Relativistic quantum kinematics
in the Moyal representation

José F. Cariñena,1 José M. Gracia-Bondı́a2 and Joseph C. Várilly2

1 Departamento de Fı́sica Teórica, Universidad de Zaragoza, 50009 Zaragoza, Spain

2 Escuela de Matemática, Universidad de Costa Rica, 11501 San José, Costa Rica

J. Phys. A 23 (1990), 901–933

Abstract

In this paper, we obtain the phase-space quantization for relativistic spinning particles. The
main tool is what we call a “Stratonovich–Weyl quantizer” which relates functions on phase space
to operators on a suitable Hilbert space, and has the essential properties of covariance (under
a group representation) and traciality. Our phase spaces are coadjoint orbits of the restricted
Poincaré group; we compute and explicitly coordinatize the orbits corresponding to massive
particles, with or without spin. Some orbits correspond to unitary irreducible representations
of the Poincaré group; we show that there is a unique Stratonovich–Weyl quantizer from each
of these phase spaces to operators on the corresponding representation spaces, and compute it
explicitly.

We develop the formalism by computing relativistic Wigner functions and twisted products
for Klein–Gordon particles; these Wigner functions are supported on the mass shell. We thereby
obtain an expression for the position probability density which is local, that is, free from the
difficulty of supraluminal propagation of the usual position probability density. It is shown
explicitly how observables on phase space may be quantized; for example, we prove that the
canonical position coordinate corresponds to the Newton–Wigner position operator, irrespective
of spin.

We show how relativistic phase-space quantization applies to particles governed by the
Dirac equation. In effect, we construct a Stratonovich–Weyl quantizer whose associated Hilbert
space is the space of positive-energy solutions of the Dirac equation.

1 Introduction
The work of Moyal [1] did much to clarify that the “Weyl correspondence” [2] and the “Wigner
distribution” [3] are elements of the fourth formulation – historically speaking – of nonrelativistic
quantum mechanics. The Moyal formulation has not had, by far, a success comparable to those of
Heisenberg, Schrödinger or Feynman. Let us note, however, that the difficulty of extending it to
cover spinning or relativistic particles was one of the reasons for that relative lack of success.

We consider in this paper elementary systems. The phrase “elementary quantum system” is
usually taken to mean an irreducible (projective) representation of some invariance group of physical

1



interest, such as the Galilei or Poincaré groups; the rays of the Hilbert space of the representation
are taken as the states of the quantum system, and its observables are operators on this Hilbert
space. The concept became firmly established following the landmark paper by Wigner [4] about
the representations of the Poincaré group.

In the Moyal formulation of quantum mechanics, a different point of view is adopted: both
states and observables are real functions (or generalized functions) on the classical phase space, and
expected values are computed, as in classical statistical physics, by averaging over the phase space.

To be precise, the rule for computing the expected values of an observable 𝑓 in a state 𝜌 remains
the classical rule:

⟨ 𝑓 ⟩𝜌 =
∫
𝑀
𝑓 (𝑢)𝜌(𝑢) 𝑑𝑢∫
𝑀
𝜌(𝑢) 𝑑𝑢

(1)

where, in the ordinary case, 𝑀 = ℝ2𝑛 and 𝑑𝑢 is Lebesgue measure on ℝ2𝑛, 𝑛 being the number of
degrees of freedom of the system.

Observables are composed via the noncommutative twisted product, since the classical pointwise
product of observables is excluded by the uncertainty principle, which forbids localization at a point
of phase space. States are defined as the positive functionals of the twisted product algebra, i.e., 𝜌
represents a state if ∫

𝑀

𝑓 × 𝑓 (𝑢)𝜌(𝑢) 𝑑𝑢 ⩾ 0

for any 𝑓 . This is parallel to what is done in classical statistical mechanics, with the twisted
product substituting for the ordinary product, but here the states no longer need to correspond to
nonnegative functions (since the twisted product of a function with its complex conjugate can take
negative values). It turns out that Wigner distributions are essentially the pure states of the twisted
product algebra; and so, an intrinsically autonomous theory can be established, equivalent to, but
independent of, conventional quantum mechanics.

In this paper, we construct the Moyal representation of relativistic quantum theory. This arises
out of a program for the Moyal quantization of general phase spaces. Although we want to consider
here only the physically relevant problem of relativistic mechanics, some points of principle con-
cerning this program must be made. Its backbone is the same as that of the “geometric quantization”
program, namely, the coadjoint orbit picture introduced by Kirillov, Kostant and Souriau [5].

In modern renditions of classical mechanics, one considers a symplectic manifold 𝑀 (or, more
generally, a manifold with a Poisson bracket structure); the invariance group is a Lie group𝐺 acting
on 𝑀 by transformations which preserve this structure. We say we have an elementary classical
system [6] if this action is transitive, i.e., if 𝑀 is a homogeneous symplectic manifold (HSM) for
the group 𝐺.

The elementary systems whose invariance group is the given (connected) Lie group 𝐺 appear
in the conventional approach to quantum mechanics as projective unitary representations of 𝐺. It
is convenient to find and to present the projective unitary representations of 𝐺 as linear unitary
representations of another group 𝐺 which is a “splitting group” for 𝐺 [7, 8], which we shall
describe in more detail below. The connection with the classical framework arises from the work of
Kirillov [5], according to which there should be a correspondence between the unitary irreducible
representations of 𝐺 and the orbits of the coadjoint action of 𝐺 on the dual space of its Lie algebra.
Experience suggests that not all coadjoint orbits are eligible, but only those which satisfy certain
integrality conditions. (A simple formulation of integrality conditions is found in [9].)

2



The phase-space quantization program may be formulated as follows. Let 𝐺 be the physical
invariance group whose elementary systems we want to study. Construct the appropriate splitting
group 𝐺 and assume that Kirillov’s paradigm works for 𝐺 (this is the case for most invariance
groups of physical interest). The orbits of the coadjoint action of 𝐺 that are also HSM for 𝐺 are the
“phase spaces”.

At this point we part company with the outlook and techniques of geometric quantization and
we seek to extend Kirillov’s theory in a new direction. A Moyal quantum elementary system is a
classical elementary system 𝑀 plus a 𝐺-equivariant twisted product on spaces of functions on 𝑀 ,
such that through (1) the physical expectations of the theory coincide with the ones on the Hilbert
space of the representation associated to 𝑀 . In practice, one frequently knows beforehand the
representation theory of 𝐺, so that the simplest approach seems to be to link the coadjoint orbits
of 𝐺 directly with the operatorial theory, by means of an appropriate correspondence rule which
yields the twisted product as the image of the usual composition of operators.

The program just outlined was carried out by two of us for the group 𝑆𝑈 (2), yielding the Moyal
representation of spin, entirely equivalent to the conventional one, but with some interpretative and
computational advantages [10]. The ordinary Moyal theory was also reinterpreted as a theory of
Galilean elementary systems in our sense and extended to particles with arbitrary spin [11].

Section 2 serves as a guideline for the whole paper. We first deal with geometric preliminaries
pertaining to Kirillov theory; this part, together with Section 3, may be taken as a primer on coadjoint
orbit techniques; we think these belong in the toolkit of every theoretical physicist. Thereafter we
introduce the key concept of Stratonovich–Weyl quantizer, which gives the aforementioned link
between the phase spaces and the operatorial theory on Hilbert spaces. Finally, we go through two
important examples of its use. The results of the first example – concerning pure spin systems
– are employed throughout the paper. In section 3 the coadjoint action for the Poincaré group is
computed. Section 4 is the heart of the paper: here we derive the (unique) Stratonovich–Weyl
quantizer for relativistic spinning particles in the Wigner realization.

Section 5 deals with the resulting Moyal formulation in the spinless (Klein–Gordon) case. In
section 6, the operators corresponding to several important phase-space observables are computed
in the Wigner realization, for any spin. Section 7 constructs the Stratonovich–Weyl quantizer for
particles described by the Dirac equation.

In the concluding Section 8, we briefly review some recent attempts to derive “relativistic
Wigner functions” or “relativistic Weyl transforms” and compare the results with those of our
group-theoretic approach.

Throughout the paper, units of measure are taken so that ℏ = 1 and 𝑐 = 1.

2 Phase-space quantization in general
2.1 Geometric preliminaries

Let 𝐺 be a connected Lie group, g its Lie algebra. If one wishes to determine the projective unitary
representations of 𝐺, the classical method proposed by Bargmann [12] may be used. It employs a
family of extensions of the covering group of𝐺, and in general different extensions must be used for
different projective unitary representations. Some years ago, one of us [7] showed how this method
could be improved, in the following sense: given 𝐺, a uniquely determined “splitting group” 𝐺 can
be found such that all projective unitary representations of𝐺 can be lifted to unitary representations

3



of 𝐺. Actually, there exist in general several groups 𝐺′ and morphisms ` : 𝐺′→ 𝐺 such that every
projective representation of 𝐺 can be lifted to a unitary representation of 𝐺′ mapping the elements
of ker ` into multiples of the identity. The particular construction proposed in [7] is very easy to
handle but may be not “minimal”: see [8].

We recall that construction. Let 𝐻2(g,ℝ) ≡ ℝ𝑚 be the second cohomology group of g for the
trivial representation of g onℝ. Then we consider the central extension g = ℝ𝑚⊕g, with appropriate
commutation relations; the connected and simply connected Lie group with Lie algebra g is the
splitting group. For details, see [7, 8], where the somewhat inadequate name “projective covering
group” was used to refer to this “splitting group”. Therein it is proved that there is a homomorphism
` : 𝐺 → 𝐺 such that each unitary representation of 𝐺 mapping the kernel of ` into the circle group
𝑈 (1) induces a projective unitary representation of 𝐺; and conversely, each projective unitary
representation of 𝐺 can be lifted to a unitary representation of 𝐺. If 𝐻2(g,ℝ) = 0, as happens
for all semisimple groups, then 𝐺 reduces to the (universal) covering group 𝐺 of 𝐺. Well-known
examples of covering groups in physics are 𝑆𝑈 (2), which covers the rotation group, and 𝑆𝐿 (2,ℂ),
which covers the group L

↑
+ of proper orthochronous Lorentz transformations.

We recall that the adjoint representation of 𝐺 is the map Ad: 𝐺 → 𝐺𝐿 (g) defined by

exp[𝑡 (Ad 𝑔)𝑋] = 𝑔(exp 𝑡𝑋)𝑔−1

for 𝑔 ∈ 𝐺, 𝑋 ∈ g, 𝑡 ∈ ℝ. The coadjoint representation of 𝐺 on g∗, the dual space of g, is the
contragredient of the adjoint representation, namely if ⟨𝑢, 𝑋⟩ := 𝑢(𝑋) for 𝑢 ∈ g∗, 𝑋 ∈ g, then

⟨(Coad 𝑔)𝑢, 𝑋⟩ := ⟨𝑢, (Ad 𝑔−1)𝑋⟩. (2)

This defines an action of 𝐺 on g∗ and we will write 𝑔 · 𝑢 := (Coad 𝑔)𝑢 to denote this coadjoint
action.

Every 𝑋 ∈ g defines a linear coordinate function b𝑋 on g∗ by b𝑋 (𝑢) := ⟨𝑢, 𝑋⟩. Moreover,
from (2), we have b𝑋 (𝑔 · 𝑢) = b(Ad 𝑔−1)𝑋 (𝑢).

The “elementary classical systems” with invariance group 𝐺 are connected homogeneous sym-
plectic 𝐺-manifolds. It is known [13, 14] that these are, up to a covering symplectomorphism,
identifiable to orbits of an affine action 𝐴\ of 𝐺 on g∗. The linear part of 𝐴\ is always the coadjoint
action of 𝐺; classification of these homogeneous 𝐺-spaces reduces thus to the classification of the
inhomogeneous part \, which in turn is given by its cohomology class [\] in 𝐻1(𝐺; g∗). Note that
to each element of 𝐻1(𝐺; g∗) we can associate an element of 𝐻2(g,ℝ).

Ideally, we would like to treat with coadjoint actions only, instead of affine actions, since the
former are simple to describe and easy to calculate. Martı́nez-Alonso [6] noticed that the splitting
group 𝐺 could serve that purpose. Let 𝑀 be a homogeneous symplectic 𝐺-space. The group 𝐺
acts on 𝑀 via the projection ` through the𝐺-action and so the kernel of ` : 𝐺 → 𝐺 acts identically
on 𝑀; and conversely, the actions of 𝐺 on 𝑀 for which ker ` acts trivially correspond to actions
of 𝐺 on 𝑀 . All the symplectic actions of 𝐺 so considered are Poisson actions and therefore the
HSM for 𝐺 are simply orbits of the coadjoint action of 𝐺 on which the kernel of ` : 𝐺 → 𝐺 acts
identically; then the associated action of 𝐺 on 𝑀 is transitive (i.e., 𝑀 is an elementary classical
system for 𝐺), and conversely.

In short, the classification of “classical” and “quantum” actions of a given group has a single
unifying principle.

4



We also recall that the coalgebra g∗ carries a natural 𝐺-invariant Poisson structure, which can
be defined as follows. For any 𝑢 ∈ g∗, the tangent space 𝑇𝑢g∗ ≃ g∗, so if 𝑓 ∈ 𝐶∞(g∗) we can regard
(𝑑𝑓 )𝑢 : 𝑇𝑢g∗ → ℝ as an element 𝑑𝑓 (𝑢) of g. The Poisson bracket is then given by [15]:

{ 𝑓 , 𝑔}𝑃 (𝑢) := ⟨𝑢, [𝑑𝑓 (𝑢), 𝑑𝑔(𝑢)]⟩. (3)

In particular, since 𝑑b𝑋 (𝑢) = 𝑋 for 𝑋 ∈ g, we have

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

Writing 𝑥𝑖 = b𝑋𝑖 where the 𝑋𝑖 fill out a basis for g, we thus have from the Lie algebra commutation
relations [𝑋𝑖, 𝑋 𝑗 ] = 𝑐𝑖 𝑗 𝑘𝑋𝑘 :

{ 𝑓 , 𝑔}𝑃 =
𝜕 𝑓

𝜕𝑥𝑖

𝜕𝑔

𝜕𝑥 𝑗
{𝑥𝑖, 𝑥 𝑗 }𝑃 =

𝜕 𝑓

𝜕𝑥𝑖

𝜕𝑔

𝜕𝑥 𝑗
b[𝑋𝑖 ,𝑋 𝑗 ] = 𝑐𝑖 𝑗

𝑘 𝜕 𝑓

𝜕𝑥𝑖

𝜕𝑔

𝜕𝑥 𝑗
𝑥𝑘 . (4)

It is worth noting that a 𝐺-invariant Poisson structure on g∗ is not necessarily unique. Another
such structure is given by

{ 𝑓 , 𝑔}𝛽 (𝑢) := { 𝑓 , 𝑔}𝑃 (𝑢) − 𝛽
(
𝑑𝑓 (𝑢), 𝑑𝑔(𝑢)

)
,

where 𝛽 is a two-cocycle on g; this is essentially equivalent to (3) only if 𝛽 is of the form
𝛽(𝑋,𝑌 ) = 𝛼( [𝑋,𝑌 ]) for some 𝛼 ∈ g∗, that is, 𝛽 is a coboundary. As hinted above, the multiplicity
of Poisson structures on g∗ is thus classified by the cohomological properties of g. We remark that
the Lie algebra of the Galilei group admits nontrivial 2-cocycles [16], whereas that of the Poincaré
group does not. In the former case, one may recover uniqueness of the invariant Poisson structure
on the coalgebra by extending to the eleven-dimensional splitting group of which the Galilei group
is a quotient; and the same procedure allows one to obtain projective representations of the Galilei
group from linear representations.

The natural symplectic structure on any orbit 𝑀 of the coadjoint action can be obtained from
the Poisson structure. Indeed, the orbits are the symplectic leaves of the natural Poisson structure
defined in the coalgebra. If, for each 𝑋 ∈ g, 𝑋 denotes the “fundamental vector field” on g∗ given
by

(𝑋 𝑓 ) (𝑢) :=
𝑑

𝑑𝑡
𝑓
(
exp(−𝑡𝑋) · 𝑢

) ���
𝑡=0
, (5)

then (𝑋b𝑌 ) (𝑢) = ⟨𝑢, [𝑋,𝑌 ]⟩ = {b𝑋 , b𝑌 }𝑃 (𝑢). The fields 𝑋 are tangent to the orbit 𝑀; if 𝑗 : 𝑀 → g∗

is the inclusion, then the symplectic 2-form 𝜔 on 𝑀 is given [17] by

𝜔( 𝑗∗𝑋, 𝑗∗𝑌 ) (𝑢) := ⟨𝑢, [𝑋,𝑌 ]⟩. (6)

Here 𝑗∗𝑋 , 𝑗∗𝑌 are the fundamental vector fields of the action of 𝐺 restricted to 𝑀 . This symplectic
structure is automatically 𝐺-invariant. By (6) the Poisson bracket associated to 𝜔 on the orbit is
simply the restriction of {−,−}𝑃. The associated volume form is a 𝐺-invariant measure on 𝑀 (the
Liouville measure); after a suitable normalization, this measure will be denoted by _𝑀 , or simply
by _, if a fixed orbit 𝑀 is understood.

