The Stratonovich-Weyl correspondence:
A general approach to Wigner functions

Joseph C. Várilly

Forschungszentrum BiBoS, Universität Bielefeld, D–4800 Bielefeld 1, Germany∗

BiBoS preprint # 345 (March 1989)

Abstract

A formalism is proposed for developing phase-space representations of elementary quantum
systems under general invariance groups. Several examples are discussed, including the usual
Weyl calculus, the Moyal formulation for spin, Poincaré disk quantizations, and the phase-space
calculus for Galilean spinning particles.

Introduction
Ever since the 1949 paper of Moyal [1], which showed how the Weyl correspondence [2] enables
one to develop Quantum Mechanics as a theory of functions on phase space, composed according
to the “twisted” or Moyal product, with states being represented by their Wigner functions [3], it has
been thought useful to extend this formalism beyond the arena of nonrelativistic spinless particles.
The case of spinning particles seemed for some time to be particularly troublesome. In fact, an
early suggestion of Stratonovich [4] for the spin case contains the seed of a Moyal theory for spin,
as has recently been shown [5].

In this paper, I develop the main idea of [5] as a general recipe, which I call the “Stratonovich–
Weyl correspondence”, linking elementary classical systems to the elementary quantum systems
with the same invariance group. The basic property of a Moyal formulation, namely, that quantum
expectation values should be computed “classically” by integrating over the phase space, turns out
to be enough (together with group covariance) to identify the twisted products (and hence, the
symbol calculus) for many invariance groups.

Examples are given to show how the Stratonovich–Weyl correspondence works for the “ordinary”
Weyl calculus, for pure spins, for Poincaré-disk quantizations, and for Galilean spinning particles.

1 Moyal quantum mechanics
The Moyal approach to Quantum Mechanics, as a theory which sticks as closely as possible to
classical mechanical formulations, may be considered to have five essential aspects.

∗Permanent address: Escuela de Matemática, Universidad de Costa Rica, 11501 San José, Costa Rica

1



1. Both observables and states are (generalized) functions on a classical phase space 𝑀 . For
example, in the simplest case of a flat phase space 𝑀 = ℝ2𝑁 , with coordinates 𝑢 = (𝒒, 𝒑),
where 𝒒, 𝒑 give the position and momentum coordinates in ℝ𝑁 , we consider observables of
the form 𝐻 = | 𝒑 |2/2𝑚 +𝑉 (𝒒), and states such as 𝜌 = 𝑒−(𝒒·𝒒+ 𝒑· 𝒑)/2.

2. Expected values should be computed “classically”, that is, by integrating over the phase space:

⟨ 𝑓 ⟩𝜌 =
∫
𝑀

𝑓 (𝑢) 𝜌(𝑢) 𝑑𝑢
/ ∫

𝑀

𝜌(𝑢) 𝑑𝑢.

3. Quantum mechanics enters via the rule for the composition of observables, which is given by
a nonlocal “twisted product”:

( 𝑓 × ℎ) (𝑢) =
∫
𝑀

∫
𝑀

𝐿 (𝑢, 𝑣, 𝑤) 𝑓 (𝑣) ℎ(𝑤) 𝑑𝑣 𝑑𝑤.

Of course, if one takes 𝐿 (𝑢, 𝑣, 𝑤) = 𝛿(𝑢 − 𝑣) 𝛿(𝑢 − 𝑤), one recovers the ordinary pointwise
product on 𝑀; but since the uncertainty principle forbids localization at points of phase space,
we must exclude this option and search for another trikernel 𝐿, which will not be local.

4. The twisted product is equivariant under canonical symmetries of 𝑀 . That is, one is given a
Lie group 𝐺, acting on the phase space 𝑀 by symplectic transformations 𝑢 ↦→ 𝑔 · 𝑢, in such
a way that

𝑓 𝑔 × ℎ𝑔 = ( 𝑓 × ℎ)𝑔, with 𝑓 𝑔 (𝑢) := 𝑓 (𝑔−1 · 𝑢).
In terms of the trikernel 𝐿, this condition becomes

𝐿 (𝑔 · 𝑢, 𝑔 · 𝑣, 𝑔 · 𝑤) = 𝐿 (𝑢, 𝑣, 𝑤) for all 𝑔 ∈ 𝐺,

which, together with Condition 2, restricts the possibilities for 𝐿. For instance, in the case
𝑀 = ℝ2𝑁 , we can take 𝐺 = ISp(2𝑁,ℝ), the group of inhomogeneous linear symplectic
transformations of ℝ2𝑁 , in which case one obtains:

𝐿 (𝒒1, 𝒑1; 𝒒2, 𝒑2; 𝒒3, 𝒑3)

= (constant) exp
{

2𝑖
ℏ

[
𝒒1 · 𝒑2 − 𝒒2 · 𝒑1 + 𝒒2 · 𝒑3 − 𝒒3 · 𝒑2 + 𝒒3 · 𝒑1 − 𝒒1 · 𝒑3

]}
,

where the only remaining freedom is the choice of the Planck constant ℏ.

5. Some correspondence rule 𝑓 ↦→ Op( 𝑓 ), of functions on 𝑀 to Hilbert-space operators,
links the theory to ordinary Quantum Mechanics, and establishes an equivalence of the two
formalisms. For the flat case, this is of course the Weyl correspondence rule:

Op( 𝑓 ) =
∫
ℝ2𝑁

𝑓 (𝒙, 𝒚)𝑊 (𝒙, 𝒚) 𝑑
𝑁𝒙 𝑑𝑁 𝒚

(2𝜋ℏ)𝑁
(1)

where𝑊 (𝒙, 𝒚) = exp{𝑖(𝒙 · 𝑸 + 𝒚 · 𝑷)/ℏ} are the Weyl operators.

2



The reason for outlining the Moyal formalism in so abstract a setting is to provide a means of
going beyond the “flat case” to incorporate spinning and relativistic particles as well. Since, as
Moyal noted [1], the “opposite” of the Weyl correspondence is simply the assignment of a transition
operator |Ψ⟩⟨Φ| to its Wigner function 𝑓ΦΨ, where

𝑓ΦΨ (𝒒, 𝒑) =
∫
ℝ𝑁

𝑒𝑖 𝒑·𝒙/ℏΦ(𝒒 − 1
2𝒙) Ψ(𝒒 + 1

2𝒙) 𝑑
𝑁𝒙

=

∫
ℝ𝑁

𝑒𝑖𝒒·𝒚/ℏΦ̂( 𝒑 + 1
2 𝒚) Ψ̂( 𝒑 − 1

2 𝒚) 𝑑
𝑁 𝒚, (2)

we may obtain “Wigner functions” for more general systems by inverting the generalized Weyl
correspondence rule. This can be done for spinning particles in an essentially unique way [5, 6], as
I shall indicate shortly.

