Wigner functions for gauge equivalence classes of unitary irreducible representations of noncommutative quantum mechanics

0.23 MB·

by  S. Hasibul Hassan Chowdhury

Wigner functions for gauge equivalence classes of unitary
irreducible representations of noncommutative quantum
7 mechanics
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0
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S. Hasibul Hassan Chowdhury∗1 and Hishamuddin Zainuddin†2

n
a
J 1,2Laboratory of Computational Sciences and Mathematical Physics, Institute for

1
Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang,
3
Selangor, Malaysia

]
h 2Malaysia-Italy Centre of Excellence for Mathematical Sciences, Universiti Putra
p
- Malaysia, Malaysia
h
t
a
m
[

Abstract
1
v
0 While Wigner functions forming phase space representation of quantum states
3 is a well-known fact, their construction for noncommutative quantum mechanics
9
8 (NCQM) remains relatively lesser known, in particular with respect to gauge de-
0 pendencies. This paper deals with the construction of Wigner functions of NCQM
.
1 for asystemof2-degreesoffreedomusing2-parameterfamilies ofgaugeequivalence
0
classesofunitaryirreduciblerepresentations(UIRs)oftheLiegroupG whichhas
7 NC
1 been identified as the kinematical symmetry group of NCQM in an earlier paper.
:
v This generalconstructionofWignerfunctions for NCQM,inturn, yieldsthe special
Xi cases of Landau and symmetric gauges of NCQM.
r
a
I Introduction
Noncommutative quantummechanics (NCQM)hasbeenstudiedextensivelyasapossible
modification ofquantummechanics atthePlanck scaleforwhich thepositionobservables
are noncommuting (hence dimension of configuration space is at least two). The moti-
vation of the introduction of NCQM directs us back to the seminal work of Snyder [23]
where he studied the quantized structure of space-time in a Lorentz invariant fashion.
Such a model of space-time in scales as small as Planck length is also proposed, among
others by Doplicher et al. [12] in order to avoid creation of microscopic black holes to the
∗[email protected]
†[email protected]
1
effect of losing the operational meaning of localization in space-time. NCQM can also
be realized as nonrelativistic approximation of noncommutative field theory (see [16]). A
noncommutative quantum field theory (NCQFT) is the onewhere the fields are functions
of space-time coordinates with spatial coordinates failing to commute with each other. A
multitude of articles appearing in the literature (see, for example, [13, 9, 22, 7, 8, 2, 21])
focus on how NCQFT is related to the studies of noncommutative geometry and string
theory. On the other hand, how noncommutative geometry can be studied independently
from the unitary irreducible representations of NCQM are all explored in [4].
Byinstitutingnoncommutativity ofthepositionobservables,onecanalsotakeittobe
in the context of restoring the symmetry between phase space coordinates when treating
aquantum systemofparticles inamagnetic fieldwherethe momenta arenoncommuting.
TheunderlyinggeneralkinematicalsymmetrygroupG ofNCQMinitssmallestdimen-
NC
sion was understood only relatively recently as the triply extended groups of translations
inR4[3]. Oneofus[4]hasexploredexhaustivelythedifferentinequivalentrepresentations
of G and showed that they fall into three classes of 4-dimensional, 2-dimensional and
NC
0-dimensional coadjoint orbits. It was observed that within the 4-dimensional coadjoint
orbit, there exists families of representations labelled by three parameters of which one
case (with two parameters tending to zero) corresponds to standard quantum mechanics.
In what follows, we will limit our interest to the case of where all three parameters are
nonzero and a certain constraint between these parameters hold.