5



2.2 The Stratonovich–Weyl correspondence

The coadjoint orbits are connected with irreducible unitary representations of 𝐺 by the Kirillov
orbit method [5, 9]. Suppose then that we are given a connected Lie group 𝐺, a linear irreducible
unitary representation 𝑈 of 𝐺 on a Hilbert space H, and an associated coadjoint orbit 𝑀 ⊂ g∗.
For instance, in the case of the Galilei and Poincaré groups, the equivalence classes of projective
unitary irreducible representations and the coadjoint orbits which can be physically interpreted as
massive particles are identically parametrized. We wish to define a generalization of the Weyl
correspondence [2] which associates an operator 𝐴 on H to a (generalized) function 𝑊𝐴 on 𝑀 in
a linear one-to-one way; thus, we conjecture the existence of an operator-valued kernel Ω : 𝑀 →
{ operators on H } such that𝑊𝐴 (𝑢) = Tr[𝐴Ω(𝑢)].

The general requirements for such a kernel were first sketched in a remarkable paper by Strato-
novich [18]. We condense them in the following definition.

Definition. Let 𝐺 be a Lie group and 𝑈 a linear irreducible unitary representation of 𝐺 on a
Hilbert space H. Let 𝑀 be a symplectic homogeneous𝐺-space and let _ be a (suitably normalized)
𝐺-invariant measure on 𝑀 . A Stratonovich–Weyl quantizer for the triple (𝐺,𝑈, 𝑀) is a function
Ω : 𝑀 → {operators on H} which satisfies, for all 𝑢 ∈ 𝑀:

(i) Ω(𝑢) is self-adjoint; (7a)
(ii) Tr[Ω(𝑢)] = 1; (7b)

(iii) 𝑈 (𝑔)Ω(𝑢)𝑈 (𝑔)−1 = Ω(𝑔 · 𝑢), for all 𝑔 ∈ 𝐺; (7c)

(iv)
∫
𝑀

Tr[Ω(𝑢)Ω(𝑣)] Ω(𝑣) 𝑑_(𝑣) = Ω(𝑢). (7d)

The correspondence 𝐴 ↦→ 𝑊𝐴 determined by (𝐺,𝑈, 𝑀,Ω) is given by

𝑊𝐴 (𝑢) := Tr[𝐴Ω(𝑢)] . (8a)

We finally require that:

(v) 𝐴 ↦→ 𝑊𝐴 be one-to-one.

Let us write, provisionally, 𝐵 =
∫
𝑀
𝑊𝐴 (𝑢)Ω(𝑢) 𝑑_(𝑢). Then (7d) gives

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),

and thus 𝐵 = 𝐴. This gives the Stratonovich–Weyl correspondence inverting (8a):

𝐴 =

∫
𝑀

𝑊𝐴 (𝑢)Ω(𝑢) 𝑑_(𝑢). (8b)

It is important to remark that the postulate (7d) thus implies that both directions (8) of the
correspondence 𝐴 ⇄ 𝑊𝐴 are implemented by the same kernel Ω(𝑢). In other words, our quantizer
is the “dequantizer”, too.

6



We have omitted to specify the exact conditions which guarantee convergence and free inter-
change of the various integrals and traces, preferring at this stage to illuminate the general scheme
by physically relevant examples. For instance, for noncompact groups Ω(𝑢) will not generally be
trace-class, but (7a) should hold in a weak sense.

Now let us spell out the consequences of (7) for the Stratonovich–Weyl correspondence. Us-
ing (8a), we obtain at once:

(a) 𝐴 selfadjoint =⇒ 𝑊𝐴 is real, and𝑊𝐴† = 𝑊 𝐴 in general;

(b) 𝑊𝐼 is the constant function 1;

(c) 𝑊𝑈 (𝑔)𝐴𝑈 (𝑔)−1 (𝑔 · 𝑢) ≡ 𝑊𝐴 (𝑢).

Furthermore, we have the tracial property of the correspondence:

(d)
∫
𝑀

𝑊𝐴 (𝑢)𝑊𝐵 (𝑢) 𝑑_(𝑢) = Tr[𝐴𝐵] . (9)

Indeed, using (8a) and (8b), both sides of (9) are equal to Tr
[
𝐴

∫
𝑀
Ω(𝑢)𝑊𝐵 (𝑢) 𝑑_(𝑢)

]
.

If 𝐸 denotes an appropriate function space on 𝑀 , then (7d) can be rephrased as:

(e) 𝐾 (𝑢, 𝑣) := Tr[Ω(𝑢)Ω(𝑣)] is the reproducing kernel for 𝐸. (10)

The twisted product 𝑓 × 𝑔 of two functions 𝑓 , 𝑔 in 𝐸 is defined by

( 𝑓 × ℎ) (𝑢) :=
∫
𝑀

∫
𝑀

𝐿 (𝑢, 𝑣, 𝑤) 𝑓 (𝑣)ℎ(𝑤) 𝑑_(𝑣) 𝑑_(𝑤),

where 𝐿 (𝑢, 𝑣, 𝑤) remains to be determined. By definition, this should correspond to the composition
of operators on H, that is, we require that𝑊𝐴 ×𝑊𝐵 = 𝑊𝐴𝐵 for any 𝐴, 𝐵. From (8) we obtain

(𝑊𝐴 ×𝑊𝐵) (𝑢) = 𝑊𝐴𝐵 (𝑢) = Tr[Ω(𝑢)𝐴𝐵]

=

∫
𝑀

∫
𝑀

Tr[Ω(𝑢)Ω(𝑣)Ω(𝑤)]𝑊𝐴 (𝑣)𝑊𝐵 (𝑤) 𝑑_(𝑣) 𝑑_(𝑤),

and we conclude that

(f) 𝐿 (𝑢, 𝑣, 𝑤) = Tr[Ω(𝑢)Ω(𝑣)Ω(𝑤)] is the trikernel for the twisted product on 𝑀 .

Since𝑊𝐼 = 1, it follows from (9) that

Tr 𝐴 =

∫
𝑀

𝑊𝐴 (𝑢) 𝑑_(𝑢), (11)

which, together with (10), yields the tracial identity for the twisted product:∫
𝑀

( 𝑓 × ℎ) (𝑢) 𝑑_(𝑢) =
∫
𝑀

𝑓 (𝑢)ℎ(𝑢) 𝑑_(𝑢), (12)

7



on account of∫
𝑀

( 𝑓 × ℎ) (𝑢) 𝑑_(𝑢) =
∫
𝑀

∫
𝑀

∫
𝑀

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

=

∫
𝑀

∫
𝑀

∫
𝑀

𝑓 (𝑣)𝑊Ω(𝑣)Ω(𝑤) (𝑢)ℎ(𝑤) 𝑑_(𝑢) 𝑑_(𝑣) 𝑑_(𝑤)

=

∫
𝑀

∫
𝑀

𝑓 (𝑣)𝐾 (𝑣, 𝑤)ℎ(𝑤) 𝑑_(𝑣) 𝑑_(𝑤) =
∫
𝑀

𝑓 (𝑣)ℎ(𝑣) 𝑑_(𝑣).

Comparing equations (11) and (12) with (1), one sees that this twisted product does indeed comply
with the classical rule for computing expected values of observables. It is apparent now that (9)
or equivalently (10) or equivalently (12) is the key property for the equivalence of expected values
calculated in the Moyal representation and those computed using the formulation in Hilbert space.
The twisted product is automatically equivariant:

( 𝑓 × ℎ)𝑔 (𝑢) = ( 𝑓 𝑔 × ℎ𝑔) (𝑢) for all 𝑔 ∈ 𝐺,

where 𝑓 𝑔 (𝑢) := 𝑓 (𝑔−1 · 𝑢).
We may now state more precisely what we mean by an “elementary quantum system”.

Definition. An elementary quantum system in the Moyal representation is a homogeneous sym-
plectic 𝐺-manifold 𝑀 , together with an equivariant algebra of functions on 𝑀 , where the algebra
product satisfies the tracial identity (12).

The definition is in principle fairly weak, in the sense that it leaves open the possibility of
the existence of Moyal quantum systems not arrived at via a Stratonovich–Weyl correspondence.
However, as far as we know, the availability of a Moyal representation for an elementary quantum
system given in the conventional formulation by a pair (𝐺,𝑈) depends on the existence of a
SW quantizer Ω. In Section 4, we construct such a quantizer for the Poincaré group.

The question of the uniqueness (up to unitary equivalence) of the quantizer is obviously of
importance. We claim at present no general results on uniqueness; however, in all the cases which
have so far been examined: the Heisenberg groups, 𝑆𝑈 (2), and the massive orbits for the Galilei
and Poincaré groups, the Stratonovich–Weyl quantizer is essentially unique. It is likely that there is
a deep group-theoretic connection here.

2.3 Example 1: pure spin systems

Consider the group 𝐺 = 𝑆𝑈 (2), the invariance group for pure spins. Its coadjoint orbits, apart
from the origin, are spheres (since 𝑆𝑈 (2) acts on g∗ ≃ ℝ3 by rotations), and its irreducible unitary
representations are the well-known D 𝑗 , for 𝑗 a half-integer. The integrality conditions select a
discrete set of spheres in g∗ corresponding to the various D 𝑗 . We identify all the spheres for
convenience. For this case, the Stratonovich–Weyl quantizer has been determined by two of us [10],
following the outline of Stratonovich [18].

Let 𝒏 be a point on the sphere of radius 1, and let Δ 𝑗 (𝒏) :=
∑ 𝑗

𝑟,𝑠=− 𝑗 𝑍
𝑗
𝑟𝑠 (𝒏) | 𝑗 𝑠⟨⟩ 𝑗𝑟 | be the

quantizer, with the matrix elements 𝑍 𝑗𝑟𝑠 as yet undetermined. The covariance condition (7c) can be
written as

[𝑚 𝑗 (𝑅)Δ 𝑗 ] (𝒏) := D 𝑗 (𝑅)Δ 𝑗 (𝑅−1𝒏)D 𝑗 (𝑅)−1 = Δ 𝑗 (𝒏),

8



for 𝑅 ∈ 𝑆𝑈 (2), 𝑅 being the corresponding rotation. The action𝑚 𝑗 of 𝑆𝑈 (2) has an easily determined
set of fixed points: one obtains [10] that Δ 𝑗 =

∑2 𝑗
𝑙=0 _

𝑗

𝑙
𝐹
𝑗

𝑙
, where

𝐹
𝑗

𝑙
(𝒏) :=

𝑗∑︁
𝑟,𝑠=− 𝑗

2𝑙 + 1
2 𝑗 + 1

1/2〈
𝑗 𝑙 𝑠(𝑟−𝑠) | 𝑗 𝑟

〉
𝑌𝑙,𝑟−𝑠 (𝒏) | 𝑗 𝑠⟨⟩ 𝑗𝑟 |, (13)

where the
〈
𝑗 𝑙 𝑠(𝑟−𝑠) | 𝑗 𝑟

〉
are Clebsch-Gordan coefficients and the 𝑌𝑙,𝑟−𝑠 are the usual spherical

harmonics on the sphere. (The notation here corrects an error in [10]; in order to obtain correctly
formulas (2.22) and (2.23) of that paper, the indices 𝑟 and 𝑠 must be permuted.)

The Δ 𝑗 are hermitian matrices (7a) only if the _ 𝑗
𝑙

are real. The reproducing kernel for the space
of spherical harmonics of degree ⩽ 2 𝑗 is

𝐾 𝑗 (𝒎, 𝒏) = 4𝜋
2 𝑗 + 1

2 𝑗∑︁
𝑙=0

𝑙∑︁
𝑚=−𝑙

𝑌𝑙𝑚 (𝒎)𝑌 𝑙𝑚 (𝒏)

if one adopts, as one must,

𝑑_(𝒏) = 2 𝑗 + 1
4𝜋

𝑑𝒏 =
2 𝑗 + 1

4𝜋
sin \ 𝑑\ 𝑑𝜙

as the Liouville measure. One verifies that the 𝐹 𝑗
𝑙

have the orthogonality property

Tr[𝐹 𝑗
𝑘
(𝒎)𝐹 𝑗

𝑙
(𝒏)] = 𝛿𝑘𝑙

𝑙∑︁
𝑚=−𝑙

𝑌𝑙𝑚 (𝒎)𝑌 𝑙𝑚 (𝒏).

Now, using (10), the tracial condition (7d) gives (_ 𝑗
𝑙
)2 = 4𝜋/(2 𝑗 +1). Since Tr(𝐹 𝑗

𝑙
(𝒏)) = 0 if 𝑙 ≠ 0,

the condition (7a) yields also that _ 𝑗0 > 0. Hence we find

_
𝑗

0 =

√︄
4𝜋

2 𝑗 + 1
; _

𝑗

𝑙
= ±

√︄
4𝜋

2 𝑗 + 1
for 𝑙 = 1, 2, . . . , 2 𝑗 . (14)

The sign ambiguities in (14) are the only measure of non-uniqueness in the SW quantizer Δ 𝑗 .
Physically, it makes sense to select all signs positive [10]. Thus we finally arrive at

Δ 𝑗 (𝒏) =
𝑗∑︁

𝑟,𝑠=− 𝑗
𝑍
𝑗
𝑟𝑠 (𝒏) | 𝑗 𝑠⟨⟩ 𝑗𝑟 | =

𝑗∑︁
𝑟,𝑠=− 𝑗

2 𝑗∑︁
𝑙=0

[4𝜋(2𝑙 + 1)]1/2
2 𝑗 + 1

〈
𝑗 𝑙 𝑠(𝑟−𝑠) | 𝑗 𝑟

〉
𝑌𝑙,𝑟−𝑠 (𝒏) | 𝑗 𝑠⟨⟩ 𝑗𝑟 |.

(15)
The kernel Δ 𝑗 is the Stratonovich–Weyl quantizer for the 𝑗-spin.

The twisted product of two functions 𝑓 , 𝑔 in the space spanned by the matrix elements 𝑍 𝑗𝑟𝑠 is
given by

𝑓 × 𝑔(𝒏) =
∫
𝕊2

∫
𝕊2
𝑓 (𝒎)𝑔(𝒌)𝐿 (𝒏,𝒎, 𝒌) 𝑑_(𝒎) 𝑑_(𝒌),

where 𝐿 (𝒏,𝒎, 𝒌) = Tr[Δ 𝑗 (𝒏)Δ 𝑗 (𝒎)Δ 𝑗 (𝒌)]. The functions 𝑍 𝑗𝑟𝑠 (𝒏) have the orthogonality and
product properties: ∫

𝕊2
𝑍
𝑗
𝑟𝑠 (𝒏)𝑍 𝑗𝑡𝑢 (𝒏) 𝑑_(𝒏) = 𝛿𝑟𝑢𝛿𝑠𝑡 , 𝑍

𝑗
𝑟𝑠 × 𝑍 𝑗𝑡𝑢 = 𝛿𝑠𝑡 𝑍

𝑗
𝑟𝑢 ,

9



as may be verified directly.
We have in particular the spin eigenstates:

𝑍
𝑗
𝑚𝑚 (𝒏) =

2 𝑗∑︁
𝑙=0

2𝑙 + 1
2 𝑗 + 1

〈
𝑗 𝑙 𝑚0 | 𝑗 𝑚

〉
𝑃𝑙 (cos \),

where the 𝑃𝑙 are the Legendre polynomials. If 𝑊𝑧 :=
∑ 𝑗

𝑚=− 𝑗 𝑚𝑍
𝑗
𝑚𝑚 is the symbol associated to the

𝐽𝑧 spin operator, then

𝑊𝑧 (𝒏) =
𝑗∑︁

𝑚=− 𝑗

2 𝑗∑︁
𝑙=0

𝑚
2𝑙 + 1
2 𝑗 + 1

〈
𝑗 𝑙 𝑚0 | 𝑗 𝑚

〉
𝑃𝑙 (cos \) =

√︁
𝑗 ( 𝑗 + 1) cos \.

By means of the quantizer (15), the dynamics of spin was revisited in [10], and Fourier analysis on
𝑆𝑈 (2) was recast in scalar form. Applications to special function theory are given in [19].

2.4 Example 2: nonrelativistic elementary quantum systems

One seeks the projective unitary irreducible representations of 𝐺 = ℝ4 ⋉ (ℝ3 ⋉ 𝑆𝑂 (3)), the identity
component of the Galilean group, acting on ℝ4 by (𝑏, 𝒂, 𝒗, 𝑅) : (𝒙, 𝑡) ↦→ (𝑅𝒙 + 𝒗𝑡 + 𝒂, 𝑡 + 𝑏). To
obtain linear representations, one replaces 𝐺 by its splitting group 𝐺 [7], which is 11-dimensional
and may be described as follows. Let g be the Lie algebra generated by {𝐻, 𝑃𝑖, 𝐽𝑖, 𝐾𝑖, 𝑀} (for
𝑖 = 1, 2, 3) with the commutation relations:

[𝐽𝑖, 𝐽 𝑗 ] = Y𝑖 𝑗𝑘𝐽𝑘 , [𝐽𝑖, 𝐾 𝑗 ] = Y𝑖 𝑗𝑘𝐾 𝑘 , [𝐽𝑖, 𝑃 𝑗 ] = Y𝑖 𝑗𝑘𝑃𝑘 ,
[𝐾 𝑖, 𝐻] = 𝑃𝑖, [𝐾 𝑖, 𝑃 𝑗 ] = 𝛿𝑖 𝑗𝑀,

with all other commutators zero.
Elimination of the central element 𝑀 gives the usual Galilean commutation relations for g, so

that g is a central extension of g; letting 𝐺 be the connected and simply connected Lie group with
Lie algebra g, we have constructed 𝐺 as a central extension of the covering group 𝐺 of 𝐺 by ℝ.
Elements of 𝐺 may be written as (\, 𝑔) =

(
exp(−\𝑀), exp(−𝑏𝐻) exp(𝒂 · 𝑷 + 𝒗 · 𝑲)𝑅

)
with \ ∈ ℝ

and 𝑔 = (𝑏, 𝒂, 𝒗, 𝑅) ∈ 𝐺. The composition law of𝐺 obtained from the above commutation relations
is

(\, 𝑔) · (\′, 𝑔′) = (\ + \′ + 𝜔(𝑔, 𝑔′), 𝑔𝑔′),
where 𝜔(𝑔, 𝑔′) = 1

2 (−𝑏
′𝒗 · 𝑅𝒗′ − 𝒗 · 𝑅𝒂′ + 𝒂 · 𝑅𝒗′) is the factor system. As before, 𝑅 ∈ 𝑆𝑈 (2) and

𝑅 is the 𝑆𝑂 (3) element by which 𝑅 acts on ℝ3.
The unitary irreducible representations of 𝐺 which interest us may be obtained by the induced

representation method and act on the momentum space H 𝑗 = 𝐿2(ℝ3, 𝑑3𝝃) ⊗ ℂ2 𝑗+1, where 𝑗 is a
half-integer number, by

[𝑈𝑚𝑢 𝑗 (\, 𝑏, 𝒂, 𝒗, 𝑅)Φ] (𝝃) := exp
(
𝑖

[
\+𝑏𝑢+ 𝑏 |𝝃 |

2

2𝑚
−𝝃 · 𝒂− 1

2𝑚𝒂 · 𝒗
] )
D 𝑗 (𝑅)Φ

(
𝑅−1(𝝃 +𝑚𝒗)

)
, (16)

where D 𝑗 is the unitary irreducible representation of 𝑆𝑈 (2) on ℂ2 𝑗+1.

