Wigner–Weyl isomorphism for quantum mechanics on Lie
groups
N. Mukundaa)
Centre for Theoretical Studies, Indian Institute of Science, Bangalore 560 012, India
G. Marmob) and A. Zampinic)
Dipartimento di Scienze Fisiche, Universita di Napoli Federico II and INFN,
Via Cinzia, 80126 Napoli, Italy
S. Chaturvedid)
School of Physics, University of Hyderabad, Hyderabad 500 046, India
R. Simone)
The Institute of Mathematical Sciences, C. I. T. Campus, Chennai 600 113, India
The Wigner–Weyl isomorphism for quantum mechanics on a compact simple Lie
group G is developed in detail. Several features are shown to arise which have no
counterparts in the familiar Cartesian case. Notable among these is the notion of a
semiquantized phase space, a structure on which the Weyl symbols of operators
turn out to be naturally defined and, figuratively speaking, located midway between
the classical phase space T * G and the Hilbert space of square integrable functions
on G. General expressions for the star product for Weyl symbols are presented and
explicitly worked out for the angle-angular momentum case.
I. INTRODUCTION
It is well known that the method of Wigner distributions,1 which describes every state of a
quantum mechanical system by a corresponding real quasiprobability density on the classical
phase space, is dual to the Weyl mapping2 of classical dynamical variables to quantum mechanical
operators. Together they provide the Wigner–Weyl isomorphism, whereby both states and operators in quantum mechanics can be given c-number descriptions on the classical phase space. The
trace of the product of two operators is then calculable as the integral of the product of the two
corresponding Weyl symbols or phase space functions. Combined with the work of Moyal,3 which
shows how products and commutators of operators are expressed in phase space language, this
entire development may be called the Wigner–Weyl–Moyal or WWM method in quantum mechanics and has been instrumental in giving rise to the fertile subject of deformation quantization.4
An important feature of the Wigner distribution is that while it is not by itself a phase space
probability density, its marginals obtained by, respectively, integrating over momenta or over
coordinates do reproduce the quantum mechanical expressions for probability densities in coordinate and in momentum space, respectively.
The WWM method has been studied most extensively in the case of Cartesian systems in
quantum mechanics. By this we mean those systems whose configuration space Q is Rn for some
integer n 艌 1. The classical phase space is then T * Q ⯝ R2n. While Schrödinger wave functions are
a)
Electronic mail: nmukunda@cts.iisc.ernet.in
Electronic mail: giuseppe.marmo@na.infn.it
c)
Electronic mail: alessandro.zampini@na.infn.it
d)
Electronic mail: scsp@uohyd.ernet.in
e)
Electronic mail: simon@imsc.res.in
b)
square integrable functions on Rn, both Wigner distributions and Weyl symbols are functions on
R2n. Quantum kinematics can be expressed via the Heisenberg canonical commutation relations
for Cartesian coordinates and their conjugate momenta, or via the exponentiated Weyl form using
families of unitary operators. An important feature in this case is that as far as their eigenvalue
spectra are concerned, the momenta do not experience any quantization on their own; they account
for the second factor in T * Q ⯝ R2n ⯝ Rn ⫻ Rn. Furthermore we have in this case the Stone–von
Neumann theorem on the uniqueness of the irreducible representation of the Heisenberg commutation relations, and the important roles of the groups Sp共2n , R兲 and Mp共2n兲 corresponding to
linear canonical transformations on coordinates and momenta.
There has been for some time considerable interest in developing the Wigner–Weyl isomorphism for other kinds of quantum systems, that is, for non-Cartesian systems.5–16 In these cases,
typically the underlying quantum kinematics cannot be expressed by Heisenberg-type commutation relations. The situations studied include the quantum mechanics of an angle-angular momentum pair, where the configuration space is Q = S1,17,18 and finite state quantum systems corresponding to a finite dimensional Hilbert space.19,20 More recently, the method of Wigner distributions
has been developed for quantum systems whose configuration space is a compact simple Lie
group; and in the discrete case when it is a finite group of odd order.21,22 In all these departures
from the Cartesian situation, an important aspect is the occurrence of new features which do not
show up at all with Cartesian variables.
The aim of the present work is to develop in detail the Wigner–Weyl isomorphism for quantum mechanics on a compact simple Lie group. Here the configuration space Q is a (compact
simple) Lie group G, so the corresponding classical phase space is T * G ⯝ G ⫻ G
គ *, where G
គ * is
the dual to the Lie algebra G
គ of G. In the quantum situation, Schrödinger wave functions are
complex square integrable functions on G, and observables or dynamical variables are linear
Hermitian operators acting on such functions. The replacements for the canonical Heisenberg
commutation relations are best formulated using the (commutative) algebra of suitable smooth
functions on G, and (say) the left regular representation of G acting on functions on itself. The
natural question that arises in trying to set up a Wigner–Weyl isomorphism in this case is whether
quantum states and operators are to be described using suitable functions on the classical phase
space T * G. In Ref. 21 an overcomplete Wigner distribution formalism for quantum states, which
transforms in a reasonable way under left and right group actions and also reproduces the natural
marginal probability distributions, has been developed. The methods developed there are here
exploited to set up a Wigner–Weyl isomorphism in full detail, disclosing many interesting differences compared to the Cartesian case. In particular we find that this isomorphism does not directly
utilize c-number functions on T * G at all, but instead uses a combination of functions on G and
operators on a simpler Hilbert space, standing in a sense midway between T * G and the Hilbert
space of the quantum system. This feature is traceable to the non-Abelian nature of G, something
which is absent in the Cartesian case when Q is the Abelian group Rn.
The material of this paper is organized as follows. In Sec. II we briefly recapitulate key
features of the Wigner–Weyl isomorphism for the Cartesian and angle-angular momentum cases.
This sets the stage for Sec. III where we develop the quantum kinematics for situations where the
configuration space is a compact Lie group and thus go beyond the Abelian cases discussed in Sec.
I. This analysis leads to a proper identification of the analogues of the momenta of the Cartesian
case and helps set up the Wigner distribution for such situations possessing properties expected of
a Wigner distribution. The Wigner distributions so defined have a certain degree of overcompleteness about them, a circumstance forced by the non-Abelian nature of the underlying group G. A
key ingredient in this construction is the notion of the midpoint of two group elements introduced
in an earlier work.21 In Sec. IV a more compact description in terms of Weyl symbols devoid of
any redundances is developed and correspondences facilitating transition from the Cartesian case
to more general situations are established. The results of Sec. IV are exploited in Sec. V towards
defining a star product between Weyl symbols for operators and the general expression for the star
product is explicitly worked out for the non-Cartesian, albeit Abelian case of angle-angular momentum. Section VI is devoted to analyzing the minimal structure on which the Weyl symbols for
operators find their natural definition. This leads to the concept of a noncommutative cotangent
space or a semiquantized phase space the ramifications of which are examined further towards
highlighting the structural similarity between classical phase space functions and the Weyl symbols. A short appendix contains some technical details concerning results used in Sec. V.
II. THE WIGNER–WEYL ISOMORPHISM: CARTESIAN AND ANGLE-ANGULAR
MOMENTUM CASES
In this section we recall briefly the relevant structures needed to set up the Wigner–Weyl
isomorphism for Cartesian quantum mechanics. This is to facilitate comparison with the Lie group
case later on. For simplicity we choose one degree of freedom only, as the extension to Q = Rn is
straightforward. We also recall the angle-angular momentum case, Q = S1, where we already see
significant differences from the Cartesian case; these increase when we go to Q = G.
One-dimensional Cartesian quantum mechanics: The canonical Heisenberg commutation relation between Hermitian coordinate and momentum operators q̂ and p̂, fixing the kinematics, is
关q̂,p̂兴 = i.
