The Schwinger SU„3… construction. I. Multiplicity problem
and relation to induced representations
S. Chaturvedia)
School of Physics, University of Hyderabad, Hyderabad 500046, India
N. Mukundab)
Centre for Theoretical Studies, Indian Institute of Science, Bangalore 560012, India
The Schwinger oscillator operator representation of SU共3兲 is analyzed with particular reference to the problem of multiplicity of irreducible representations. It is
shown that with the use of an Sp(2,R) unitary representation commuting with the
SU共3兲 representation, the infinity of occurrences of each SU共3兲 irreducible representation can be handled in complete detail. A natural ‘‘generating representation’’
for SU共3兲, containing each irreducible representation exactly once, is identified
within a subspace of the Schwinger construction, and this is shown to be equivalent
to an induced representation of SU共3兲.
I. INTRODUCTION
The well known Schwinger representation of the Lie algebra of SU共2兲,1 constructed using the
annihilation and creation operators of two independent quantum mechanical harmonic oscillators,
has played an important role in many widely differing contexts. Within the quantum theory of
angular momentum it has made the calculation of various quantities somewhat easier than by other
methods. Beyond this, it has been very effectively exploited in the physics of strongly correlated
systems,2 in quantum optics of two mode radiation fields,3 and in the study of certain classes of
partially coherent optical beams,4 namely to obtain the coherent mode decomposition of anisotropic Gaussian Schell model beams. It arises quite naturally in the context of a classical description of particles with non-Abelian charges5 and has also been used in a recent investigation of the
Pauli spin-statistics theorem.6
Bargmann has presented an entire function Hilbert space analog of the Schwinger construction, which is extremely elegant and possesses special merits of its own.7 This may be viewed as
a counterpart to the Fock space description of quantum mechanical oscillator systems.
Certain specially attractive features of the Schwinger SU共2兲 construction should be mentioned. It leads upon exponentiation to a unitary representation 共UR兲 of SU共2兲 in which each
unitary irreducible representation 共UIR兲, labeled as usual by the spin quantum number j with
possible values 0, 21 ,1,..., appears exactly once. In other words, it is complete in the sense that no
UIR of SU共2兲 is missed, and also economical in the sense of being multiplicity free. Thus,
reflecting these two features, it may be regarded as a ‘‘generating representation’’ for SU共2兲, a
concept that has been effectively used in understanding the structures of various kinds of Clebsch–
Gordan series for UIRs of the noncompact group SU共1,1兲.8 In addition, of course, the use of boson
operator methods makes many operator and state vector calculations relatively easy to carry out.
It is of considerable interest to extend the Schwinger construction to other compact Lie
groups, the next natural case after SU共2兲 being SU共3兲. The aims behind any such attempt would be
a兲
Electronic mail: scsp@uohyd.ernet.in
Honorary Professor, Jawaharlal Nehru Center for Advanced Scientific Research, Jakkur, Bangalore 560064. Electronic
mail: nmukunda@cts.iisc.ernet.in
b兲
to preserve the simplicity of the boson calculus, to cover all UIRs of the concerned group, and to
do it in a multiplicity free manner.
The case of SU共3兲 has been studied by several authors since the work of Moshinsky.9 The aim
of the present article is somewhat different from previous studies, being motivated by the particular points of view mentioned above. In particular our aim is to see to what extent the attractive
features of the SU共2兲 construction survive when we consider SU共3兲, and which ones have to be
given up.
A brief overview of this article is as follows. In Sec. II we collect together some relevant facts
regarding unitary representations of compact Lie groups with special attention to SU共3兲. In particular, we highlight the fact that the theory of induced representations leads to a unitary representation of SU共3兲 which has all the properties becoming of a ‘‘generating representation’’ of
SU共3兲 in that it contains all the UIRs of SU共3兲 exactly once each. The Hilbert space carrying this
unitary representation turns out to be the Hilbert space of functions on unit sphere in C 3 . In Sec.
III, we turn to the Schwinger oscillator construction for SU共3兲 and show that a naive extension of
the Schwinger SU共2兲-construction making use of six oscillators leads to a very ‘‘fat’’ UR of SU共3兲
containing each UIR of SU共3兲 infinitely many times. We then show how the group Sp(2,R)
enables us to completely handle this multiplicity and also neatly isolate from this rather large
space a subspace carrying a UR of SU共3兲 of a ‘‘generating representation’’ type. At this stage, we
have two ‘‘generating representations’’ of SU共3兲, one based on the Hilbert space of functions on a
unit sphere in C 3 and the other based on the Fock space of six oscillators, and a natural question
to ask is how the two are related. To this end, in Sec. IV, we make use of the Bargmann representation, to transcribe the Fock space description into a description based on a Hilbert space of
square integrable functions in six complex variables satisfying certain conditions. This transcription enables us to establish an equivalence map between the Hilbert spaces supporting the two
incarnations of the ‘‘generating representation’’ for SU共3兲, details of which are given in Secs. V
and VI. Section VII contains concluding remarks and further outlook, and an appendix gives the
details of the construction of SU(3)⫻Sp(2,R) basis states.
II. UNITARY REPRESENTATIONS OF COMPACT LIE GROUPS, THE SU„3… CASE
It is useful to first recall some basic facts concerning the representation theory of any compact
simple Lie group G. The basic building blocks are the UIRs of G. Each UIR carries certain
identifying labels 共eigenvalues of Casimir operators兲, such as j for SU共2兲. It is of a characteristic
dimension, such as 2 j⫹1 for SU共2兲. In addition, we may set up some convenient orthonormal
basis in the space of the UIR, as simultaneous eigenvectors of some complete commuting set of
Hermitian operators. The eigenvalue sets labeling the basis vectors are generalizations of the
single magnetic quantum number m for SU共2兲.
A general UR of G is reducible into UIRs, each occurring with some multiplicity. Thus the
UR as a whole is in principle completely determined upto equivalence by these multiplicities.
However, certain URs have special significance, reflecting the way they are constructed, and so
deserve special attention. We consider two cases—the regular representation, and representations
induced from various Lie subgroups of G.