10



The coadjoint orbits of 𝐺 have been described in [6] and may be obtained as follows. One first
computes the adjoint action of 𝐺 on the generators 𝐻, 𝑷, 𝑱, 𝑲, 𝑀 of g; denoting the coordinates on
g ∗ by ℎ = b𝐻 , 𝒑 = b𝑷, 𝒋 = b𝑱, 𝒌 = b𝑲 , 𝑚 = b𝑀 , which transform according to Ad(𝑔−1), one finds
(\, 𝑏, 𝒂, 𝒗, 𝑅) · (ℎ, 𝒑, 𝒋, 𝒌, 𝑚) explicitly. (We carry out the analogous calculation for the Poincaré
group in detail in the next Section.) Three invariant quantities appear: 𝑚 itself, 𝑢 = 2𝑚ℎ − | 𝒑 |2 and
|𝑚 𝒋 + 𝒑× 𝒌 |2; these are the “Casimir functions” for the canonical Poisson structure of the coalgebra
of 𝐺, which are constant on the maximal-dimensional orbits, which thus have dimension 8.

Fix 𝑚 > 0 and 𝑢 ∈ ℝ; fix also 𝑠 = | 𝒋 + (1/𝑚) 𝒑 × 𝒌 |; then one obtains a coadjoint orbit O𝑚𝑢𝑠
on which one may introduce coordinates 𝒒 := 𝒌/𝑚, 𝒑, and if 𝑠 > 0, 𝒔 := 𝒋 + (1/𝑚) 𝒑 × 𝒌. The
coadjoint action of 𝐺 on O𝑚𝑢𝑠 reduces to [6]:

(\, 𝑏, 𝒂, 𝒗, 𝑅) · (𝒒, 𝒑, 𝒔) =
(
𝑅(𝒒 − 𝑏

𝑚
𝒑) + 𝒂 + 𝑏𝒗, 𝑅 𝒑 − 𝑚𝒗, 𝑅𝒔

)
. (17)

It can be checked that (4) reduces to

{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 𝑗𝑘 𝑠
𝑘 . (18)

From (17) and (18) it is clear that O𝑚𝑢𝑠 is isomorphic to ℝ6 × 𝕊2 as a symplectic manifold, if
𝑠 > 0, and that O𝑚𝑢0 is symplectically isomorphic to ℝ6, with canonical coordinates (𝒒, 𝒑). We
interpret O𝑚𝑢𝑠 as the elementary classical nonrelativistic particle of mass 𝑚, spin 𝑠 and internal
energy 𝑢 (for most practical purposes, 𝑢 can be taken to be zero).

We now take up the question of quantization. For massive spinless Galilean particles, the ap-
propriate triple is (𝐺,𝑈𝑚00,O𝑚00) where 𝑚 > 0. The desired quantizer is given by the Grossmann–
Royer reflection operators [11, 20]:[

Π(𝒒, 𝒑)Φ
]
(𝝃) := 23 exp

[
2𝑖𝒒 · ( 𝒑 − 𝝃)

]
Φ(2 𝒑 − 𝝃),

acting on H = 𝐿2(ℝ3, 𝑑3𝝃). These operators are self-adjoint and satisfy:

Tr
[
Π(𝒒, 𝒑)

]
= 1; Tr

[
Π(𝒒, 𝒑)Π(𝒒′, 𝒑′)

]
= (2𝜋)3𝛿(𝒒 − 𝒒′) 𝛿( 𝒑 − 𝒑′).

Moreover, it is readily checked, using (16) and (17), that

𝑈𝑚00(𝑔)Π(𝒒, 𝒑)𝑈𝑚00(𝑔)−1 = Π
(
𝑔 · (𝒒, 𝒑)

)
.

(In fact, this holds for 𝑔 belonging to a much larger group than the Galilei group [21].) Thus,
if 𝑑_(𝒒, 𝒑) := (2𝜋)−3 𝑑3𝒒 𝑑3 𝒑, then Π satisfies (7) and so is a quantizer. As was first noticed
by Grossmann and Royer, formula (8b), with the present Stratonovich–Weyl kernel and Liouville
measure, is equivalent to the old Weyl correspondence rule. We recall the formula for the Wigner
functions:

𝑊Φ(𝒒, 𝒑) := ⟨Φ | Π(𝒒, 𝒑) | Φ⟩ = 23
∫
ℝ3

Φ̂(𝒒 + 𝒗)Φ̂(𝒒 − 𝒗)𝑒2𝑖 𝒑·𝒗 𝑑3𝒗, (19)

where Φ̂ denotes the Fourier transform of Φ.
Finally, in order to quantize massive Galilean particles with spin, we may consider the triple

(𝐺,𝑈𝑚0 𝑗 ,O𝑚0𝑠) with 𝑗 > 0 a half-integer. It is shown in [11], and can be verified directly from the
preceding paragraphs, that Ω 𝑗 (𝒒, 𝒑, 𝒏) := Π(𝒒, 𝒑) ⊗ Δ 𝑗 (𝒏), acting on H 𝑗 = 𝐿2(ℝ3, 𝑑3𝝃) ⊗ ℂ2 𝑗+1,
satisfies the properties (7), and thus provides a Stratonovich–Weyl quantizer for Galilean spinning
particles.

11



3 Relativistic classical elementary systems
In this Section, we describe coadjoint orbits for the Poincaré group P. As usual, 𝑀4 denotes
Minkowski space, and if 𝑥 = (𝑥0, 𝒙), 𝑦 = (𝑦0, 𝒚) are 4-vectors in 𝑀4, their Lorentz product is
denoted (𝑥𝑦) = −𝑥0𝑦0 + 𝒙 · 𝒚.

The group P is the semidirect product 𝑇4 ⋉ L where L is the Lorentz group and

(𝑎,Λ) · (𝑎′,Λ′) = (𝑎 + Λ𝑎′,ΛΛ′) for 𝑎 ∈ 𝑇4, Λ ∈ L.

The identity component is P0 = 𝑇4 ⋉ L
↑
+ = 𝑇4 ⋉ 𝑆𝑂0(3, 1), the proper orthochronous Poincaré

group; and in order to assure that only linear representations need be considered, we will work with
its splitting group; this turns out to be just the simply connected double cover P̃0 = 𝑇4 ⋉ 𝑆𝐿 (2,ℂ),
which does not have nontrivial extensions.

If Λ̃ ∈ 𝑆𝐿 (2,ℂ), let Λ be its natural image in 𝑆𝑂0(3, 1): let 𝑋 := 𝑥0𝐼 + 𝒙 · 𝝈 for 𝑥 ∈ 𝑇4, where
𝝈 = (𝜎1, 𝜎2, 𝜎3) is the set of Pauli matrices in ℂ2×2, be the corresponding hermitian matrix in
ℂ2×2; then Λ𝑥 is the 4-vector corresponding to Λ̃𝑋Λ̃†. The product in P̃0 obeys

(𝑎, Λ̃) · (𝑎′, Λ̃′) = (𝑎 + Λ𝑎′, Λ̃Λ̃′) for 𝑎 ∈ 𝑇4, Λ̃ ∈ 𝑆𝐿 (2,ℂ). (20)

The Lie algebra of P̃0 (or of P0 or P) is generated by ten elements 𝐻, 𝑃𝑖, 𝐽𝑖, 𝐾𝑖 (for 𝑖 = 1, 2, 3)
corresponding respectively to time translations, space translations, rotations and pure boosts. Any
element of P̃0 may be written in the standard form

𝑔 = exp(−𝑎0𝐻 + 𝒂 · 𝑷) exp(Z𝒏 · 𝑲) exp(𝛼𝒎 · 𝑱), (21)

where 𝑎 ∈ 𝑇4, 𝒏 and 𝒎 are unit 3-vectors, Z ⩾ 0 and 0 ⩽ 𝛼 ⩽ 2𝜋, with the convention that
exp(2𝜋𝒎 · 𝑱) = −𝐼 in 𝑆𝐿 (2,ℂ) for all 𝒎.

The commutation relations for the generators are:

[𝐽𝑖, 𝐽 𝑗 ] = Y𝑖 𝑗𝑘𝐽𝑘 , [𝐽𝑖, 𝐾 𝑗 ] = Y𝑖 𝑗𝑘𝐾 𝑘 , [𝐽𝑖, 𝑃 𝑗 ] = Y𝑖 𝑗𝑘𝑃𝑘 ,
[𝐾 𝑖, 𝐾 𝑗 ] = −Y𝑖 𝑗𝑘𝐽𝑘 , [𝐾 𝑖, 𝑃 𝑗 ] = 𝛿𝑖 𝑗𝐻, [𝐾 𝑖, 𝐻] = 𝑃𝑖, (22)

as may be verified directly from (20) together with:

exp(Z𝒏 · 𝑲) = cosh 1
2 Z + sinh 1

2 Z𝒏 · 𝝈,
exp(𝛼𝒎 · 𝑱) = cos 1

2𝛼 − 𝑖 sin 1
2𝛼𝒎 · 𝝈.

The adjoint action of P̃0 on its Lie algebra g may be computed as follows. Writing (ad 𝑋)𝑌 :=
[𝑋,𝑌 ] for 𝑋,𝑌 ∈ g, we have

Ad(exp 𝑋)𝑌 = (𝑒ad 𝑋)𝑌 = 𝑌 + [𝑋,𝑌 ] + 1
2!
[𝑋, [𝑋,𝑌 ]] + · · · . (23)

From (23) it is easy to obtain (Ad(exp 𝑋))𝑌 whenever 𝑋 = −𝑎0𝐻, 𝒂 · 𝑷, 𝛼𝒎 · 𝑱 or Z𝒏 · 𝑲, and
𝑌 = 𝐻, 𝑃𝑖, 𝐽𝑖 or 𝐾 𝑖. For instance, if 𝑋 = Z𝒏 · 𝑲, 𝑌 = 𝐻, then

Ad(exp(Z𝒏 · 𝑲))𝐻

= 𝐻 + Z [𝒏 · 𝑲, 𝐻] + Z
2

2!
[𝒏 · 𝑲, [𝒏 · 𝑲, 𝐻]] + Z

3

3!
[𝒏 · 𝑲, [𝒏 · 𝑲, [𝒏 · 𝑲, 𝐻]]] + · · ·

= 𝐻 + Z𝒏 · 𝑷 + Z
2

2!
𝐻 + Z

3

3!
𝒏 · 𝑷 + · · · = 𝐻 cosh Z + 𝒏 · 𝑷 sinh Z .

12



Table 1: The adjoint action Ad(exp 𝑋)𝑌

𝑌⧹𝑋 −𝑎0𝐻 𝒂 · 𝑷 𝛼𝒎 · 𝑱 Z𝒏 · 𝑲
𝐻 𝐻 𝐻 𝐻 𝐻 cosh Z + 𝒏 · 𝑷 sinh Z
𝑷 𝑷 𝑷 𝑅−1

𝛼,𝒎𝑷 𝑷 + 𝐻𝒏 sinh Z + (𝒏 · 𝑷)𝒏(cosh Z − 1)
𝑱 𝑱 𝑱 − 𝒂 × 𝑷 𝑅−1

𝛼,𝒎𝑱 𝑱 cosh Z − 𝒏 × 𝑲 sinh Z − (𝒏 · 𝑱)𝒏(cosh Z − 1)
𝑲 𝑲 + 𝑎0𝑷 𝑲 − 𝐻𝒂 𝑅−1

𝛼,𝒎𝑲 𝑲 cosh Z + 𝒏 × 𝑱 sinh Z − (𝒏 · 𝑲)𝒏(cosh Z − 1)

Table 2: The coadjoint action Coad(exp 𝑋)𝑦

𝑦⧹𝑋 −𝑎0𝐻 𝒂 · 𝑷 𝛼𝒎 · 𝑱 Z𝒏 · 𝑲
ℎ ℎ ℎ ℎ ℎ cosh Z − 𝒏 · 𝒑 sinh Z
𝒑 𝒑 𝒑 𝑅𝛼,𝒎 𝒑 𝒑 − ℎ𝒏 sinh Z + (𝒏 · 𝒑)𝒏(cosh Z − 1)
𝒋 𝒋 𝒋 + 𝒂 × 𝒑 𝑅𝛼,𝒎 𝒋 𝒋 cosh Z + 𝒏 × 𝒌 sinh Z − (𝒏 · 𝒋)𝒏(cosh Z − 1)
𝒌 𝒌 − 𝑎0 𝒑 𝒌 + ℎ𝒂 𝑅𝛼,𝒎 𝒌 𝒌 cosh Z − 𝒏 × 𝒋 sinh Z − (𝒏 · 𝒌)𝒏(cosh Z − 1)

In this way we obtain Table 1, which, together with (21), exhibits the adjoint action of P̃0 in a fully
explicit manner. (We use the notation Λ = 𝑅𝛼𝒎 for the rotation obtained from Λ̃ = exp(𝛼𝒎 · 𝑱) ∈
𝑆𝑈 (2).)

The coadjoint action of P̃0 on g∗ can now be derived immediately. Let ℎ be the linear coordinate
on g∗ associated with 𝐻, and similarly let 𝑝𝑖, 𝑗 𝑖, 𝑘 𝑖 be the coordinates associated to 𝑃𝑖, 𝐽𝑖, 𝐾𝑖

(𝑖 = 1, 2, 3). The coadjoint action is given in these coordinates by Table 2.
The orbits arise from systems of differential equations 𝑋 𝑓 = 0, where 𝑋 is the fundamental vector

field (5), and 𝑋 runs over a basis of g. Due to the commutation relations (22), this yields an involutive
system of differential equations, and the coadjoint orbits are the integral manifolds given by the
Stefan–Sussmann generalization of the Frobenius theorem [14]. Explicitly, for 𝑓 = 𝑓 (ℎ, 𝒑, 𝒋, 𝒌),
we have:

𝒑 · 𝜕 𝑓
𝜕𝒌

= 0, 𝒑 × 𝜕 𝑓
𝜕 𝒋
+ ℎ𝜕 𝑓

𝜕𝒌
= 0,

𝒑 × 𝜕 𝑓
𝜕 𝒑
+ 𝒋 × 𝜕 𝑓

𝜕 𝒋
+ 𝒌 × 𝜕 𝑓

𝜕𝒌
= 0,

𝒑
𝜕 𝑓

𝜕ℎ
+ ℎ 𝜕 𝑓

𝜕 𝒑
− 𝒌 × 𝜕 𝑓

𝜕 𝒋
+ 𝒋 × 𝜕 𝑓

𝜕𝒌
= 0.

In principle, we have to solve these differential equations to find the orbits. In the present case,
this system of equations, while not of constant rank, is generically of rank 8, so that the maximal-
dimensional orbits arise as level sets of two “Casimir functions” 𝐶1, 𝐶2 on g∗. These Casimir
functions are easy to guess from other treatments and to obtain explicitly. Let 𝑝 := (ℎ, 𝒑) be the
“energy-momentum” 4-vector and let 𝑤 := (𝑤0, 𝒘) be the Pauli–Lubański 4-vector given by

𝑤0 := 𝒋 · 𝒑, 𝒘 := 𝒑 × 𝒌 + ℎ 𝒋 . (24)

13



From Table 2 one verifies that 𝑤0 transforms like ℎ and 𝒘 like 𝒑 under the coadjoint action; in
particular, under Coad(exp(Z𝒏 · 𝑲)):

𝑤0 ↦→ 𝑤0 cosh Z − 𝒏 · 𝒘 sinh Z,
𝒘 ↦→ 𝒘 − 𝑤0𝒏 sinh Z + (𝒏 · 𝒘)𝒏(cosh Z − 1). (25)

Thus the Casimir functions we seek are

𝐶1 := (𝑝𝑝) = −ℎ2 + 𝒑 · 𝒑,
𝐶2 := (𝑤𝑤) = −( 𝒋 · 𝒑)2 + | 𝒑 × 𝒌 + ℎ 𝒋 |2.

