# Weyl correspondence

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A mapping between a class of (generalized) functions on the phase space and the set of closed densely defined operators on the Hilbert space [a1] (cf. also Generalized function; Hilbert space). It is defined as follows: Let be an arbitrary point of (called phase space) and let be an arbitrary vector on . For a point in , the Grossmann–Royer operator is defined as [a2], [a3]: Now, take a function . The Weyl mapping is defined as [a4], [a5]: 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 is bounded, the operator is also bounded.

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

iv) Let , the Schwartz space, and define the Weyl product as [a6]: The Weyl product defines an algebra structure on , which admits a closure with the topology of the space of tempered distributions, (cf. also Generalized functions, space of). The algebra includes the space and the Weyl mapping can be uniquely extended to .

v) Obviously, .

vi) If is the multiplication operator on and , then , where denotes the symmetric product of factors and factors .

vii) For any positive trace-class operator on , there exists a signed measure on , such that for any , 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 .

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 [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 of the representation group of a certain Lie group of symmetries of a given physical system as phase space. The Hilbert space used here supports a linear unitary irreducible representation of the group associated to . Then, a generalization of the Grossmann–Royer operator is needed, associating each point of the orbit with a self-adjoint operator . Then, for a suitable class of measurable functions on , one defines: , where is a measure on that is invariant under the action of ; 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 is the Heisenberg group, [a1], [a5].

How to Cite This Entry:
Weyl correspondence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weyl_correspondence&oldid=18852
This article was adapted from an original article by M. Gadella (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article