Having restored the symmetry of the phase-space observables through their non-
commutativities, the plausible natural setting for the full-fledged NCQM is the Wigner
function representations. Wigner function representations [24] are known to be the
quantum-mechanical analogue of the phase space distribution of classical statistical me-
chanics [19] that includes quantum mechanical corrections and hence serve many com-
putational purposes in quantum settings. It is also known that the Wigner function can
be related to variation of information measures and entropies (see [20]). General fea-
tures of time independent Wigner functions are studied in [10]. The quasi-probabilistic
nature of Wigner function may pose problems in interpretation but one can smoothen
the function over phase space area elements to form the more well-behaved Husimi dis-
tribution (see [18]). Another problem discussed in constructing Wigner functions is its
gauge-dependency whentreating noncommutativity ofmomenta operators intheLandau
system of (charged) particle in magnetic field. For the full NCQM, it is expected that
more gauge dependency will appear with further noncommuting positions. The gauge
dependencies of the Wigner functions are rendered harmless in the group-theoretic for-
mulation since there will be unitary operators transforming from one equivalent gauged
representation to another. In the case of NCQM, the NC Wigner functions have been
calculated for the cases of the Landau and symmetric gauges in [5]. It is thus of interest
tous toexplorethe NCWigner functions inthe general gaugesetting andshow explicitly
2
a two-gauge parameter dependence, allowing access to more computational possibilities.
II On the Lie group G , the kinematical symmetry group
NC
of NCQM
The group G , being a 7-dimensional connected, simply connected nilpotent Lie group
NC
was first introduced in [6], the role of which in 2-dimensional NCQM runs parallel to that
of the 5-dimensional Weyl-Heisenberg group for quantum mechanics in 2-dimensions. To
have abrief comparative reviewof these two nilpotent Lie groups consult section IIof [4].
The group G is a seven parameter, real Lie group. We shall write a general element of
NC
the group as
g = (θ,φ,ψ,q,p), θ,φ,ψ R, q = (q ,q ) R2, p = (p ,p ) R2, (2.1)
1 2 1 2
∈ ∈ ∈
with the group multiplication given by
(θ,φ,ψ,q,p)(θ′,φ′,ψ′,q′,p′)
α β γ
= (θ+θ′+ [ q p′ p q′ ],φ+φ′+ [p p′],ψ+ψ′+ [q q′],
2 h · i−h · i 2 ∧ 2 ∧
q+q′,p+p′). (2.2)
where, for two 2-vectors, a = (a ,a ),b = (b ,b ), a b = a b + a b and a b =
1 2 1 2 1 1 2 2
· ∧
a b a b . The quantities α,β and γ are dimensional constants. The three parameters
1 2 2 1
−
θ,φ,ψ constitute the centre of the group, while the q and p, are the parameters of R4,
the three-fold central extension of which leads to G . Denoting the centre by , we see
NC Z
that G / R4. Note that if we denote the dimension of the position coordinate by
NC Z ≃
[q] and that of the momentum coordinate by [p], then we immediately see that in order
to have θ, φ and ψ to be all dimensionless, we must have [α] = 1 , [β] = 1 and
pq p2
[γ] = 1 . h i h i
q2
Whe kniow from [3] that G admits coadjoint orbits of dimension 4, 2 and 0 which are
NC
completely determined by therealtriple (ρ,σ,τ). Itwas alsofoundtherethattheunitary
dual of G is in1-1 correspondence with its various coadjoint orbits and hence slabelled
NC
by the sametriple (ρ,σ,τ). Respectingthenotations usedin [3], ageneric coadjoint orbit
of G will be denoted by ρ,σ,τ. In this paper, the 4 dimensional coadjoint orbits with
NC O
nonzero ρ, σ and τ satisfying ρ2α2 σβγτ = 0 will only concern us. Such orbits will be
− 6
ρ,σ,τ
denoted as .