The next step is to find a suitable set of coordinates on a particular orbit. We shall now restrict
ourselves to orbits for which 𝐶1 < 0, and write 𝐶1 = −𝑚2 with 𝑚 > 0, in order to deal with massive
particles only. (It is clear that the other orbits correspond to the zero-mass and tachyon cases. For a
classification of orbits and an early view on the subject, see [22].) They further subdivide according
to whether ℎ = (𝑚2 + 𝒑 · 𝒑)1/2 or ℎ = −(𝑚2 + 𝒑 · 𝒑)1/2. For convenience, we restrict for the moment
to the positive-energy orbits: ℎ > 0.

Let𝐻+𝑚 denote the forward hyperboloid (𝑥𝑥) = −𝑚2, 𝑥0 > 0, and let ^ := (𝑚, 0) be its vertex. Let
𝐿𝑝 denote the Lorentz boost which takes ^ to 𝑝. We identify 𝐿𝑝 with the boost 𝐵(Z𝒏) := exp(Z𝒏 ·𝑲)
in P0, where

cosh Z =
ℎ

𝑚
, sinh Z =

√
ℎ2 − 𝑚2

𝑚
, 𝒏 = − 𝒑

(ℎ2 − 𝑚2)1/2
.

By a slight abuse of notation, we also identify 𝐿𝑝 with the element

ℎ + 𝑚 − 𝒑 · 𝝈
[2𝑚(ℎ + 𝑚)]1/2

∈ P̃0.

If 𝑎 is a 4-vector, we thus have

𝐿𝑝𝑎 =

(
ℎ𝑎0 + 𝒑 · 𝒂

𝑚
, 𝒂 + 𝑎

0

𝑚
𝒑 + 𝒑 · 𝒂

𝑚(𝑚 + ℎ) 𝒑
)
. (26)

Let us also write 𝑝 for the image of 𝑝 under spatial reflection: 𝑝 := (ℎ,− 𝒑). Then, since
𝐿𝑝𝑝 = ^, it follows from (26) that

𝐿−1
𝑝 = 𝐿𝑝 , and 𝐿𝑝𝑎 = 𝐿𝑝 �̄�.

From the definition (24), 𝑝 and 𝑤 are orthogonal: (𝑝𝑤) = 0, and so

0 = (𝑝𝑤) = (𝐿𝑝𝑝 𝐿𝑝𝑤) = (^ 𝐿𝑝𝑤),

which means that
𝐿𝑝𝑤 = (0, 𝑚𝒔) (27)

for some 3-vector 𝒔. Since
𝐶2 = (𝑤𝑤) = (𝐿𝑝𝑤 𝐿𝑝𝑤) = 𝑚2 |𝒔 |2

14



is constant on any orbit, we may interpret 𝒔 as a “spin” vector.
From (0, 𝑚𝒔) = 𝐿𝑝𝑤, (26) yields:

𝒔 =
𝒘

𝑚
− 1
𝑚

(
𝑤0

𝑚
− 𝒑 · 𝒘
𝑚(𝑚 + ℎ)

)
𝒑 =

𝒘

𝑚
− 𝑤0 𝒑

𝑚(𝑚 + ℎ) . (28)

We write, as usual, 𝑠 = |𝒔 | (not a 4-vector!) to denote the spin modulus.
For fixed 𝑚 and 𝑠 and positive ℎ, we obtain a single orbit O𝑚𝑠+. If 𝑠 > 0, we may take as

coordinates on O𝑚𝑠+ the “momenta” 𝑝1, 𝑝2, 𝑝3 and two spherical coordinates arising from 𝒔; three
coordinates remain to be determined. A possible choice is 𝑞1, 𝑞2, 𝑞3, where

𝒒 :=
𝒌

ℎ
− 𝒑 × 𝒘

𝑚ℎ(𝑚 + ℎ) =
𝒌

ℎ
− 𝒑 × 𝒔

ℎ(𝑚 + ℎ) . (29)

The set (𝒒, 𝒑, 𝒔), where 𝑠 = |𝒔 | > 0 is fixed, gives a system of eight coordinates on the orbit
O𝑚𝑠+. Here 𝒑 ranges over ℝ3, 𝒔/𝑠 over the sphere 𝕊2 and, for fixed 𝒑 and 𝒔, 𝒒 ranges over ℝ3. So
the coadjoint orbit O𝑚𝑠+ is homeomorphic to ℝ6 × 𝕊2, as in the Galilean case.

The Poisson structure on the relevant submanifold of g∗ may be obtained from (4) and (22),
where we use the basis { ℎ, 𝑝𝑖, 𝑗 𝑖, 𝑘 𝑖 : 𝑖 = 1, 2, 3 } of coordinate functions on g∗. Using (24) and (28),
the Poisson brackets for the O𝑚𝑠+ coordinate functions are readily obtained. For instance,

{𝑝𝑖, 𝑝 𝑗 }𝑃 = 0,

{𝑝𝑖, 𝑠 𝑗 }𝑃 =
1
𝑚
{𝑝𝑖, 𝑤 𝑗 }𝑃 −

𝑝 𝑗

𝑚(𝑚 + ℎ) {𝑝
𝑖, 𝑤0}𝑃

=
1
𝑚

(
Y 𝑗𝑘𝑙 𝑝

𝑘 {𝑝𝑖, 𝑘 𝑙}𝑃 + ℎ{𝑝𝑖, 𝑗 𝑗 }𝑃
)
− 𝑝 𝑗 𝑝𝑘

𝑚(𝑚 + ℎ) {𝑝
𝑖, 𝑗 𝑘 }𝑃

=
1
𝑚
(−Y 𝑗𝑘𝑖𝑝𝑘ℎ + Y𝑖 𝑗𝑘ℎ𝑝𝑘 ) −

𝑝 𝑗

𝑚(𝑚 + ℎ) Y
𝑖𝑘
𝑙 𝑝
𝑘 𝑝𝑙 = 0,

{𝑝𝑖, 𝑞 𝑗 }𝑃 = {𝑝𝑖, ℎ−1𝑘 𝑗 − ℎ−1(𝑚 + ℎ)−1Y 𝑗𝑘𝑙 𝑝
𝑘 𝑠𝑙}𝑃

=
1
ℎ
{𝑝𝑖, 𝑘 𝑗 }𝑃 −

Y 𝑗𝑘𝑙

ℎ(𝑚 + ℎ)
(
{𝑝𝑖, 𝑝𝑘 }𝑃𝑠𝑙 + 𝑝𝑘 {𝑝𝑖, 𝑠𝑙}𝑃

)
=

1
ℎ
(−𝛿𝑖 𝑗ℎ) = −𝛿𝑖 𝑗 .

The Poisson brackets of the 𝑞𝑖 and 𝑠𝑖 are similarly computed; the full results are:

{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 𝑗𝑘 𝑠
𝑘 .

Thus we see that { 𝑞𝑖, 𝑝𝑖 } are (part of a set of) canonical coordinates, and that, just as in the
Galilean case, O𝑚𝑠+ is isomorphic to ℝ6 × 𝕊2 as a symplectic manifold, if 𝑠 > 0. It follows that
𝑑3𝒒 𝑑3 𝒑 𝑑2𝒔 is a Liouville measure on O𝑚𝑠+. The case 𝑠 = 0 gives a 6-dimensional orbit O𝑚0+,
isomorphic to ℝ6.

15



It is useful to have at hand the expressions of the g∗ coordinates (ℎ, 𝒑, 𝒋, 𝒌) over the orbit in
terms of the O𝑚𝑠+ coordinates (𝒒, 𝒑, 𝒔). Inverting (24), (28) and (29) yields:

𝑤0 = 𝒑 · 𝒔, 𝒘 = 𝑚𝒔 + 𝒑 · 𝒔
𝑚 + ℎ 𝒑,

𝒋 = 𝒒 × 𝒑 + 𝒔, 𝒌 = ℎ𝒒 + 𝒑 × 𝒔

𝑚 + ℎ ,

as functions on O𝑚𝑠+.
Finally, we could recover from Table 2 the expression of the coadjoint action of P̃0 on O𝑚𝑠+ in

terms of the coordinates (𝒒, 𝒑, 𝒔). From (28) and (29) we readily obtain:

exp(−𝑎0𝐻) · (𝒒, 𝒑, 𝒔) =
(
𝒒 − 𝑎

0

ℎ
𝒑, 𝒑, 𝒔

)
,

exp(𝒂 · 𝑷) · (𝒒, 𝒑, 𝒔) = (𝒒 + 𝒂, 𝒑, 𝒔),
exp(𝛼𝒎 · 𝑱) · (𝒒, 𝒑, 𝒔) = (𝑅𝛼𝒎𝒒, 𝑅𝛼𝒎 𝒑, 𝑅𝛼𝒎 𝒔). (30)

These formulas conform to our “intuition” as to how a relativistic particle should behave.
The effect of the boost is more cumbersome to express. First we note that it acts on the spin

coordinates by a rotation. Indeed, if 𝒔 ↦→ 𝒔′, 𝑤 ↦→ 𝑤′ under the boost 𝐵 = exp(Z𝒏 · 𝑲), then (27)
gives (0, 𝑚𝒔′) = 𝐿−1

𝐵𝑝
𝑤′ = 𝐿−1

𝐵𝑝
𝐵𝑤 = 𝐿−1

𝐵𝑝
𝐵𝐿𝑝 (0, 𝑚𝒔), so that 𝒔′ = 𝑅𝒔 where 𝑅 = 𝐿−1

𝐵𝑝
𝐵𝐿𝑝 is the

“Wigner rotation” corresponding to 𝐵 and 𝑝.
From (25) and (26) we derive the explicit expression

𝒔′ = 𝒔 cos 𝛽 − sin 𝛽
|𝒏 × 𝒑 | (𝒏 × 𝒑) × 𝒔 + (𝒏 × 𝒑) · 𝒔(1 − cos 𝛽)

|𝒏 × 𝒑 |2
𝒏 × 𝒑,

which is a rotation with axis 𝒏 × 𝒑 and angle 𝛽, where

cos 𝛽 = 1 − |𝒏 × 𝒑 |2
(𝑚 + ℎ) (𝑚 + ℎ′) (cosh Z − 1), sin 𝛽 =

−(𝑚 + ℎ) sinh Z + 𝒏 · 𝒑(cosh Z − 1)
(𝑚 + ℎ) (𝑚 + ℎ′) |𝒏 × 𝒑 |,

with ℎ′ := ℎ cosh Z − 𝒏 · 𝒑 sinh Z , as in Table 2. This coincides with the transformation formula
derived previously by Sudarshan and Mukunda [23]. Note in particular that whenever 𝒑 = 0, the
Wigner rotation reduces to the identity and 𝒔′ = 𝒔.

To see how the boost acts on the position coordinates, let us temporarily suppress 𝒔 and write
(𝒒, 𝒑) as coordinates for the orbit O𝑚0+. Let us write

(𝒒′, 𝒑′) := exp(Z𝒏 · 𝑲) · (𝒒, 𝒑).

Then
𝒑′ = 𝒑 − ℎ𝒏 sinh Z + (𝒏 · 𝒑)𝒏(cosh Z − 1),

as in Table 2; again with ℎ′ := ℎ cosh Z − 𝒏 · 𝒑 sinh Z , we obtain

𝒒′ = 𝒒 + sinh Z
ℎ′
(𝒏 · 𝒒) 𝒑 − ℎ(cosh Z − 1)

ℎ′
(𝒏 · 𝒒)𝒏. (31)

Formula (31) corresponds to a covariant transformation of the position coordinate, in the
following precise sense: it gives the rule of transformation of the initial conditions (for free

16



motion) on changing from one Lorentz frame (with unprimed coordinates) to another (with primed
coordinates). Write:

𝒒′(𝑡′) := 𝒒′ + 𝒑′

ℎ′
𝑡′, (32)

and substitute the preceding formulas for 𝒒′, 𝒑′, ℎ′ together with

𝑡′ := 𝑡 cosh Z − 𝒒(𝑡) · 𝒏 sinh Z . (33)

Using 𝒒(𝑡) = 𝒒 + ( 𝒑/ℎ)𝑡 to eliminate 𝒒 in the result, one finally obtains:

𝒒′(𝑡′) = 𝒒(𝑡) − 𝒏𝑡 sinh Z + (𝒏 · 𝒒(𝑡))𝒏(cosh Z − 1), (34)

stipulating that (𝑡, 𝒒(𝑡)) transforms under boosts like a 4-vector. Conversely, starting from (33)
and (34) and the free motion conditions, one eliminates 𝑡 and 𝑡′ from the formulation and recov-
ers (31). To repeat, this last formula does not relate two different coordinatizations of the same set
of events but rather two simultaneity hyperplanes; and as such, it is the expression of the Lorentz
covariance under boosts in a formulation in which time has been eliminated; canonical transforma-
tions can be considered as transformations of the initial conditions of a covariant formulation. The
same point has been made convincingly in the field theory context [24]. One could consider the
following double bundle:

O𝑚0+
𝜋1←− 𝐵 𝜋2−→𝑀4

where the manifold 𝐵 = P̃0/𝑆𝑈 (2) ≈ ℝ7 has the global coordinates (𝑡, 𝒃, 𝒑), so that 𝜋1(𝑡, 𝒃, 𝒑) =
(𝒃 − 𝑡 𝒑/ℎ, 𝒑) and 𝜋2(𝑡, 𝒃, 𝒑) = (𝑡, 𝒃). On 𝐵 the Poincaré group acts in the natural manner; by
quotienting over the fibres we recover the formulae of the coadjoint action. Physically, 𝐵 represents
the set of trajectories of free particles with initial conditions given by the points of O𝑚0+.

Now we turn to the general case of nonzero spin. The transformation formula of 𝒒 under boosts
is an involved expression with spin-dependent terms. Thus we seek to replace 𝒒 by another set of
position coordinates (at the price of losing the canonical property, of course). It is better to work at
the infinitesimal level: we look for a new set of coordinates 𝒙 = (𝑥1, 𝑥2, 𝑥3) on the orbit O𝑚𝑠+ such
that

{𝑘 𝑖, 𝑥 𝑗 }𝑃 = −𝑥𝑖𝑣 𝑗 (𝑖, 𝑗 = 1, 2, 3), (35)

where 𝑣 𝑗 := {𝑥 𝑗 , ℎ}𝑃 = 𝑝 𝑗/ℎ. This is but the infinitesimal form of (31). This equality does not
hold for 𝒙 = 𝒒 when 𝒔 ≠ 0. Introduce, however:

𝒙 :=
𝒌

ℎ
− 𝒑 × 𝒔

𝑚ℎ
= 𝒒 − 𝒑 × 𝒔

𝑚(𝑚 + ℎ) .

Then one easily finds that (35) is verified. A straightforward but tedious computation then allows
one to check that 𝒙 transforms under boosts as desired:

𝒙′ = 𝒙 + sinh Z
ℎ′
(𝒏 · 𝒙) 𝒑 − ℎ(cosh Z − 1)

ℎ′
(𝒏 · 𝒙)𝒏.

We will use 𝒙 instead of 𝒒 for labelling the Ω-kernel in the next section; some simplification is
thereby effected. The fortunate fact here is that 𝑑3𝒙 𝑑3 𝒑 𝑑2𝒔 is still a Liouville measure on O𝑚𝑠+.

We summarize the coadjoint action of P̃0 on the orbit O𝑚𝑠+ in Table 3.

17



Table 3: The coadjoint action Coad(exp 𝑋) (𝒙, 𝒑, 𝒔)

𝑢⧹𝑋 −𝑎0𝐻 𝒂 · 𝑷 𝛼𝒎 · 𝑱 Z𝒏 · 𝑲

𝒙 𝒙 − 𝑎
0

ℎ
𝒑 𝒙 + 𝒂 𝑅𝛼,𝒎 𝒙 𝒙 + 𝒏 · 𝒙

ℎ′
𝒑 sinh Z − ℎ(𝒏 · 𝒙)

ℎ′
𝒏(cosh Z − 1)

𝒑 𝒑 𝒑 𝑅𝛼,𝒎 𝒑 𝒑 − ℎ𝒏 sinh Z + (𝒏 · 𝒑)𝒏(cosh Z − 1)

𝒔 𝒔 𝒔 𝑅𝛼,𝒎 𝒔 𝒔 cos 𝛽 − sin 𝛽
|𝒏 × 𝒑 | (𝒏 × 𝒑) × 𝒔 + [𝒏, 𝒑, 𝒔] (1 − cos 𝛽)

|𝒏 × 𝒑 |2
𝒏 × 𝒑

The existence of the covariant position vector 𝒙 and its distinction from the canonical position
vector 𝒒 were noticed by Pauri and Prosperi [25] and by Bel and Martı́n [26]. The latter obtained
the coadjoint action (without the name) by studying the transformation of the initial conditions for
Poincaré-invariant systems of second-order differential equations. The characterizations of [25]
and [26] are in principle only local, whereas ours is global; but this is a moot point here. Neglect
of these simple facts has obscured the parallel discussion on relativistic “position operators”; we
return to that question later.