共2.1兲
In the unitary Weyl form this is expressed as follows:
U共p兲 = exp共ipq̂兲,
V共q兲 = exp共− iqp̂兲,
q,p 僆 R.
U共p兲V共q兲 = V共q兲U共p兲eiqp,
共2.2兲
In the Cartesian case the exponentials can be combined to define a phase space displacement
operator
D共q,p兲 = U共p兲V共q兲e−iqp/2 = V共q兲U共p兲eiqp/2 = exp共ipq̂ − iqp̂兲.
共2.3兲
However this cannot be done even in the single angle-angular momentum pair case, and also
when we treat the Lie group case. We therefore use expressions in which the exponentials are kept
separate.
The standard form of the unique irreducible representation of Eqs. (2.1) and (2.2) uses the
Hilbert space of square integrable functions ␺共q兲 on R. Introducing as usual an ideal basis of
eigenvectors of q̂ we have
再
H = L2共R兲 = ␺共q兲兩 储␺储2 =
␺共q兲 = 具q兩␺典,
冕
R
冎
dq兩␺共q兲兩2 ⬍ ⬁ ,
q̂兩q典 = q兩q典,
共2.4兲
具q⬘兩q典 = ␦共q⬘ − q兲.
On such ␺共q兲 (subject to relevant domain conditions) the actions of q̂, p̂, U共p兲, V共q兲 are
共q̂␺兲共q兲 = q␺共q兲,
共p̂␺兲共q兲 = − i
共U共p⬘兲␺兲共q兲 = eip⬘q␺共q兲,
d
␺共q兲,
dq
共V共q⬘兲␺兲共q兲 = ␺共q − q⬘兲.
共2.5兲
The momentum space description of 兩␺典 uses the Fourier transform of ␺共q兲; in terms of the
ideal eigenstates 兩p典 for p̂,
˜␺共p兲 = 具p兩␺典 =
冕冑
dq
R
2␲
exp共− ipq兲␺共q兲,
储␺储2 =
冕
R
dp兩˜␺共p兲兩2 .
共2.6兲
The displacement operators (2.2) form a complete trace orthonormal set (in the continuum
sense) in the space of operators on H,
Tr共共U共p⬘兲V共q⬘兲兲†U共p兲V共q兲兲 = 2␲␦共q⬘ − q兲␦共p⬘ − p兲.
共2.7兲
The completeness property will be used later.
The definitions of the Wigner distribution for a normalized pure state 兩␺典 苸 H, or more generally for a mixed state with density operator ␳ˆ , are
W共q,p兲 =
1
2␲
W共q,p兲 =
冉
冕
冊冉
1
1
dq⬘ ␺ q − q⬘ ␺ q + q⬘
2
2
R
1
2␲
冕
冓
冊
*
exp 共ipq⬘兲,
冔
1
1
dq⬘ q − q⬘兩␳ˆ 兩q + q⬘ exp共ipq⬘兲.
2
2
R
共2.8兲
(The dependences on 兩␺典, ␳ˆ are left implicit.) While W共q , p兲 is real though not always nonnegative, the recovery of the marginal position and momentum space probability densities is
assured by
冕
R
dp W共q,p兲 = 具q兩␳ˆ 兩q典,
冕
R
dq W共q,p兲 = 具p兩␳ˆ 兩p典.
共2.9兲
It is possible to express W共q , p兲 in a more compact form by introducing a family of Hermitian
operators Ŵ共q , p兲 on H with interesting algebraic properties. They are essentially the double
Fourier transforms of the displacement operators (2.2),
W共q,p兲 = Tr共␳ˆ Ŵ共q,p兲兲,
Ŵ共q,p兲 = Ŵ共q,p兲† =
1
共2␲兲2
冕冕
R
R
dq⬘ dp⬘ U共p⬘兲V共q⬘兲eipq⬘−ip⬘共q+共1/2兲q⬘兲 .
共2.10兲
It has been shown in Ref. 17 that, apart from sharing the trace orthonormality property (2.7)
which is preserved by the Fourier transformation,
Tr共Ŵ共q⬘,p⬘兲Ŵ共q,p兲兲 =
1
␦共q⬘ − q兲␦共p⬘ − p兲,
2␲
共2.11兲
we have the following behaviors under anticommutation with q̂ and p̂:
1
2 兵q̂,Ŵ共q,p兲其
= qŴ共q,p兲,
1
2 兵p̂,Ŵ共q,p兲其
= pŴ共q,p兲.
共2.12兲
Thus we may regard Ŵ共q , p兲 as operator analogues of Dirac delta functions concentrated at
individual phase space points. In Ref. 19 they have been called phase point operators.
Turning to the Weyl map, it takes a general classical dynamical variable, a (square integrable)
function a共q , p兲 on the classical phase space, to a corresponding (Hilbert–Schmidt) operator  on
H:
冕冕
冕冕 ⬘
a共q,p兲 → ã共p⬘,q⬘兲 =
→ Â =
1
2␲
R
dq dp a共q,p兲ei共pq⬘−qp⬘兲
R
R
R
dq dp⬘ ã共p⬘,q⬘兲U共p⬘兲V共q⬘兲e−iq⬘p⬘/2 .
共2.13兲
The important property of this map is that traces of operators on H go into integrals over
phase space,
Tr共†B̂兲 =
冕冕
R
共2.14兲
dq dp a共q,p兲 * b共q,p兲.
R
One can immediately see that the relation between a共q , p兲 and  is given by
 = 2␲
冕冕
R
共2.15兲
dq dp a共q,p兲Ŵ共q,p兲,
R
thus establishing that the Wigner and Weyl maps are inverses of one another. Indeed extending the
definition of the Wigner distribution (2.10) to a general operator  on H, we have
共2.16兲
a共q,p兲 = Tr共ÂŴ共q,p兲兲.
It is this kind of isomorphism that we wish to develop when R is replaced by a compact simple Lie
group G.
The angle-angular momentum case: We now trace the changes which appear if we replace the
Cartesian variable q 苸 R by an angle ␪ 苸 共−␲ , ␲兲. The corresponding Hermitian operator is denoted by ␪ˆ , with eigenvalues ␪; its canonical conjugate M̂ has integer eigenvalues m
= 0 , ± 1 , ± 2 , . . .. Thus m 苸 Z unlike the Cartesian p, so M̂ is already quantized. The replacements
for Eqs. (2.4) and (2.6) are
再
H = L2共S1兲 = ␺共␪兲兩 储␺储2 =
␺共␪兲 = 具␪兩␺典,
冕
␲
−␲
冎
d␪兩␺共␪兲兩2 ⬍ ⬁ ,
具␪⬘兩␪典 = ␦共␪⬘ − ␪兲,
␺m = 具m兩␺典 =
1
2␲
储␺储2 =
M̂兩m典 = m兩m典,
冕
␲
−␲
␪ˆ 兩␪典 = ␪兩␪典,
d␪ e−im␪␺共␪兲,
共2.17兲
兩 ␺ m兩 2 ,
兺
m苸Z
具␪兩m典 =
1
冑2␲ e
im␪
.
In place of the Heisenberg commutation relation (2.1), we have only the exponentiated Weyl
version,
U共m兲 = exp共im␪ˆ 兲,
V共␪兲 = exp共− i␪ M̂兲,
U共m兲V共␪兲 = V共␪兲U共m兲eim␪ .
共2.18兲
With the actions
共U共m⬘兲␺兲共␪兲 = eim⬘␪␺共␪兲,
共V共␪⬘兲␺兲共␪兲 = ␺共关␪ − ␪⬘兴兲,
共2.19兲
关␪ − ␪⬘兴 = ␪ − ␪⬘ mod 2␲ ,
we have an irreducible system on H = L2共S1兲. The analogues of the displacement operators (2.3)
are now
U共m兲V共␪兲e−im␪/2 = V共␪兲U共m兲eim␪/2 ,
共2.20兲
but here the exponents cannot be combined. They do however form a complete trace orthonormal
system,
Tr共共U共m⬘兲V共␪⬘兲兲†U共m兲V共␪兲兲 = 2␲␦mm⬘␦共␪⬘ − ␪兲.