A key observation, made independently by Grossmann [7] and Royer [8], is that the Weyl
correspondence can be recast more simply by, so to speak, applying the Fourier inversion formula
to (1) to obtain:

Op( 𝑓 ) =
∫
ℝ2𝑁

𝑓 (𝒒, 𝒑) Π(𝒒, 𝒑) 𝑑
𝑁𝒒 𝑑𝑁 𝒑

(2𝜋ℏ)𝑁
(3)

where the “parity operators” Π(𝒒, 𝒑) are defined by:[
Π(𝒒, 𝒑)Ψ

]
(𝝃) := 2𝑁 exp

{2𝑖
ℏ
𝒒 · ( 𝒑 − 𝝃)

}
Ψ(2 𝒑 − 𝝃) (4)

in the momentum-space representation. Henceforth we will consider only the form (3) of the Weyl
correspondence rule.

The rule (3) is also the starting point for a “discrete approximation” to the Weyl calculus,
considered by Cohendet et al. [9] (see also Wootters [10]). Here one takes 𝑀 = ℤ𝑁 × ℤ𝑁 , where
ℤ𝑁 is the cyclic group of order 𝑁; if 𝑁 is odd, one may define

Op( 𝑓 ) :=
1
𝑁

∑︁
𝑞,𝑝∈ℤ𝑁

𝑓 (𝑞, 𝑝) Δ𝑞𝑝

where the parity operators Δ𝑞𝑝 act on ℂ𝑁 = 𝐿2(ℤ𝑁 ) by[
Δ𝑞𝑝Ψ

]
(𝑘) := exp

{4𝜋𝑖
𝑁
𝑞(𝑝 − 𝑘)

}
Ψ(2𝑝 − 𝑘).

The underlying symmetry group is the finite Heisenberg group 𝐴𝑁 [11, 12]; however, as 𝑀 is not
a symplectic manifold, this case does not lie completely within the class of Moyal quantizations
which we consider here.

2 Elementary classical systems
The underlying phase space𝑀 is taken, as is usual in classical mechanics, to be a symplectic manifold
with a nondegenerate closed 2-form (symplectic form) 𝜔𝑀 . We also assume that a symmetry group
𝐺 is given, which acts on 𝑀 by symplectomorphisms (i.e., diffeomorphisms leaving 𝜔𝑀 invariant).
An elementary classical system is a pair (𝐺, 𝑀) where the action of 𝐺 on 𝑀 is transitive, that is,
(𝑀,𝜔𝑀) is a 𝐺-homogeneous symplectic manifold.

3



We shall henceforth assume that 𝐺 is a connected Lie group. The classification of 𝐺-homoge-
neous symplectic manifolds is known: see [13, 14], for example. (𝑀,𝜔𝑀) is locally isomorphic to
an orbit of some affine action of𝐺 on the dual space g∗ of its Lie algebra g, and the linear part of this
action is always the coadjoint action of 𝐺. The inhomogeneous part of the action is classified by
the cohomology space 𝐻2(g,ℝ); this part can be removed by lifting to a suitable central extension
of 𝐺, which can be taken simply connected, so that (𝑀,𝜔𝑀) will appear as just a coadjoint orbit of
the extended group.

A different role of central extensions of 𝐺 occurs in the study of projective unitary irreducible
representations of 𝐺. These are also classified by 𝐻2(g,ℝ). Indeed, if 𝐺 is the simply connected
covering group of 𝐺, the required extension 𝐺 is given by the exact sequence [15]:

0−→𝐻2(g,ℝ) −→𝐺 −→𝐺 −→ 0. (5)

𝐺 is called the “splitting group” of 𝐺, since any projective unitary irreducible representation of 𝐺
can be lifted to a linear unitary irreducible representation of 𝐺.

As an example of a splitting group, we may take 𝐺 to be the abelian group ℝ2𝑁 ; then g = ℝ2𝑁

and 𝐻2(g,ℝ) = ℝ. If {𝑋1, . . . , 𝑋𝑁 , 𝑌1, . . . , 𝑌𝑁 } is a basis for g, then a basis for the extended Lie
algebra g is {𝑋1, . . . , 𝑋𝑁 , 𝑌1, . . . , 𝑌𝑁 , 𝑍}, where 𝑍 is central and [𝑋𝑖, 𝑌 𝑗 ] = 𝛿𝑖 𝑗𝑍 . The splitting
group 𝐺 is thus the (2𝑁 + 1)-dimensional Heisenberg group ℍ2𝑁+1.

A related example is the Galilean group 𝐺 = ℝ4 ⋉ (ℝ3 ⋉ 𝑆𝑂 (3)), where ⋉ denotes a semidirect
product, parametrized by (𝑏, 𝒂, 𝒗, 𝑅) with 𝑏 ∈ ℝ, 𝒂, 𝒗 ∈ ℝ3, 𝑅 ∈ 𝑆𝑂 (3), acting on ℝ4 by
(𝑏, 𝒂, 𝒗, 𝑅) · (𝑡, 𝒙) = (𝑡 + 𝑏, 𝑅𝒙 − 𝒗𝑡 + 𝒂). Again 𝐻2(g,ℝ) = ℝ, so the Lie algebra g of 𝐺 is
11-dimensional and has a basis {𝐻, 𝑃1, 𝑃2, 𝑃3, 𝐽1, 𝐽2, 𝐽3, 𝐾1, 𝐾2, 𝐾3, 𝑀} satisfying:

[𝐽𝑖, 𝐽 𝑗 ] = Y𝑖 𝑗𝑘𝐽𝑘 , [𝐽𝑖, 𝑃 𝑗 ] = Y𝑖 𝑗𝑘𝑃𝑘 , [𝐽𝑖, 𝐾 𝑗 ] = Y𝑖 𝑗𝑘𝐾𝑘 ,
[𝐾𝑖, 𝐻] = 𝑃𝑖, [𝐾𝑖, 𝑃 𝑗 ] = 𝛿𝑖 𝑗𝑀, (6)

where 𝑀 is the central basis element. Notice that the subalgebra which is generated by the elements
{𝐾1, 𝐾2, 𝐾3, 𝑃1, 𝑃2, 𝑃3, 𝑀} is the Heisenberg algebra h7, so that the Heisenberg group is a subgroup
of the splitting group 𝐺 of the Galilei group.

A further example arises from the restricted Poincaré group 𝐺 = ℝ4 ⋉ 𝑆𝑂0(3, 1), with 𝐺 =

ℝ4 ⋉ 𝑆𝐿 (2,ℂ). Here 𝐻2(g,ℝ) = 0, so that 𝐺 = 𝐺 is again a 10-dimensional group.

