View source for Wigner-Weyl 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

<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>

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.

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>

(<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" />):

<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>

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

<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>

<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>

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,

<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>

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]]]

<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>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120190/w12019042.png" /> 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>

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]]].

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]]].

====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>