O4
3
III Gauge equivalence classes of unitary irreducible repre-
sentations of noncommutative quantum mechanics
In [4], 2-parameter (l,m) gauge equivalence classes of UIRs of G were computed. It
NC
was also shown there that they, in turn, give rise to vector potentials labeled by m. The
2-parameter family of unitarily equivalent irreducible representations (UIRs) of G was
NC
due to a fixed coadjoint orbit determined by ρ = σ = τ = 1 where the unitary dual
of G is labeled by the triple (ρ,σ,τ). When one fixes this triple, by varying (l,m),
NC
one can choose a representative of the underlying equivalence class of unitary irreducible
representations of G . The setof UIRsUρ,σ,τ forρ = 0, σ = 0, τ = 0and ρ2 γβστ = 0
NC l,m 6 6 6 − 6
with fixed m R and l Rr ρ2α2 will be denoted by N. These UIRs read as follows
∈ ∈ {γβστ}
ρ,σ,τ
(U (θ,φ,ψ,q~,p~)f)(r ,r )
l,m 1 2
= e−iρθ−iσφ−iτψeiραp1r1+iραp2r2+iτργ2σαβ2lτ−γρ(12−α2l)q1r2+ilτγq2r1+ihρ2α+ρταγτσγβσlβ−mρ(21α−2l)ip1q1
eihρ2α−lτγσβρ(α1−m)ip2q2+i(m−12)σβp1p2+i(cid:20)τ2γ−τγ(1−l)(ττγγσσββl−l−τργ2σαβ2lm−ρ2α2)(cid:21)q1q2
×
×f(cid:18)r1−(1−ρmα)σβp2+τγσβτ(lγ+σmβl−−lρm2)α−2ρ2α2q1,r2+mρσαβp1−τγσβl(1ρ−2αm2)−ρ2α2q2(cid:19), (3.1)
where f L2(R2,dr dr ).
1 2
∈
The corresponding self adjoint representation of g acting on the smooth vectors of
NC
L2(R2,dr dr ) is given by
1 2
iσβ ∂
Qˆm = r m ,
1 1− ρ2α2∂r
2
iσβ ∂
Qˆm = r +(1 m) ,
2 2 − ρ2α2∂r
1
(3.2)
τγρα(1 l) i τγσβ(l+m lm) ρ2α2 ∂
Pˆl,m = − r − − ,
1 τγσβl ρ2α2 2− ρα τγσβl ρ2α2 ∂r
− (cid:20) − (cid:21) 1
lτγ τγσβl(1 m) ρ2α2 ∂
Pˆl,m = r +i − − .
2 ρα 1 ρ3α3 ∂r
(cid:20) (cid:21) 2
It is interesting note that these operators are slightly different from the usual notion
ofWigneroperators[17]forwhich thepolarized formof operators areinplace asexpected
in a quantization scheme. Rearranging the terms in the last two equations of (3.2) using
the first two then leads us to
iσβ ∂
Qˆm = r m ,
1 1− ρ2α2∂r
2
iσβ ∂
Qˆm = r +(1 m) ,
2 2 − ρ2α2∂r
1
(3.3)
τγρα(1 l) i τγσβ(1 l) ∂
Pˆl,m = − Qˆm − +1 ,
1 τγσβl ρ2α2 2 − ρα τγσβl ρ2α2 ∂r
− (cid:20) − (cid:21) 1
lτγ i lτγσβ ∂
Pˆl,m = Qˆm 1 .
2 ρα 1 − ρα − ρ2α2 ∂r
(cid:18) (cid:19) 2
4
One is then motivated by (3.3) to define the underlying vector potential Aρ,σ,τ
≡
ρ,σ,τ ρ,σ,τ
(A ,A ) for the 2-dimensional system of NCQM as
1 2
τγρα(1 l) lτγ
Aρ,σ,τ − Qˆm, Qˆm . (3.4)
≡ −τγσβl ρ2α2 2 −ρα 1
(cid:18) − (cid:19)
Now,iftheconstantmagneticfieldappliedperpendiculartotheunderlying2-dimensional
system is denoted by B, then from the discussion conducted at p. 14 in [3], the magnetic
field reads off as B = τγ. It becomes evident then that, as in a quantum mechanical
−ρα
system in the presence of a vertical constant magnetic field B (without spatial noncom-
mutativity taken into consideration), the relation between the magnetic field and the
ρ,σ,τ ρ,σ,τ ρ,σ,τ
underlying vector potential, i.e. ∂ A ∂ A = B, no longer holds, where A ’s,
1 2 − 2 1 i
for i = 1,2, are the 2-components of the vector potential Aρ,σ,τ. The limiting expression
of Aρ,σ,τ, as σ 0, denoted by Aρ,τ (Aρ,τ,Aρ,τ) then is easily found to restore the
→ ≡ 1 2
ρ,τ ρ,τ
relation ∂ A ∂ A = B.