The Hilbert space carrying the regular representation of G is the space L 2 (G) of all complex
square integrable functions on G, the integration being with respect to the 共left and right兲 translation invariant volume element on G. On this space there are in fact two 共mutually commuting兲
regular representations of G, the left and the right regular representations. Upon reduction into
UIRs each of these contains every UIR of G without exception, the multiplicity of occurrence of
a particular UIR is just its dimension. Thus the regular representations possess the completeness
property of the Schwinger SU共2兲 construction, but not its economy.
Next we look at the family of induced URs of G. 10 Let H be some Lie subgroup of G, and let
D(h), h苸H, be the operators of a UIR of H on some Hilbert space V. Then a certain unique UR
(ind,D)
of G, with operators D H
(g) for g苸G, can be constructed. As the labels indicate, this UR is
(ind,D)
of this UR consists of functions on
induced from the UIR D(•) of H. The Hilbert space H H
G with values in V obeying a covariance condition and having finite norm:
␺ 苸H H(ind,D) : ␺ 共 g 兲 苸V,g苸G,
␺ 共 gh 兲 ⫽D 共 h ⫺1 兲 ␺ 共 g 兲 ,h苸H,
储 ␺ 储 2⫽
冕
G
共2.1兲
dg 共 ␺ 共 g 兲 , ␺ 共 g 兲兲 V⬍⬁.
Here dg is the 共suitably normalized兲 invariant volume element on G, and the integrand is the
squared norm of ␺ (g)苸V. The covariance condition means that ␺ (g) is essentially a function on
the coset space G/H, in the sense that the ‘‘values’’ of ␺ (g) all over a coset are determined by its
‘‘value’’ at any one representative point. Correspondingly, due to unitarity of D(h), ( ␺ (g), ␺ (g)) V
is constant over each coset; so, the expression for 储 ␺ 储 2 can be simplified and expressed in terms
(ind,D)
(g) on ␺ is then given by
of a G-invariant volume element on G/H. The action of D H
(ind,D)
g苸G:D H
共 g 兲␺ ⫽ ␺ ⬘,
␺ ⬘ 共 g ⬘ 兲 ⫽ ␺ 共 g ⫺1 g ⬘ 兲 .
共2.2兲
(ind,D)
.
It is clear that G action preserves the covariance condition, and we have a UR of G on H H
(ind,D)
(•) is in general reducible; so it is a
Whereas D(•) was assumed to be a UIR of H, D H
direct sum of the various UIRs of G, each occurring with some multiplicity. These multiplicities
(ind,D)
(•) as often
are determined by the reciprocity theorem:10 Each UIR D(•) of G appears in D H
as D(•) contains D(•) upon restriction from G to H.
With this general background we now take up the specific case of SU共3兲. The defining
representation of this group is
SU共 3 兲 ⫽ 兵 A⫽3⫻3 complex matrix兩 A † A⫽I 3⫻3 ,det A⫽1 其 ,
共2.3兲
with the group operation given by matrix multiplication. In this representation the eight Hermitian
generators are 21 ␭ ␣ , ␣ ⫽1,2,...,8, where the matrices ␭ ␣ and the structure constants f ␣␤␥ occurring in the commutation relations
关 ␭ ␣ ,␭ ␤ 兴 ⫽2i f ␣␤␥ ␭ ␥ ,
␣ , ␤ , ␥ ⫽1,2,...,8,
共2.4兲
are all very well known.11
A general UIR of SU共3兲 is determined by two independent nonnegative integers p and q, so
it may be denoted as ( p,q). It is of dimension d(p,q)⫽ 21 (p⫹1)(q⫹1)(p⫹q⫹2). The defining
three-dimensional UIR in 共2.3兲 is 共1,0兲, while the inequivalent complex conjugate UIR is 共0,1兲. In
general the complex conjugate of (p,q) is (q,p), and the adjoint UIR is 共1,1兲 of dimension eight.
Various choices of ‘‘magnetic quantum numbers’’ within a UIR may be made. The one corresponding to the canonical subgroup SU(2)⫻U(1)/Z 2 ⫽U(2)傺SU(3) leads to the three quantum
numbers I, M , Y in standard notation. Here I and M are the isospin and magnetic quantum
number labels for a general UIR of SU共2兲, while Y is the eigenvalue of the 共suitably normalized兲
U共1兲 or hypercharge generator. The subgroups SU共2兲 and U共1兲 commute, and for definiteness we
take SU共2兲 to be the one acting on the first two dimensions of the three dimensions in the UIR
共1,0兲. The spectrum of ‘‘I⫺Y ’’ multiplets present in the UIR (p,q) can be described thus:
I⫽ 12 共 r⫹s 兲 , Y ⫽r⫺s⫹ 32 共 q⫺ p 兲 , 0⭐r⭐p, 0⭐s⭐q.
共2.5兲
Thus for each pair of integers (r,s) in the above ranges, we have one I⫺Y multiplet, with M
going over the usual 2I⫹1 values I,I⫺1,...,⫺I⫹1,⫺I. Then the orthonormal basis vectors for
the UIR (p,q) of SU共3兲 may be written as 兩 p,q;IM Y 典 . This UIR can be realized via suitably
constructed irreducible tensors. A tensor T with p indices belonging to the UIR 共1,0兲 and q indices
j ¯j
to the UIR 共0,1兲 is a collection of complex components T k1 ¯kp , j and k⫽1,2,3, transforming
1
q
under A苸SU(3) by the rule
j ¯j
j
j
k
k
l ¯l p
.
1 ¯m q
T ⬘ k1 ¯kp ⫽A l 1 ¯A l p A m1 * ¯A mq * T m1
1
q
1
p
1
q
共2.6兲
If in addition T is completely symmetric separately in the superscripts and in the subscripts, and
is traceless, i.e., contraction of any upper index with any lower index leads to zero, then all these
properties are maintained under SU共3兲 action and T is an irreducible tensor. It then has precisely
d(p,q) independent components 共in the complex sense兲, and the space of all such tensors carries
j ¯j
the UIR ( p,q). The explicit transition from the tensor components T k1 ¯kp to the canonical com1
q
ponents T (p,q)
IM Y may be found in Ref. 12
The regular representations of SU共3兲 act on the space L 2 (SU(3)), and in each of them the
UIR ( p,q) appears d(p,q) times. We shall not be concerned with this UR of SU共3兲 in our work.