4 Construction of the Stratonovich–Weyl quantizer
The unitary irreducible representations of the Poincaré group are constructed by the induced rep-
resentation method. Since P̃0 = 𝑇4 ⋉ 𝑆𝐿 (2,ℂ), the representation space may be realized as a
(multicomponent) function space on an orbit of 𝑆𝐿 (2,ℂ) in the dual space of 𝑇4. Again we restrict
consideration to orbits 𝐻+𝑚 corresponding to massive particles of positive energy: b = (b0, 𝝃) ∈ 𝐻+𝑚
iff (bb) = −𝑚2 and b0 > 0. The corresponding representations are given by [4, 27]:

[𝑈𝑚 𝑗+(𝑎, Λ̃)Φ] (b) := 𝑒−𝑖(𝑎b)D 𝑗 (𝐿−1
b Λ̃𝐿Λ−1b)Φ(Λ−1b),

where 𝐿−1
b
Λ̃𝐿Λ−1b ∈ 𝑆𝑈 (2) is a “Wigner rotation”. In the case Λ̃ = 𝐿𝑝, we shall denote the

Wigner rotation by 𝑅(𝑝, b) := 𝐿−1
b
𝐿𝑝𝐿𝐿−1

𝑝 b
. Thus 𝑗 is a half-integer and the representation space is

H
𝑗 ,+
𝑚 = ℂ2 𝑗+1 ⊗ 𝐿2(𝐻+𝑚, 𝑑`(b)), where ` is the Lorentz-invariant measure: 𝑑`(b) := 𝑑3𝝃/b0.

The coadjoint orbits corresponding to these representations are O𝑚𝑠+, with the same 𝑚 and
corresponding discrete values of spin. To reduce notational clutter, we fix 𝑚 > 0 and a half-
integer 𝑗 , and write simply H 𝑗 , 𝑈 𝑗 , O 𝑗 rather than H

𝑗 ,+
𝑚 , 𝑈𝑚 𝑗+ and O𝑚𝑠+. We shall identify the

sphere of radius 𝑠 with the unit sphere if 𝑠 > 0, and use coordinates (𝒙, 𝒑, 𝒏) on the orbit O 𝑗 .
We are now ready to introduce the Stratonovich–Weyl quantizer satisfying (7). The measure

on O 𝑗 will be of the form 𝑑_ 𝑗 (𝒙, 𝒑, 𝒏) := 𝐶 𝑗 𝑑3𝒙 𝑑3 𝒑 𝑑2𝒏 = 𝐶 𝑗 𝑑
3𝒒 𝑑3 𝒑 𝑑2𝒏; the precise value of

the constant 𝐶 𝑗 must be determined in the process.
A suitable family of kets for H 𝑗 is { |b, 𝑟⟩ : b ∈ 𝐻+𝑚, 𝑟 = − 𝑗 ,− 𝑗 + 1, . . . , 𝑗 }, subject to the

closure relation:
𝑗∑︁

𝑟=− 𝑗

∫
𝐻+𝑚

|b, 𝑟 ⟨⟩b, 𝑟 | 𝑑`(b) = 𝐼,

and the corresponding orthogonality properties:

⟨b, 𝑟 | b′, 𝑟′⟩ = b0 𝛿𝑟𝑟 ′ 𝛿(𝝃 − 𝝃′).

18



Traces of operators on H 𝑗 are then computed as:

Tr 𝐴 =

𝑗∑︁
𝑟=− 𝑗

∫
𝐻+𝑚

⟨b, 𝑟 | 𝐴 | b, 𝑟⟩ 𝑑`(b).

Before deriving the SW quantizer, it is useful to establish a few notational conventions. We will
write, for 𝑝, b ∈ 𝑀4,

{𝑝b} :=
(𝑝b)
(𝑝𝑝) .

In particular, {𝑝𝑝} = 1 and {^b} = b0/𝑚. Moreover, we define the hyperbolic reflection:

𝑀𝑝b := 2{𝑝b}𝑝 − b.

𝑀𝑝 is an (improper) Lorentz transformation, 𝑀𝑝 (𝑀𝑝b) = b, 𝑀𝑝𝑝 = 𝑝 and (𝑝 𝑀𝑝b) = (𝑝b).
Moreover, 𝑀^b = b̄. We shall write 𝑀𝑝𝝃 for the 3-vector component of 𝑀𝑝b. Furthermore, if Λ is
any Lorentz transformation, then Λ𝑀𝑝Λ

−1 = 𝑀Λ𝑝 . Finally, note the relation 𝑀𝑝b = 𝐿𝑝𝐿𝑝 b̄.

Theorem. The unique Stratonovich–Weyl quantizer Ω 𝑗 for the triple (P̃0,𝑈 𝑗 ,O 𝑗 ) is given by:

[Ω 𝑗 (𝒙, 𝒑, 𝒏)Φ] (b)
:= 23{𝑝b}3/2 exp

[
𝑖𝒙 · (𝑀𝑝𝝃 − 𝝃)

]
D 𝑗 (𝑅(𝑝, b)) Δ 𝑗 (𝒏)D 𝑗 (𝑅(𝑝, 𝑀𝑝b)−1)Φ(𝑀𝑝b) (36)

for Φ ∈ H 𝑗 .

Proof. We must verify that this definition and no other satisfies the properties (7).
First we note that (7c) defines a transitive system of covariance; such a system is determined

by specifying an operator Ω 𝑗 (𝑢0) which commutes with the representatives 𝑈 𝑗 (𝑔) of the isotropy
group of 𝑢0. We take 𝑢0 = (0, 0, 𝒏0). Somewhat more generally, it is necessary and sufficient to
establish (7c) for some subgroup which contains the isotropy subgroup of 𝑢0.

To this end, we abbreviate Ω 𝑗 (𝒏) := Ω 𝑗 (0, 0, 𝒏) and consider the subgroup of P̃0 generated
by 𝑆𝑈 (2) and the time translations. We assume 𝑗 > 0. For 𝑅 ∈ 𝑆𝑈 (2), we require Ω 𝑗 to satisfy

[𝑀 𝑗 (𝑅)Ω 𝑗 ] (𝒏) := 𝑈 𝑗 (𝑅)Ω 𝑗 (𝑅−1𝒏)𝑈 𝑗 (𝑅)−1 = Ω 𝑗 (𝒏).

Since 𝑈 𝑗 (𝑅) = D 𝑗 (𝑅) ⊗ _(𝑅), where _ is the left regular representation of 𝑆𝑈 (2) on 𝐿2(𝐻+𝑚, 𝑑`),
any operator fixed by 𝑀 𝑗 (𝑆𝑈 (2)) is of the form

Ω 𝑗 (𝒏) =
2 𝑗∑︁
𝑙=0

_
𝑗

𝑙
𝐹
𝑗

𝑙
(𝒏) ⊗ 𝑃 𝑗

𝑙
, (37)

where { 𝐹 𝑗0 , . . . , 𝐹
𝑗

2 𝑗 } generate the fixed points (13) of 𝑚 𝑗 (𝑆𝑈 (2)), the _ 𝑗
𝑙

are constants given
by (14), and 𝑃 𝑗

𝑙
are operators on 𝐿2(𝐻+𝑚, 𝑑`) such that

⟨𝑅b | 𝑃 𝑗
𝑙
| 𝑅b′⟩ = ⟨b | 𝑃 𝑗

𝑙
| b′⟩ for all 𝑅 ∈ 𝑆𝑈 (2).

19



Furthermore, each 𝑃 𝑗
𝑙

must commute with the representatives of the time translations, so that

⟨b | 𝑃 𝑗
𝑙
| b′⟩ = 𝛿

(
|𝝃 |2 − |𝝃′|2

)
𝑘
𝑗

𝑙
(𝝃, 𝝃′). (38)

We turn now to the tracial condition (7d). We first observe that, on account of (7c), we can write

Ω 𝑗 (𝒙, 𝒑, 𝒏) = 𝑈 (𝑇𝒙)𝑈 (𝐿𝑝)Ω 𝑗 (𝒏)𝑈 (𝐿𝑝)−1𝑈 (𝑇𝒙)−1, (39)

where 𝑇𝒙 denotes the space translation exp(𝒙 · 𝑷). Indeed, it suffices to note that

(𝒙, 𝒑, 𝒏) = 𝑇𝒙 · (0, 𝒑, 𝒏) = (𝑇𝒙𝐿𝑝) · (0, 0, 𝒏),

which follows from Table 3.
From (39), the trace Tr[Ω 𝑗 (𝑢)Ω 𝑗 (𝑣)], where 𝑢 = (𝒙, 𝒑, 𝒏) and 𝑣 = (𝒙′, 𝒑′, 𝒏′) are any two

points of the orbit O 𝑗 , simplifies to:

Tr[Ω 𝑗 (𝒙, 𝒑, 𝒏)Ω 𝑗 (𝒙′, 𝒑′, 𝒏′)] = Tr[𝑈 (𝑇𝒙)Ω 𝑗 (0, 𝒑, 𝒏)𝑈 (𝑇−𝒙)𝑈 (𝑇𝒙′)Ω 𝑗 (0, 𝒑′, 𝒏′)𝑈 (𝑇−𝒙′)]
= Tr[Ω 𝑗 (𝒙 − 𝒙′, 𝒑, 𝒏)Ω 𝑗 (0, 𝒑′, 𝒏′)] . (40)

Thus we can take 𝑣 = (0, 𝒑′, 𝒏′) without loss of generality. Since the𝑇𝒙 and the 𝐿𝑝 do not commute,
no further simplification is possible.

Taking account of (40), the tracial condition can now be written as

Tr[Ω 𝑗 (𝒙, 𝒑, 𝒏)Ω 𝑗 (0, 𝒑′, 𝒏′)] = 𝐶 𝑗𝛿(𝒙) 𝛿( 𝒑 − 𝒑′) 𝐾 𝑗 (𝒏, 𝒏′), (41)

where 𝐶 𝑗 := (2 𝑗 + 1)/4𝜋𝐶 𝑗 . The left hand side of (41) can be expanded as

𝑗∑︁
𝑟,𝑟 ′=− 𝑗

∫
𝐻+𝑚

∫
𝐻+𝑚

⟨b, 𝑟 | Ω 𝑗 (𝒙, 𝒑, 𝒏) | b′, 𝑟′⟩ ⟨b′, 𝑟′ | Ω 𝑗 (0, 𝒑′, 𝒏′) | b, 𝑟⟩ 𝑑`(b) 𝑑`(b′)

=

𝑗∑︁
𝑟,𝑟 ′=− 𝑗

∫
𝐻+𝑚

∫
𝐻+𝑚

exp[𝑖𝒙 · (𝝃 − 𝝃′)] ⟨b, 𝑟 | 𝑈 (𝐿𝑝)Ω 𝑗 (𝒏)𝑈 (𝐿𝑝) | b′, 𝑟′⟩

× ⟨b′, 𝑟′ | 𝑈 (𝐿𝑝′)Ω 𝑗 (𝒏′)𝑈 (𝐿𝑝′) | b, 𝑟⟩ 𝑑`(b) 𝑑`(b′).

Using (37) and the identity 𝑈 (𝐿𝑝) |b, 𝑟⟩ =
∑ 𝑗

𝑠=− 𝑗 D
𝑗
𝑠𝑟 (𝑅(𝑝, 𝐿𝑝b)) |𝐿𝑝b, 𝑠⟩, the integrand is a

sum of terms of the form ⟨𝐿𝑝b | 𝑃 𝑗𝑙 | 𝐿𝑝b
′⟩ ⟨𝐿𝑝′b′ | 𝑃 𝑗𝑙′ | 𝐿𝑝′b⟩. From (26), we note that if |𝝃′| = |𝝃 |

and 𝑝 ≠ ^, then |𝐿𝑝𝝃′| = |𝐿𝑝𝝃 | only if 𝝃′ = ±𝝃. Since the right hand side of (41) is not zero, we
conclude that (38) may be refined to

⟨b | 𝑃 𝑗
𝑙
| b′⟩ = 𝑓

𝑗

𝑙
(b0) 𝛿(𝝃 + 𝝃′) + 𝑔 𝑗

𝑙
(b0) 𝛿(𝝃 − 𝝃′),

or alternatively
𝑃
𝑗

𝑙
= 𝑓

𝑗

𝑙
(b0)𝑃 + 𝑔 𝑗

𝑙
(b0)𝐼,

with 𝑃 denoting the parity operator |b⟩ ↦→ |b̄⟩. The normalization condition (7a), together with
Tr 𝐼 = +∞, demands that 𝑔 𝑗

𝑙
(b0) = 0. Hence

Ω 𝑗 (𝒏) =
2 𝑗∑︁
𝑙=0

_
𝑗

𝑙
𝐹
𝑗

𝑙
(𝒏) ⊗ 23 𝑓

𝑗

𝑙
(b0)𝑃,

20



where we have renormalized the 𝑓 𝑗
𝑙

for convenience; by (7a), each 𝑓
𝑗

𝑙
must be a real function.

Thus we can write

[Ω 𝑗 (𝒙, 𝒑, 𝒏)Φ] (b) = [𝑈 (𝑇𝒙)𝑈 (𝐿𝑝)Ω 𝑗 (𝒏)𝑈 (𝐿𝑝)𝑈 (𝑇−𝒙)Φ] (b)

= 23
2 𝑗∑︁
𝑙=0

exp[𝑖𝒙 · (𝑀𝑝𝝃 − 𝝃)]_ 𝑗𝑙 𝑓
𝑗

𝑙
(𝑚{𝑝b})D 𝑗 (𝑅(𝑝, b))

× 𝐹 𝑗
𝑙
(𝒏)D 𝑗 (𝑅(𝑝, 𝐿𝑝 b̄))Φ(𝑀𝑝b). (42)

Since 𝑅(𝑝, 𝐿𝑝 b̄) = 𝑅(𝑝, 𝑀𝑝b)−1, we conclude that

𝐶 𝑗𝛿(𝒙) 𝛿( 𝒑 − 𝒑′) 𝐾 𝑗 (𝒏, 𝒏′) =
𝑗∑︁

𝑟=− 𝑗

∫
𝐻+𝑚

⟨b, 𝑟 | Ω(𝒙, 𝒑, 𝒏)Ω(0, 𝒑′, 𝒏′) | b, 𝑟⟩ 𝑑`(b)

= 26
∫
𝐻+𝑚

2 𝑗∑︁
𝑘,𝑙=0

_
𝑗

𝑘
_
𝑗

𝑙
𝑓
𝑗

𝑘
(𝑚{𝑝b}) 𝑓 𝑗

𝑙
(𝑚{𝑝′b}) exp[𝑖𝒙 · (𝑀𝑝𝝃 − 𝝃)]⟨𝑀𝑝b | 𝑀𝑝′b⟩ (43)

× Tr
[
D 𝑗 (𝑅(𝑝, b))𝐹 𝑗

𝑘
(𝒏)D 𝑗 (𝑅(𝑝, 𝑀𝑝b)−1)

{
D 𝑗 (𝑅(𝑝′, b))𝐹 𝑗

𝑙
(𝒏′)D 𝑗 (𝑅(𝑝′, 𝑀𝑝′b)−1)

}†]
𝑑`(b).

Now ⟨𝑀𝑝b |𝑀𝑝′b⟩ = (𝑀𝑝b)0 𝛿
(
2{𝑝b} 𝒑−2{𝑝′b} 𝒑′

)
= 2−3ℎ{𝑝b}−2 𝛿( 𝒑− 𝒑′), and so (43) reduces

to

𝐶 𝑗 𝛿(𝒙) 𝐾 𝑗 (𝒏, 𝒏′) = 4𝜋
2 𝑗 + 1

∫
𝐻+𝑚

23ℎ{𝑝b}−2
2 𝑗∑︁
𝑙=0
[ 𝑓 𝑗
𝑙
(𝑚{𝑝b})]2

×
𝑙∑︁

𝑚=−𝑙
𝑌𝑙𝑚 (𝒏)𝑌 𝑙𝑚 (𝒏′) exp[𝑖𝒙 · (𝑀𝑝𝝃 − 𝝃)] 𝑑`(b).

Thus the functions ( 𝑓 𝑗
𝑙
)2 coincide for 𝑙 = 0, 1, . . . , 2 𝑗 . Since∫

ℝ3
23 ℎ

b0 {𝑝b} exp[𝑖𝒙 · (𝑀𝑝𝝃 − 𝝃)] 𝑑3𝝃 =

∫
ℝ3
𝑒𝑖𝒙·𝜼 𝑑3𝜼 = (2𝜋)3 𝛿(𝒙),

and since (7a) now gives 𝑓
𝑗

0 (𝑚) = 1, we conclude that 𝑓 𝑗
𝑙
(b0) = (b0/𝑚)3/2 for all 𝑙 and that

𝐶 𝑗 = (2𝜋)3 for all 𝑗 . Substituting 𝑓
𝑗

𝑙
(𝑚{𝑝b}) = {𝑝b}3/2 in (42) gives the desired result (36).

It remains to check that (7a) holds; by the established covariance, we need only check that Ω 𝑗 (𝒏)
is selfadjoint. Since [Ω 𝑗 (𝒏)Φ] (b) = 23(b0/𝑚)3/2Δ 𝑗 (𝒏)Φ(b̄), it is clear from the invariance of the
measure 𝑑`(b) under b ↦→ b̄ that Ω 𝑗 (𝒏) is symmetric. It has moreover a bounded inverse, and
hence is a selfadjoint operator. □

The Liouville measure on O 𝑗 should thus be normalized to

𝑑_(𝒙, 𝒑, 𝒏) = 𝑑3𝒙 𝑑3 𝒑

(2𝜋)3
2 𝑗 + 1

4𝜋
𝑑2𝒏 for 𝑗 > 0.

For the case 𝑗 = 0, where 𝒙 = 𝒒, the proof is similar (indeed simpler) and the corresponding
Liouville measure is just 𝑑_(𝒒, 𝒑) := (2𝜋)−3 𝑑3𝒒 𝑑3 𝒑.