The diagram (5) is also useful in a purely classical theory, as was noticed by Martı́nez-
Alonso [16]: it enables us to lift affine symplectic actions of 𝐺 to linear symplectic actions
of 𝐺. Thus, the 𝐺-homogeneous symplectic manifolds appear as coadjoint orbits of 𝐺, which may
then be called elementary classical systems [16].

A final ingredient of the theory should be a bridge between coadjoint orbits and linear unitary
irreducible representations of 𝐺. Indeed, for nilpotent groups, the Kirillov correspondence [17]
provides this link.

For other classes of groups, such as semidirect products of abelian and semisimple groups, the
situation is not so clear; suffice it to say that the Kirillov recipe for assigning orbits to representations
is in general neither onto (the orbits must satisfy certain “integrality” conditions) nor one-to-one.
The approach to be outlined here attempts to finesse this problem by assuming that in each particular
case, an orbit and a representation have already been “matched” by some Kirillov-type recipe.

4



We turn now to explicit definitions. Let 𝐺 be a connected Lie group (we will not consider
discrete symmetries here), and let g be its Lie algebra. Let g∗ denote the dual vector space of g. We
take note of the adjoint and coadjoint actions of 𝐺, which are the representations:

Ad: 𝐺 → End(g), determined by exp[𝑡 (Ad 𝑔)𝑋] = 𝑔(exp 𝑡𝑋)𝑔−1,

Coad: 𝐺 → End(g∗), defined by ⟨(Coad 𝑔)𝑢, 𝑋⟩ = ⟨𝑢, (Ad 𝑔−1)𝑋⟩, (7)

for 𝑔 ∈ 𝐺, 𝑋 ∈ g, 𝑢 ∈ g∗. We will abbreviate 𝑔 · 𝑢 ≡ (Coad 𝑔)𝑢.
In terms of the linear coordinate functions on g∗:

b𝑋 (𝑢) := ⟨𝑢, 𝑋⟩, for 𝑋 ∈ g, (8)

the coadjoint action is determined by

b𝑋 (𝑔 · 𝑢) = b(Ad 𝑔−1)𝑋 (𝑢). (9)

Thus, in practice, one computes Ad by (7) and uses (9) to determine Coad in terms of the coordinate
functions on g∗.

Now g∗ carries a natural Poisson bracket structure, given by

{ 𝑓 , ℎ}𝑃 (𝑢) := ⟨𝑢, [𝑑𝑓 (𝑢), 𝑑ℎ(𝑢)]⟩,

since a covector 𝑑𝑓 (𝑢) at 𝑢 ∈ g∗ may be identified with an element of g. In particular,

{b𝑋 , b𝑌 }𝑃 (𝑢) = ⟨𝑢, [𝑋,𝑌 ]⟩ = b[𝑋,𝑌 ] (𝑢). (10)

If we write 𝑥𝑖 := b𝑋𝑖 as 𝑋𝑖 runs through a basis for g, then the Poisson brackets may be explicitly
computed by

{ 𝑓 , ℎ}𝑃 =
𝜕 𝑓

𝜕𝑥𝑖

𝜕ℎ

𝜕𝑥 𝑗
{b𝑋𝑖 , b𝑋 𝑗

}𝑃

=
𝜕 𝑓

𝜕𝑥𝑖

𝜕ℎ

𝜕𝑥 𝑗
b[𝑋𝑖 ,𝑋 𝑗 ] = 𝑐

𝑘
𝑖 𝑗

𝜕 𝑓

𝜕𝑥𝑖

𝜕ℎ

𝜕𝑥 𝑗
𝑥𝑘 (11)

where 𝑐 𝑘
𝑖 𝑗

are the structure constants for g.
The Poisson bracket structure is 𝐺-equivariant: if 𝑓 𝑔 (𝑢) := 𝑓 (𝑔−1 · 𝑢), then(

{ 𝑓 , ℎ}𝑃
)𝑔

= { 𝑓 𝑔, ℎ𝑔}𝑃 ,

as can be seen directly from (9) and (10). Thus this structure is foliated by the coadjoint orbits, and
its restriction to any orbit 𝑀 defines there a symplectic form 𝜔𝑀 (in particular, the dimension of 𝑀
is always even); see [14, 18] for details.

The corresponding volume form

_ = (constant) 𝜔𝑀 ∧ 𝜔𝑀 ∧ · · · ∧ 𝜔𝑀 ( 1
2 dim𝑀 times)

is a 𝐺-invariant measure (the “Liouville measure”) on 𝑀 . We are free to choose the normalization
constant as we think fit; it will be convenient to hold this choice in reserve for the moment.

5



3 Stratonovich–Weyl kernels
We shall now take as given: (i) a connected Lie group 𝐺 (the splitting group of the original
invariance group); (ii) a linear unitary irreducible representation𝑈 of 𝐺 on some separable Hilbert
space H; and (iii) a coadjoint orbit 𝑀 of 𝐺.

A Stratonovich–Weyl kernel for the triple (𝐺,𝑈, 𝑀) is a function Ω from 𝑀 to a space of
operators on H, such that, for all 𝑢 ∈ 𝑀:

Ω(𝑢) is selfadjoint; (12a)
Tr[Ω(𝑢)] = 1; (12b)
𝑈 (𝑔)Ω(𝑢)𝑈 (𝑔)−1 = Ω(𝑔 · 𝑢), for all 𝑔 ∈ 𝐺; (12c)∫
𝑀

Tr[Ω(𝑢)Ω(𝑣)] Ω(𝑣) 𝑑_(𝑣) = Ω(𝑢). (12d)

In general, as we shall see, Ω(𝑢) need not be trace-class, so that (12b) and (12d) should be
understood in a weak sense, e.g., by considering distributions over 𝑀 . However, we may evade this
technical problem by using an alternative formulation. Write:

𝑊𝐴 (𝑢) := Tr[𝐴Ω(𝑢)] . (13)

The operator-function correspondence 𝐴 ↦→ 𝑊𝐴 is what I shall call the Stratonovich–Weyl corre-
spondence. We may supplement (12) with the requirement that 𝐴 ↦→ 𝑊𝐴 be one-to-one (which
follows automatically in many cases). Then (13) has an inversion formula:

𝐴 =

∫
𝑀

𝑊𝐴 (𝑢)Ω(𝑢) 𝑑_(𝑢). (14)

To see this, let 𝐵 be the right-hand side of (14), and notice that (13) and (12d) give

𝑊𝐵 (𝑢) = Tr[Ω(𝑢) 𝐵] =
∫
𝑀

𝑊𝐴 (𝑣) Tr[Ω(𝑢)Ω(𝑣)] 𝑑_(𝑣)

= Tr
[
𝐴

∫
𝑀

Ω(𝑣) Tr[Ω(𝑢)Ω(𝑣)] 𝑑_(𝑣)
]

= Tr[𝐴Ω(𝑢)] = 𝑊𝐴 (𝑢),

so that 𝐵 = 𝐴. It turns out that the Weyl correspondence (3) is a particular case of (14); and it was
Stratonovich [4] who emphasized the importance of going back and forth between functions and
operators with the same operator kernel, as in (13) and (14).