Instead we give now the UIR contents of some selected induced URs of SU共3兲. For illustrative
purposes we consider the following four subgroups:
U共 1 兲 ⫻U共 1 兲 ⫽ 兵 A⫽diag共 e i( ␪ 1 ⫹ ␪ 2 ) ,e i( ␪ 1 ⫺ ␪ 2 ) ,e ⫺2i ␪ 1 兲 兩 0⭐ ␪ 1 , ␪ 2 ⭐2 ␲ 其 ;
再 冉 冊冏 冎
冊冏 冎
再 冉
SU共 2 兲 ⫽ A⫽
U共 2 兲 ⫽ A⫽
a
0
0
1
a苸SU共 2 兲 ;
u
0
0
共 det u 兲 ⫺1
u苸U共 2 兲 ;
SO共 3 兲 ⫽ 兵 A苸SU共 3 兲 兩 A * ⫽A 其 .
共2.7a兲
共2.7b兲
共2.7c兲
共2.7d兲
In each case, we look at the induced UR of SU共3兲 arising from the trivial one-dimensional UIR of
the subgroup. In the first two cases, in order to apply the reciprocity theorem, we can use the
information in 共2.5兲 giving the SU(2)⫻U(1)/Z 2 content of the UIR (p,q) of SU共3兲. Defining by
a zero in the superscript the trivial UIR of the relevant subgroup, we have the results:
⬁
(ind,0)
⫽
DU(1)⫻U(1)
兺
p,q⫽0,1,¯
p⫽qmod3
丣 n p,q 共 p,q 兲 ,
n p,q ⫽min共 p⫹1,q⫹1 兲 ;
共2.8a兲
⬁
(ind,0)
⫽
DSU(2)
兺
p,q⫽0,1,¯
丣 共 p,q 兲 ;
共2.8b兲
⬁
(ind,0)
⫽
DU(2)
兺
p⫽0,1,¯
丣 共 p,p 兲 .
共2.8c兲
The real dimensions of the corresponding coset spaces SU(3)/U(1)⫻U(1), SU(3)/SU(2)
and SU(3)/U(2) are 6, 5 and 4, respectively. In the case of induction from the trivial UIR of
SO共3兲, we need to use the fact that the UIR ( p,q) of SU共3兲 does not contain an SO共3兲 invariant
state if either p or q or both are odd, while it contains one such state if both p and q are even.
Then we arrive at the reduction
⬁
(ind,0)
DSO(3)
⫽
兺
r,s⫽0,1,¯
with SU(3)/SO(3) being of real dimension 5.
丣 共 2r,2s 兲 ,
共2.9兲
(ind,0)
From the above discussion we see that the induced UR DSU(2)
of SU共3兲 is particularly
interesting in that it captures both the completeness and the economy properties of the Schwinger
SU共2兲 construction: each UIR of SU共3兲 is present, exactly once. Thus we may call this a generating representation of SU共3兲; it is much leaner than the regular representations.
III. THE MINIMAL SU„3… SCHWINGER OSCILLATOR CONSTRUCTION
An elementary oscillator operator construction of the SU共3兲 generators is based on three
independent pairs of annihilation and creation operators â j , â †j obeying
关 â j ,â †k 兴 ⫽ ␦ jk ,
关 â j ,â k 兴 ⫽ 关 â †j ,â †k 兴 ⫽0,
j,k⫽1,2,3.
共3.1兲
We write H (a) for the Hilbert space on which these operators act irreducibly. The individual and
total number operators are
†
N̂ (a)
1 ⫽â 1 â 1 ,
†
N̂ (a)
2 ⫽â 2 â 2 ,
†
N̂ (a)
3 ⫽â 3 â 3 ,
N̂ (a) ⫽â †j â j .
共3.2兲
If we now define the bilinear operators
Q ␣(a) ⫽ 21 â † ␭ ␣ â,
␣ ⫽1,2,...,8,
共3.3兲
each Q ␣(a) is Hermitian, and they obey the SU共3兲 Lie algebra commutation relations
关 Q ␣(a) ,Q ␤(a) 兴 ⫽i f ␣␤␥ Q ␥(a) .
共3.4兲
In addition they conserve the total number operator:
关 Q ␣(a) ,N̂ (a) 兴 ⫽0.
共3.5兲
Upon exponentiation of these generators we obtain a particular UR, U (a) (A) say, of SU共3兲 acting
on H (a) , under which the creation 共annihilation兲 operators â †j (â j ) transform via the UIR 共1,0兲
共共0,1兲兲:
U (a) 共 A 兲 â †j U (a) 共 A 兲 ⫺1 ⫽A kj â †k ,
U
(a)
共 A 兲 â j U
(a)
共A兲
⫺1
⫽A kj * â k .
共3.6兲
However, upon reduction, U (a) (A) contains only the ‘‘triangular’’ UIRs ( p,0) of SU共3兲, once each.
In that sense this UR may be regarded as the ‘‘generating representation’’ for this subset of UIRs.
For any given p⭓0, the UIR (p,0) is realized on that subspace H (p,0) of H (a) over which the total
number operator N̂ (a) takes the eigenvalue p; and the connection between the tensor and the Fock
space descriptions is given in this manner:
兵 T j 1 ¯ j p 其 → 兩 T 典 ⫽T j 1 ¯ j p â †j 1 ¯â †j p 兩 0គ 典 苸H (p,0) 傺H (a) ,
â j 兩 0គ 典 ⫽0;
U (a) 共 A 兲 兩 T 典 ⫽ 兩 T ⬘ 典 ,
T ⬘ j 1 ¯ j p ⫽A l 1 ¯A l p T l 1 ¯l p .
j
j
1
p
Therefore we have the 共orthogonal兲 direct sum decompositions
共3.7兲
⬁
兺
H (a) ⫽
p⫽0,1,¯
丣 H (p,0) ,
H (p,0) ⫽Sp兵 â †j ¯â †j 兩 0គ 典 其 ,
1
共3.8兲
p
⬁
U
(a)
⫽
兺
p⫽0,1,¯
丣 共 p,0兲 .
To be able to obtain the other UIRs as well, we bring in another independent triplet of
oscillator operators b̂ j and b̂ †j obeying the same commutation relations 共3.1兲 and commuting with
â’s and â † ’s:
关 b̂ j ,b̂ †k 兴 ⫽ ␦ jk ,
关 b̂ j ,b̂ k 兴 ⫽ 关 b̂ †j ,b̂ †k 兴 ⫽0,
j,k⫽1,2,3,
共3.9兲
关 â j or â †j , b̂ k or b̂ †k 兴 ⫽0.