21



Remark. The SW quantizer is given uniquely by the formula (36), but there is the residual ambiguity
in the 𝑆𝑈 (2)-quantizer Δ 𝑗 , as the signs of the _ 𝑗1, . . . , _

𝑗

2 𝑗 may be chosen freely in (14). We will
keep our choice of only positive _ 𝑗

𝑙
and will continue to speak, with a slight abuse of language, of a

“unique” Stratonovich–Weyl quantizer in the Poincaré group case.

5 The phase-space formalism for Klein–Gordon particles
There is a remarkable dichotomy in relativistic quantum theory between the presentation of ele-
mentary systems by means of unitary irreducible representations of the Poincaré group and their
presentation by means of covariant “wave” equations of various sorts. The objects associated to the
latter were historically introduced first, and are easier to handle because of their manifest covariance
properties; also, they lend themselves to the introduction of interactions. On the other hand, the
theory of unitary irreducible representations introduced by Wigner [4] for reasons of principle,
treats all the particles in a unified way, allowing a more systematic classification.

In practice, this state of affairs indicates that we have by no means finished our task. The
relation between the “Wigner” and “covariant” approaches is by now well understood. One must
adapt the Stratonovich–Weyl construct to the more usual context of wave equations; of course, the
new families of Stratonovich–Weyl operators will be essentially equivalent to that given by (36).
We carry out this adaptation in Section 7 for the case of spin-half particles; meanwhile, in order
to gain familiarity with the phase-space formalism, it will be useful to treat the case of spinless
particles, where the dichotomy becomes vacuous. In this case the representation space becomes
𝐿2(𝐻+𝑚, 𝑑`(b)), which may be immediately identified with the momentum wavefunction space.

The (𝒒, 𝒑) labelling of observables and states can be considered phase-space coordinates for
any Lorentzian observer; by construction the formalism is invariant, although not in a manifest way.

Now suppose that we have a state prepared at 𝑡 = 0 in the configuration 𝜌0(𝒒, 𝒑) = 𝜌0(𝑢). Its
free evolution is given by the classical formula:

𝜌𝑡 (𝒒, 𝒑) = Tr
[
𝑈 (𝑇𝑡)−1(2𝜋)−3

(∫
ℝ6
𝜌0(𝒒′, 𝒑′)Ω0(𝒒′, 𝒑′) 𝑑3𝒒′ 𝑑3 𝒑′

)
𝑈 (𝑇𝑡)Ω0(𝒒, 𝒑)

]
= Tr

[
(2𝜋)−3

∫
ℝ6
𝜌0(𝒒′, 𝒑′)Ω0(𝒒′, 𝒑′)Ω0(𝒒 − 𝑡 𝒑/ℎ, 𝒑) 𝑑3𝒒′ 𝑑3 𝒑′

]
=

∫
ℝ6
𝜌0(𝒒′, 𝒑′) 𝛿(𝒒 − 𝑡 𝒑/ℎ − 𝒒′) 𝛿( 𝒑 − 𝒑′) 𝑑3𝒒′ 𝑑3 𝒑′

= 𝜌0(𝒒 − 𝑡 𝒑/ℎ, 𝒑) = 𝜌0(𝒒 − 𝒗𝑡, 𝒑), (44)

with 𝒗 denoting the velocity 𝒑/ℎ; and the expected value of the observable 𝑓 in the state 𝜌 is:

⟨ 𝑓 ⟩𝜌𝑡 = (2𝜋)−3
∫
ℝ6
𝜌𝑡 (𝒒, 𝒑) 𝑓 (𝒒, 𝒑) 𝑑3𝒒 𝑑3 𝒑.

As
∫
𝜌𝑡 (𝒒, 𝒑) 𝑑3𝒒 𝑑3 𝒑 = (2𝜋)3, this is the promised (1)!

The inherent simplicity of these formulas lies in the fact that there is no need to “quantize” 𝑓 ,
that is, to express it in operatorial form, a hopeless task in general. In this sense, phase space
quantization is an ab initio quantization.

22



Not every configuration 𝜌0 qualifies as a state, however. The more important states are the
Wigner functions:

𝑊Φ(𝒒, 𝒑) := ⟨Φ | Ω0(𝒒, 𝒑) | Φ⟩

= 23
∫
ℝ3
{𝑝b}3/2 exp

[
𝑖𝒒 · (𝑀𝑝𝝃 − 𝝃)

]
Φ(b)Φ(𝑀𝑝b) 𝑑3𝝃/b0. (45)

Note that𝑊Φ ×𝑊Φ = 𝑊Φ. Thus it is clear from (12) that we have

(2𝜋)−3
∫
ℝ6
𝑊Φ(𝒒, 𝒑) ( 𝑓 × 𝑓 ) (𝒒, 𝒑) 𝑑3𝒒 𝑑3 𝒑 = (2𝜋)−3

∫
ℝ6
|𝑊Φ × 𝑓 |2(𝒒, 𝒑) 𝑑3𝒒 𝑑3 𝒑 ⩾ 0,

so 𝑊Φ qualifies as a state. The simplest example is the “plane wave” Φ𝒌 (𝝃) = 𝑘0 𝛿(𝝃 − 𝒌).
Substituting in (45) we get:

𝑊Φ𝒌 (𝒒, 𝒑) = 23
∫
ℝ3
{𝑝b}3/2 exp

[
𝑖𝒒 · (𝑀𝑝𝝃 − 𝝃)

]
𝛿(𝝃 − 𝒌) 𝛿(𝑀𝑝𝝃 − 𝒌) (𝑘0)2/b0 𝑑3𝝃

= 23
∫
ℝ3
{𝑝b}3/2 exp

[
2𝑖𝒒 · ({𝑝b} 𝒑 − 𝝃)

]
𝛿(𝝃 − 𝒌) 𝛿(2( 𝒑 − 𝒌)) (𝑘0)2/b0 𝑑3𝝃

= 𝑘0 𝛿( 𝒑 − 𝒌).

Let us denote generally by Op( 𝑓 ) the operator corresponding by (8b) to a function 𝑓 ; by
definition, Op(𝑊𝐴) = 𝐴.

Now we prove that (2𝜋)−3
∫
𝒒Ω0(𝒒, 𝒑) 𝑑3𝒒 𝑑3 𝒑 is the Newton–Wigner operator 𝑸op; that is,

Op(𝒒) = 𝑸op. This one can expect from the proof by O’Connell and Wigner [28] that 𝑸op is unique
in fulfilling ⟨𝑸op⟩Φ(𝑡) = ⟨𝑸op⟩Φ(0) + 𝑡⟨𝐻−1

op 𝑷op⟩Φ, similarly to our (32).
We have:

(2𝜋)−3
∫

𝒒 ⟨Φ | Ω0(𝒒, 𝒑) | Φ⟩ 𝑑3𝒒 𝑑3 𝒑

= 𝜋−3
∫
ℝ9
{𝑝b}3/2𝒒 exp

[
𝑖𝒒 · (𝑀𝑝𝝃 − 𝝃)

]
Φ(b)Φ(𝑀𝑝b) (b0)−1 𝑑3𝝃 𝑑3𝒒 𝑑3 𝒑

= (2𝜋)−3
∫
ℝ9

𝒒 exp[𝑖𝒒 · (𝒗 − 𝝃)]Φ(b)Φ(𝑣)
[
𝑣0 + b0

2b0𝑣0

(1 + {𝑣b}
2

)−3/4
]
𝑑3𝝃 𝑑3𝒒 𝑑3𝒗 (46)

where we have made the change of variable 𝒑 ↦→ 𝒗 := 𝑀𝑝𝝃.
Introducing new variables 𝒚 := 1

2 (𝒗 + 𝝃), 𝒛 := 𝒗 − 𝝃, let ℎ(𝒚, 𝒛) be the bracketed expression in
the integrand (46) in terms of 𝒚, 𝒛; due to this expression’s symmetry in 𝒗 and 𝝃, we have 𝜕ℎ

𝜕𝒛

��
𝒛=0 = 0.

Integrating now over the 𝒒 variables, we obtain

(2𝜋)−3
∫

𝒒⟨Φ | Ω(𝒒, 𝒑) | Φ⟩ 𝑑3𝒒 𝑑3 𝒑 = −𝑖
∫
ℝ6

𝜕

𝜕𝒛
(𝛿(𝒛))Φ(𝒚 − 1

2 𝒛)Φ(𝒚 +
1
2 𝒛)ℎ(𝒚, 𝒛) 𝑑

3𝒛 𝑑3𝒚

= 𝑖

∫
ℝ6

𝜕

𝜕𝒛

(
Φ(𝒚 − 1

2 𝒛)Φ(𝒚 +
1
2 𝒛)ℎ(𝒚, 𝒛)

) ��
𝒛=0 𝑑

3𝒚

=

∫
ℝ3

Φ(𝒚)
(
𝑖
𝜕

𝜕𝒚
− 𝑖𝒚

2(𝑦0)2

)
Φ(𝒚) 𝑑

3𝒚

𝑦0 .

23



From our remarks at the end of Section 3, it follows that the Newton–Wigner operator for spinless
particles corresponds to a covariant position observable. This is not made altogether clear in the
original paper by Newton and Wigner [29].

Other relevant observables are associated to the generators of the representation:

𝐻op = b0, 𝑷op = 𝝃, 𝑱op = 𝑖
𝜕

𝜕𝝃
× 𝝃, 𝑲op = 𝑖b0 𝜕

𝜕𝝃
.

Similar routine calculations allow one to check

⟨Φ | 𝐻op | Φ⟩ = (2𝜋)−3
∫
ℝ6
ℎ𝑊Φ(𝒒, 𝒑) 𝑑3𝒒 𝑑3 𝒑,

⟨Φ | 𝑷op | Φ⟩ = (2𝜋)−3
∫
ℝ6

𝒑𝑊Φ(𝒒, 𝒑) 𝑑3𝒒 𝑑3 𝒑,

⟨Φ | 𝑱op | Φ⟩ = (2𝜋)−3
∫
ℝ6

𝒋𝑊Φ(𝒒, 𝒑) 𝑑3𝒒 𝑑3 𝒑,

⟨Φ | 𝑲op | Φ⟩ = (2𝜋)−3
∫
ℝ6

𝒌𝑊Φ(𝒒, 𝒑) 𝑑3𝒒 𝑑3 𝒑.

One of the typical properties of the ordinary Wigner function is that its marginals give the
correct probabilities for finding the particle localized at a given point or with a given momentum.
In the nonrelativistic case, these properties come for free from our basic postulates of traciality
and covariance. But here we do not have an a priori reason for expecting that to happen; nor is it
necessary for a consistent phase-space theory.

In fact, we do have

(2𝜋)−3
∫
ℝ3
𝑊Φ(𝒒, 𝒑) 𝑑3𝒒 =

|Φ( 𝒑) |2
ℎ

so the marginal with respect to 𝒑 indeed gives the standard probability of finding a particle with the
given momentum. But the analogous property is not true for the integral over 𝒑.

The standard formulation of the probability for locating the position of a particle is somewhat
involved, so it may be worth recalling it. Let 𝜙(𝒒, 𝑡) be the spacetime wavefunction corresponding
to Φ:

𝜙(𝒒, 𝑡) := (2𝜋)−3/2
∫
ℝ3

exp[−𝑖(b0𝑡 − 𝝃 · 𝒒)]Φ(b) 𝑑3𝝃/b0.

By definition, it satisfies the Klein–Gordon equation:(
−𝜕

2𝜙

𝜕𝑡2
+ Δ − 𝑚2

)
𝜙 = 0.

Now, letting
𝜒 := 𝐻1/2

KG𝜙 := (−Δ + 𝑚2)1/4𝜙,

and 𝑑KG(𝒒, 𝑡) := |𝜒(𝒒, 𝑡) |2, one can check that 𝑑KG has properties required to be interpreted as a
probability density in configuration space:∫

ℝ3
𝑑KG(𝒒, 𝑡) 𝑑3𝒒 = 1;

∫
ℝ3

𝒒 𝑑KG(𝒒, 𝑡) 𝑑3𝒒 = ⟨𝑸op⟩(𝑡).

24



The trouble with this 𝑑KG (as with its higher spin counterparts) lies in the nonlocal properties
pointed out by Hegerfeldt [30].

On the other hand, we can define a different density function from the Wigner function 𝑊Φ

corresponding to Φ:

𝑑 (𝒒, 𝑡) := (2𝜋)−3
∫
ℝ3
𝑊Φ(𝒒 − 𝑡 𝒑/ℎ, 𝒑) 𝑑3 𝒑.

It is immediately clear, from the facts already proved, that∫
ℝ3
𝑑 (𝒒, 𝑡) 𝑑3𝒒 = 1;

∫
ℝ3

𝒒 𝑑 (𝒒, 𝑡) 𝑑3𝒒 = ⟨𝑸op⟩(𝑡);

but it follows from (44) that 𝑑 is local (that is, its support does not grow or shrink supraluminally).
The higher moments of 𝑑KG and 𝑑 are of course different. In fact, one has:∫

ℝ3
(𝑞1)𝑛1 (𝑞2)𝑛2 (𝑞3)𝑛3𝑑KG(𝒒, 𝑡) 𝑑3𝒒 =

∫
ℝ3
(𝑞1)×𝑛1 (𝑞2)×𝑛2 (𝑞3)×𝑛3 𝑑 (𝒒, 𝑡) 𝑑3𝒒

= ⟨(𝑄1
op)𝑛1 (𝑄2

op)𝑛2 (𝑄3
op)𝑛3⟩(𝑡), (47)

where (𝑞𝑖)×𝑛𝑖 denotes the 𝑛𝑖-fold twisted product 𝑞𝑖 × 𝑞𝑖 × · · · × 𝑞𝑖. The fact that 𝑑KG ≠ 𝑑 is related
to (𝑞𝑖)×𝑛 ≠ (𝑞𝑖)𝑛, whereas, as we prove below, (𝑝𝑖)×𝑛 = (𝑝𝑖)𝑛. We do not know, however, whether
𝑑 is always nonnegative.

We argue that the rule expressed by (47) is a natural one. In effect, it is universally true that

(2𝜋)−3
∫
ℝ6
𝑓 ×(𝑊𝐴) (𝒒, 𝒑)𝑊Φ(𝒒, 𝒑) 𝑑3𝒒 𝑑3 𝒑 = ⟨ 𝑓 (𝐴)⟩Φ

for an operator 𝐴, provided 𝑓 × is the twisted function corresponding to 𝑓 . In the ordinary nonrela-
tivistic Moyal mechanics, as long as𝑊𝐴 corresponds to a canonical coordinate, 𝑓 × and 𝑓 coincide.
For examples of nontrivial twisted function computations with the harmonic oscillator Hamiltonian
and other important operators, see [19].

We turn our attention now to the twisted product. Its trikernel is given by:

𝐿 (𝒒1, 𝒑1; 𝒒2, 𝒑2; 𝒒3, 𝒑3) = Tr
[
Ω0(𝒒1, 𝒑1)Ω0(𝒒2, 𝒑2)Ω0(𝒒3, 𝒑3)

]
= 29

∫
ℝ12
{𝑝1b1}3/2{𝑝2b2}3/2{𝑝3b3}3/2

× exp
{
𝑖
[
𝒒1 · (𝑀𝑝1𝝃1 − 𝝃1) + 𝒒2 · (𝑀𝑝2𝝃2 − 𝝃2) + 𝒒3 · (𝑀𝑝3𝝃3 − 𝝃3)

]}
× 𝛿(𝝃 − 𝝃1) 𝛿(𝑀𝑝1𝝃1 − 𝝃2) 𝛿(𝑀𝑝2𝝃2 − 𝝃3) 𝛿(𝑀𝑝3𝝃3 − 𝝃) 𝑑3𝝃1 𝑑

3𝝃2 𝑑
3𝝃3 𝑑

3𝝃

= 29
∫
ℝ3
{𝑝1b}3/2{𝑝2𝑀𝑝1b}3/2{𝑝3𝑀𝑝2𝑀𝑝1b}3/2

× exp
{
𝑖
[
𝒒1 · (𝑀𝑝1𝝃 − 𝝃) + 𝒒2 · (𝑀𝑝2𝑀𝑝1𝝃 − 𝑀𝑝1𝝃) + 𝒒3 · (𝝃 − 𝑀𝑝2𝑀𝑝1𝝃)

]}
× 𝛿(𝑀𝑝3𝑀𝑝2𝑀𝑝1𝝃 − 𝝃) 𝑑3𝝃 . (48)

The equation 𝑀𝑝3𝑀𝑝2𝑀𝑝1b = b has the unique solution

b = 𝐴
(
{𝑝2𝑝3}𝑝1 − {𝑝3𝑝1}𝑝2 + {𝑝1𝑝2}𝑝3

)
,

25



where
𝐴 =

(
{𝑝1𝑝2}2 + {𝑝2𝑝3}2 + {𝑝3𝑝1}2 − 2{𝑝1𝑝2}{𝑝2𝑝3}{𝑝3𝑝1}

)−1/2
.

For simplicity, we abbreviate 𝑎 = {𝑝2𝑝3}, 𝑏 = {𝑝3𝑝1}, 𝑐 = {𝑝1𝑝2}. Substituting in (48), we obtain,
after a laborious computation,

𝐿 (𝒒1, 𝒑1; 𝒒2, 𝒑2; 𝒒3, 𝒑3)
= 26𝐴13/2(𝑎𝑏𝑐)3/2 exp

(
2𝑖𝐴[𝑏𝒒1 · 𝒑2 + 𝑐𝒒2 · 𝒑3 + 𝑎𝒒3 · 𝒑1 − 𝑐𝒒1 · 𝒑3 − 𝑎𝒒2 · 𝒑1 − 𝑏𝒒3 · 𝒑2]

)
.