1 2 − 2 1
The well-known symmetric gauge representation of NCQM corresponds to the choice
l = ρα(ρα−√ρ2α2−γβστ) := l and m = 1 in (3.2) as is given below:
γβστ s 2
1 iσβ ∂
Qˆ2 = r ,
1 1− 2ρ2α2∂r
2
1 iσβ ∂
Qˆ2 = r + ,
2 2 2ρ2α2∂r
1
(3.5)
Pˆls,21 = ( ρ2α2−γβστ −ρα)r i (ρα+ ρ2α2 γβστ) ∂ ,
1 σβ 2− 2ρ2α2 − ∂r
p 1
p
Pˆls,21 = (ρα− ρ2α2 −γβστ)r i (ρα+ ρ2α2 γβστ) ∂ .
2 σβ 1− 2ρ2α2 − ∂r
p 2
p
The corresponding vector potential Aρ,σ,τ, from (3.4), then reads off as
sym
(ρα ρ2α2 γβστ) 1 ( ρ2α2 γβστ ρα) 1
Aρ,σ,τ − − Qˆ2, − − Qˆ2 . (3.6)
sym ≡ σβ 2 σβ 1
p p !
Denotethe2-componentsofthevectorpotentialAρ,σ,τ byAρ,στ withi= 1,2andobserve
sym i,sym
that the following holds
2~ Bϑ
∂ Aρ,σ,τ ∂ Aρ,σ,τ = 1 1 := B¯, (3.7)
1 2,sym− 2 1,sym ϑ − ~ −
r !
where we chose B = τγ, ϑ = σβ and ~ = 1 as in p. 14 of [3]. Compare (3.7) with
−ρα −ρ2α2 ρα
(43) at p. 14 of [11].
Also, of considerable interest inPhysics literature, is theLandau gauge representation
of NCQM that corresponds to l = 1, m = 0 in (3.2). The self adjoint representation of
5
g in the Landau gauge representation can be written as
NC
Qˆ0 = r ,
1 1
iσβ ∂
Qˆ0 = r + ,
2 2 ρ2α2∂r
1
i ∂ (3.8)
Pˆ1,0 = ,
1 −ρα∂r
1
τγ i(τγσβ ρ2α2) ∂
Pˆ1,0 = r + − .
2 ρα 1 ρ3α3 ∂r
2
Substituting l = 1, m = 0 in (3.4) then yields the Landau gauge vector potential
τγ
Aρ,σ,τ (0, Qˆ0) = (0,Br ), (3.9)
Landau ≡ −ρα 1 1
ρ,σ,τ ρ,σ,τ ρ,σ,τ
recovering ∂ A ∂ A = B where A ’s with i= 1,2, being the compo-
1 2,Landau− 2 1,Landau i,Landau
nents of the vector potential Aρ,σ,τ . This is again in agreement with what Delduc et al.
Landau
found in [11] (see p. 15).
IV Wigner functions for equivalence classes of UIRs of G
NC
In this section, we will be dealing with he construction of Wigner function associated
with a 2-dimensional system of NCQM using the general method developed in [1]. The
UIRs of G that we will be focussing on for this purpose are given by (3.1). For this
NC
purpose, we first need to compute the Plancherel measure that the unitary dual of G
NC
is equipped with. We will then employ this measure to compute the NCQM Wigner
function (see (4.14)) for these gauge equivalence classes of UIRs of G .
NC
ˆ
IV.1 Plancherel measure associated with the sector N of G
NC
Let us denote by Hρ,σ,τ, a copy of L2(R2,dr dr ) and consider the Hilbert bundle based
1 2
on the unitary dual Gˆ , a generic point of which has coordinates (ρ,σ,τ) and the fiber
NC
at each (ρ,σ,τ) being the Hilbert space of Hilbert-Schmidt operators on Hρ,σ,τ denoted
by (ρ,σ,τ). Now consider the measurable fields of operators (ρ,σ,τ) (ρ,σ,τ)
2
B 7−→ A
with each (ρ,σ,τ) (ρ,σ,τ). The space of such measurable fields of operators is
2
A ∈ B
endowed with an inner product structure and is called a direct integral Hilbert space.
