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 ![]() |
[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) |
Weyl correspondence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weyl_correspondence&oldid=18852