The corresponding Hilbert space is H (b) , and the b-type number operators are
†
N̂ (b)
1 ⫽b̂ 1 b̂ 1 ,
†
N̂ (b)
2 ⫽b̂ 2 b̂ 2 ,
†
N̂ (b)
3 ⫽b̂ 3 b̂ 3 ,
N̂ (b) ⫽b̂ †j b̂ j .
共3.10兲
We define the b-type SU共3兲 generators as
Q ␣(b) ⫽⫺ 21 b̂ † ␭ ␣* b̂,
␣ ⫽1,2,...,8,
共3.11兲
and they obey
关 Q ␣(b) ,Q ␤(b) 兴 ⫽i f ␣␤␥ Q ␥(b) ,
共3.12兲
关 Q ␣(b) ,N̂ (b) 兴 ⫽0.
Exponentiation of these generators leads to a UR U (b) (A) acting on H (b) , under which the
creation 共annihilation兲 operators b̂ †j (b̂ j ) transform via the UIR 共0,1兲 共共1,0兲兲:
U (b) 共 A 兲 b̂ †j U (b) 共 A 兲 ⫺1 ⫽A kj * b̂ †k ,
U (b) 共 A 兲 b̂ j U (b) 共 A 兲 ⫺1 ⫽A kj b̂ k .
共3.13兲
Now this UR of SU共3兲 contains each of the triangular UIRs (0,q) for q⭓0 once each, so it is a
generating representation for this family of UIRs. For each q⭓0, the UIR (0,q) is realized on that
subspace H (0,q) of H (b) over which the total number operator N̂ (b) takes the eigenvalue q.
Analogous to 共3.7兲, the tensor-Fock space connection is now
兵 T k 1 ¯k q 其 → 兩 T 典 ⫽T k 1 ¯k q b̂ k†1 ¯b̂ k†q 兩 0គ 典 苸H (0,q) 傺H (b) ,
b̂ k 兩 0គ 典 ⫽0;
U (b) 共 A 兲 兩 T 典 ⫽ 兩 T ⬘ 典 ,
k
k
T ⬘ k 1 ¯k q ⫽A m1 * ¯A mq * T m 1 ¯m q .
1
q
共3.14兲
关The use of a common symbol 兩 0គ 典 for the Fock ground states in H (a) and H (b) , and 兩 T 典 in 共3.7兲
and 共3.14兲, should cause no confusion as the meanings are always clear from the context.兴 In place
of 共3.8兲 we now have
⬁
H (b) ⫽
兺
q⫽0,1,¯
丣 H (0,q) ,
H (0,q) ⫽Sp兵 b̂ k† ¯b̂ k† 兩 0គ 典 其 ,
1
共3.15兲
q
⬁
U
(b)
⫽
兺
q⫽0,1,¯
丣 共 0,q 兲 .
From these considerations it is clear that if we want to obtain all the UIRs ( p,q) of SU共3兲,
missing none, the minimal scheme is to use all six independent oscillators â j , â †j , b̂ j , b̂ †j and
define the SU共3兲 generators13
Q ␣ ⫽Q ␣(a) ⫹Q ␣(b) .
共3.16兲
They act on the product Hilbert space H⫽H (a) ⫻H (b) , of course obey the SU共3兲 commutation
relations, and upon exponentiation lead to the UR U(A)⫽U (a) (A)⫻U (b) (A). However, as we see
in a moment, while each UIR (p,q) is certainly present in U(A), it occurs infinitely many times.
A systematic group theoretic procedure to handle this multiplicity, based on the noncompact group
Sp(2,R), will be set up below. The tensor-Fock space connection is now given as follows. To an
j ¯j
irreducible tensor T k1 ¯kp which is symmetric and traceless and so ‘‘belongs’’ to the UIR (p,q) we
1
q
associate the vector 兩 T 典 苸H by
j ¯j
兩 T 典 ⫽T k1 ¯kp â †j ¯â †j b̂ k† ¯b̂ k† 兩 0គ ,0គ 典 苸H (p,0) ⫻H (0,q) 傺H,
1
1
q
p
1
q
â j 兩 0គ ,0គ 典 ⫽b̂ j 兩 0គ ,0គ 典 ⫽0,
共3.17兲
U共 A 兲 兩 T 典 ⫽ 兩 T ⬘ 典 ,
the components of T ⬘ being given by 共2.6兲. While this vector 兩 T 典 is certainly a simultaneous
eigenvector of the two number operators N̂ (a) , N̂ (b) with eigenvalues p, q, respectively, the
j ¯j
tracelessness of the tensor T k1 ¯kp implies that 共unless at least one of p and q vanishes兲 we do not
1
q
get all such independent vectors in H. This aspect is further clarified below. On the other hand, if
we drop the tracelessness condition and retain only symmetry, we do span all of H (p,0) ⫻H (0,q) via
共3.17兲.
The decomposition of U(A) into UIRs, and the counting of multiplicities, is accomplished by
appealing to the Clebsch–Gordan series for the product of two triangular UIRs (p,0) and (0,q): 14
共 p,0兲 ⫻ 共 0,q 兲 ⫽ 共 p,q 兲 丣 共 p⫺1,q⫺1 兲 丣 共 p⫺2,q⫺2 兲 丣 ... 丣 共 p⫺r,q⫺r 兲 , r⫽min共 p,q 兲 .
共3.18兲
Therefore, at the Hilbert space level one has the orthogonal subspace decomposition
H⫽H (a) ⫻H (b) ⫽
冉
⬁
兺
p⫽0,1,¯
丣 H (p,0)
冊冉
⫻
⬁
兺
q⫽0,1,¯
丣 H (0,q)
冊
⬁
⫽
兺
p,q⫽0,1,¯
丣 H (p,0) ⫻H (0,q) ,
r
H (p,0) ⫻H (0,q) ⫽
兺
␳ ⫽0,1,¯
丣 H (p⫺ ␳ ,q⫺ ␳ ; ␳ ) ,
r⫽min共 p,q 兲 .
