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A mapping between a class of (generalized) functions on the phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w1200801.png" /> and the set of closed densely defined operators on the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w1200802.png" /> [[#References|[a1]]] (cf. also [[Generalized function|Generalized function]]; [[Hilbert space|Hilbert space]]). It is defined as follows: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w1200803.png" /> be an arbitrary point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w1200804.png" /> (called phase space) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w1200805.png" /> be an arbitrary vector on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w1200806.png" />. For a point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w1200807.png" />, the Grossmann–Royer operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w1200808.png" /> is defined as [[#References|[a2]]], [[#References|[a3]]]:
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<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/w120080/w1200809.png" /></td> </tr></table>
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Now, take a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008010.png" />. The Weyl mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008011.png" /> is defined as [[#References|[a4]]], [[#References|[a5]]]:
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A mapping between a class of (generalized) functions on the phase space $\mathbf{R} ^ { 2 n }$ and the set of closed densely defined operators on the Hilbert space $L ^ { 2 } ( \mathbf{R} ^ { n } )$ [[#References|[a1]]] (cf. also [[Generalized function|Generalized function]]; [[Hilbert space|Hilbert space]]). It is defined as follows: Let $( q , p )$ be an arbitrary point of $\mathbf{R} ^ { 2 n }$ (called phase space) and let $\psi ( x )$ be an arbitrary vector on $L ^ { 2 } ( \mathbf{R} ^ { n } )$. For a point in $\mathbf{R} ^ { 2 n }$, the Grossmann–Royer operator $\Omega ( q , p )$ is defined as [[#References|[a2]]], [[#References|[a3]]]:
  
<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/w120080/w12008012.png" /></td> </tr></table>
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\begin{equation*} \Omega ( q , p ) \psi ( x ) = 2 ^ { n } \operatorname { exp } \{ 2 i p \cdot ( x - q ) \} \psi ( 2 q - x ). \end{equation*}
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Now, take a function $f ( q , p ) \in L ^ { 2 } ( \mathbf{R} ^ { 2 n } )$. The Weyl mapping $W$ is defined as [[#References|[a4]]], [[#References|[a5]]]:
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\begin{equation*} W ( f ) = \frac { 1 } { 2 \pi } \int _ { R ^ { 2 n } } f ( q , p ) \Omega ( q , p ) d q d p. \end{equation*}
  
 
The Weyl mapping defines the Weyl correspondence between functions and operators. It has the following properties:
 
The Weyl mapping defines the Weyl correspondence between functions and operators. It has the following properties:
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i) It is linear and one-to-one.
 
i) It is linear and one-to-one.
  
ii) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008013.png" /> is bounded, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008014.png" /> is also bounded.
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ii) If $f$ is bounded, the operator $W ( f )$ is also bounded.
  
iii) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008015.png" /> is real, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008016.png" /> is self-adjoint (cf. also [[Self-adjoint operator|Self-adjoint operator]]).
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iii) If $f$ is real, $W ( f )$ is self-adjoint (cf. also [[Self-adjoint operator|Self-adjoint operator]]).
  
iv) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008017.png" />, the Schwartz space, and define the Weyl product as [[#References|[a6]]]:
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iv) Let $f ( q , p ) , g ( q , p ) \in S ( {\bf R} ^ { 2 n } )$, the Schwartz space, and define the Weyl product as [[#References|[a6]]]:
  
<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/w120080/w12008018.png" /></td> </tr></table>
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\begin{equation*} ( f \times g ) ( q , p ) : = W ^ { - 1 } ( W ( f ) .W ( g ) ). \end{equation*}
  
The Weyl product defines an algebra structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008019.png" />, which admits a closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008020.png" /> with the topology of the space of tempered distributions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008021.png" /> (cf. also [[Generalized functions, space of|Generalized functions, space of]]). The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008022.png" /> includes the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008023.png" /> and the Weyl mapping can be uniquely extended to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008024.png" />.
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The Weyl product defines an algebra structure on $S ( \mathbf{R} ^ { 2 n } )$, which admits a closure $\mathcal{M} ( \mathbf{R} ^ { 2 n } )$ with the topology of the space of tempered distributions, $S ^ { \prime } ( \mathbf{R} ^ { 2 n } )$ (cf. also [[Generalized functions, space of|Generalized functions, space of]]). The algebra $\mathcal{M} ( \mathbf{R} ^ { 2 n } )$ includes the space $L ^ { 2 } ( \mathbf{R} ^ { 2 n } )$ and the Weyl mapping can be uniquely extended to $\mathcal{M} ( \mathbf{R} ^ { 2 n } )$.
  