Given two such fields (ρ,σ,τ) 1(ρ,σ,τ) and (ρ,σ,τ) 2(ρ,σ,τ), the inner
7−→ A 7−→ A
product between them is given by
1 2 = Tr( 1(ρ,σ,τ)∗ 2(ρ,σ,τ))dν (ρ,σ,τ), (4.1)
hA |A iB2⊕ ZGˆNC A A GNC
where dν is the well-known Plancherel measure that the unitary dual G can be
GNC NC
endowed with and we denote the direct integral Hilbert space of measurable fields of
6
Hilbert-Schmidt operators by ⊕ = ⊕ (ρ,σ,τ)dν (ρ,σ,τ) rendering the fact that
B2 GˆNCB2 GNC
the inner product of such Hilbert space is given by (4.1).
R
Now the Plancherel measure for Gˆ can be computed using a general orthogonality
NC
relation (see, for example, [1]) given by
Tr(Uρ,σ,τ(g)∗A1(ρ,σ,τ)C−1 )dν (ρ,σ,τ)
l,m ρ,σ,τ GNC
ZGNC(cid:20)ZGˆNC
Tr(Uρ′,σ′,τ′(g)∗A2(ρ′,σ′,τ′)C−1 )dν (ρ′,σ′,τ′) dµ(g) = A1 A2
× GˆNC l,m ρ′,σ′,τ′ GNC h | iB2⊕
R i (4.2)
Thegroup G isunimodular and dµ istheHaarmeasureonit; C istheDuflo-Moore
NC ρ,σ,τ
ρ,σ,τ
operator [14], for the representation U (g) given by (3.1).
l,m
Now for the measurable vector fields (ρ,σ,τ) λ and (ρ,σ,τ) χ with
ρ,σ,τ ρ,σ,τ
7−→ 7−→
the vectors λ ,χ L2(R2,dr dr ), one can choose the operator fields 1 = 2
ρ,σ,τ ρ,σ,τ 1 2
∈ A A
in (4.2) using the rank 1-operators 1(ρ,σ,τ) = 2(ρ,σ,τ) = χ λ . Then
ρ,σ,τ ρ,σ,τ
A A | ih |
writing the Plancherel measure dν with the help of a suitable density κ(ρ,σ,τ) as
GNC
dν (ρ,σ,τ) = κ(ρ,σ,τ)dρdσdτ, one computes the right side of (4.2) restricted to the
GNC
sector N of Gˆ using the inner product defined by (4.1).
NC
A1 A1 = Tr(A1(ρ,σ,τ)∗A1(ρ,σ,τ))κ(ρ,σ,τ)dρdσdτ
h | iB2⊕ ZN
= Tr(λ χ χ λ )κ(ρ,σ,τ)dρdσdτ
ρ,σ,τ ρ,σ,τ ρ,σ,τ ρ,σ,τ
| ih | ih |
ZN
= χ 2 λ 2κ(ρ,σ,τ)dρdσdτ. (4.3)
ρ,σ,τ ρ,σ,τ
k k k k
ZN
In [5], the Plancherel measure of the unitary dual Gˆ restricted to the sector with
NC
ρ = 0, σ = 0, τ = 0 and ρ2α2 γβστ = 0 has already been computed (see proposition
6 6 6 − 6
1, p. 4). There, the unitary irreducible representation of G , i.e. the representative
NC
chosen for each distinct value of the triple (ρ,σ,τ) was the one associated with l = 1 and
m = 1 in (3.1). But the Plancherel measure of the unitary dual Gˆ restricted to the
NC
sector ρ = 0, σ = 0, τ = 0 and ρ2α2 γβστ = 0 should be independent of the choice of
6 6 6 − 6
the representative for each (ρ,σ,τ) as has been verified in the following proposition:
Proposition 1. If one considers the unitary dual Gˆ restricted to the sector ρ = 0,
NC
6
σ = 0, τ = 0 and ρ2α2 γβστ = 0 where each equivalence class of UIRs of G is
NC
6 6 − 6
represented by the representation (3.1) for a fixed ordered pair (l,m), then the Plancherel
measure of Gˆ restricted to such family of UIRs of G is given by
NC NC
ρ2α2 γβστ
dν (ρ,σ,τ) = | − |dρdσdτ, (4.4)
GNC α2
and the corresponding Duflo-Moore operator reads
Cρ,σ,τ = (2π)52I, (4.5)
7
where I is the identity operator on L2(R2,dr dr ).