共3.19兲
Here H (p⫺ ␳ ,q⫺ ␳ ; ␳ ) is that unique subspace of H (p,0) ⫻H (0,q) carrying the UIR (p⫺ ␳ ,q⫺ ␳ )
present on the right hand side of 共3.18兲. All vectors in H (p⫺ ␳ ,q⫺ ␳ ; ␳ ) are eigenvectors of N̂ (a) and
N̂ (b) with eigenvalues p and q, respectively; and if the tensor T in 共3.17兲 is assumed traceless,
only vectors in H (p,q;0) 傺H (p,0) ⫻H (0,q) are obtained on the right in that equation.
Focusing on a given UIR ( p,q), we see that it appears once each in H (p,0)
⫻H (0,q) ,H (p⫹1,0) ⫻H (0,q⫹1) ,..., in the respective irreducible subspaces H (p,q;0) ,H (p,q;1) ,... .
Thus it is the leading piece in H (p,0) ⫻H (0,q) , the next to the leading piece in H (p⫹1,0)
⫻H (0,q⫹1) , and so on. Therefore, the decomposition 共3.19兲 of H can be presented in the alternative manner
⬁
⬁
H⫽
兺
丣
p,q⫽0,1.¯
兺
␳ ⫽0,1.¯
丣 H (p,q; ␳ ) ,H (p,q; ␳ ) 傺H (p⫹ ␳ ,0) ⫻H (0,q⫹ ␳ ) ,
共3.20兲
each H (p,q; ␳ ) carrying the same UIR ( p,q). Thus the index ␳ is an 共orthogonal兲 multiplicity label
with an infinite number of values. For ␳ ⫽ ␳ ⬘ , H (p,q; ␳ ⬘ ) and H (p,q; ␳ ) are mutually orthogonal. This
is also evident as N̂ (a) ⫽p⫹ ␳ ⬘ , N̂ (b) ⫽q⫹ ␳ ⬘ in the former and N̂ (a) ⫽ p⫹ ␳ , N̂ (b) ⫽q⫹ ␳ in the
latter.
We now introduce the group Sp(2,R) to handle in a systematic way the multiplicity index ␳.
The Hermitian generators of Sp(2,R) and their commutation relations are15
J 0 ⫽ 12 共 N̂ (a) ⫹N̂ (b) ⫹3 兲 ,
K 1 ⫽ 21 共 â †j b̂ †j ⫹â j b̂ j 兲 ,
i
K 2 ⫽⫺ 共 â †j b̂ †j ⫺â j b̂ j 兲 ;
2
关 J 0 ,K 1 兴 ⫽iK 2 ,
关 J 0 ,K 2 兴 ⫽⫺iK 1 ,
共3.21兲
关 K 1 ,K 2 兴 ⫽⫺iJ 0 .
Using the raising and lowering combinations K ⫾ ⫽K 1 ⫾iK 2 we have
K ⫹ ⫽â †j b̂ †j ,
关 J 0 ,K ⫾ 兴 ⫽⫾K ⫾ ,
†
K ⫺ ⫽K ⫹
⫽â j b̂ j ;
关 K ⫹ ,K ⫺ 兴 ⫽⫺2J 0 .
共3.22兲
The significance of this construction is that the two groups SU共3兲 and Sp(2,R), both acting
unitarily on H, commute with one another:
关 J 0 or K 1 or K 2 ,Q ␣ 兴 ⫽0.
共3.23兲
It is this that helps us handle the multiplicity of occurrences of each SU共3兲 UIR ( p,q) in H:␳
becoming a ‘‘magnetic quantum number’’ within a suitable UIR of Sp(2,R).
The family of 共infinite dimensional兲 UIRs of Sp(2,R) relevant here is the positive discrete
family D (⫹)
, labeled by k⫽ 21,1, 32 ,2,2... . 共Actually we encounter only k⭓ 23.) Within the UIR D (⫹)
k
k
we have an orthonormal basis 兩 k,m 典 on which the generators act as follows:16
J 0 兩 k,m 典 ⫽m 兩 k,m 典 ,
m⫽k,k⫹1,k⫹2,...,
K ⫾ 兩 k,m 典 ⫽ 冑共 m⫾k 兲共 m⫿k⫾1 兲 兩 k,m⫾1 典 .
共3.24兲
From these follow the useful results
K 21 ⫹K 22 ⫺J 20 ⫽k 共 1⫺k 兲 ,
共3.25a兲
兩 k,m 典 ⫽
冑
共 2k⫺1 兲 !
K m⫺k 兩 k,k 典 ,
共 m⫺k 兲 ! 共 m⫹k⫺1 兲 ! ⫹
m⫺k m⫺k
K⫹
K ⫺ 兩 k,m 典 ⫽
共3.25b兲
共 m⫺k 兲 ! 共 m⫹k⫺1 兲 !
兩 k,m 典 .
共 2k⫺1 兲 !
共3.25c兲
Going back to the generators 共3.21兲 it is clear that on all of H (p,0) ⫻H (0,q) , and so on each
H (p⫺ ␳ ,q⫺ ␳ ; ␳ ) , J 0 has the eigenvalue 21 (p⫹q⫹3); therefore on H (p,q; ␳ ) it has the eigenvalue
1
(p,0)
⫻H (0,q) leads to a subspace of
2 (p⫹q⫹3)⫹ ␳ . It is also clear that action by K ⫾ on H
(p⫾1,0)
(0,q⫾1)
⫻H
. Therefore, because of 共3.23兲, we see that K ⫾ acting on H (p,q; ␳ ) yield
H
(p,q; ␳ ⫾1)
H
. Of course H (p,q;0) is annihilated by K ⫺ .
Reflecting all this we see that an orthonormal basis for H can be set up labeled as follows:
兩 p,q;IM Y ;m 典 :p,q⫽0,1,2,...;
m⫽k,k⫹1,k⫹2,...,
共3.26兲
k⫽ 共 p⫹q⫹3 兲 ;
1
2
N (a) ⫽ p⫹m⫺k, N (b) ⫽q⫹m⫺k.