From (12), (13) and (14), it is clear that the “symbols” 𝑊𝐴 of self-adjoint operators 𝐴 must be
real functions on 𝑀; that𝑊1 is the constant function 1 (so that Tr[Ω(𝑢)] = 1 can be replaced by the
equivalent distributional identity

∫
𝑀
Ω(𝑢) 𝑑_(𝑢) = 1); that (12c) yields the covariance condition:

𝑊𝑈 (𝑔)𝐴𝑈 (𝑔)−1 (𝑔 · 𝑢) ≡ 𝑊𝐴 (𝑢);

and that (12d) yields the tracial property:∫
𝑀

𝑊𝐴 (𝑢)𝑊𝐵 (𝑢) 𝑑_(𝑢) = Tr[𝐴𝐵] . (15)

6



The equation (15) is the centerpiece of a Moyal formulation: it means that the quantum expec-
tation value Tr[𝐴𝐵] is computed by integrating the product of the corresponding symbols𝑊𝐴,𝑊𝐵

over phase space. If 𝐵 is a density matrix, 𝑊𝐵 is its “Wigner function”; and in general we may
expect it to take some negative values, which emphasizes its non-classical nature. (For a discussion
of the nonnegativity of Wigner functions in the “flat case”, see [19–21].)

The traces of products of the operators Ω(𝑢) define interesting functions [4]. Postulate (12d)
says that

𝐾 (𝑢, 𝑣) := Tr[Ω(𝑢)Ω(𝑣)]
is a reproducing kernel for a space of symbols𝑊𝐴. It turns out that the trikernel

𝐿 (𝑢, 𝑣, 𝑤) := Tr[Ω(𝑢)Ω(𝑣)Ω(𝑤)] (16)

is all we need to define a twisted product over 𝑀:

( 𝑓 × ℎ) (𝑢) :=
∫
𝑀

∫
𝑀

𝐿 (𝑢, 𝑣, 𝑤) 𝑓 (𝑣) ℎ(𝑤) 𝑑_(𝑣) 𝑑_(𝑤). (17)

The cyclicity of the kernel (16), together with (15), gives the required associativity of the twisted
product. The covariance postulate (12c) gives equivariance of the twisted product:

( 𝑓 × ℎ)𝑔 = 𝑓 𝑔 × ℎ𝑔 .

The remaining postulates of (12) give supplementary information about the twisted product [22].
Thus, (12a) yields 𝐿 (𝑢, 𝑣, 𝑤) = 𝐿 (𝑢, 𝑤, 𝑣), and (12b) gives∫

𝑀

𝐿 (𝑢, 𝑣, 𝑤) 𝑑_(𝑢) = 𝐾 (𝑣, 𝑤),

which in turn yields the tracial identity for the twisted product:∫
𝑀

( 𝑓 × ℎ) (𝑢) 𝑑_(𝑢) =
∫
𝑀

𝑓 (𝑢) ℎ(𝑢) 𝑑_(𝑢).

This tracial identity is the essential property of the twisted product – indeed, it is a version of the
Moyal property (15) – and has many interesting mathematical consequences, such as the possibility
of extending the twisted product to distributions on 𝑀 [23, 24], or developing harmonic analysis in
the phase-space framework [5]. We shall not explore this issue further here.

4 Examples
Example 1 (The flat case). Here𝐺 = ℍ2𝑁+1, the (2𝑁 +1)-dimensional Heisenberg group; a typical
element is 𝑔 = (𝒂, 𝒃, 𝑐) with 𝒂, 𝒃 ∈ ℝ𝑁 , 𝑐 ∈ ℝ, and

(𝒂1, 𝒃1, 𝑐1) (𝒂2, 𝒃2, 𝑐2) =
(
𝒂1 + 𝒂2, 𝒃1 + 𝒃2, 𝑐1 + 𝑐2 + 1

2 (𝒂1 · 𝒃2 − 𝒂2 · 𝒃1)
)
.

One takes a basis {𝑋1, . . . , 𝑋𝑁 , 𝑌1, . . . , 𝑌𝑁 , 𝑍} for g, so that [𝑋𝑖, 𝑌 𝑗 ] = 𝛿𝑖 𝑗𝑍 . By writing 𝑔 =

exp(𝒂 · 𝑿) exp(𝒃 · 𝒀) exp(𝑐𝑍), one computes, using (9), that

(𝒂, 𝒃, 𝑐) · (𝒙, 𝒚, 𝑧) = (𝒙 + 𝑧𝒃, 𝒚 − 𝑧𝒂, 𝑧) (18)

7



where 𝑥1, . . . , 𝑥𝑁 , 𝑦1, . . . , 𝑦𝑁 , 𝑧 are the coordinate functions (8) on g∗ for the chosen basis of g.
Thus, as is well known, the coadjoint orbits of ℍ2𝑁+1 are either points (𝒙, 𝒚, 0) or hyperplanes 𝑀𝑧

with 𝑧 = nonzero constant.
Fix 𝑧 ≠ 0, and define orbit coordinates (𝒒, 𝒑) ∈ 𝑀𝑧 by

𝒑 := 𝒚, 𝒒 :=
𝒙

𝑧
.

Then (11) gives the Poisson brackets:

{𝑞𝑖, 𝑞 𝑗 }𝑃 = {𝑝𝑖, 𝑝 𝑗 }𝑃 = 0, {𝑞𝑖, 𝑝 𝑗 }𝑃 = 1,

so that (𝒒, 𝒑) are canonical coordinates, and 𝑀𝑧 ≃ ℝ2𝑁 as a symplectic manifold. The coadjoint
action (18) is expressed in the orbit coordinates as:

(𝒂, 𝒃, 𝑐) · (𝒒, 𝒑) = (𝒒 + 𝒃, 𝒑 − 𝑧𝒂). (19)

For simplicity, we shall take the “Planck constant” 𝑧 = 1. Then, if we consider the unitary
irreducible representation𝑈 of ℍ2𝑁+1 acting on H = 𝐿2(ℝ𝑁 , 𝑑𝑁𝝃) by[

𝑈 (𝒂, 𝒃, 𝑐)Ψ
]
(𝝃) := exp{−𝑖(𝑐 + 𝒃 · 𝝃 + 1

2 𝒂 · 𝒃)}Ψ(𝝃 + 𝒂),

it may be verified directly that the Grossmann–Royer operators (4), with ℏ = 1, satisfy (12): that is,
Ω(𝒒, 𝒑) = Π(𝒒, 𝒑). Thus the Weyl correspondence is a special case of (14).