共2.21兲
With this preparation we can turn to the definition of the Wigner distribution and the Weyl
map. For a given density operator ␳ˆ on H, the former is
W共␪,m兲 =
1
2␲
冕
冓
␲
冔
1
1
d␪⬘ ␪ − ␪⬘兩␳ˆ 兩␪ + ␪⬘ exp共im␪⬘兲.
2
2
−␲
共2.22兲
We see immediately that this is not a function on the classical phase space T * S1 ⯝ S1 ⫻ R, which
is a cylinder, but on a partially quantized space S1 ⫻ Z. We may regard this space as standing
somewhere in between T * S1 and the fully quantum mechanical Hilbert space and operator setup.
The marginals are properly reproduced in the sense that
冕
␲
−␲
d␪ W共␪,m兲 = 具m兩␳ˆ 兩m典,
兺 W共␪,m兲 = 具␪兩␳ˆ 兩␪典.
共2.23兲
m苸Z
We can display W共␪ , m兲 as
W共␪,m兲 = Tr共␳ˆ Ŵ共␪,m兲兲,
Ŵ共␪,m兲 = Ŵ共␪,m兲† =
1
共2␲兲2 m
兺
⬘苸Z
冕
␲
−␲
d␪⬘ U共m⬘兲V共␪⬘兲eim␪⬘−im⬘共␪+1/2␪⬘兲 ,
共2.24兲
and like their Cartesian counterparts these operators form a trace orthonormal system,
Tr共Ŵ共␪⬘,m⬘兲Ŵ共␪,m兲兲 =
1
␦共␪⬘ − ␪兲␦mm⬘ .
2␲
共2.25兲
In a similar spirit, the Weyl map now takes any classical function a共␪ , m兲 on S1 ⫻ Z into an
operator on L2共S1兲,
兺
m苸Z
a共␪,m兲 → ã共m⬘, ␪⬘兲 =
→ Â =
1
2␲ m
兺
⬘苸Z
冕
冕
␲
−␲
␲
d␪ a共␪,m兲ei共m␪⬘−m⬘␪兲
d␪⬘ ã共m⬘, ␪⬘兲U共m⬘兲V共␪⬘兲e−im⬘␪⬘/2 .
−␲
共2.26兲
Then the trace operation becomes, as in (2.14),
Tr共†B̂兲 =
兺
m苸Z
冕
␲
−␲
d␪ a共␪,m兲 * b共␪,m兲.
共2.27兲
Combining Eqs. (2.24) and (2.26) we are able to get the analogue to (2.15),
 = 2␲
兺
m苸Z
冕
␲
−␲
d␪ a共␪,m兲Ŵ共␪,m兲.
共2.28兲
In this way the similarities as well as important differences compared to the Cartesian case are
easily seen.
III. QUANTUM KINEMATICS IN THE LIE GROUP CASE AND THE WIGNER DISTRIBUTION
Let G be a (non-Abelian) compact simple Lie group of order n, with elements g , g⬘ , . . . and
composition law g⬘ , g → g⬘g. To set up the kinematics appropriate for a quantum system with
configuration space Q = G, it is simplest to begin with the Hilbert space of Schrödinger wave
functions. The normalized left and right invariant volume element on G is written as dg. For
suitable functions f共g兲 on G we have the invariances and normalization condition
冕
dg f共g兲 =
G
冕
dg 共f共g⬘g兲
or f共gg⬘兲 or f共g−1兲兲,
G
冕
共3.1兲
dg = 1.
G
Correspondingly we can introduce a Dirac delta function on G characterized by
冕
dg共␦共g⬘−1g兲
or ␦共gg⬘−1兲
or ␦共g−1g⬘兲
or ␦共g⬘g−1兲兲f共g兲 = f共g⬘兲.
共3.2兲
G
Thus ␦共g兲 is a delta function concentrated at the identity element e 苸 G.
We take the Hilbert space H for the quantum system to be made up of all complex square
integrable functions on G:
再
H = L2共G兲 = ␺共g兲 苸 C兩 储␺储2 =
冕
冎
dg兩␺共g兲兩2 ⬍ ⬁ .
G
共3.3兲
A convenient basis of ideal vectors 兩g典 can be introduced such that for a general 兩␺典 苸 H we may
write
␺共g兲 = 具g兩␺典,
具g⬘兩g典 = ␦共g⬘g−1兲.
共3.4兲
The notion of position coordinates is intrinsically captured by the commutative algebra representing real valued smooth functions f共g兲 on G, i.e., f 苸 F共G兲. To each such function we
associate a Hermitian multiplicative operator f̂ on H:
f 苸 F共G兲 → f̂ =
冕
dg f共g兲兩g典具g兩,
G
共f̂ ␺兲共g兲 = f共g兲␺共g兲.
共3.5兲
Thus all these operators commute with one another, being diagonal in the position description
␺共g兲 of 兩␺典.
To complete the kinematics and to obtain an irreducible system of operators on H we have to
adjoin suitable momenta. Here we have two choices, corresponding to the left and right translations of G on itself by group action. We choose the former, and so define a family of unitary
operators V共g兲 to give the left regular representation of G:
共V共g⬘兲␺兲共g兲 = ␺共g⬘−1g兲,
V共g⬘兲兩g典 = 兩g⬘g典.
共3.6兲
They obey
V共g⬘兲V共g兲 = V共g⬘g兲,
V共g兲†V共g兲 = I.
共3.7兲
គ of G. Using
To identify their Hermitian generators, we introduce a basis 兵er其 in the Lie algebra G
the exponential map G
គ → G, we write a general g 苸 G as
g = exp共␣rer兲,
共3.8兲
the sum on r being from 1 to n. The generators Ĵr of V共g兲 are then identified by
V共exp共␣rer兲兲 = exp共− i␣rĴr兲.
共3.9兲
These are Hermitian operators on the Hilbert space H, obeying commutation relations involving
t
of G:
the structure constants Crs
关Ĵr,Ĵs兴 = iCrstĴt .
共3.10兲
On Schrödinger wave functions ␺共g兲 each Ĵr acts as a first order partial differential operator;
indeed if the (right invariant) vector fields generating the left action of G on itself are written as
Xr, then we have
Ĵr␺共g兲 = iXr␺共g兲.
共3.11兲
The commutation relations (3.10) are direct consequences of similar commutation relations among
the vector fields Xr.
The analogue of the Cartesian Heisenberg–Weyl system (2.1) and (2.2) is now obtained by
setting together the following ingredients:
f 1, f 2 苸 F共G兲 → f̂ 1 f̂ 2 = f̂ 2 f̂ 1 ,
f 苸 F共G兲,
g⬘ 苸 G → V共g⬘兲f̂V共g⬘兲−1 = f̂ ⬘ ,
共3.12兲
f ⬘共g兲 = f共g⬘−1g兲,
along with the representation property (3.7) for V共g兲. This is in the spirit of the unitary Weyl
system (2.2). In infinitesimal terms we have
ˆ
关Ĵr, f̂兴 = i共X
r f兲 ,
共3.13兲
combined with (3.10). The space H is indeed irreducible with respect to the family of operators
兵f̂ , V共·兲其 or equivalently 兵f̂ , Ĵr其.
We can express functions of position also via unitary operators in the Weyl spirit as follows:
for each real f 苸 F共G兲, we define the unitary operator U共f兲 by
ˆ
U共f兲 = eif : 共U共f兲␺兲共g兲 = eif共g兲␺共g兲.