1 2
Proof. GiventhefactthatG isunimodular, theunderlyingDuflo-Mooreoperatorreads
NC
C = NI, where N is a real number and I is the identity operator on L2(R2,dr dr ).
ρ,σ,τ 1 2
Now the left side of (4.2) can be read off as
1
χ Uρ,σ,τ(θ,φ,ψ,q,p)λ κ(ρ,σ,τ)dρdσdτ
N2 ZR7(cid:20)ZR∗×R∗×R∗h ρ,σ,τ| l,m ρ,σ,τi
× R∗×R∗×R∗hχρ′,σ′,τ′|Ulρ,m′,σ′,τ′(θ,φ,ψ,q,p)λρ′,σ′,τ′iκ(ρ′,σ′,τ′)dρ′dσ′dτ′ dθdφdψdqdp
R i
= 1 ei(ρ−ρ′)θ+i(σ−σ′)φ+i(τ−τ′)ψ
N2 (ρ,σ,τ) (ρ′,σ′,τ′) R7 (r1,r2)∈R2 (r1′,r2′)∈R2
R e−iαRp1(ρr1−ρ′rh1′R)−iαnp2R(ρr2−ρ′r2′)−Riα2γ(1−l)q1(cid:18)τγσρβ2lτ−rρ22α2−τ′γσρ′′β2τl−′rρ2′′2α2(cid:19)
×
e−ilγq2(τr1−τ′r1′)−ih(ρ−2ρ′)α+αβγm(1−l)(cid:16)τγσβρlσ−τρ2α2−τ′γσρ′β′σl−′τρ′′2α2(cid:17)ip1q1
×
e−i (ρ−2ρ′)α−lγβ(α1−m) τρσ−τρ′σ′′ p2q2−i(m−21)β(σ−σ′)p1p2
h (cid:16) (cid:17)i
×
i (τ−τ′)γ−γ(1−l) (τ2γσβl−τ2γσβlm−τρ2α2)−(τ′2γσ′βl−τ′2γσ′βlm−τ′2ρ′2α2)
e(cid:20) 2 (cid:26) τγσβl−ρ2α2 τ′γσ′βl−ρ′2α2 (cid:27)(cid:21)
×
[1−m]σβ [τγσβ(l+m−lm)−ρ2α2] mσβ [τγσβl(1−m)−ρ2α2]
×χρ,σ,τ(r1,r2)λρ,σ,τ r1− ρα p2+ τγσβl−ρ2α2 q1,r2+ ρα p1− ρ2α2 q2
(cid:18) (cid:19)
×χρ′,σ′,τ′(r1′,r2′)λρ′,σ′,τ′ r1′−[1−ρm′α]σ′βp2+τ′γσ′τβ′(γlσ+′mβl−−lρm′2)α−2ρ′2α2q1,r2′+mρσ′α′βp1−[τ′γσ′βl(ρ1′−2αm2)−ρ′2α2]q2!
×dr1dr2dr1′dr2′}dθdφdψdq1dq2dp1dp2]κ(ρ,σ,τ)κ(ρ′,σ′,τ′)dρdσdτdρ′dσ′dτ′
= (2Nπ2)3 (ρ,σ,τ) R4 (r1,r2)∈R2 (r1′,r2′)∈R2eiαρp1(r1−r1′)+iαρp2(r2−r2′)
×Re−iρτ2γασ2hβγRlτ−(1ρ−2αnl)2qR1(r2−r2′)−ilγRτq2(r1−r1′)χρ,σ,τ(r1,r2)
[1−m]σβ [τγσβ(l+m−lm)−ρ2α2] mσβ [τγσβl(1−m)−ρ2α2]
×λρ,σ,τ r1− ρα p2+ τγσβl−ρ2α2 q1,r2+ ρα p1− ρ2α2 q2
(cid:18) (cid:19)
×χρ,σ,τ(r1′,r2′)λρ,σ,τ r1′−[1−ρmα]σβp2+τγσβτ(lγ+σmβ−l−lmρ2)α−2ρ′2α2q1,r2′+mρσαβp1−[τγσβl(1ρ−2αm2)−ρ2α2]q2!
