Weyl correspondence

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].

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