共3.14兲
It is then easy to see that we have the relations
共U共f兲V共g⬘兲␺兲共g兲 = eif共g兲␺共g⬘−1g兲,
共V共g⬘兲U共f兲␺兲共g兲 = eif共g⬘
−1g兲
␺共g⬘−1g兲,
共U共f兲V共g⬘兲共V共g⬘兲U共f兲兲†␺兲共g兲 = eif共g兲−if共g⬘
−1g兲
共3.15兲
␺共g兲,
which is in the spirit of Eqs. (2.2) and (2.18), except that f is not restricted to be linear in any
coordinate variables.
We see here that unlike in the n-dimensional Cartesian case the canonical momenta are a
noncommutative system. Therefore the analogue or generalization of the single momentum eigenstate 兩pគ 典 in the Cartesian situation will turn out to be a generally multidimensional Hermitian
irreducible representation of (3.10), namely the generators of some unitary irreducible representation (UIR) of G. We will see this in detail as we proceed.
For completeness we should mention the operators giving the right regular representation of
G. These are, say, Ṽ共g兲, defined by and obeying
共Ṽ共g⬘兲␺兲共g兲 = ␺共gg⬘兲,
Ṽ共g⬘兲兩g典 = 兩gg⬘−1典,
共3.16兲
Ṽ共g⬘兲Ṽ共g兲 = Ṽ共g⬘g兲,
V共g⬘兲Ṽ共g兲 = Ṽ共g兲V共g⬘兲.
ˆ
However as is well known their generators J̃r are determined by Ĵr and the matrices 共Drs共g兲兲 of the
adjoint representation of G, by
ˆ
J̃r = − Drs共g兲Ĵs .
共3.17兲
Therefore it suffices to regard the collection of operators 兵f̂ , V共·兲其 as providing the replacement for
the Heisenberg–Weyl system in the present case.
Complementary to the position basis 兩g典 for H is a momentum basis. This can be set up using
the Peter–Weyl theorem involving all the UIR’s of G. We denote the various UIR’s by j, with
dimension N j; we label rows and columns within the jth UIR by magnetic quantum numbers m , n.
Thus the unitary matrix representing g 苸 G in the jth UIR is
j
g → 共Dmn
共g兲兲.
共3.18兲
In general each of j , m , n is a collection of several independent discrete quantum numbers, and
there is a freedom of unitary changes in the choice of m , n. In addition to unitarity and the
composition law,
j
j
共g兲 * Dm⬘n共g兲 = ␦mm⬘ ,
兺n Dmn
j
j
j
共g⬘兲Dnn⬘共g兲 = Dmn⬘共g⬘g兲,
兺n Dmn
共3.19兲
we have orthogonality and completeness properties,
冕
G
j
dg Dmn
共g兲Dm⬘⬘n⬘共g兲 * = ␦ jj⬘␦mm⬘␦nn⬘/N j ,
j
j
j
N jDmn
共g兲Dmn
共g⬘兲 * = ␦共g−1g⬘兲.
兺
jmn
共3.20兲
Then a simultaneous complete reduction of both representations V共·兲, Ṽ共·兲 of G is achieved by
passing to a new orthonormal basis 兩jmn典 for H. Its definition and basic properties are
兩jmn典 = N1/2
j
冕
j
dg Dmn
共g兲兩g典,
G
具j⬘m⬘n⬘兩jmn典 = ␦ j⬘ j␦m⬘m␦n⬘n ,
V共g兲兩jmn典 =
兺
m⬘
Ṽ共g兲兩jmn典 =
共3.21兲
j
Dmm⬘共g−1兲兩jm⬘n典,
兺 Dnj ⬘n共g兲兩jmn⬘典.
n⬘
Therefore in 兩jmn典 the index n counts the multiplicity of occurrence of the jth UIR in the reduction
of V共·兲 and m performs a similar function in the reduction of Ṽ共·兲.
We now regard the sets of N2j states 兵兩jmn典其 for each fixed j as momentum eigenstates in the
present context. This means that the n-dimensional real momentum eigenvalue pគ in Cartesian
quantum mechanics is now replaced by a collection of (discrete) quantum numbers jmn. A vector
兩␺典 苸 H with wave function ␺共g兲 is given in the momentum description by a set of expansion
coefficients ␺ jmn,
␺ 苸 H → ␺ jmn = 具jmn兩␺典 = N1/2
j
冕
j
dg Dmn
共g兲 * ␺共g兲,
G
储␺储2 =
兩␺ jmn兩2 .
兺
jmn
共3.22兲
A normalized 兩␺典 then determines two complementary probability distributions, 兩␺共g兲兩2 on G and
兩␺ jmn兩2 on momentum space.
In this situation a (provisional and overcomplete) Wigner distribution W̃共g ; jmn m⬘n⬘兲 can be
defined for each 兩␺典 苸 H (or for any mixed state ␳ˆ as well). (Here we depart slightly from the
notation in Ref. 21, so that our later expressions are more concise.) It transforms in a reasonable
manner when 兩␺典 is acted upon by V共·兲 or Ṽ共·兲; and it reproduces in a simple and direct way the
two probability distributions determined by 兩␺典, as marginals. We give only the latter property
here,
W̃共g; jmn mn兲 = 兩␺共g兲兩2 ,
兺
jmn
冕
G
dg W̃共g; jmn m⬘n⬘兲 = ␺ jm⬘n⬘ ␺*jmn .
共3.23兲
The right-hand side of the second relation is a natural generalization of 兩␺ jmn兩2, to allow for
freedom in the choice of labels m , n within each UIR j. The expression for this Wigner distribution
involves a function s, G ⫻ G → G obeying certain conditions and is
W̃共g; jmn m⬘n⬘兲 = N j
冕 ⬘冕
dg
G
G
j
dg⬙ ␺共g⬙兲␺共g⬘兲 * Dm⬘n⬘共g⬙兲 * Dmn
共g⬘兲␦共g−1s共g⬘,g⬙兲兲.
j
共3.24兲
Reality in the Cartesian or single angle-angular momentum cases is replaced here by Hermiticity,
W̃共g; jmn m⬘n⬘兲 * = W̃共g; jm⬘n⬘ mn兲.
共3.25兲
The conditions on s共g⬘ , g⬙兲 to ensure that all the above properties are secured are
g⬘,g⬙ 苸 G → s共g⬘,g⬙兲 = s共g⬙,g⬘兲 苸 G,
s共g1g⬘g2,g1g⬙g2兲 = g1 s共g⬘,g⬙兲g2 ,
共3.26兲
s共g⬘,g⬘兲 = g⬘ .
We can simplify the problem of constructing such a function by exploiting the second of these
relations to write
s共g⬘,g⬙兲 = g⬘s共e,g⬘−1g⬙兲 = g⬘s0共g⬘−1g⬙兲,
共3.27兲
so the function s0共g兲 of a single group element must satisfy
s0共e兲 = e,
s0共g−1兲 = g−1s0共g兲 = s0共g兲g−1 ,
共3.28兲
s0共g⬘gg⬘−1兲 = g⬘ s0共g兲g⬘−1 .
The solution proposed in Ref. 21 is to take s0共g兲 to be the midpoint along the geodesic from the
identity e 苸 G to g. These geodesics are determined starting from the invariant Cartan–Killing
metric on G, and have the necessary behaviors under left and right group actions to ensure that all
of Eqs. (3.26) and (3.28) are obeyed. In the exponential notation of Eq. (3.8) we have
s0共exp共␣r er兲兲 = exp共 21 ␣r er兲,
共3.29兲
since it is true that geodesics passing through the identity are one parameter subgroups. With this
explicit construction we have the additional relation
s0共g−1兲 = s0共g兲−1 ,
i.e.,
s0共g兲 s0共g兲 = g.