Since k is determined in terms of p and q, we do not include it as an additional label in the basis
kets above. 关The ranges for I, M , Y within the SU共3兲 UIR (p,q) are given in 共2.5兲.兴 The SU共3兲
of Sp(2,R). For fixed p, q as I, M ,
UIR labels p, q determine k and so the associated UIR D (⫹)
k
of SU(3)⫻Sp(2,R). We can now
Y , m vary we get a set of states carrying the UIR ( p,q)⫻D (⫹)
k
appreciate the following relationships:
H (p,q; ␳ ) ⫽Sp兵 兩 p,q;IM Y ;k⫹ ␳ 典 兩 IM Y varying其 ,
␳ ⫽0,1,2,...;
共3.27a兲
␳
H (p,q; ␳ ) ⫽K ⫹
H (p,q;0) ;
共3.27b兲
K ⫺ H (p,q;0) ⫽0.
共3.27c兲
Therefore, the null space of K ⫺ within H is the subspace
⬁
H0 ⫽
兺
p,q⫽0,1,...
丣 H (p,q;0) ⫽Sp兵 兩 p,q;IM Y ;k 典 兩 p,q,IM Y
varying其 ,
共3.28兲
and we see that the UR U(A) of SU共3兲 on H when restricted to H0 gives a UR D0 which is
multiplicity free and includes every UIR of SU共3兲. It is thus identical in structure to the induced
(ind,0)
representation DSU(2)
in 共2.8b兲. We see how the use of Sp(2,R) helps us isolate H0 in a neat
manner.
In addition to the subspaces H (p,q; ␳ ) , H0 of H defined above, it is also useful to define the
series of mutually orthogonal infinite dimensional subspaces
⬁
H
(p,q)
⫽
兺
␳ ⫽0
丣 H (p,q; ␳ ) ⫽Sp兵 兩 p,q;IM Y ;m 典 兩 IM Y m
varying其 ,
p,q⫽0,1,2,... .
Thus the infinity of occurrences of the SU共3兲 UIR (p,q) are collected together in H (p,q) .
共3.29兲
In the Appendix we give explicit formulas for the state vectors 兩 p,q;IM Y ;m 典 as functions of
the operators â †j , b̂ †j acting on the Fock vacuum 兩 0គ ,0គ 典 .
IV. THE BARGMANN REPRESENTATION
For some purposes the use of the Bargmann representation of the canonical commutation
relations is more convenient than the Fock space description.17 We outline the definitions of H and
the SU共3兲 UR U(A)⫽U (a) (A)⫻U (b) (A) in this language, and then turn to the problem of isolating
the subspace H0 in H.
Vectors in H correspond to entire functions f (zគ ,w
គ ) in six independent complex variables zគ
គ ⫽(w j ), j⫽1,2,3, with the squared norm defined as
⫽(z j ), w
冕 兿 冉 ␲ 冊冉 ␲ 冊
3
储f储 ⫽
2
d 2z j
d 2w j
e ⫺z
† z⫺w † w
j⫽1
兩 f 共 zគ ,w
គ 兲兩 2.
共4.1兲
Any such f (zគ ,w
គ ) has a unique Taylor series expansion
⬁
គ 兲⫽
f 共 zគ ,w
兺 f kj ¯¯kj z j ¯z j w k ¯w k ,
p,q⫽0,1,¯
1
p
1
q
1
p
1
q
共4.2兲
j ¯j
involving the tensor components f k1 ¯kp separately symmetric in the superscripts and the sub1
q
scripts. In terms of these the squared norm is
⬁
储 f 储 2⫽
兺
p,q⫽0,1,¯
j ¯j
*
j ¯j
p!q! f k1 ¯kp f k1 ¯kp .
1
q
1
q
共4.3兲
គ ) as follows:
The operators â j , â †j , b̂ j , b̂ †j act on f (zគ ,w
â j →
⳵
⳵
, â †j →z j , b̂ j →
, b̂ †j →w j .
⳵z j
⳵w j
共4.4兲
The UR U(A) of SU共3兲 acts very simply via point transformations:
គ 兲 ⫽ f 共 A ⫺1 zគ ,A ⫺1 * wគ 兲 .
共 U共 A 兲 f 兲共 zគ ,w
共4.5兲
The Sp(2,R) generators are particularly simple:
J 0⫽
冉
冊
⳵
⳵
1
zj
⫹w j
⫹3 ,
2
⳵z j
⳵w j
គ,
K ⫹ ⫽z j w j ⬅zគ •w
K ⫺⫽
共4.6兲
⳵2
⳵ ⳵
⬅ •
.
⳵ z j ⳵ w j ⳵ zគ ⳵ wគ
We will use these below.
It is clear that the terms in 共4.2兲 and 共4.3兲 for fixed p and q are contributions from H (p,0)
⫻H (0,q) . The action by K ⫹ obeys
គ 兲 苸H (p,0) ⫻H (0,q) →K ⫹ f 共 zគ ,wគ 兲 ⫽zគ •w
គ f 共 zគ ,wគ 兲 苸H (p⫹1,0) ⫻H (0,q⫹1) .
f 共 zគ ,w
共4.7兲
On the other hand, action by K ⫺ is the analytic equivalent of taking the trace: starting with 共4.2兲
we get
⬁
K ⫺ f 共 zគ ,w
គ 兲⫽
兺
jj ¯j
p,q⫽0,1,¯
pq f jk1 ¯kp⫺1 z j 1 ¯z j p⫺1 w k 1 ¯w k q⫺1 .
1
q⫺1
共4.8兲
From these and earlier remarks we can see that the correspondences between 共symmetric, traceless兲 tensors, entire functions, and subspaces of H are
j ¯j
H (p,0) ⫻H (0,q) ↔ 兵 f k1 ¯kp 其 ↔ f 共 zគ ,wគ 兲 :
1
q
f 共 ␭zគ , ␮ wគ 兲 ⫽␭ p ␮ q f 共 zគ ,wគ 兲 ;
f 共 zគ ,w
គ 兲 苸H (p,q; ␳ ) ⇔ f 共 zគ ,w
គ 兲 ⫽ 共 zគ •w
គ 兲 ␳ f 0 共 zគ ,wគ 兲 , f 0 共 zគ ,wគ 兲 苸H (p,q;0) 傺H (p,0) ⫻H (0,q) ,
⳵ ⳵
•
f 共 zគ ,wគ 兲 ⫽0.