×dr1dr2dr1′dr2′}dq1dq2dp1dp2][κ(ρ,σ,τ)]2dρdσdτ
= (2π)3 e−iρα(cid:20)σβρ(ταγ(τσγβσl−β−ρ2ρα22α)2()1q−1m)+1Π−2m(cid:21)(r2−r2′)
N2 Z(ρ,σ,τ)"ZR4(Z(r1,r2)∈R2Z(r1′,r2′)∈R2
e−iρα(cid:20)(τγσmβlρ−αρσ2βα2)q2+Πm1(cid:21)(r1−r1′)χ (r ,r )λ r σβΠ ,r + σβΠ
ρ,σ,τ 1 2 ρ,σ,τ 1 2 2 1
× − ρα ρα
(cid:18) (cid:19)
σβ σβ
χ (r′,r′)λ r′ Π ,r′ + Π dr dr dr′dr′
× ρ,σ,τ 1 2 ρ,σ,τ 1− ρα 2 2 ρα 1 1 2 1 2
(cid:18) (cid:19) (cid:27)
1
dΠ dΠ dq dq [κ(ρ,σ,τ)]2dρdσdτ, (4.6)
1 2 1 2
× m(1 m)
| − | (cid:21)
where we have made the following changes of variables:
mσβ [τγσβl(1 m) ρ2α2] σβ
p − − q = Π
ρα 1− ρ2α2 2 ρα 1
(4.7)
(1 m)σβ [τγσβ(l+m lm) ρ2α2] σβ
− p − − q = Π .
ρα 2− τγσβl ρ2α2 1 ρα 2
−
8
Now (4.6) reduces to
(2π)5 σβγ(τγσβl ρ2α2)(1 m) mσβγ 1
| − − | | |
N2 ρ2α2 τγσβ ρ2α2 × τγσβl ρ2α2 × m(1 m)
Z(ρ,σ,τ) | − | | − | | − |
σβ σβ
χ (r ,r )λ r Π ,r + Π
ρ,σ,τ 1 2 ρ,σ,τ 1 2 2 1
×"Z(Π1,Π2)∈R2(Z(r1,r2)∈R2 (cid:18) − ρα ρα (cid:19)
σβ σβ
χ (r ,r )λ r Π ,r + Π dr dr dΠ dΠ [κ(ρ,σ,τ)]2dρdσdτ
ρ,σ,τ 1 2 ρ,σ,τ 1 2 2 1 1 2 1 2
× − ρα ρα
(cid:18) (cid:19) (cid:27) (cid:21)
(2π)5 σ2β2γ2
= χ (r ,r )χ (r ,r )
N2 Z(ρ,σ,τ) ρ2α2|τγσβ −ρ2α2| "Z(r1,r2)∈R2 ρ,σ,τ 1 2 ρ,σ,τ 1 2
ρ2α4
λ (Π ,Π )λ (Π ,Π ) dΠ dΠ dr dr
×(Z(Π2,Π1)∈R2 ρ,σ,τ 2 1 ρ,σ,τ 2 1 σ2β2γ2 1 2) 1 2#
[κ(ρ,eσ,τe)]2dρdσdτ, e e e e e e (4.8)
×
where, in (4.8), we have introduced the following change of variables:
σβ α
r Π = Π
1 2 2
− ρα γ
(4.9)
σβ α
r Π = Πe ,
2 1 1
− ρα γ
so that (4.8) now takes the following simple form: e
(2π)5 α2
χ 2 λ 2[κ(ρ,σ,τ)]2dρdσdτ. (4.10)
N2 τγσβ ρ2α2 || ρ,σ,τ|| || ρ,σ,τ||
Z(ρ,σ,τ) | − |
Now comparing (4.10) with the right side of (4.3) then yields
5
N = (2π)2
τγσβ ρ2α2 (4.11)
κ(ρ,σ,τ) = | − |,
α2
proving the proposition.