共3.30兲
Thus s0共g兲 is the (almost everywhere unique) square root of g and s共g⬘ , g⬙兲 is a kind of symmetric
square root of g⬘ and g⬙.
We shall explore the properties of W̃共g ; jmn m⬘n⬘兲 in the next section, especially the sense in
which it contains information about 兩␺典具␺兩 in an overcomplete manner. This will then lead to the
Wigner–Weyl isomorphism for quantum mechanics on a (compact simple) Lie group.
IV. THE WIGNER–WEYL ISOMORPHISM IN THE LIE GROUP CASE
The definition (3.24) can be immediately extended to associate an object W̃Aˆ共g ; jmn m⬘n⬘兲
with every linear operator  on H (of Hilbert–Schmidt class). In terms of the integral kernel
具g⬙兩Â兩g⬘典 of  we have
W̃Aˆ共g; jmn m⬘n⬘兲 = N j
冕 ⬘冕
j
dg⬙具g⬙兩Â兩g⬘典Dm⬘n⬘共g⬙兲 * Dmn
共g⬘兲␦共g−1s共g⬘,g⬙兲兲.
j
dg
G
G
共4.1兲
It is indeed the case that this expression describes or determines  completely, however this
happens in an overcomplete manner. There are certain linear relations obeyed by W̃Aˆ共g ; jmn m⬘n⬘兲
which have an  independent form. We now obtain these relations, then proceed to construct a
simpler expression which contains complete information about  without redundancy.
The Dirac delta function in the integral on the right-hand side of Eq. (4.1) means that the only
contributions to the integral are from the points where
s共g⬘,g⬙兲 = g.
共4.2兲
s0共g⬘−1g⬙兲 = g⬘−1g,
共4.3兲
Writing this as
and then using Eq. (3.30), we see that, say, in the g⬙ integration the delta function picks out the
single point determined by
g⬘−1g⬙ = 共g⬘−1g兲2 ,
i.e.,
g⬙ = gg⬘−1g.
共4.4兲
This means that ␦共g−1s共g⬘ , g⬙兲兲 is some Jacobian factor times ␦共g⬙−1gg⬘−1g兲. We are therefore
permitted to use this value for g⬙ elsewhere in the integrand, so
W̃Aˆ共g; jmn m⬘n⬘兲 = N j
冕 ⬘冕
dg
G
G
j
dg⬙具gg⬘−1g兩Â兩g⬘典Dm⬘n⬘共gg⬘−1g兲 * Dmn
共g⬘兲␦共g−1s共g⬘,g⬙兲兲.
j
共4.5兲
Transferring the g-dependent representation matrices from the right-hand side to the left-hand side
and using unitarity, we get
兺 Dmj ⬘m⬙共g兲Dnj ⬙n⬘共g兲W̃Aˆ共g; jmn m⬘n⬘兲 = N j
m⬘n⬘
冕 ⬘冕
dg
G
dg⬙ ␦共g−1s共g⬘,g⬙兲兲
G
j
⫻具gg⬘−1g兩Â兩g⬘典Dn⬙m⬙共g⬘兲Dmn
共g⬘兲.
j
共4.6兲
It is now clear we have symmetry of the expression on the left-hand side under the simultaneous
interchanges m ↔ n⬙, n ↔ m⬙, a statement independent of Â. This is the sense in which
W̃Aˆ共g ; jmn m⬘n⬘兲 contains information about  in an overcomplete manner, and this happens only
when G is non-Abelian.
Taking advantage of this, we now associate to  the simpler quantity
WAˆ共g; jmm⬘兲 = N−1
j
兺n W̃Aˆ共g; jmn m⬘n兲 =
冕 ⬘冕
dg
G
G
dg⬙具g⬙兩Â兩g⬘典Dmm⬘共g⬘g⬙−1兲␦共g−1s共g⬘,g⬙兲兲.
j
共4.7兲
We shall call this the Weyl symbol corresponding to the operator Â. The passage  → † results in
WAˆ†共g; jmm⬘兲 = WAˆ共g; jm⬘m兲 * .
共4.8兲
It is easy to obtain the transformation properties of the Weyl symbol under conjugation of  by
either the left or the right regular representation,
Â⬘ = V共g0兲ÂV共g0兲−1 ,
WAˆ⬘共g; jmm⬘兲 =
兺
j
Dmm
共g 兲Dm m⬘共g0兲 * WAˆ共g−1
0 g; jm1m1⬘兲,
1 0
⬘ 1
j
m1m1⬘
共4.9兲
Â⬙ = Ṽ共g0兲ÂṼ共g0兲 ,
−1
WAˆ⬙共g; jmm⬘兲 = WAˆ共gg0 ; jmm⬘兲.
Next we can verify that if  and B̂ are any two Hilbert–Schmidt operators on H, then Tr共ÂB̂兲 can
be simply expressed in terms of their Weyl symbols,
Tr共ÂB̂兲 =
兺
jmm⬘
Nj
冕
dg WAˆ共g; jmm⬘兲WBˆ共g; jm⬘m兲.
共4.10兲
G
The proof exploits the completeness relation in (3.20) and the properties (3.26) of s共g⬘ , g⬙兲. This
key result proves that  is indeed completely determined by its Weyl symbol:  is certainly
determined by the values of Tr共ÂB̂兲 for all B̂, and the latter are known once the Weyl symbols are
known.
Before expressing the Weyl symbol of  in a form analogous to Eq. (2.16), we give examples
for some simple choices of Â,
f̂ =
WAˆ共g; jmm⬘兲
Â
冕
f共g兲␦mm⬘
dg f共g兲兩g典具g兩
G
j
V共g0兲
Dmm⬘共g−1
0 兲
Ṽ共g0兲
Dmm⬘共gg0g−1兲
f̂V共g0兲
f共s0共g0兲g兲Dmm⬘共g−1
0 兲
V共g0兲f̂
f共s0共g0兲−1g兲Dmm⬘共g−1
0 兲
j
共4.11兲
j
j
We shall comment later on the structure of these Weyl symbols. However it is already instructive to compare these results with the Cartesian situation
Â
W共q,p兲
f̂ = f共q̂兲
f共q兲
V共q⬘兲
exp共− ipq⬘兲
f̂V共q⬘兲 = f共q̂兲V共q⬘兲 f共q + q⬘/2兲exp共− ipq⬘兲
V共q⬘兲f共q̂兲
共4.12兲
f共q − q⬘/2兲exp共− ipq⬘兲
Now we turn to the problem of expressing the Weyl symbol of  in the form
WAˆ共g; jmm⬘兲 = Tr共ÂŴ共g; jmm⬘兲兲
共4.13兲
for a suitable operator Ŵ共g ; jmm⬘兲. This would be the analogue of Ŵ共q , p兲 in Eq. (2.10). Since the
kernel 具g⬙兩Â兩g⬘典 is quite general, Eq. (4.13) and Eq. (4.7) imply
具g⬘兩Ŵ共g; jmm⬘兲兩g⬙典 = Dmm⬘共g⬘g⬙−1兲␦共g−1s共g⬘,g⬙兲兲 = Dmm⬘共g⬘g⬙−1兲␦共g−1s0共g⬙g⬘−1兲g⬘兲.
j
j
共4.14兲
We shall synthesize Ŵ共g ; jmm⬘兲 in steps. We begin by defining a family of commuting operators
U共jmn兲 in the manner of Eq. (3.5), all of them diagonal in the position basis,
j
共U共jmn兲␺兲共g兲 = Dmn
共g兲␺共g兲.
共4.15兲
These are analogous to the Cartesian U共p⬘兲, labeled by a momentum eigenvalue jmn, functions of
position alone. They are unitary in the matrix sense,
兺m U共jmn兲†U共jmn⬘兲 = 兺m U共jnm兲†U共jn⬘m兲 = ␦n⬘nI.
