Wigner-Weyl transform

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, [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 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 [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 [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 [a4] for a definition of "appropriate" ). Then the Weyl transform of $f$, $\Omega f$, is defined in terms of the Fourier transform $\phi$ of $f$ as [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 [a2]. The Weyl and Wigner transforms are mutual inverses: $( \Omega f ) _ { \operatorname{w} } = f$ and $\Omega A _ { W } = A$ [a5].

Serious mathematical interest in the Wigner transform revived in 1985, when H. Neunzert published [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, [a7], [a8] the more practical latter situation is still under study (1998), the main problem being the question of appropriate boundary conditions [a9].

References

[a1]	E. Wigner, "On the quantum correction for thermodynamic equilibrium" Phys. Rev. , 40 (1932) pp. 749–759
[a2]	J. von Neumann, "Mathematical foundations of quantum mechanics" , Princeton Univ. Press (1955)
[a3]	H. Weyl, "The theory of groups and quantum mechanics" , Dover (1950)
[a4]	G.B. Folland, "Harmonic analysis in phase space" , Princeton Univ. Press (1989)
[a5]	P.F. Zweifel, "The Wigner transform and the Wigner–Poisson system" Trans. Theor. Stat. Phys. , 22 (1993) pp. 459–484
[a6]	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
[a7]	F. Brezzi, P. Markowich, "The three-dimensional Wigner–Poisson problem: existence, uniqueness and approximation" Math. Meth. Appl. Sci. , 14 (1991) pp. 35
[a8]	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
[a9]	P.F. Zweifel, B. Toomire, "Quantum transport theory" Trans. Theor. Stat. Phys. , 27 (1998) pp. 347–359