⳵ zគ ⳵ wគ 0
共4.9a兲
共4.9b兲
Thus traceless symmetric tensors of type (p,q) are in correspondence with entire functions
គ ) of degrees of homogeneity p and q, respectively, obeying the partial differential equation
f 0 (zគ ,w
共4.9b兲. Alternatively, given any f (zគ ,wគ )苸H (p,0) ⫻H (0,q) , there is a unique ‘‘traceless’’ part
f 0 (zគ ,w
គ ) belonging to the leading subspace H (p,q;0) and annihilated by K ⫺ . Thus ‘‘trace removal’’
can be accomplished by analytical means. We now give the procedure to pass from f (zគ ,wគ ) to
គ ).
f 0 (zគ ,w
For any f (zគ ,w
គ )苸H (p,0) ⫻H (0,q) we can easily establish the general formula
n
n
n⫹1
K ⫺ 兵 共 zគ •w
គ 兲nK ⫺
f 共 zគ ,w
គ 兲 其 ⫽n 共 p⫹q⫹2⫺n 兲共 zគ •w
គ 兲 n⫺1 K ⫺
f 共 zគ ,wគ 兲 ⫹ 共 zគ •w
គ 兲nK ⫺
f 共 zគ ,wគ 兲 .
共4.10兲
គ ) the expression
We try for f 0 (zគ ,w
f 0 共 zគ ,w
គ 兲 ⫽ f 共 zគ ,w
គ 兲⫺
兺
n⫽1,2,¯
n
␣ n 共 zគ •wគ 兲 n K ⫺
f 共 zគ ,wគ 兲 ,
共4.11兲
and get, using 共4.10兲 共and omitting the arguments zគ ,wគ ),
K ⫺ f 0 ⫽K ⫺ f ⫺ 共 p⫹q⫹1 兲 ␣ 1 K ⫺ f ⫺
兺
n⫽1,2,¯
n⫹1
f.
兵 ␣ n ⫹ 共 n⫹1 兲共 p⫹q⫹1⫺n 兲 ␣ n⫹1 其 共 zគ •wគ 兲 n K ⫺
共4.12兲
We can therefore attain K ⫺ f 0 ⫽0 by choosing
␣ n ⫽ 共 ⫺1 兲 n⫺1
共 p⫹q⫹1⫺n 兲 !
,
n! 共 p⫹q⫹1 兲 !
n⫽1,2,... .
共4.13兲
Therefore, for any 共bihomogeneous兲 polynomial f (zគ ,wគ )苸H (p,0) ⫻H (0,q) the leading traceless part
annihilated by K ⫺ is an element f 0 (zគ ,wគ ) in H (p,q;0) :
គ 兲 ⫽ f 共 zគ ,w
គ 兲⫺
f 0 共 zគ ,w
共 p⫹q⫹1⫺n 兲 !
兺 共 ⫺1 兲 n⫺1 n! 共 p⫹q⫹1 兲 ! 共 zគ •wគ 兲 n K ⫺n f 共 zគ ,wគ 兲 .
n⫽1,2,¯
共4.14兲
This result can be extended and expressed in the Fock space language. Any 兩 ␺ 典 苸H (p,0)
⫻H (0,q) has a unique orthogonal decomposition into various parts belonging to various UIRs of
SU共3兲; using 共3.25c兲 this reads
兩 ␺ 典 苸H (p,0) ⫻H (0,q) ⫽H (p,q;0)
丣 H (p⫺1,q⫺1;1) 丣 H (p⫺2,q⫺2;2) 丣 ¯ :
兩 ␺ 典 ⫽ 兩 ␺ 0 典 ⫹ 兩 ␺ 1 典 ⫹ 兩 ␺ 2 典 ⫹¯ ,
兩 ␺ 0 典 苸H (p,q;0) , K ⫺ 兩 ␺ 0 典 ⫽0;
兩 ␺ 1 典 ⫽K ⫹ 兩 ␾ 1 典 苸H (p⫺1,q⫺1;1) ,
兩 ␾ 1典 ⫽
共 p⫹q 兲 !
K 兩 ␺ 典 苸H (p⫺1,q⫺1;0) ,
1! 共 p⫹q⫹1 兲 ! ⫺ 1
共4.15兲
2
兩 ␺ 1 典 ⫽0;
K⫺
2
兩 ␺ 2 典 ⫽K ⫹
兩 ␾ 2 典 苸H (p⫺2,q⫺2;2) ,
兩 ␾ 2典 ⫽
共 p⫹q⫺2 兲 ! 2
K 兩 ␺ 典 苸H (p⫺2,q⫺2;0) ,
2! 共 p⫹q 兲 ! ⫺ 2
3
兩 ␺ 2 典 ⫽0;¯ .
K⫺
The ‘‘leading’’ piece in 兩 ␺ 典 is thus
兩 ␺ 0 典 ⫽ 兩 ␺ 典 ⫺ 兩 ␺ 1 典 ⫺ 兩 ␺ 2 典 ⫺¯⫽ 兩 ␺ 典 ⫺aគ̂ † •bគ̂ † 兩 ␾ 典 ,
共4.16兲
兩 ␾ 典 ⫽ 兩 ␾ 1 典 ⫹aគ̂ † •bគ̂ † 兩 ␾ 2 典 ⫹¯苸H (p⫺1,0) ⫻H (0,q⫺1) .
We can now infer that if to begin with we had 兩 ␺ 典 ⫽aគ̂ † •bគ̂ † 兩 ␾ 典 for some ␾ 苸H (p⫺1,0)
⫻H (0,q⫺1) , then 兩 ␺ 0 典 necessarily vanishes:
兩 ␺ 典 ⫽aគ̂ † •bគ̂ † 兩 ␾ 典 ⇔ 兩 ␺ 0 典 ⫽0.
共4.17兲
In the Bargmann description this means in terms of 共4.14兲
គ 兲 ⫽zគ •w
គ g 共 zគ ,wគ 兲 ⇔ f 0 共 zគ ,wគ 兲 ⫽0,
f 共 zគ ,w
共4.18兲
a result which can be directly verified with some effort.
The subspace H0 傺H identified in 共3.28兲 is describable in the Bargmann language as follows:
再
冏
H0 ⫽ f 共 zគ ,wគ 兲 苸H
冎
⳵ ⳵
•
f 共 zគ ,wគ 兲 ⫽0 .
⳵ zគ ⳵ wគ
共4.19兲
j ¯j
In the Taylor series expansion 共4.2兲 for such f (zគ ,w
គ ), the tensors f k1 ¯kp are traceless and vice
1
q
versa. The squared norm and SU共3兲 action are given for H0 by 共4.3兲 and 共4.5兲, respectively.