共4.16兲
These operators allow us to express the map f 苸 F共G兲 → f̂ of Eq. (3.5) more explicitly as
follows:
f共g兲 =
j
f jmnDmn
共g兲 ⇒ f̂ = 兺 f jmnU共jmn兲.
兺
jmn
jmn
共4.17兲
Upon conjugation by V共g兲 we have
V共g兲−1U共jmn兲V共g兲 =
j
共g兲U共jm⬘n兲.
兺 Dmm
⬘
共4.18兲
m⬘
Combining Eqs. (3.16), (3.20), and (4.15) we easily obtain the trace orthonormality property
−1
Tr共共U共j⬘m⬘n⬘兲V共g⬘兲兲†U共jmn兲V共g兲兲 = N−1
j ␦ j⬘ j␦m⬘m␦n⬘n␦共g g⬘兲,
共4.19兲
analogous to Eqs. (2.7) and (2.5). The action of U共j⬘m⬘n⬘兲 on the momentum eigenstates 兩jmn典
can be worked out; it involves the Clebsch–Gordan coefficients for the reduction of direct products
of two general UIR’s of G and reads
U共j ⬘m⬘n⬘兲兩jmn典 =
兺
j ⬙m⬙n⬙␭
冑
N j j⬘ jj⬙␭
j jj ␭
C
* Cn⬘⬘nn⬙⬙ 兩j⬙m⬙n⬙典.
N j⬙ m⬘mm⬙
共4.20兲
Here ␭ is a multiplicity index keeping track of the possibly multiple occurrences of the UIR D j⬙
in the reduction of the direct product D j⬘ ⫻ D j. The significance of this relation is similar in spirit
to the statement in the Cartesian case that U共p⬘兲 = exp共ip⬘q̂兲 generates a translation in p̂, in other
words that in the momentum description q̂ is given by the differential operator i共d / dp兲. The result
(4.20) however involves discrete labels since G is compact, unlike continuous Cartesian variables,
and incorporates non-Abelianness as well. Therefore translating the momentum jmn by the
amount j⬘m⬘n⬘ yields several final momenta j⬙m⬙n⬙ according to the contents of the direct product
D j⬘ ⫻ D j of UIR’s of G.
j
Now multiply both sides of Eq. (4.14) by Dm1 m⬘共g兲 and integrate with respect to g, this is
1 1
Fourier transformation with respect to g and gives
具g⬘兩
冕
j
G
j
dg Dm1 m⬘共g兲Ŵ共g; jmm⬘兲兩g⬙典 = Dmm⬘共g⬘g⬙−1兲Dm1 m⬘共s0共g⬙g⬘−1兲g⬘兲.
j
1 1
1 1
共4.21兲
Now perform an inverse Fourier transformation with respect to the momenta jmm⬘ to get
兺
jmm⬘
N jDmm⬘共g1兲 * 具g⬘兩
j
冕
j
G
dg Dm1 m⬘共g兲Ŵ共g; jmm⬘兲兩g⬙典
1 1
j
j
−1
= Dm1 m⬘共s0共g⬙g⬘−1兲g⬘兲␦共g1g⬙g⬘−1兲 = Dm1 m⬘共s0共g−1
1 兲g⬘兲␦共g1g⬙g⬘ 兲
1 1
=
1 1
j
具g⬘兩g1g⬙典Dm1 m⬘共s0共g−1
1 兲g⬘兲
1 1
=
具g⬘兩U共j1m2m⬘1兲V共g1兲兩g⬙典Dmj m 共s0共g−1
兺
1 兲兲.
m
1
1 2
共4.22兲
2
Comparing the two sides and peeling off 具g⬘兩 and 兩g⬙典 gives
兺
jmm⬘
j
N jDmm⬘共g1兲 *
冕
G
j
dg Dm1 m⬘共g兲Ŵ共g; jmm⬘兲 =
1 1
⬘兲V共g1兲.
Dmj m 共s0共g−1
兺
1 兲兲U共j 1m2m1
m
1
1 2
2
共4.23兲
Then Fourier inversion twice yields the result
Ŵ共g; jmm⬘兲 =
兺
j m m
N j1
1 1 2
冕
G
j
dg1 U共j1m2m1兲V共g1兲Dmm⬘共g1兲Dmj1 m 共g−1s0共g−1
1 兲兲.
1 2
共4.24兲
This may be compared in every detail with the Cartesian result in Eq. (2.10), the correspondence
of arguments and integration/summation variables is (including the factors representing momentum eigenfunctions)
p → jmm⬘,
q → g,
eipq⬘ → Dmm⬘共g1兲,
j
q ⬘ → g 1,
p ⬘ → j 1m 1m 2 ,
e−ip⬘共q+q⬘/2兲 → Dmj1 m 共g−1s0共g−1
1 兲兲.
1 2
Giving due attention to the new matrix features, the correspondence is quite remarkable.
Combining Eqs. (3.28) and (4.14) we obtain the relation
共4.25兲
Ŵ共g; jmn兲† = Ŵ共g; jnm兲.
共4.26兲
Similarly combining Eqs. (4.24) and (4.19) and carrying out quite elementary operations leads to
analogues to the Cartesian relations (2.11) and (2.15) in the forms
−1
Tr共Ŵ共g⬘ ; j⬘m⬘n⬘兲†Ŵ共g; jmn兲兲 = N−1
j ␦ jj⬘␦mm⬘␦nn⬘␦共g g⬘兲,
 =
Nj
兺
jmn
冕
dg WAˆ共g; jnm兲Ŵ共g; jmn兲.
共4.27兲
G
We may thus conclude that we have succeeded in setting up a Wigner–Weyl isomorphism for
quantum mechanics on a compact simple Lie group with reasonable properties.
V. THE STAR PRODUCT FOR WEYL SYMBOLS
In this section we sketch the derivation of the expression for noncommutative operator multiplication in terms of the corresponding Weyl symbols, relegating some details to the Appendix.
Thus, for two operators  and B̂ we seek an expression for the Weyl symbol of ÂB̂ in terms of
those of  and B̂ in the form
WAˆBˆ共g; jmn兲 = 共WAˆ 쐓 WBˆ兲共g; jmn兲.
共5.1兲
共WAˆ 쐓 WBˆ兲共g; jmn兲 = Tr共ÂB̂Ŵ共g; jmn兲兲,
共5.2兲
From Eq. (4.13) we have
so using Eq. (4.27) for  as well as for B̂ we have
共WAˆ 쐓 WBˆ兲共g; jmn兲 =
兺
j ⬘m⬘n⬘
N j ⬘N j ⬙
冕 ⬙冕
dg
G
dg⬘ WAˆ共g⬙ ; j⬙n⬙m⬙兲WBˆ共g⬘ ; j⬘n⬘m⬘兲
G
j ⬙m⬙n⬙
⫻ Tr共Ŵ共g⬙ ; j⬙m⬙n⬙兲Ŵ共g⬘ ; j⬘m⬘n⬘兲Ŵ共g; jmn兲兲.
共5.3兲
We therefore need to compute the trace of the product of three Ŵ’s, which is a nonlocal integral
kernel defining the (associative but noncommutative) star product on the left-hand side. The two
ingredients for this calculation are expressions for the product U共jmn兲V共g兲 in terms of
Ŵ共g⬘ ; j⬘m⬘n⬘兲, and for the product U共j⬘m⬘n⬘兲V共g⬘兲U共jmn兲V共g兲 in terms of similar products UV.
These are
U共jmn兲V共g兲 =
兺
j ⬘m⬘n⬘
N j⬘Dm⬘⬘n⬘共g兲 *
j
U共j ⬘m⬘n⬘兲V共g⬘兲U共jmn兲V共g兲 =
兺
j ⬙m⬙n⬙k
冕
j
dg⬘ Dmn
共s0共g兲g⬘兲Ŵ共g⬘ ; j⬘m⬘n⬘兲,
共5.4a兲
G
Cm⬘⬘n⬘
j
j j⬙
j
−1
kn m⬙n⬙Dmk共g⬘ 兲U共j ⬙m⬙n⬙兲V共g⬘g兲.