Using (30) and (31), one verifies directly that this trikernel has the desired equivariance property:

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

In the nonrelativistic limit, 𝑎, 𝑏, 𝑐, 𝐴 → 1. Thus we recover the trikernel for the nonrelativistic
twisted product:

𝐿 (𝒒1, 𝒑1; 𝒒2, 𝒑2; 𝒒3, 𝒑3) = 26 exp
[
2𝑖

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

) ]
.

Let us illustrate how twisted products may be computed by two simple examples. Let 𝑓 ( 𝒑) be
a function of 𝒑 alone. Then if ℎ(𝒒, 𝒑) = 𝑝 𝑗 × 𝑓 ( 𝒑), we find that

ℎ(𝒒1, 𝒑1) = 𝜋−6
∫
ℝ12
(𝑎𝑏𝑐)3/2𝐴13/2𝑝 𝑗2 𝑓 ( 𝒑3)

× exp
{
−2𝑖𝐴[𝑎 𝒑1 · (𝒒2 − 𝒒3) + 𝑏 𝒑2 · (𝒒3 − 𝒒1) + 𝑐 𝒑3 · (𝒒1 − 𝒒2)]

}
𝑑3𝒒2 𝑑

3 𝒑2 𝑑
3𝒒3 𝑑

3 𝒑3

= 𝜋−3
∫
ℝ9
(𝑎𝑏𝑐)3/2𝐴13/2𝑝 𝑗2 𝑓 ( 𝒑3) · 𝛿(𝑏 𝒑2 − 𝑎 𝒑1)

× exp
{
−2𝑖𝐴[(𝑎 𝒑1 − 𝑐 𝒑3) · 𝒒2 − (𝑏 𝒑2 − 𝑐 𝒑3) · 𝒒1]

}
𝑑3𝒒2 𝑑

3 𝒑2 𝑑
3 𝒑3

= 𝜋−3
∫
ℝ9
𝑝
𝑗

2 𝑓 ( 𝒑3)
𝑏ℎ1
ℎ3

𝛿( 𝒑2 − 𝒑1) exp[−2𝑖(𝑏 𝒑1 − 𝒑3) · (𝒒2 − 𝒒1)] 𝑑3𝒒2 𝑑
3 𝒑2 𝑑

3 𝒑3

= 𝜋−3𝑝
𝑗

1

∫
ℝ6
𝑓 ( 𝒑3)

𝑏ℎ1
ℎ3

exp[−2𝑖(𝑏 𝒑1 − 𝒑3) · (𝒒2 − 𝒒1)] 𝑑3𝒒2 𝑑
3 𝒑3

= 𝑝
𝑗

1

∫
ℝ3
𝑓 ( 𝒑3)

𝑏ℎ1
ℎ3

𝛿(𝑏 𝒑1 − 𝒑3) exp[−2𝑖𝒒1 · (𝑏 𝒑1 − 𝒑3)] 𝑑3 𝒑3

= 𝑝
𝑗

1

∫
ℝ3
𝑓 ( 𝒑3) 𝛿( 𝒑3 − 𝒑1) exp[−2𝑖𝒒1 · ( 𝒑1 − 𝒑3)] 𝑑3 𝒑3 = 𝑝

𝑗

1 𝑓 ( 𝒑1),

as expected. Taking 𝑓 ( 𝒑) := (𝑝 𝑗 )𝑛−1 confirms our previous assertion that (𝑝 𝑗 )×𝑛 = (𝑝 𝑗 )𝑛.
Furthermore, if 𝑘 (𝒒, 𝒑) = 𝑞 𝑗 × 𝑓 ( 𝒑), a similar calculation gives

𝑘 (𝒒1, 𝒑1) = 𝜋−3
∫
ℝ6
𝑞
𝑗

2 𝑓 ( 𝒑3)
𝑏ℎ1
ℎ3

exp[−2𝑖(𝑏 𝒑1 − 𝒑3) · (𝒒2 − 𝒒1)] 𝑑3𝒒2 𝑑
3 𝒑3

=
1
2𝑖

∫
ℝ3
𝑓 ( 𝒑3)

𝑏ℎ1
ℎ3

𝜕𝑗𝛿(𝑏 𝒑1 − 𝒑3) exp[2𝑖𝒒1 · (𝑏 𝒑1 − 𝒑3)] 𝑑3 𝒑3

=
1
2𝑖

∫
ℝ3
𝑓 ( 𝒑3(𝒓)) 𝜕𝑗𝛿(𝒓) exp[−2𝑖𝒒1 · 𝒓] 𝑑3𝒓

=
𝑖

2
𝜕

𝜕𝑟 𝑗

(
𝑓 ( 𝒑3(𝒓)) exp[−2𝑖𝒒1 · 𝒓]

) ����
𝒓=0

= 𝑞
𝑗

1 𝑓 ( 𝒑1) + 𝑖
2 𝜕𝑗 𝑓 ( 𝒑1),

26



using the change of variable 𝒓 := 𝒑3 − 𝑏 𝒑1 and noting that 𝒓 = 0 only when 𝒑3 = 𝒑1. We may
summarize these results as:

𝑝 𝑗 × 𝑓 ( 𝒑) = 𝑝 𝑗 𝑓 ( 𝒑), 𝑞 𝑗 × 𝑓 ( 𝒑) = 𝑞 𝑗 𝑓 ( 𝒑) + 𝑖
2
𝜕 𝑓

𝜕𝑝 𝑗
( 𝒑).

In an analogous manner, it is easy to show that

𝑓 ( 𝒑) × 𝑞 𝑗 = 𝑞 𝑗 𝑓 ( 𝒑) − 𝑖
2
𝜕 𝑓

𝜕𝑝 𝑗
( 𝒑),

so that the Moyal bracket of 𝑞 𝑗 and 𝑓 ( 𝒑), defined by

{𝑞 𝑗 , 𝑓 ( 𝒑)}𝑀 :=
1
𝑖
(𝑞 𝑗 × 𝑓 ( 𝒑) − 𝑓 ( 𝒑) × 𝑞 𝑗 ),

satisfies
{𝑞 𝑗 , 𝑓 ( 𝒑)}𝑀 =

𝜕 𝑓

𝜕𝑝 𝑗
( 𝒑) = {𝑞 𝑗 , 𝑓 ( 𝒑)}𝑃 .

6 Quantized observables
We now “quantize” the main observables in the Wigner realization for 𝑗 > 0. As before, we write
𝐻op, 𝑷op, 𝑱op, 𝑲op,𝑸op, 𝑿op for the operators corresponding to the coordinates ℎ, 𝒑, 𝒋, 𝒌, 𝒒, 𝒙.

First we check that Op( 𝑓 ( 𝒑)) = 𝑓 (𝑷op), where 𝑷op is of course the multiplication operator 𝝃.
To simplify the notation, summation over repeated indices of the Wigner spinors is to be understood:

⟨Φ | 𝑓 ( 𝒑)Ω 𝑗 (𝒙, 𝒑, 𝒏) | Ψ⟩ = 23
∫

Φ(b){𝑝b}3/2 exp[𝑖𝒙 · (𝑀𝑝𝝃 − 𝝃)] 𝑓 ( 𝒑)D 𝑗 (𝑅(𝑝, b))Δ 𝑗 (𝒏)

×D 𝑗 (𝑅(𝑝, 𝑀𝑝b)−1)Ψ(𝑀𝑝b)
𝑑3𝝃

b0
2 𝑗 + 1

4𝜋
𝑑2𝒏

𝑑3𝒙 𝑑3 𝒑

(2𝜋)3

= 23
∫

Φ(b){𝑝b}3/2 𝛿(2 𝒑 − 2𝝃) 𝑓 ( 𝒑)D 𝑗 (𝑅(𝑝, b))Δ 𝑗 (𝒏)

×D 𝑗 (𝑅(𝑀𝑝b, 𝑝))Ψ(𝑀𝑝b)
𝑑3𝝃

b0
2 𝑗 + 1

4𝜋
𝑑2𝒏 𝑑3 𝒑

=

∫
Φ(b) 𝑓 (𝝃)Ψ(b) 𝑑`(b) = ⟨Φ | 𝑓 (𝑷op) | Ψ⟩.

It can be seen that the calculations made in [10] for pure functions of spin are essentially
unchanged here and we obtain

Op
(√︁
𝑗 ( 𝑗 + 1) 𝒏

)
= 𝑺op

where 𝑺op is the angular momentum generator for spin 𝑗 . Moreover, it is obvious that if we quantize√︁
𝑗 ( 𝑗 + 1) 𝒏 𝑓 ( 𝒑), we get 𝑺op 𝑓 (𝑷op) = 𝑓 (𝑷op)𝑺op.

Now we compute 𝑿op:

⟨Φ | 𝒙Ω 𝑗 (𝒙, 𝒑, 𝒏) | Ψ⟩ = 23
∫

Φ(b){𝑝b}3/2𝒙 exp[𝑖𝒙 · (𝑀𝑝𝝃 − 𝝃)] 𝑓 ( 𝒑)D 𝑗 (𝑅(𝑝, b))Δ 𝑗 (𝒏)

×D 𝑗 (𝑅(𝑀𝑝b, 𝑝))Ψ(𝑀𝑝b)
𝑑3𝝃

b0
2 𝑗 + 1

4𝜋
𝑑2𝒏

𝑑3𝒙 𝑑3 𝒑

(2𝜋)3
. (49)

27



We now change variables as in (46): first 𝒑 ↦→ 𝒗 := 𝑀𝑝𝝃, then 𝒚 := 1
2 (𝒗 +𝝃), 𝒛 := 𝒗−𝝃. Integrating

over the 𝒒 and 𝒏 variables and then integrating by parts, (49) yields

𝑖

∫
ℝ3

𝜕

𝜕𝒛

[
Φ(𝒚 − 1

2 𝒛)Ψ(𝒚 +
1
2 𝒛)ℎ(𝒚, 𝒛)D

𝑗 (𝑅′(𝒚, 𝒛))D 𝑗 (𝑅′′(𝒚, 𝒛))
] ����
𝒛=0

𝑑3𝒚

where ℎ(𝒚, 𝒛) is the same function as before and 𝑅′, 𝑅′′ denote respectively 𝑅(𝑝, b) and 𝑅(𝑀𝑝b, 𝑝)
as functions of 𝒚, 𝒛. The explicit expression for 𝑅(𝑝, b) is

𝑅(𝑝, b) = (b
0 + 𝑚) (ℎ + 𝑚) − 𝒑 · 𝝃 − 𝑖𝝈 · (𝝃 × 𝒑)
[2(b0 + 𝑚) (ℎ + 𝑚) (𝑚2 − (b𝑝))]1/2

.

We have then, for 𝑝 ≈ b:

𝑅(𝑝, b) ≈ 𝑅(𝑀𝑝b, 𝑝) ≈ 1 − 𝑖
4
𝝈 · (𝝃 × 𝒗)
𝑚(b0 + 𝑚)

,

and
D 𝑗 (𝑅′(𝒚, 𝒛)) ≈ D 𝑗 (𝑅′′(𝒚, 𝒛)) ≈ 1 − 𝑖

2
(𝑺op × 𝒚) · 𝒛
𝑚(𝑦0 + 𝑚)

.

We conclude that
𝑿op = 𝑖

(
𝜕

𝜕𝝃
− 𝝃

2(b0)2

)
+

𝑺op × 𝝃
𝑚(b0 + 𝑚)

.

Now (𝑺op × 𝝃)/𝑚(b0 + 𝑚) is the “quantization” of (𝒔 × 𝒑)/𝑚(ℎ + 𝑚), where 𝒔 =
√︁
𝑗 ( 𝑗 + 1) 𝒏.

It follows that
𝑸op = 𝑖

(
𝜕

𝜕𝝃
− 𝝃

2(b0)2

)
for all 𝑗 : the Newton–Wigner operator has a form which is independent of 𝑗 (in the Wigner
realization). This often forgotten fact has been recalled recently by Chakrabarti [31].

Routine calculations now establish:

𝑱op = 𝑖
𝜕

𝜕𝝃
× 𝝃 + 𝑺op , 𝑲op = 𝑖b0 𝜕

𝜕𝝃
+
𝝃 × 𝑺op

b0 + 𝑚
.

7 The phase-space formalism for Dirac particles
We begin by fixing some notation. In the spirit of [32], we think of 4-vectors 𝑦 = (𝑦0, 𝒚) as the
corresponding matrices 𝑦0𝐼 + 𝒚 · 𝝈, and freely multiply them:

𝑦1𝑦2 = (𝑦0
1𝑦

0
2 + 𝒚1 · 𝒚2, 𝑦

0
1𝒚2 + 𝑦0

2𝒚1 + 𝑖𝒚1 × 𝒚2).

This leads to a remarkable simplification in several formulas. As(
(b0 + 𝑚, 𝝃)√︁

2(b0 + 𝑚)

)2
= b,

we shall write simply √︁
b :=

(b0 + 𝑚, 𝝃)√︁
2(b0 + 𝑚)

.

28



(We shall not have occasion to use the other square roots of b.) Our Dirac matrices are

𝛾0 :=
(
0 𝐼

𝐼 0

)
, 𝜸 :=

(
0 −𝝈
𝝈 0

)
,

i.e., we choose the chiral representation. Define

/b := −(b𝛾) =
(
0 b

b̄ 0

)
.

It is well-known and easy to check that if 𝐷𝑖, 𝑗 denotes the standard finite-dimensional represen-
tations of 𝑆𝐿 (2,ℂ) and

𝑆(Λ̃) :=
(
Λ̃ 0
0 Λ̃†−1

)
=

(
𝐷1/2,0(Λ̃) 0

0 𝐷0,1/2(Λ̃)

)
with Λ̃ ∈ 𝑆𝐿 (2,ℂ), then

𝑆(Λ̃) /b 𝑆(Λ̃)−1 = ⧸Λb.

The method of relating Dirac’s and Wigner’s realizations for spin-half particles is also well
known. We shall proceed from the latter to the former. Consider the representation space H1/2,+

𝑚 :=
ℂ2 ⊗ 𝐿2(𝐻+𝑚, 𝑑`(b)). We shall introduce a Hilbert space Ĥ1/2,+

𝑚 of 4-spinors isomorphic to H
1/2,+
𝑚 .

If Φ ∈ H1/2,+
𝑚 , define

𝑣𝐴 (b) :=
1
√

2
𝐷1/2,0(𝐿b)Φ(b) =

√︁
b/2𝑚Φ(b),

𝑣𝐵 (b) :=
1
√

2
𝐷0,1/2(𝐿b)Φ(b) =

√︃
b̄/2𝑚Φ(b).

We have then:
𝑣𝐵 (b) =

b̄

𝑚
𝑣𝐴 (b), 𝑣𝐴 (b) =

b

𝑚
𝑣𝐵 (b). (50)

If Φ,Φ′ ∈ H1/2,+
𝑚 , we abbreviate Φ(b) · Φ′(b) = Φ1(b)Φ′1(b) + Φ2(b)Φ′2(b). Then their inner

product is given by

⟨Φ | Φ′⟩ =
∫
𝐻+𝑚

Φ(b) · Φ′(b) 𝑑`(b)

= 2
∫
𝐻+𝑚

�̄�𝐴 (b) · 𝑣′𝐵 (b) 𝑑`(b) = 2
∫
𝐻+𝑚

�̄�𝐵 (b) · 𝑣′𝐴 (b) 𝑑`(b)

=

∫
𝐻+𝑚

(
�̄�𝐴 (b) · 𝑣′𝐵 (b) + �̄�𝐵 (b) · 𝑣′𝐴 (b)

)
𝑑`(b).

If we now set
Ψ(b) :=

(
𝑣𝐴 (b)
𝑣𝐵 (b)

)
=

1
√

2
(𝐷1/2,0 ⊕ 𝐷0,1/2) (𝐿b)

(
Φ(b)
Φ(b)

)
, (51)

we get

(Ψ | Ψ′) :=
∫
𝐻+𝑚

Ψ(b)𝛾0Ψ′(b) 𝑑`(b) = ⟨Φ | Φ′⟩.

29



(Again we sum over the repeated indices of the spinors.) The Hilbert space Ĥ
1/2,+
𝑚 of 4-spinors of

the form (51), with this inner product, is then unitarily equivalent to H
1/2,+
𝑚 .

Now, Ĥ1/2,+
𝑚 is the space of solutions of the Dirac equation. In effect, from (50) we see that

𝑚

(
𝑣𝐴 (b)
𝑣𝐵 (b)

)
=

(
0 b

b̄ 0

) (
𝑣𝐴 (b)
𝑣𝐵 (b)

)
,

that is, (/b − 𝑚)Ψ = 0, the Dirac equation in momentum space.
The “basis” {Φ𝒌,±1/2(b)} = { 𝑘0 𝛿(𝝃 − 𝒌)

(1
0
)
, 𝑘0 𝛿(𝝃 − 𝒌)

(0
1
)

: 𝒌 ∈ ℝ3 } transforms under the
given isomorphism 𝑇 : H1/2,+

𝑚 → Ĥ
1/2,+
𝑚 into

{Ψ𝒌,±1/2(b)} :=
{
𝑘0 𝛿(𝝃 − 𝒌)

(√︁
𝑘/2𝑚

(1
0
)√︁

�̄�/2𝑚
(1
0
)) , 𝑘0 𝛿(𝝃 − 𝒌)

(√︁
𝑘/2𝑚

(0
1
)√︁

�̄�/2𝑚
(0
1
)) : 𝒌 ∈ ℝ3

}
and so (Ψ𝒌,𝑟 | Ψ𝒌′,𝑟 ′) = 𝑘0 𝛿(𝒌 − 𝒌′) 𝛿𝑟𝑟 ′ . One naturally uses this basis to compute traces in Ĥ

1/2,+
𝑚 .