Example 2 (The finite Heisenberg group). The finite Heisenberg group 𝐴𝑁 [11, 12] is a central
extension of the abelian group ℤ𝑁 ×ℤ𝑁 by ℤ𝑁 . The coadjoint orbits have no place here, of course,
but one may consider the group 𝐴𝑁 acting on the finite torus ℤ𝑁 × ℤ𝑁 by

(𝑎, 𝑏, 𝑐) · (𝑥, 𝑦) := (𝑥 + 𝑏, 𝑦 − 𝑐𝑎)

in a formal analogy with (19). Then, for 𝑐 = 1 and for odd 𝑁 , one may obtain a solution
Ω(𝑥, 𝑦) = Δ𝑥𝑦 ∈ ℂ𝑁×𝑁 , where Δ𝑥𝑦 are the “Fano operators”:

Δ𝑥𝑦 (𝑘) := exp
{4𝜋𝑖
𝑁
𝑥(𝑦 − 𝑘)

}
Ψ(2𝑦 − 𝑘).

However, there are several other solutions, obtained by including an extra factor of the form
exp{2𝜋𝑖[(𝑥, 𝑦)/𝑁} on the right hand side, so that the postulates (12) do not single out a unique
“best” kernel in this discrete case [25].

Example 3 (The pure spin case). The details for 𝐺 = SU(2), the invariance group of spin, have
been worked out by J. M. Gracia-Bondı́a and myself [5]. The Moyal formulation thereby obtained
allows one to study spin dynamics as a flow on the 2-sphere acting on suitable Wigner functions.
Here I shall summarize the method to obtain a Stratonovich–Weyl kernel.

We first note that SU(2)/{±1} = SO(3) and the covering homomorphism SU(2) → SO(3) can
be identified with the coadjoint action, since g∗ ≃ ℝ3. If 𝑅 ∈ SU(2), and if 𝑅 ∈ SO(3) denotes
the corresponding rotation, then 𝑅 · 𝒙 = 𝑅𝒙 ∈ ℝ3. Thus the coadjoint orbits are the point {0} and
spheres centred at 0. Fix the sphere 𝕊2 of radius 1, writing its elements as 𝒏 = (\, 𝜙) in spherical
coordinates.

8



Let D 𝑗 be the unitary irreducible representation of SU(2) on ℂ2 𝑗+1 ( 𝑗 = 1
2 , 1,

3
2 , . . . ). Then

the Stratonovich–Weyl kernel Ω 𝑗 for (SU(2),D 𝑗 ,𝕊2) will be a (2 𝑗 + 1) × (2 𝑗 + 1)-matrix-valued
function, with matrix elements 𝑍 𝑗𝑟𝑠 (𝒏) (𝑟, 𝑠 = − 𝑗 ,− 𝑗 + 1, . . . , 𝑗). The covariance condition (12c)
gives

𝑍
𝑗
𝑟𝑠 (𝑅𝒏) =

𝑗∑︁
𝑝,𝑞=− 𝑗

D
𝑗
𝑟 𝑝 (𝑅) 𝑍 𝑗𝑝𝑞 (𝒏)D

𝑗

𝑠𝑞 (𝑅).

Writing

𝑌 ′
𝑙𝑚 (𝒏) :=

𝑗∑︁
𝑘=− 𝑗

√︂
2𝑙 + 1

4𝜋

〈
𝑗 𝑙

𝑘 𝑚

���� 𝑗

𝑘 + 𝑚

〉
𝑍
𝑗

𝑘,𝑘+𝑚 (𝒏),

where the
〈
𝑗 𝑙

𝑘 𝑚

���� 𝑗

𝑘 + 𝑚

〉
are Clebsch–Gordan coefficients, we find that

𝑌 ′
𝑙𝑚 (𝑅𝒏) :=

𝑙∑︁
𝑝=−𝑙

D
𝑙

𝑚𝑝 (𝑅)𝑌 ′
𝑙 𝑝 (𝒏),

so that 𝑌 ′
𝑙𝑚
(𝒏) = Y 𝑗

𝑙
𝑌𝑙𝑚 (𝒏), where the latter are the usual spherical harmonics. Since 𝐾 𝑗 (𝒎, 𝒏) =∑2 𝑗

𝑙=0
∑𝑙
𝑝=−𝑙 𝑌𝑙 𝑝 (𝒎)𝑌 𝑙𝑚 (𝒏) is the appropriate reproducing kernel, the tracial postulate (12d) reduces

to |Y 𝑗
𝑙
|2 = 𝑙. Now (12a) says that Y 𝑗

𝑙
is real, so Y 𝑗

𝑙
= ±1, and (12b) shows that Y 𝑗0 = +1. For

convenience, we select Y 𝑗
𝑙
= +1 for all 𝑙, and arrive at

𝑍
𝑗
𝑟𝑠 (𝒏) =

2 𝑗∑︁
𝑙=0

√︄
4𝜋(2𝑙 + 1)

2 𝑗 + 1

〈
𝑗 𝑙

𝑟 𝑠 − 𝑟

���� 𝑗𝑠〉𝑌𝑙,𝑠−𝑟 (𝒏), (20)

which are the symbols 𝑊| 𝑗𝑟⟩⟨ 𝑗 𝑠 | of the spin transitions | 𝑗𝑟⟩⟨ 𝑗 𝑠 |. For 𝑟 = 𝑠, we obtain the Wigner
functions for the pure spin states:

𝑍
𝑗
𝑟𝑟 (𝒏) =

2 𝑗∑︁
𝑙=0

2𝑙 + 1
2 𝑗 + 1

〈
𝑗 𝑙

𝑟 0

���� 𝑗𝑟〉 𝑃𝑙 (cos \). (21)

The formulas (20) and (21) may be taken as the starting point for a theory of spin which makes
no explicit mention of the operator formalism. For instance, the symbol of the 𝐽𝑧 spin operator is

𝑗∑︁
𝑚=− 𝑗

𝑚𝑍
𝑗
𝑚𝑚 (𝒏) =

√︁
𝑗 ( 𝑗 + 1) cos \.