共5.4b兲
The derivations are given in the Appendix, and the C-symbol on the right-hand side in the second
equation is a sum of products of Clebsch–Gordan coefficients of the type occurring in Eq. (4.20).
Starting from Eq. (4.24) and using Eq. (5.4b) we have for the product of two Ŵ’s,
Ŵ共g⬘ ; j⬘m⬘n⬘兲Ŵ共g; jmn兲
=
兺
j0,m0,n0
N j 0N j ⬘
0
冕 冕
G
j
dg⬘0 Dmn
共g0兲Dm⬘⬘n⬘共g⬘0兲
j
dg0
G
j0⬘m⬘0 n0⬘
j⬘
−1
−1
0
⬘兲V共g0⬘兲U共j0m0n0兲V共g0兲
⫻Dnj0 m 共g−1s0共g−1
0 兲兲Dn⬘m⬘共g⬘ s0共g0⬘ 兲兲U共j 0⬘m0⬘n0
0 0
=
0 0
j⬘
兺
N j0N j⬘Cm0⬘n⬘
0
j 0m0n0k0
0 0
冕 冕
j0⬙
j0
k0n0 m⬙n⬙
0 0 G
j
dg⬘0 Dmn
共g0兲Dm⬘⬘n⬘共g⬘0兲
j
dg0
G
j0⬘m⬘0 n0⬘
j0⬙m0⬙n0⬙
j⬘
−1
−1
0
⫻Dmj0 k 共g⬘0−1兲Dnj0 m 共g−1s0共g−1
0 兲兲Dn⬘m⬘共g⬘ s0共g0⬘ 兲兲 ⫻ U共j ⬙
0 m0⬙n⬙
0 兲V共g0⬘g0兲.
0 0
0 0
0 0
共5.5兲
If here we use Eq. (5.4a) and then Eq. (4.27) we obtain for the kernel in Eq. (5.3),
Tr共Ŵ共g⬙ ; j⬙m⬙n⬙兲Ŵ共g⬘ ; j⬘m⬘n⬘兲Ŵ共g; jmn兲兲
=
冕 冕
j⬘
兺
j 0m0n0k0
j⬙0
j0
k0n0 m⬙n⬙
0 0
0 0 G
N j0N j⬘Cm0⬘n⬘
0
dg0
G
j
dg0⬘ Dmn
共g0兲Dm⬘⬘n⬘共g0⬘兲Dn⬙⬙m⬙共g0⬘g0兲 *
j
j
j0⬘m0⬘n0⬘
j⬙0 m0⬙n⬙0
j⬘
j⬙
−1
0
⬘−1兲兲Dm0⬙n⬙共s0共g0⬘g0兲g⬙兲.
⫻Dmj0 k 共g0⬘−1兲Dnj0 m 共g−1s0共g−1
0 兲兲Dn⬘m⬘共g⬘ s0共g0
0 0
0 0
0 0
0 0
共5.6兲
The star product of Eq. (5.3) is then obtained by inserting this integral kernel on the right-hand
side.
A slightly simpler expression—which amounts to trading four of the D-functions for Dirac
delta functions—results from direct use of Eq. (4.14),
Tr共Ŵ共g⬙ ; j⬙m⬙n⬙兲Ŵ共g⬘ ; j⬘m⬘n⬘兲Ŵ共g; jmn兲兲
=
冕 冕 ⬘冕
冕 冕 ⬘冕
dg0
G
=
dg0
G
dg0
G
dg0
G
dg⬙0具g0兩Ŵ共g⬙ ; j⬙m⬙n⬙兲兩g0⬘典具g0⬘兩Ŵ共g⬘ ; j⬘m⬘n⬘兲兩g⬙0典具g⬙0兩Ŵ共g; jmn兲兩g0典
G
G
dg0⬙ Dm⬙⬙n⬙共g0g0⬘−1兲Dm⬘⬘n⬘共g0⬘g0⬙−1兲
j
j
j
−1
⬘兲兲␦共g⬘−1s共g0⬘,g0⬙兲兲␦共g−1s共g0⬙,g0兲兲.
共g0⬙g−1
⫻Dmn
0 兲␦共g⬙ s共g0,g0
共5.7兲
These expressions for the star product show an unavoidable complexity for general compact
non-Abelian G. In the one-dimensional Abelian (but non-Cartesian) case Q = S1, there are some
simplifications. Referring to Sec. II, we have the rule for Weyl symbols given by Eq. (2.25) and
(2.28),
a共␪ ;m兲 = Tr共ÂŴ共␪ ;m兲兲,
 = 2␲
兺
m苸Z
The star product then appears as
冕
␲
−␲
d␪ a共␪ ;m兲Ŵ共␪ ;m兲.
共5.8兲
共a 쐓 b兲共␪ ;m兲 =
兺
m⬘,m⬙苸Z
冕 冕
␲
−␲
d␪⬙
␲
−␲
Tr共Ŵ共␪⬙ ;m⬙兲Ŵ共␪⬘ ;m⬘兲Ŵ共␪ ;m兲兲 =
d␪⬘ Tr共Ŵ共␪⬙ ;m⬙兲Ŵ共␪⬘ ;m⬘兲Ŵ共␪ ;m兲兲a共␪⬙ ;m⬙兲b共␪⬘ ;m⬘兲,
1
4␲2
兺
m0,m0⬘苸Z
冕 ⬘冕
␲
−␲
d␪0
␲
−␲
d␪0ei/2共m0⬘␪0−m0␪0⬘兲exp关i共m␪0 − m0␪
+ m⬘␪⬘0 − m⬘0␪⬘ + 共m0 + m0⬘兲␪⬙ − m⬙共␪0 + ␪⬘0兲兲兴.
共5.9兲
This expression for the kernel results from Eq. (5.6) if we first drop the magnetic quantum
numbers m , n , m⬘ , n⬘ , m⬙ , n⬙ , m0 , n0 , k0 , m⬘0 , n0⬘ , m⬙0 , n0⬙; then set the dimensionalities N j0, N j⬘ equal
0
to unity; next make the replacements j → m , j⬘ → m⬘ , j⬙ → m⬙ , j0 → m0 , j0⬘ → m0⬘ , g0 → ␪0 , g⬘0 → ␪0⬘,
and use for the C coefficient the Kronecker delta ␦ j⬙,m0+m⬘. Even with some simplifications, the
0
0
kernel in Eq. (5.9) remains nonlocal because of (among other things) the occurrence of half-angles
in the exponent.
VI. DISCUSSION AND CONCLUDING REMARKS
The characteristic feature revealed by our analysis is that for quantum mechanics on a Lie
group G as configuration space, the concept of canonical momentum is a collection of noncommuting operators Ĵr, in fact constituting the Lie algebra of the left regular representation of G on
L2共G兲. This in itself is known, but it results in the analogues of momentum eigenvalue being a set
of discrete labels jmn, and the single Cartesian momentum eigenvector 兩p典 being replaced by a
multidimensional set of vectors 兵兩jmn典其. Other consequences of this non-Abelianness should be
noted. One needs to work with both overcomplete and with complete nonredundant Weyl symbols
for general operators Â: the former are useful for reproducing in a simple manner the two complementary marginal probability distributions associated with a pure or mixed quantum state from its
Wigner distribution as shown in Eq. (3.23); while the latter lead to the Wigner–Weyl isomorphism
in a reasonable manner.