v) Obviously, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008025.png" />.
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v) Obviously, $W ( f \times g ) = W ( f ) . W ( g )$.
  
vi) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008026.png" /> is the multiplication operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008028.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008030.png" /> denotes the symmetric product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008031.png" /> factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008033.png" /> factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008034.png" />.
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vi) If $Q$ is the multiplication operator on $L ^ { 2 } ( \mathbf{R} ^ { n } )$ and $P = - i \overset{\rightharpoonup}{ \nabla }$, then $W ( q ^ { r } p ^ { s } ) = ( Q ^ { r } P ^ { s } )_S $, where $S$ denotes the symmetric product of $r$ factors $Q$ and $s$ factors $P$.
  
vii) For any positive trace-class operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008035.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008036.png" />, there exists a signed measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008037.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008038.png" />, such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008039.png" />,
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vii) For any positive trace-class operator $\rho$ on $L ^ { 2 } ( \mathbf{R} ^ { n } )$, there exists a signed measure $d\mu ( q , p )$ on $\mathbf{R} ^ { 2 n }$, such that for any $a , b \in \mathbf{R} ^ { n }$,
  
<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/w120080/w12008040.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008040.png"/></td> </tr></table>
  
This measure has a Radon–Nikodým derivative (cf. also [[Radon–Nikodým theorem|Radon–Nikodým theorem]]) with respect to the [[Lebesgue measure|Lebesgue measure]], which is called the Wigner function associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008041.png" />.
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This measure has a Radon–Nikodým derivative (cf. also [[Radon–Nikodým theorem|Radon–Nikodým theorem]]) with respect to the [[Lebesgue measure|Lebesgue measure]], which is called the Wigner function associated to $\rho$.
  
The Weyl correspondence is used by physicists to formulate quantum mechanics of non-relativistic systems without spin or other constraints on the flat phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008042.png" /> [[#References|[a5]]].
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The Weyl correspondence is used by physicists to formulate quantum mechanics of non-relativistic systems without spin or other constraints on the flat phase space $\mathbf{R} ^ { 2 n }$ [[#References|[a5]]].
  
The Stratonovich–Weyl correspondence [[#References|[a4]]], [[#References|[a5]]], [[#References|[a7]]] or Stratonovich–Weyl mapping generalizes the Weyl mapping to other types of phase spaces. Choose a co-adjoint orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008043.png" /> of the representation group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008044.png" /> of a certain [[Lie group|Lie group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008045.png" /> of symmetries of a given physical system as phase space. The [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008046.png" /> used here supports a linear unitary irreducible representation of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008047.png" /> associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008048.png" />. Then, a generalization of the Grossmann–Royer operator is needed, associating each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008049.png" /> of the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008050.png" /> with a self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008051.png" />. Then, for a suitable class of measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008052.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008053.png" />, one defines: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008055.png" /> is a [[Measure|measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008056.png" /> that is invariant under the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008057.png" />; such a measure is uniquely defined, up to a multiplicative constant.
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The Stratonovich–Weyl correspondence [[#References|[a4]]], [[#References|[a5]]], [[#References|[a7]]] or Stratonovich–Weyl mapping generalizes the Weyl mapping to other types of phase spaces. Choose a co-adjoint orbit $X$ of the representation group $\overline { G }$ of a certain [[Lie group|Lie group]] $G$ of symmetries of a given physical system as phase space. The [[Hilbert space|Hilbert space]] $\mathcal{H} ( X )$ used here supports a linear unitary irreducible representation of the group $\overline { G }$ associated to $X$. Then, a generalization of the Grossmann–Royer operator is needed, associating each point $u$ of the orbit $X$ with a self-adjoint operator $\Omega ( u )$. Then, for a suitable class of measurable functions $f ( u )$ on $X$, one defines: $W ( f ) = \int _ { X } f ( u ) \Omega ( u ) d \mu _ { X } ( u )$, where $d \mu _ { X } ( u )$ is a [[Measure|measure]] on $X$ that is invariant under the action of $\overline { G }$; such a measure is uniquely defined, up to a multiplicative constant.
  
