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''Weyl–Wigner transform''
 
''Weyl–Wigner transform''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w1201901.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w1201902.png" />, be a ray in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w1201903.png" />. Then, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w1201904.png" />, the Wigner transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w1201905.png" /> is
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w1201906.png" /></td> </tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w1201907.png" /> is Planck's constant. The quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w1201908.png" /> is called the Wigner function. It was introduced by E.P. Wigner in 1932, [[#References|[a1]]], who interpreted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w1201909.png" /> as a quasi-probability density in the phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019010.png" /> and showed that it obeyed a kinetic pseudo-differential equation (the Wigner equation) of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019012.png" /> is a [[Pseudo-differential operator|pseudo-differential operator]] with symbol defined by the potential energy of the system. Wigner went on to discuss how <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019013.png" /> might be used to calculate quantities of physical interest. In particular, the density is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019014.png" />. Since, in general, the potential energy depends on the density, the Wigner equation is non-linear.
+
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]]]
 
Generalizing to a mixed state, described not by a wave function but by a von Neumann density matrix [[#References|[a2]]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019015.png" /></td> </tr></table>
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\begin{equation*} \rho = \sum \lambda _ { i } P _ { i } , \quad 0 \leq \lambda _ { i } \leq 1 , \sum \lambda _ { i } = 1 \end{equation*}
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019016.png" /> is the projection onto the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019017.png" />):
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($P_ i$ is the projection onto the vector $\psi _ { i }$):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019018.png" /></td> </tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019019.png" /> be a (bounded) operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019020.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019021.png" /> be a basis for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019022.png" /> and write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019023.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019025.png" /> is the inner product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019026.png" />. Then the Wigner transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019027.png" /> of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019028.png" /> is
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019029.png" /></td> </tr></table>
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\begin{equation*} A _ { \text{w} } ( x , p ) = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019030.png" /></td> </tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019031.png" /> is a trace-class operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019033.png" /> is bounded as above,
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In particular, if $B$ is a trace-class operator on $\mathcal{H}$ and $A$ is bounded as above,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019034.png" /></td> </tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019035.png" /> be an appropriate function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019036.png" /> (see [[#References|[a4]]] for a definition of  "appropriate" ). Then the Weyl transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019038.png" />, is defined in terms of the [[Fourier transform|Fourier transform]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019039.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019040.png" /> as [[#References|[a5]]]
+
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]]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019041.png" /></td> </tr></table>
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\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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019042.png" /> is the operator
+
Here, $\Omega f = F$ is the operator
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019043.png" /></td> </tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019044.png" /> is the multiplication operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019045.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019047.png" />. These are the usual position and momentum operators of quantum mechanics [[#References|[a2]]]. The Weyl and Wigner transforms are mutual inverses: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019049.png" /> [[#References|[a5]]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019050.png" /> and, more recently, in a closed proper subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019052.png" />. While the situation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019053.png" /> 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]]].
+
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====
 
====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>
+
<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>

Latest revision as of 16:59, 1 July 2020

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
How to Cite This Entry:
Wigner-Weyl transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wigner-Weyl_transform&oldid=50338
This article was adapted from an original article by P.F. Zweifel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article