The trikernel for the twisted product is

𝐿 𝑗 (𝒎, 𝒏, 𝒌) =
𝑗∑︁

𝑟,𝑠,𝑡=− 𝑗
𝑍
𝑗
𝑟𝑠 (𝒎) 𝑍 𝑗𝑠𝑡 (𝒏) 𝑍

𝑗
𝑡𝑟 (𝒌)

9



and the Liouville measure on 𝕊2 should be normalized as 𝑑_(𝒏) = 2 𝑗+1
4𝜋 sin \ 𝑑\ 𝑑𝜙, in order that

(20) and (12d) be compatible. For instance, if 𝑗 = 1
2 , we get

𝐿1/2(𝒎, 𝒏, 𝒌) = 𝜋2 (1 + 3(𝒎 · 𝒏 + 𝒏 · 𝒌 + 𝒌 · 𝒏) + 3
√

3 𝑖[𝒎, 𝒏, 𝒌]
)
.

In practice, the twisted product is computed from the observation that 𝑍 𝑗𝑟𝑠 × 𝑍 𝑗𝑡𝑢 = 𝛿𝑠𝑡𝑍
𝑗
𝑟𝑢, which

makes the explicit evaluation of 𝐿 𝑗 superfluous. Several applications of the formalism to spin
calculations are developed in [5]; in particular, the proof of the Majorana formula reduces to a few
lines.

Example 4 (Poincaré disk quantization). A favourite test case for those who propose quantization
schemes is to quantize the Poincaré disk. For this case, the invariance group is usually taken to be
SL(2,ℝ). The Stratonovich–Weyl recipe also applies here. The material of this subsection is due
to Héctor Figueroa [26].

Take 𝐺 = SL(2,ℝ). The commutation relations for g, with the basis:

𝑍 =
1
2

(
1 0
0 −1

)
, 𝑋 =

1
2

(
0 1
1 0

)
, 𝑊 =

1
2

(
0 1
−1 0

)
,

are
[𝑍, 𝑋] = 𝑊, [𝑍,𝑊] = 𝑋, [𝑋,𝑊] = −𝑍.

Any 𝑔 ∈ 𝐺 can be written as 𝑔 = exp(𝑠𝑍) exp(𝑡𝑋) exp(\𝑊), with 𝑠, 𝑡 ∈ ℝ, −𝜋 < \ ⩽ 𝜋. Letting
𝑧 = b𝑍 , 𝑥 = b𝑋 , 𝑤 = b𝑊 denote the coordinate functions on g∗ ≃ ℝ3, we compute from (9) that

exp(𝑠𝑍) · (𝑧, 𝑥, 𝑤) = (𝑧, 𝑥 cosh 𝑠 − 𝑤 sinh 𝑠, 𝑤 cosh 𝑠 − 𝑥 sinh 𝑠),
exp(𝑡𝑋) · (𝑧, 𝑥, 𝑤) = (𝑧 cosh 𝑡 + 𝑤 sinh 𝑡, 𝑥, 𝑤 cosh 𝑡 + 𝑧 sinh 𝑡),

exp(\𝑊) · (𝑧, 𝑥, 𝑤) = (𝑧 cos \ + 𝑥 sin \, 𝑥 cos \ − 𝑧 sin \, 𝑤).

From this it is clear that Coad is the homomorphism SL(2,ℝ) → SO(2, 1), and that the “Casimir
function” 𝐶 = 𝑧2 + 𝑥2 − 𝑤2 is invariant. The coadjoint orbits of SL(2,ℝ) are thus: halves of two-
sheeted hyperboloids, for 𝐶 < 0; one-sheeted hyperboloids, for 𝐶 > 0; a point and two half-cones,
for 𝐶 = 0. The Kirillov correspondence [27] associates two principal-series representations to
each orbit with 𝐶 > 0, and a discrete-series representation to each half-hyperboloid with 𝐶 = −𝑛2

(𝑛 = 1, 2, 3, . . . ).
Let 𝑀+

𝑛 be the orbit given by 𝐶 = −𝑛2, 𝑤 ⩾ 𝑛, for 𝑛 = 1, 2, 3, . . . . One may choose coordinates
on 𝑀+

𝑛 as 𝑢 = (𝑡, 𝜙), where

𝑢 = (𝑛 sinh 𝑡 cos 𝜙, 𝑛 sinh 𝑡 sin 𝜙, 𝑛 cosh 𝑡) = exp(−𝜙𝑊) exp(𝑡𝑋) · (0, 0, 𝑛) ∈ g∗.

Alternatively, one could choose (𝑤, 𝜙) as coordinates, since they form a canonical pair: {𝑤, 𝜙}𝑃 = 1.
The stereographic projection centred at (0, 0,−1):

𝑟 =
𝑛 sinh 𝑡

𝑛 cosh 𝑡 + 1
, 𝜙 = 𝜙

identifies (𝑡, 𝜙) ∈ 𝑀+
𝑛 with the point with polar coordinates (𝑟, 𝜙) ∈ 𝔻, the Poincaré disk, so that

the disk is indeed seen as a “phase space” for the action of SL(2,ℝ), and there is a whole series of
distinct quantizations of 𝔻.

10



The discrete series representations of SL(2,ℝ) are more conveniently expressed using the Cayley

transform 𝐶 =
1
√

2

(
1 −𝑖
1 𝑖

)
to identify SL(2,ℝ) with SU(1, 1) = 𝐶 SL(2,ℝ) 𝐶−1. Then

exp(\𝑊) ↦→ 𝐶 exp(\𝑊)𝐶−1 =

(
𝑒𝑖\/2 0

0 𝑒−𝑖\/2

)
.

The representation𝑈𝑛 acts on H = 𝐿2
hol

(
𝔻, (1 − |Z |2)𝑛−1 𝑑Z 𝑑Z̄

)
according to[

𝑈𝑛

(
𝛼 𝛽

𝛽 �̄�

)
Ψ

]
(Z) := (−𝛽Z + 𝛼)−𝑛−1Ψ

(
�̄�Z − 𝛽
−𝛽Z + 𝛼

)
.

Note that [𝑈𝑛 (exp(\𝑊))Ψ] (Z) = 𝑒−𝑖(𝑛+1)\/2Ψ(𝑒𝑖\Z), so that Ψ𝑚 : Z ↦→ Z𝑚 is a joint eigenvector of
this one-parameter subgroup (𝑚 = 0, 1, 2, . . . ); traces are computed by Tr 𝐴 =

∑∞
𝑚=0⟨Ψ𝑚 | 𝐴 | Ψ𝑚⟩.