The Weyl correspondence is a particular case of the Stratonovich–Weyl correspondence for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008058.png" /> is the Heisenberg group, [[#References|[a1]]], [[#References|[a5]]].
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The Weyl correspondence is a particular case of the Stratonovich–Weyl correspondence for which $G$ is the Heisenberg group, [[#References|[a1]]], [[#References|[a5]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Weyl,  "The theory of groups and quantum mechanics" , Dover  (1931)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Grossmann,  "Parity operator and quantization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120080/w12008059.png" /> functions"  ''Comm. Math. Phys.'' , '''48'''  (1976)  pp. 191</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Royer,  "Wigner function as the expectation value of a parity operator"  ''Phys. Rev. A'' , '''15'''  (1977)  pp. 449</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.M. Gracia-Bondia,  J.C. Varilly,  "The Moyal representation of spin"  ''Ann. Phys. (NY)'' , '''190'''  (1989)  pp. 107</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Gadella,  "Moyal formulation of quantum mechanics"  ''Fortschr. Phys.'' , '''43'''  (1995)  pp. 229</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.M. Gracia-Bondia,  J.C. Varilly,  "Algebras of distributions suitable for phase space quantum mechanics"  ''J. Math. Phys.'' , '''29'''  (1988)  pp. 869</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.C. Varilly,  "The Stratonovich–Weyl correspondence: a general approach to Wigner functions"  ''BIBOS preprint 345 Univ. Bielefeld, Germany''  (1988)</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  H. Weyl,  "The theory of groups and quantum mechanics" , Dover  (1931)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  A. Grossmann,  "Parity operator and quantization of $\delta$ functions"  ''Comm. Math. Phys.'' , '''48'''  (1976)  pp. 191</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  A. Royer,  "Wigner function as the expectation value of a parity operator"  ''Phys. Rev. A'' , '''15'''  (1977)  pp. 449</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  J.M. Gracia-Bondia,  J.C. Varilly,  "The Moyal representation of spin"  ''Ann. Phys. (NY)'' , '''190'''  (1989)  pp. 107</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  M. Gadella,  "Moyal formulation of quantum mechanics"  ''Fortschr. Phys.'' , '''43'''  (1995)  pp. 229</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  J.M. Gracia-Bondia,  J.C. Varilly,  "Algebras of distributions suitable for phase space quantum mechanics"  ''J. Math. Phys.'' , '''29'''  (1988)  pp. 869</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  J.C. Varilly,  "The Stratonovich–Weyl correspondence: a general approach to Wigner functions"  ''BIBOS preprint 345 Univ. Bielefeld, Germany''  (1988)</td></tr></table>

Revision as of 16:55, 1 July 2020

A mapping between a class of (generalized) functions on the phase space $\mathbf{R} ^ { 2 n }$ and the set of closed densely defined operators on the Hilbert space $L ^ { 2 } ( \mathbf{R} ^ { n } )$ [a1] (cf. also Generalized function; Hilbert space). It is defined as follows: Let $( q , p )$ be an arbitrary point of $\mathbf{R} ^ { 2 n }$ (called phase space) and let $\psi ( x )$ be an arbitrary vector on $L ^ { 2 } ( \mathbf{R} ^ { n } )$. For a point in $\mathbf{R} ^ { 2 n }$, the Grossmann–Royer operator $\Omega ( q , p )$ is defined as [a2], [a3]:

\begin{equation*} \Omega ( q , p ) \psi ( x ) = 2 ^ { n } \operatorname { exp } \{ 2 i p \cdot ( x - q ) \} \psi ( 2 q - x ). \end{equation*}

Now, take a function $f ( q , p ) \in L ^ { 2 } ( \mathbf{R} ^ { 2 n } )$. The Weyl mapping $W$ is defined as [a4], [a5]:

\begin{equation*} W ( f ) = \frac { 1 } { 2 \pi } \int _ { R ^ { 2 n } } f ( q , p ) \Omega ( q , p ) d q d p. \end{equation*}

The Weyl mapping defines the Weyl correspondence between functions and operators. It has the following properties:

i) It is linear and one-to-one.

ii) If $f$ is bounded, the operator $W ( f )$ is also bounded.

iii) If $f$ is real, $W ( f )$ is self-adjoint (cf. also Self-adjoint operator).