It is interesting that the Weyl symbols WAˆ共g ; jmm⬘兲 are not complex valued functions on the
classical phase space T * G. They may be more compactly viewed as follows. Whereas by the
Peter–Weyl theorem the Hilbert space H = L2共G兲 carries each UIR D共j兲共·兲 of G as often as its
dimension N j, the structure of Eq. (4.7) leads us to define a smaller Hilbert space H0 carrying each
UIR of G exactly once:
H0 =
兺j
H共j兲 = Sp兵兩jm兲其,
丣
H共j兲 ,
dim H共j兲 = N j ,
共6.1兲
共j⬘m⬘兩jm兲 = ␦ j⬘ j␦m⬘m ,
with H共j兲 carrying the UIR D共j兲共·兲 of G. Then the Weyl symbol of a general operator Â,
WAˆ共g ; jmm⬘兲, may be regarded as a function of g 苸 G and an operator on H0. This is evident from
the examples of Weyl symbols given in Eq. (4.11); in the Cartesian case in Eq. (4.12) such features
are of course absent. This can be understood also from the following point of view. In the normal
quantum description an operator  on H = L2共G兲 can be given via its kernel 具g⬙兩Â兩g⬘典, or via its
complementary diagonal plus off-diagonal matrix elements 具j⬘m⬘n⬘兩Â兩jmn典. If in the latter we
trade half of the labels for a dependence on a group element g, we arrive at the Weyl symbol
WAˆ共g ; jmm⬘兲 viewed as a block diagonal operator on H0 with simultaneously a dependence on g.
Thus while the Wigner–Weyl isomorphism does not work directly with the true classical phase
space T * G, it seems to use what may be called a noncommutative cotangent space, standing
somewhere between T * G and operators on L2共G兲.
Nevertheless the link to functions on the classical phase space T * G can be established, as we
will see below.
We may use the phrase “semiquantized phase space” for the space on which the Weyl symbols
WAˆ共g ; jmn兲 of operators  are defined. It is to be understood that this phrase includes the restriction that only (g-dependent) block-diagonal operators on H0 are encountered. This may be viewed
as a superselection rule. In detail, given an operator  on H = L2共G兲, we associate with it the
g-dependent block-diagonal operator,
Ã共g兲 =
冑N jWAˆ共g; jmn兲兩jm兲共jn兩,
兺j 兺
m,n
共6.2兲
acting on H0, and we then have the connection
TrH共ÂB̂兲 =
冕
G
dg TrH0共Ã共g兲B̃共g兲兲.
共6.3兲
The Weyl symbol Ã共g兲 is simpler than  both in that it acts on the much smaller Hilbert space H0,
and in that it is block diagonal.
To finally establish the link to suitable functions on the classical phase space T * G, we exploit
both the fact that the representation of G on H0 has a multiplicity-free reduction into UIR’s, and
the fact that Ã共g兲 is block diagonal. Let us denote the generators of G on H0 by Ĵr共0兲, r
= 1 , 2 , . . . , n. The Weyl symbol Ã共g兲 may initially be written as the direct sum of symbols à j共g兲
acting within each subspace H共j兲 in H0,
Ã共g兲 =
à j共g兲 =
兺j
丣
à j共g兲,
冑N jW˜A共g; jmn兲兩jm兲共jn兩.
兺
m,n
共6.4兲
Next, using the irreducibility of 兵Ĵr共0兲其 acting on H共j兲, we can expand à j共g兲 uniquely as a sum of
symmetrized polynomials in Ĵr共0兲,
à j共g兲 =
兺
兺
N=0,1,. . . r1,r2,. . .,rN
兵Ĵr共0兲Ĵr共0兲 ¯ Ĵr共0兲其共j兲
S =
1
2
N
ar1. . .rN共g; j兲兵Ĵr共0兲Ĵr共0兲 ¯ Ĵr共0兲其共j兲
S ,
1
2
N
1
共Ĵ共0兲 ¯ Ĵr共0兲 兲共j兲 .
P共N兲
N! P苸SN rP共1兲
兺
共6.5兲
Here the upper limit of N is determined by the UIR D j; SN is the permutation group on N symbols;
and the superscript 共j兲 denotes the restriction to H共j兲. The coefficients ar1,. . .,rN共g ; j兲 are c-number
quantities symmetric in r1 , . . . , rN. If we now replace their j dependences by dependences on the
independent mutually commuting Casimir operators Ĉ of G, themselves symmetric homogeneous
polynomials in Ĵr共0兲, we can use (6.5) in (6.4) and write
⬁
Ã共g兲 =
兺 兺
N=0 r1,. . .,rN
ar1,. . .,rN共g;Ĉ兲兵Ĵr共0兲 ¯ Ĵr共0兲其S .
1
N
共6.6兲
This expression for the Weyl symbol Ã共g兲 of  can now be set into one-to-one correspondence
with the classical phase space function
⬁
a共g;J兲 =
兺 兺
N=0 r1,. . .,rN
ar1,. . .,rN共g;C兲Jr1 ¯ JrN ,
共6.7兲
where the commuting classical variables Jr are the canonical momentum coordinates of the classical phase space T * G,21 while C are invariant (Casimir) homogeneous polynomials in them. Thus
we have the two-stage sequence of correspondences
 on H = L2共G兲 ⇔ Ã共g兲 = block-diagonal operator on H0 ↔ a共g;J兲 苸 F共T * G兲.
共6.8兲
The importance of the multiplicity-free nature of the representation of G on H0, and the superselection rule, is evident. In contrast to the Cartesian case in Sec. II, the appearance of the
semiquantized phase space as an intermediate step is to be noted. We hope to return to this aspect
in a future publication.
APPENDIX
We indicate here the derivations of Eqs. (5.4a) and (5.4b). For Eq. (5.4a), we begin with Eq.
(4.23) and use the unitarity of the D-matrices to shift the D-matrix on the right-hand side to the
left-hand side. This immediately gives Eq. (5.4a). For Eq. (5.4b) we begin with the decomposition
of the product of two U’s; from Eq. (4.15), using Eq. (A29) in Ref. 21,
j
共g兲兩g典 =
U共j⬘m⬘n⬘兲U共jmn兲兩g典 = Dm⬘⬘n⬘共g兲Dmn
j
兺
j ⬙m⬙n⬙␭
Cm⬘⬘
j
j j⬙␭* j⬘ j j⬙␭ j⬙
m m⬙ Cn⬘ n n⬙ Dm⬙n⬙共g兲兩g典.
共A1兲
Here the C’s are the usual Clebsch–Gordan coefficients for the decomposition of the direct product
D j⬘ ⫻ D j of two UIR’s into UIR’s D j⬙, with a multiplicity index ␭ to keep track of multiple
occurrences of a given D j⬙. If we introduce the short-hand notation
Cm⬘⬘n⬘
j
j j⬙
mn m⬙n⬙
=
兺␭ Cmj⬘⬘ mj mj⬙⬙␭*Cnj⬘⬘ nj nj⬙⬙␭ ,
共A2兲
兺
共A3兲
we get from (A1):
U共j⬘m⬘n⬘兲U共jmn兲 =
j ⬙m⬙n⬙
Cm⬘⬘n⬘
j
j j⬙
mn m⬙n⬙U共j ⬙m⬙n⬙兲.
We can now tackle the product of four factors in Eq. (5.4b). First using Eqs. (3.7) and (4.18) and
then using (A3) above gives
U共j⬘m⬘n⬘兲V共g⬘兲U共jmn兲V共g兲 = U共j⬘m⬘n⬘兲
=
兺
j ⬙m⬙n⬙k
兺k Dmkj共g⬘−1兲U共jkn兲V共g⬘g兲
j
Dmk
共g⬘−1兲Cm⬘⬘n⬘
j
j j⬙
kn m⬙n⬙U共j ⬙m⬙n⬙兲V共g⬘g兲,
共A4兲
which is Eq. (5.4b).
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9