With these tools, it is straightforward to verify that

[Ω𝑛 (𝑡, 𝜙)Ψ] (Z)

:= 2
(cosh 𝑡

2 − 𝑖𝑒𝑖𝜙Z sinh 𝑡
2

cosh 𝑡
2 − 𝑖Z sinh 𝑡

2

)𝑛+1
(cosh 𝑡 − 𝑖𝑒𝑖𝜙Z sinh 𝑡)−𝑛−1Ψ

(
Z cosh 𝑡 + 𝑖𝑒−𝑖𝜙 sinh 𝑡
𝑖Z sinh 𝑡 − 𝑒−𝑖𝜙 cosh 𝑡

)
satisfies the conditions (12) and so is a Stratonovich–Weyl kernel for (SL(2,ℝ),𝑈𝑛, 𝑀+

𝑛 ).
In particular, notice that

[Ω𝑛 (0, 0)Ψ] (Z) = 2Ψ(−Z),
so that the parity operator of 𝔻 yields the quantization recipe. This point of view has been ex-
ploited [28] to derive quantizations with noncompact phase spaces. Indeed, starting with Grossmann
and Royer [7,8], the parity operator appears as a general talisman for quantization schemes; however,
this Ansatz breaks down in compact cases, such as the spin case of the previous subsection, where
a Stratonovich–Weyl kernel may nevertheless be constructed.

Example 5 (Galilean spinning particles). For an elementary system corresponding to a nonrela-
tivistic particle, the appropriate invariance group is the Galilean group; let 𝐺 be its splitting group.
An element of 𝐺 may be written as

𝑔 = (\; 𝑏, 𝒂, 𝒗, 𝑅) = (exp(−\𝑀); exp(−𝑏𝐻) exp(−𝒂 · 𝑷) exp(−𝒗 · 𝑷)𝑅)

with \, 𝑏 ∈ ℝ, 𝒂, 𝒗 ∈ ℝ3, 𝑅 ∈ SU(2). The multiplication law is

(\1; 𝑏1, 𝒂1, 𝒗1, 𝑅1) (\2; 𝑏2, 𝒂2, 𝒗2, 𝑅2)
=
(
\1 + \2 + 1

2 (𝒂1 · 𝑅1𝒗2 − 𝒗1 · 𝑅1𝒂2 − 𝑏2𝒗1 · 𝑅1𝒗2); (𝑏1, 𝒂1, 𝒗1, 𝑅1) (𝑏2, 𝒂2, 𝒗2, 𝑅2)
)

with
(𝑏1, 𝒂1, 𝒗1, 𝑅1) (𝑏2, 𝒂2, 𝒗2, 𝑅2) = (𝑏1 + 𝑏2, 𝒂1 + 𝑅1𝒂2 − 𝑏2𝒗1, 𝒗1 + 𝑅1𝒗2, 𝑅1𝑅2).

Once again, using (9) one can compute the coadjoint action explicitly; and it turns out that there
appear three invariant “Casimir functions” of the coordinates 𝑚, ℎ, 𝒑, 𝒋, 𝒌 corresponding to the Lie
algebra generators of (6). These are: 𝑚 itself (the Galilean mass), 𝑢 = 2𝑚ℎ− 𝒑 · 𝒑 and |𝑚 𝒋+ 𝒑×𝒌 |2.
The generic coadjoint orbits thus have dimension 8.

11



Select an orbit 𝑀 by fixing 𝑚 > 0, 𝑢 ∈ ℝ and 𝑠 := | 𝒋 + 1
𝑚
𝒑 × 𝒌 | > 0. On this orbit, coordinates

may be chosen as follows: let

𝒑 = 𝒑, 𝒒 := 𝒌/𝑚, 𝒔 := ( 𝒋 + 1
𝑚
𝒑 × 𝒌)/𝑠 = ( 𝒋 − 𝒒 × 𝒑)/𝑠.

Here 𝒑 and 𝒒 range freely over ℝ3, while 𝒔 is a unit vector in 𝕊2. Moreover, using (6) and (11), one
may compute that:

{𝑞𝑖, 𝑞 𝑗 }𝑃 = {𝑝𝑖, 𝑝 𝑗 }𝑃 = 0, {𝑞𝑖, 𝑝 𝑗 }𝑃 = 𝛿𝑖 𝑗 ,

{𝑞𝑖, 𝑠 𝑗 }𝑃 = {𝑝𝑖, 𝑠 𝑗 }𝑃 = 0, {𝑠𝑖, 𝑠 𝑗 }𝑃 = Y𝑖 𝑗
𝑘 𝑠𝑘 .

Thus 𝑀 ≃ ℝ6 × 𝕊2 as a symplectic manifold.
The corresponding unitary representation of 𝐺 may be taken to be[

𝑈 (\; 𝑏, 𝒂, 𝒗, 𝑅)Ψ
]
(𝝃)

:= exp
{
𝑖

(
\ + 𝑏𝑢 + 𝑏

2𝑚
𝝃 · 𝝃 − 𝒂 · 𝝃 − 𝑚

2
𝒂 · 𝒗

)}
D 𝑗 (𝑅) Ψ

(
𝑅−1(𝝃 + 𝑚𝒗)

)
,

acting on H = 𝐿2(ℝ3, 𝑑3𝝃) ⊗ ℂ2 𝑗+1.
It only remains to produce the Stratonovich–Weyl kernel for (𝐺,𝑈, 𝑀). In a recent paper [6],

J. M. Gracia-Bondı́a and I have established that (12) is satisfied by

[Ω(𝒒, 𝒑, 𝒔)Ψ] (𝝃) := 23 exp
{
2𝑖𝒒 · ( 𝒑 − 𝝃)

}
Ω 𝑗 (𝒔) Ψ(2 𝒑 − 𝝃),

whereΩ 𝑗 (𝒔) is the spin kernel of Example 3. Thus, for Galilean spinning particles, the Stratonovich–
Weyl kernel is a direct product of the Grossmann–Royer kernel on ℝ6 and the 𝑗-spin kernel on 𝕊2.

5 Outlook
The above set of examples indicates that the concept of a Stratonovich–Weyl kernel is the appropriate
one to build a Moyal quantum kinematics for free particles with any given connected invariance
group. To justify this hope in a general setting requires some existence (and, where possible,
uniqueness) theorems for Stratonovich–Weyl kernels. The above list, while very suggestive, does
not provide this existence proof. It is therefore comforting to know that the same scheme also
goes through for the Poincaré group: J. F. Cariñena, J. M. Gracia-Bondı́a and I have constructed
a relativistic Stratonovich–Weyl kernel, yielding twisted products and Wigner functions for Klein–
Gordon and Dirac particles [29]. In contrast to the usual procedure [30–32] of constructing
“relativistic Wigner functions” by a Minkowskian analogy with the “flat” Wigner functions (2), our
group-theoretic approach avoids the awkward phenomenon of “leaving the mass shell”. However,
several subtle points arise in the relativistic construction, so I shall not discuss it further here.

The precise role of the parity operators on noncompact phase spaces remains a puzzle. Indica-
tions exist that the postulates (12) suffice in many cases to prove a uniqueness theorem for Ω, via a
procedure which yields a family of eigenvalues of Ω which in noncompact cases alternate between
+1 and −1, from which a parity operator may be reconstructed. The compact case of Example 3,
which is a subcase of the Galilean Example 5, shows however that in general the parity-operator
Ansatz is misleading, and one must look deeper for a quantization recipe.