On Ĥ
1/2,+
𝑚 we consider the representation 𝑉𝐷 := 𝑇𝑈1/2𝑇

−1. Explicitly,

[𝑉𝐷 (𝑎, Λ̃)Ψ] (b) = 𝑒−𝑖(𝑎b)𝑆(Λ̃)Ψ(Λ−1b). (52)

We have recovered the usual expression for the relativistic invariance of the Dirac equation, in the
chiral representation. (Of course, we can make a similarity transformation to recover any other
representation we like.) Note that (52) can be considered as a (reducible) representation on the
whole space of 4-spinors; thus defined, 𝑉𝐷 commutes with the projector 𝑃𝐷 = (/b + 𝑚)/2𝑚 on the
subspace Ĥ

1/2,+
𝑚 , and preserves the form of spinors given by (50) and (51).

A well-known result in representation theory establishes that a representation leaves invariant
an inner product given by an associated Hermitian matrix (here 𝛾0) if and only if it is equivalent to
its contragredient conjugate representation. This is the reason for choosing the representation space
of 𝐷1/2,0 ⊕ 𝐷0,1/2, which is reduced to Ĥ

1/2,+
𝑚 by our definitions.

We can now define the Stratonovich–Weyl quantizer for the (positive energy) Dirac particles
Ω𝐷 (𝒙, 𝒑, 𝒏) asΩ𝐷 (𝑢) := 𝑇Ω1/2(𝑢)𝑇−1. We compute this by recalling that𝐷1/2,0(𝑅) = 𝐷0,1/2(𝑅) =
D1/2(𝑅) for 𝑅 ∈ 𝑆𝑈 (2):

[Ω𝐷 (𝒙, 𝒑, 𝒏)Ψ] (b) = 23{𝑝b}3/2 exp[𝑖𝒙 · (𝑀𝑝𝝃 − 𝝃)]𝑆(𝐿b)𝑆(𝐿−1
b 𝐿𝑝𝐿𝐿−1

𝑝 b
)Δ𝐷 (𝒏)

× 𝑆(𝐿𝐿−1
𝑝 b
𝐿−1
𝑝 𝐿𝑀𝑝b)𝑆(𝐿−1

𝑀𝑝b
)Ψ(𝑀𝑝b)

= 23{𝑝b}3/2 exp[𝑖𝒙 · (𝑀𝑝𝝃 − 𝝃)]𝑆(𝐿𝑝𝐿𝐿−1
𝑝 b
)Δ𝐷 (𝒏)𝑆(𝐿𝐿−1

𝑝 b
𝐿−1
𝑝 )Ψ(𝑀𝑝b).

(53a)

where Δ𝐷 (𝒏) := (Δ1/2 ⊕ Δ1/2) (𝒏). In particular:

[Ω𝐷 (0, 0, 𝒏)Ψ] (b) = 23(b0/𝑚)3/2 𝑆(𝐿b) Δ𝐷 (𝒏) 𝑆(𝐿b) Ψ(b̄). (53b)

From these particular operators (53b) one recovers the whole quantizer by means of the covariance
rule, using the usual matrix functions 𝑆 which express the relativistic invariance of the Dirac
equation; note that the formulas (53) are valid for any choice of the 𝛾 matrices. Note also that
Ω𝐷 (𝑢)𝑃𝐷 = 𝑃𝐷Ω𝐷 (𝑢).

30



The Wigner function corresponding to a Dirac wavefunction Ψ (in momentum space) is again
simply the expected value of the quantizer Ω𝐷 :

𝑊Ψ (𝒙, 𝒑, 𝒏) = (Ψ | Ω𝐷 (𝒙, 𝒑, 𝒏)Ψ) =
∫
𝐻+𝑚

Ψ(b)𝛾0 [Ω𝐷 (𝒙, 𝒑, 𝒏)Ψ] (b) 𝑑`(b).

For instance, if Ψ is the plane wave Ψ𝒌,1/2, we get

𝑊𝒌,1/2(𝒙, 𝒑, 𝒏) = 1
2 𝑘

0 𝛿( 𝒑 − 𝒌)
(√︁
�̄�/𝑚 (1 0)

√︁
𝑘/𝑚 (1 0)

) (√︁
𝑘/𝑚 0
0

√︁
�̄�/𝑚

)
×

( 1
2 (1,
√

3𝒏) 0
0 1

2 (1,
√

3𝒏)

) (√︁
�̄�/𝑚 0
0

√︁
𝑘/𝑚

) (√︁
𝑘/2𝑚

(1
0
)√︁

�̄�/2𝑚
(1
0
))

= 1
2 𝑘

0 𝛿( 𝒑 − 𝒌)
(
(1 0) (1 0)

) ( 1
2 (1,
√

3𝒏) 0
0 1

2 (1,
√

3𝒏)

) ((1
0
)(1

0
))

= 1
2 (1 +

√
3 𝑛3)𝑘0 𝛿( 𝒑 − 𝒌).

Analogously,𝑊𝒌,−1/2(𝒙, 𝒑, 𝒏) = 1
2 (1 −

√
3 𝑛3)𝑘0 𝛿( 𝒑 − 𝒌).

In this paper we have treated only positive-energy particles. To repeat the foregoing development
for negative-energy particles, which unfolds in a completely parallel way, one starts from the
coadjoint orbit O𝑚𝑠− determined by replacing ℎ =

√︁
𝑚2 + 𝒑 · 𝒑 by ℎ = −

√︁
𝑚2 + 𝒑 · 𝒑. In particular,

we remark that the negative-energy solutions of the Dirac equation arise in a similar manner. We
obtain a Stratonovich–Weyl quantizer with values which are operators on the representation space
H

1/2,−
𝑚 , and an isomorphism of this space with a subspace Ĥ

1/2,−
𝑚 of 4-spinors, corresponding

to (50); the analogue of (53) gives rise to the desired quantizer for the space of negative-energy
solutions. We leave the details to the reader.

As another useful exercise, we point out that, by putting back the 𝑐’s in our formulas, the whole
formulation reduces to the Galilean one with 𝑢 = 𝑚𝑐2 for the internal energy in the nonrelativistic
limit.

8 Concluding remarks
As a rule of thumb, contributions in the physical and mathematical literature have respectively tried
to “make relativistic” two different elements of phase-space quantum mechanics. On the physical
side, it so happens that “relativistic Wigner functions” have been sporadically employed for some
time [33]; they are introduced by formally extending Wigner’s definition [3] to Minkowskian phase
spaces. On the mathematical side, the Weyl quantization rule is perceived as the basic subject for
generalization and, besides the papers we comment on below, there has also been a busy Japanese
school [34] trying to establish selfadjointness of some classes of operators obtained by formal
application of the Weyl correspondence [2] in the relativistic context. Our carefully systematic
generalization allows at least a preliminary assessment of the worth of such attempts. In general, it
would seem that the other elements of a proper Moyal formulation, such as the twisted product with
its tracial property, cannot be appended to them; and no actual calculations in the Moyal spirit are
done.

31



The basic definition for “relativistic Wigner functions” from which most authors start is generally

𝑁Φ(𝑥, 𝑝) ∝
∫

Φ̂(𝑥 + 𝑣) Φ̂(𝑥 − 𝑣)𝑒2𝑖(𝑝𝑣) 𝑑4𝑣,

where Φ̂, the Fourier transform of Φ, is a wavefunction (or field) satisfying the Klein–Gordon
equation. This is immediately seen to be a simpleminded generalization of (19). Some fail to note
that such an object 𝑁Φ must then satisfy the equation(

(𝑝𝑝) + 𝑚2 + □𝑥
)
𝑁Φ = 0,

and so, unless Φ is a plane wave, in which case 𝑁Φ would equal our 𝑊Φ, 𝑁Φ is not supported on
the mass shell. In practice, this means that this “transport approach” to relativistic field theory is of
an approximate nature from the beginning, which we consider unwarranted.

In the more mathematical vein, there have been some recent attempts to generalize “Weyl
correspondences” to the relativistic context (always in the spinless case). Ali and Antoine [35]
purport to have a recipe for a relativistic Weyl transform for the (1 + 1) Poincaré group. Their
approach, for all its mathematical sophistication, stems from the “old” form of the Weyl rule, which
is notoriously difficult to generalize, instead of the newer one afforded by the Grossmann–Royer
operators. They obtain results different from ours. The Unterbergers [36] come nearer to our point
of view, as they apply covariance and the heuristic parity rule, which is similar to ours but without
the factor {𝑝b}3/2. Of course, the resulting correspondence rule has no tracial property; this is the
reason why they have to define two symbols, a “passive” and an “active” one.

We want to emphasize that the bridge between the coadjoint orbits and the representation
spaces, given by our Stratonovich–Weyl quantizer, must be carefully constructed in order to ensure
the physical equivalence with the standard quantum theory. In this respect we depart from the
démarche of the school-creating papers by Bayen et al [37] that give rise to a bewildering variety
of “twisted products” of little use in physics. What all the reviewed attempts have in common is a
methodology in which an element of the ordinary Moyal formulation is detached from the rest and
imputed an unrestricted power and significance in the generalization; we have shown, nevertheless,
that a procedure is available which combines harmoniously all the basic elements of the Moyal
approach.

Apart from [10, 11], there is not much precedent, then, in the literature for our endeavour.
We want to point out, however, the formal analogy between our work and the “discrete Quantum
Mechanics” formalism in [38]. Also, the idea to start quantization from elementary classical systems
in our sense is present in the interesting paper [39].

Summarizing, we have provided the foundations for the phase-space formulation of relativistic
quantum theory. There is much work to do before the present approach can show its usefulness as
an established method in elementary particle physics. To begin with, some obviously unfinished
business remains, such as quantization of observables in the Dirac case, constructions of SW
quantizers for higher spin wave equations, treatment of the orbits corresponding to massless particles,
and so on. To include treatments of all these topics would have excessively lengthened an article
already not very brief.

The old discussion about localization and position observables, which has sometimes been
conducted at an appalling level, can very well be clarified by employing our one-to-one quantization–
dequantization rule: proposed position operators can always be dequantized for examination of their

32



reasonableness at the classical level, and vice versa. We plan to develop in a forthcoming paper how
interactions can be introduced and the application of our theory in simplifying perturbative QED.
Finally, our proof of existence and uniqueness of the quantizer is specific to the Poincaré group. An
interesting mathematical question concerns the realization of the Stratonovich–Weyl postulates for
general classes of groups.

Acknowledgements

JFC wishes to acknowledge support from the International Centre for Theoretical Physics as a Vis-
iting Scholar in San José, during the inception of this paper. JMG-B is grateful to the Departamento
de Fı́sica Teórica of the Universidad de Zaragoza for its hospitality during the advanced stages of
writing.

We thank L. J. Boya, H. Figueroa and M. Gadella for discussions and L. M. Nieto for indicating
the error in [10] and for help with some calculations.

JMG-B and JCV acknowledge support from the Vicerrectorı́a de Investigación of the Universidad
de Costa Rica, and thank F. Hansen and G. K. Pedersen for their hospitality at the Matematiske
Institut of the University of Copenhagen.

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] E. P. Wigner, “On unitary representations of the inhomogeneous Lorentz group”, Ann. Math. 40 (1939), 149–204.

[5] A. A. Kirillov, Elements of the Theory of Representations, Springer, Berlin, 1976;
B. Kostant, “Quantization and unitary representations”, in Lectures in Modern Analysis and Applications III,
Springer, Berlin, 1970, pp. 87–207;
J. M. Souriau, Structure des Systèmes Dynamiques, Dunod, Paris, 1970.

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

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

[8] J. F. Cariñena and M. Santander, “Projective covering group versus representation groups”, J. Math. Phys. 21
(1980), 440–443.

[9] 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.

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

[11] 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.

[12] V. Bargmann, “Unitary ray representations of continuous groups”, Ann. Math. 59 (1954), 1–46.

[13] S. Sternberg, “Symplectic homogeneous spaces”, Trans. Amer. Math. Soc. 212 (1975), 113–130.

33



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

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

[16] J.-M. Lévy-Leblond, “Galilei group and Galilean invariance”, in Group Theory and its Applications, E. M. Loebl,
ed. Academic Press, New York, 1971, pp. 221–299.

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

[18] R. L. Stratonovich, “On distributions in representation space”, Sov. Phys. JETP 4 (1957), 891–898.

[19] J. C. Várilly, J. M. Gracia-Bondı́a and W. Schempp, “The Moyal representation of quantum mechanics and
special function theory”, Acta Appl. Math. 11 (1990), 225–250.

[20] A. Grossmann, “Parity operators and quantization of 𝛿-functions”, Commun. Math. Phys. 48 (1976), 191–193;
A. Royer, “Wigner function as the expectation value of a parity operator”, Phys. Rev. A 15 (1977), 449–450.

[21] M. Gadella, J. M. Gracia-Bondı́a, L. M. Nieto and J. C. Várilly, “Quadratic Hamiltonians in phase space quantum
mechanics”, J. Phys. A: Math. Gen. 22 (1989), 2709–2738.

[22] R. Arens, “Classical Lorentz invariant particles”, J. Math. Phys. 12 (1971), 2415–2442.

[23] E. C. G. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective, Wiley, New York, 1974.

[24] Č. Crnković and E. Witten, “Covariant description of canonical formalism in geometrical theories”, in Three
Hundred Years of Gravitation, S. W. Hawking and W. Israel, eds., Cambridge Univ. Press, Cambridge, 1987,
pp. 676–684.

[25] M. Pauri and G. M. Prosperi, “Canonical realizations of the Poincaré group. I. General theory”, J. Math. Phys.
16 (1975), 1503–1521.

[26] L. Bel and J. Martı́n, “Predictive relativistic mechanics of systems of 𝑁 particles with spin”, Ann. Inst. Henri
Poincaré A 33 (1980), 409–442.

[27] A. S. Wightman, “L’invariance dans la mécanique relativiste”, in Lectures, Ecole d’été de Physique Théorique,
Les Houches, C. de Witt and R. Omnes, eds. Hermann, Paris, 1960, pp. 159–226.

[28] R. F. O’Connell and E. P. Wigner, “On the relation between momentum and velocity for elementary systems”,
Phys. Lett. A 61 (1977), 353–354.

[29] T. D. Newton and E. P. Wigner, “Localized states for elementary systems”, Rev. Mod. Phys. 21 (1949), 400–406.

[30] G. C. Hegerfeldt, “Remark on causality and particle localization”, Phys. Rev. D 10 (1974), 3320–3321;
and “Violation of causality in relativistic quantum theory?”, Phys. Rev. Lett. 54 (1985), 2395–2398;
S. N. M. Ruijsenaars, “On Newton–Wigner localization and superluminal propagation speeds”, Ann. Phys. 137
(1981), 33–43.

[31] A. Chakrabarti, “Wigner rotations and precession of polarization”, Fortsch. Phys. 36 (1988), 863–880.

[32] W. E. Baylis and G. Jones, “The Pauli algebra approach to special relativity”, J. Phys. A: Math. Gen. 22 (1989),
1–15.

34



[33] P. Carruthers and F. Zachariasen, “Relativistic quantum transport theory approach to multiparticle production”,
Phys. Rev. D 13 (1976), 950–960;
and “Quantum collision theory with phase-space distributions”, Rev. Mod. Phys. 55 (1983), 245–285;
S. R. de Groot, W. A. van Leeuwen and C. G. van Weert, Relativistic Kinetic Theory, North-Holland, Amsterdam,
1980;
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;
J. Winter, “Wigner transformation in curved space-time and the curvature correction of the Vlasov equation for
semiclassical gravitating systems”, Phys. Rev. D 32 (1985), 1871–1888;
H. E. Kandrup, “Generalized Wigner functions in curved spaces: a new approach”, Phys. Rev. D 37 (1988),
2165–2169;
E. Calzetta and B. L. Hu, “Nonequilibrium quantum fields: closed-time-path effective action, Wigner functions
and Boltzmann equation”, Phys. Rev. D 37 (1988), 2878–2900.

[34] T. Ichinose, “The nonrelativistic limit problem for a relativistic spinless particle in an electromagnetic field”, J.
Funct. Anal. 73 (1987), 233–257; M. Nagase and T. Umeda, “On the essential selfadjointness of pseudodifferential
operators”, Proc. Japan Acad. A 64 (1988), 94–97.

[35] S. T. Ali and J.-P. Antoine, “Coherent states of the 1 + 1-dimensional Poincaré group: square integrability and a
relativistic Weyl transform”, Ann. Inst. Henri Poincaré A 51 (1989), 23–44.

[36] A. Unterberger and J. Unterberger, “A quantization of the Cartan domain 𝐵𝐷 𝐼 (𝑞 = 2) and operators on the light
cone”, J. Funct. Anal. 72 (1987), 279–319;
A. Unterberger, ”Analyse relativiste”, C. R. Acad. Sci. Paris 305A (1987), 415–418.

[37] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, “Deformation theory and quantization. I”,
Ann. Phys. (NY) 111 (1978), 61–110;
and “Deformation theory and quantization. II: Physical applications”, Ann. Phys. (NY) 111 (1978), 111–151.

[38] O. Cohendet, Ph. 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;
O. Cohendet, “Etude d’une mécanique stochastique dans l’espace des phases”, thèse de doctorat, Université de
Provence, 1987.

[39] I. Bakas and A. C. Kakas, “Quantum mechanics of nonlinear systems”, J. Phys. A: Math. Gen. 20 (1987),
3713–3725.

35