iv) Let $f ( q , p ) , g ( q , p ) \in S ( {\bf R} ^ { 2 n } )$, the Schwartz space, and define the Weyl product as [a6]:

\begin{equation*} ( f \times g ) ( q , p ) : = W ^ { - 1 } ( W ( f ) .W ( g ) ). \end{equation*}

The Weyl product defines an algebra structure on $S ( \mathbf{R} ^ { 2 n } )$, which admits a closure $\mathcal{M} ( \mathbf{R} ^ { 2 n } )$ with the topology of the space of tempered distributions, $S ^ { \prime } ( \mathbf{R} ^ { 2 n } )$ (cf. also Generalized functions, space of). The algebra $\mathcal{M} ( \mathbf{R} ^ { 2 n } )$ includes the space $L ^ { 2 } ( \mathbf{R} ^ { 2 n } )$ and the Weyl mapping can be uniquely extended to $\mathcal{M} ( \mathbf{R} ^ { 2 n } )$.

v) Obviously, $W ( f \times g ) = W ( f ) . W ( g )$.

vi) If $Q$ is the multiplication operator on $L ^ { 2 } ( \mathbf{R} ^ { n } )$ and $P = - i \overset{\rightharpoonup}{ \nabla }$, then $W ( q ^ { r } p ^ { s } ) = ( Q ^ { r } P ^ { s } )_S $, where $S$ denotes the symmetric product of $r$ factors $Q$ and $s$ factors $P$.

vii) For any positive trace-class operator $\rho$ on $L ^ { 2 } ( \mathbf{R} ^ { n } )$, there exists a signed measure $d\mu ( q , p )$ on $\mathbf{R} ^ { 2 n }$, such that for any $a , b \in \mathbf{R} ^ { n }$,

This measure has a Radon–Nikodým derivative (cf. also Radon–Nikodým theorem) with respect to the Lebesgue measure, which is called the Wigner function associated to $\rho$.

The Weyl correspondence is used by physicists to formulate quantum mechanics of non-relativistic systems without spin or other constraints on the flat phase space $\mathbf{R} ^ { 2 n }$ [a5].

The Stratonovich–Weyl correspondence [a4], [a5], [a7] or Stratonovich–Weyl mapping generalizes the Weyl mapping to other types of phase spaces. Choose a co-adjoint orbit $X$ of the representation group $\overline { G }$ of a certain Lie group $G$ of symmetries of a given physical system as phase space. The Hilbert space $\mathcal{H} ( X )$ used here supports a linear unitary irreducible representation of the group $\overline { G }$ associated to $X$. Then, a generalization of the Grossmann–Royer operator is needed, associating each point $u$ of the orbit $X$ with a self-adjoint operator $\Omega ( u )$. Then, for a suitable class of measurable functions $f ( u )$ on $X$, one defines: $W ( f ) = \int _ { X } f ( u ) \Omega ( u ) d \mu _ { X } ( u )$, where $d \mu _ { X } ( u )$ is a measure on $X$ that is invariant under the action of $\overline { G }$; such a measure is uniquely defined, up to a multiplicative constant.

The Weyl correspondence is a particular case of the Stratonovich–Weyl correspondence for which $G$ is the Heisenberg group, [a1], [a5].

References

[a1] H. Weyl, "The theory of groups and quantum mechanics" , Dover (1931)
[a2] A. Grossmann, "Parity operator and quantization of $\delta$ functions" Comm. Math. Phys. , 48 (1976) pp. 191
[a3] A. Royer, "Wigner function as the expectation value of a parity operator" Phys. Rev. A , 15 (1977) pp. 449
[a4] J.M. Gracia-Bondia, J.C. Varilly, "The Moyal representation of spin" Ann. Phys. (NY) , 190 (1989) pp. 107
[a5] M. Gadella, "Moyal formulation of quantum mechanics" Fortschr. Phys. , 43 (1995) pp. 229
[a6] J.M. Gracia-Bondia, J.C. Varilly, "Algebras of distributions suitable for phase space quantum mechanics" J. Math. Phys. , 29 (1988) pp. 869
[a7] J.C. Varilly, "The Stratonovich–Weyl correspondence: a general approach to Wigner functions" BIBOS preprint 345 Univ. Bielefeld, Germany (1988)
How to Cite This Entry:
Weyl correspondence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weyl_correspondence&oldid=50059
This article was adapted from an original article by M. Gadella (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article