Remark IV.1. A few remarks on the statement of the proposition 1 and its proof are
in order. In this paper, we deal with an arbitrary member of each family of equivalence
classes of the sector N of the unitary dual Gˆ labeled by the ordered pair (l,m). The
NC
underlying proposition states that for any choice of the representative (determined by
(l,m)) from each equivalence class restricted to the sector N of Gˆ , we obtain the given
NC
Plancherel measure. It is to be noted that during the proof, one has to exercise caution
while introducing change of variables (see (4.9)) where on the left side one has quantities
with dimension of length. So, to bring the dimension of momentum , i.e. the one asso-
ciated with the Π ’s, for i = 1,2 on the right side of (4.9), to that of length, we had to
i
insert a factor of α which indeed has the dimension of Length/Momentum. These careful
γ
e
change of variables yield the desired Plancherel measure.
9
IV.2 Construction of Wigner function
Recall from [5] that the Lebesgue measure dX∗ on the dual Lie algebra g∗ decomposes
NC
as
dX∗ = s (X∗ ) dν (ρ,σ,τ) dΩ (X∗ ), X∗ ρ,σ,τ (4.12)
ρ,σ,τ ρ,σ,τ GNC ρ,σ,τ ρ,σ,τ ρ,σ,τ ∈ O
where s is a positive density, dΩ is the canonical invariant measure on the coad-
ρ,σ,τ ρ,σ,τ
joint orbit ρ,σ,τ (in this case, just the Lebesgue measure on R4) and dν is the
O GNC
Plancherel measure (4.4). By X∗ and X∗ , we denote a generic point in g∗ and in
ρ,σ,τ NC
the 4-dimensional coadjoint orbit ρ,σ,τ g∗ , respectively. It then immediately follows
O4 ∈ NC
that the strictly positive density s (X∗ ) is given by
ρ,σ,τ ρ,σ,τ
α2
s (X∗ ) = , (4.13)
ρ,σ,τ ρ,σ,τ ρ2α2 γβστ
| − |
which upon substitution in the definition (see p. 6 of [5]) of the NCQM Wigner function,
i.e.
W(A; X∗ ) = [sρ,σ,τ(Xρ∗,σ,τ)]21 e−ihXρ∗,σ,τ;Xi
ρ,σ,τ 7
(2π)2 ZgNC
Tr[Uω,ν,µ(e−X)A(ω,ν,µ)C−1 ]dν (ω,ν,µ) dX, (4.14)
× l,m ω,ν,µ GNC
(cid:20)ZN (cid:21)
leads us to the following expression:
α
W(A; X∗ ) = | | e−ihXρ∗,σ,τ;Xi
ρ,σ,τ (2π)6 ρ2α2 γβστ 21 ZgNC
| − |
Tr[Uω,ν,µ(e−X)A(ω,ν,µ)] dν (ω,ν,µ) dX. (4.15)
× l,m GNC
(cid:20)ZN (cid:21)
Ithasbeenarguedin[1,15]thatfornilpotentLiegroupthingsbecomemoretractable
in the sense that the underlying Wigner functions are decomposable, meaning that the
Wigner function only picks up the contribution of the Hilbert space of Hilbert-Schmidt
operators associated with the underlying coadjoint orbit for the measurable operator
fields (ρ,σ,τ) A(ρ,σ,τ) (ρ,σ,τ), so that
2
7−→ ∈ B
W(A; X∗ )= [W A(ρ,σ,τ)](X∗ ):= W(A(ρ,σ,τ); X∗ ; ρ,σ,τ), (4.16)
ρ,σ,τ ρ,σ,τ ρ,σ,τ ρ,σ,τ
We are now in a state to provide the main theorem of the paper which is as follows
Theorem IV.2. The Wigner function for NCQM in 2-dimensions restricted to the 4-
dimensional coadjoint orbit k1,k2,k3 for nonzero k ’s satisfying k2α2 k k γβ = 0 due
O4 i 1 − 2 3 6
to the 2-parameter family of UIRs of G (see (3.1)) labeled by (l,m) can be computed
NC
10