12



The uniqueness question is important since, insofar as uniqueness holds, there will be only one
“correct” recipe for Wigner functions of a given type. (For 𝐺 = SU(2), the undetermined signs
Y
𝑗

1, . . . , Y
𝑗

2 𝑗 give a technical counterexample to uniqueness, so the general problem remains open.)
These “correct” Wigner functions may then form the basis for moving beyond kinematical issues to
develop Moyal Quantum Mechanics in more intricate contexts.

Acknowledgements

This work was supported by the generous assistance of the Deutsche akademische Austausch-
dienst (DAAD), which enabled me to visit the BiBoS research centre, and to have fruitful discus-
sions with Philippe Blanchard, Madeleine Sirugue-Collin, Philippe Combe and Olivier Cohendet.
The antecedent work, on which this report was partly based, was done jointly with José M. Gracia-
Bondı́a, seasoned with several enlightening discussions with José Cariñena and Héctor Figueroa.

Support from the Vicerrectorı́a de Investigación de la Universidad de Costa Rica is gratefully
acknowledged.

References
[1] J. E. Moyal, “Quantum mechanics as a statistical theory”, Proc. Cambridge Philos. Soc. 45 (1949), 99–124.

[2] H. Weyl, Gruppentheorie und Quantenmechanik, Hirzel, Leipzig, 1928.

[3] E. P. Wigner, “On the quantum correction for thermodynamic equilibrium”, Phys. Rev. 40 (1932), 749–759.

[4] R. L. Stratonovich, “On distributions in representation space”, Zh. Eksp. Teor. Fiz. 31 (1956) 1012–1020 (in
Russian); Sov. Phys. JETP 4 (1957), 891–898.

[5] J. C. Várilly and J. M. Gracia-Bondı́a, “The Moyal representation for spin”, Ann. Phys. 190 (1989), 107–148.

[6] J. M. Gracia-Bondı́a and J. C. Várilly, “Phase-space representation for Galilean quantum particles of arbitrary
spin”, J. Phys. A: Math. Gen. 21 (1988), L879–L893.

[7] A. Grossmann, “Parity operators and quantization of 𝛿-functions”, Commun. Math. Phys. 48 (1976), 191–193.

[8] A. Royer, “Wigner function as the expectation value of a parity operator”, Phys. Rev. A 15 (1977), 449–450.

[9] O. Cohendet, P. Combe, M. Sirugue and M. Sirugue-Collin, “A stochastic treatment of the dynamics of an integer
spin”, J. Phys. A: Math. Gen. 21 (1988), 2875–2883.

[10] W. K. Wootters, “A Wigner-function formulation of finite-state quantum mechanics”, Ann. Phys. 176 (1987),
1–21.

[11] L. Auslander and R. Tolmieri, “Is computing with the finite Fourier transform pure or applied mathematics?”,
Bull. Amer. Math. Soc. 1 (1979), 847–897.

[12] W. Schempp, “Group theoretical methods in approximation theory, elementary number theory, and computational
signal geometry”, in Approximation Theory V, C. K. Chui, L. L. Schumaker and J. D. Ward, eds., Academic
Press, Boston, 1986; pp. 129–171.

[13] P. Libermann and C. M. Marle, Symplectic Geometry and Analytic Mechanics, D. Reidel, Dordrecht, 1987.

[14] J. F. Cariñena, “Canonical group actions”, lecture notes IC/88/37, ICTP, Trieste, 1988.

13



[15] J. F. Cariñena and M. Santander, “On the projective unitary representations of connected Lie groups”, J. Math.
Phys. 16 (1975), 1416–1420.

[16] L. Martı́nez-Alonso, “Group-theoretical foundations of classical and quantum mechanics. II. Elementary sys-
tems”, J. Math. Phys. 20 (1979), 219–230.

[17] A. A. Kirillov, Elements of the Theory of Representations, Springer, Berlin, 1976.

[18] M. Vergne, “La structure de Poisson sur l’algèbre symétrique d’une algèbre de Lie nilpotente”, Bull. Soc. Math.
France 100 (1972), 301–335.

[19] R. L. Hudson, “When is the Wigner quasi-probability density nonnegative?”, Rep. Math. Phys. 6 (1974), 249–252.

[20] R. G. Littlejohn, “The semiclassical evolution of wave packets”, Phys. Reports 138 (1986), 193–291.

[21] J. M. Gracia-Bondı́a and J. C. Várilly, “Nonnegative mixed states in Weyl–Wigner–Moyal theory”, Phys. Lett. A
128 (1988), 20–24.

[22] I. Castillo, “Productos cuánticos en espacios de funciones analı́ticas”, tesis de licenciatura, Universidad de Costa
Rica, San José, 1988.

[23] M. A. Antonets, “The classical limit for Weyl quantization”, Lett. Math. Phys. 2 (1978), 241–245; and “Classical
limit of Weyl quantization”, Teor. Mat. Fiz. 38 (1979), 331–344; Theor. Math. Phys. 38 (1979), 219–228.

[24] J. M. Gracia-Bondı́a and J. C. Várilly, “Algebras of distributions suitable for phase-space quantum mechanics.
I”, J. Math. Phys. 29 (1988), 869–879.

[25] O. Cohendet, P. Combe and M. Sirugue-Collin, “Fokker–Planck equation associated to the Wigner function of a
quantum system with a finite number of states”, J. Phys. A 23 (1990), 2001–2011.

[26] H. Figueroa, private communication, 1988.

[27] M. Vergne, “Representations of Lie groups and the orbit method”, in Emmy Noether in Bryn Mawr, B. Srinivasan
and J. Sally, eds., Springer, Berlin, 1983; pp. 59–101.

[28] A. Unterberger, “Analyse harmonique et analyse pseudodifférentielle du cône de lumière”, Astérisque 156 (1987),
1–201.

[29] J. F. Cariñena, J. M. Gracia-Bondı́a and J. C. Várilly, “Relativistic quantum kinematics in the Moyal representa-
tion”, J. Phys. A 23 (1990), 901–933.

[30] R. Hakim and H. Sivak, “Covariant Wigner function approach to the relativistic quantum electron gas in a strong
magnetic field”, Ann. Phys. 139 (1982), 230–292.

[31] P. Carruthers and F. Zachariasen, “Quantum collision theory with phase-space distributions”, Rev. Mod. Phys.
55 (1983), 245–285.

[32] S. R. de Groot, W. A. van Leeuwen and C. G. van Weert, Relativistic Kinetic Theory, North-Holland, Amsterdam,
1980.

14