Weyl correspondence

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].

References

[a1]	H. Weyl, "The theory of groups and quantum mechanics" , Dover (1931)
[a2]	A. Grossmann, "Parity operator and quantization of 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)