ${\bf R} ^ { n }$, $n = 1,2,3$. While the situation in $\mathbf{R} ^ { 3 }$ is pretty well understood, [[#References|[a7]]], [[#References|[a8]]] the more practical latter situation is still under study (1998), the main problem being the question of appropriate boundary conditions [[#References|[a9]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  E. Wigner,   "On the quantum correction for thermodynamic equilibrium"  ''Phys. Rev.'' , '''40'''  (1932)  pp. 749–759</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J. von Neumann,   "Mathematical foundations of quantum mechanics" , Princeton Univ. Press  (1955)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  H. Weyl,   "The theory of groups and quantum mechanics" , Dover  (1950)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  G.B. Folland,   "Harmonic analysis in phase space" , Princeton Univ. Press  (1989)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  P.F. Zweifel,   "The Wigner transform and the Wigner–Poisson system"  ''Trans. Theor. Stat. Phys.'' , '''22'''  (1993)  pp. 459–484</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  H. Neunzert,   "The nuclear Vlasov equation: methods and results that can (not) be taken over from the  "classical"  case"  ''Il Nuovo Cimento'' , '''87A'''  (1985)  pp. 151–161</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  F. Brezzi,   P. Markowich,   "The three-dimensional Wigner–Poisson problem: existence, uniqueness and approximation"  ''Math. Meth. Appl. Sci.'' , '''14'''  (1991)  pp. 35</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  R. Illner,   H. Lange,   P.F. Zweifel,   "Global existence and asymptotic behaviour of solutions of the Wigner–Poisson and Schrödinger–Poisson systems"  ''Math. Meth. Appl. Sci.'' , '''17'''  (1994)  pp. 349–376</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  P.F. Zweifel,   B. Toomire,   "Quantum transport theory"  ''Trans. Theor. Stat. Phys.'' , '''27'''  (1998)  pp. 347–359</td></tr></table>