„IND,0…
V. THE UR DSU„2…
OF SU„3…
(ind,0)
(ind,0)
The Hilbert space HSU(2)
carrying the UR DSU(2)
of SU共3兲 consists of single component
共scalar兲 complex functions on the coset space SU共3兲/SU共2兲. This coset space is the unit sphere in
three-dimensional complex space C 3 , with the natural norm and SU共3兲 action. Temporarily omitting the superscript zero and subscript SU共2兲 for simplicity, we have
再
冏 冕兿 冉 冊
3
H (ind) ⫽ ␺ 共 ␰គ 兲 苸C, ␰គ 苸C 3 储 ␺ 储 2 ⫽
j⫽1
冎
d 2␰ j
␦ 共 ␰ † ␰ ⫺1 兲 兩 ␺ 共 ␰គ 兲 兩 2 ,
␲
共 D (ind) 共 A 兲 ␺ 兲共 ␰គ 兲 ⫽ ␺ 共 A ⫺1 ␰គ 兲 .
共5.1兲
Clearly only the values of ␺ ( ␰គ ) for ␰ † ␰ ⫽1 are relevant. For a general ␺ ( ␰គ ) with a Taylor series
expansion we write
⬁
␺ 共 ␰គ 兲 ⫽
j ¯j
␺ k ¯k ␰ j ¯ ␰ j ␰ k* ¯ ␰ k* .
兺
p,q⫽0,1,¯
1
p
1
q
1
p
1
共5.2兲
q
关Strictly speaking, such an expansion holds only for ␺ ( ␰គ ) in some dense subset of H (ind) .] We
note that here ␺ ( ␰គ ) is not an entire function of ␰ j , and since ␰ † ␰ ⫽1, the tensor components
j ¯j
␺ k1 ¯kp may be assumed to be traceless apart from being symmetric. Then they determine ␺ ( ␰គ )
1
q
uniquely and vice versa.
To express the inner product 共␾, ␺兲 for general ␾, ␺ 苸H (ind) in terms of their tensor components, we need to evaluate
冕 兿 冉 ␲␰ 冊 ␦ ␰ ␰
3
I k¯l¯
j¯m¯ ⫽
d2
j
共
j⫽1
†
⫺1 兲 ␰ j 1 ¯ ␰ j p 共 ␰ k 1 ¯ ␰ k q 兲 * 共 ␰ l 1 ¯ ␰ l p 兲 * 共 ␰ m 1 ¯ ␰ m q 兲 ,
⬘
⬘
共5.3兲
for general p, q, p ⬘ , q ⬘ and indices j, k, l, m. Using SU共3兲 invariance and symmetry, we see that
the result must be expressible in terms of products of Kronecker deltas. Combining this with the
tracelessness of the tensor components of ␾ and ␺, we can check first that we need only consider
the case p⫽p ⬘ , q⫽q ⬘ , and next that
I k¯l¯
j¯m¯ ⫽N
兺 兺
P苸S p Q苸S q
l
l
k
k
␦ jP(1) ¯ ␦ jP(p) ␦ m1
1
p
Q(1)
¯ ␦ mq
Q(q)
⫹¯ .
共5.4兲
Here N is a normalizing factor, and the dots denote terms with factors ␦ kj or ␦ m
l or both. Again the
latter can be ignored. The factor N can be computed say by setting all j⫽l⫽1 and all k⫽m
⫽2:
N⫽
1
.
共 p⫹q⫹2 兲 !
共5.5兲
We then get the result for any ␾, ␺ 苸H (ind) :
共 ␾,␺ 兲⫽
p!q!
j ¯j
j ¯j
␾ k ¯k * ␺ k ¯k .
兺
p,q⫽0,1,¯ 共 p⫹q⫹2 兲 !
1
p
1
p
1
q
1
q
共5.6兲
(ind,0)
With these results, all details of the induced UR DSU(2)
of SU共3兲 are in hand: the Hilbert space
in 共5.1兲, the expression 共5.6兲 for inner products, and the SU共3兲 action as in 共5.1兲.
(ind,0)
HSU(2)
VI. EQUIVALENCE MAP
The full equivalence of the two UR’s of SU共3兲, one on the subspace H0 傺H based on the six
(ind,0)
, can
oscillator Schwinger construction of Sec. III, and the other the induced representation DSU(2)
now be set up. The tensor component expressions 共4.2兲 and 共5.2兲 for vectors, and 共4.3兲 and 共5.6兲
for inner products, determine the one-to-one map to achieve this in full detail:
j ¯j
j ¯j
f 共 zគ ,w
គ 兲 ⫽ 兵 f k1 ¯kp 其 苸H0 ↔ ␺ 共 ␰គ 兲 ⫽ 兵 ␺ k1 ¯kp 其 苸H (ind) :
1
q
1
q
␺ k1 ¯kp ⫽ 冑共 p⫹q⫹2 兲 ! f k1 ¯kp ,p,q⫽0,1,... .
j ¯j
1
j ¯j
q
1
q
共6.1兲
The two inner products then match, and the SU共3兲 actions given in 共4.5兲 and 共5.1兲 on f (zគ ,wគ ) and
␺ ( ␰គ ) also match.
It is worth emphasizing here the two different arguments leading to the tracelessness of the
symmetric tensors on the two sides of 共6.1兲. In the case of the left hand side, the reason is that the
argument of ␺ ( ␰គ ) obeys the constraint ␰ † ␰ ⫽1. As for the right hand side, it happens because
entire functions f (zគ ,w
គ )苸H0 obey the partial differential equation in 共4.19兲. In both cases tracelessness leads to the UR being multiplicity free, apart from being complete in the sense that all
SU共3兲 UIR’s do appear.
VII. CONCLUDING REMARKS
To conclude, we have brought out the difficulties one encounters in naively extending the
Schwinger SU共2兲 construction to SU共3兲 particularly if one wishes to retain the simplicity and
economy intrinsic to the SU共2兲 case. We have shown how these difficulties can be overcome by
exploiting the group Sp(2,R) to obtain a ‘‘generating representation’’ of SU共3兲 based on six
bosonic oscillators. This UR of SU共3兲 contains all the representations of SU共3兲 exactly once. 共It
has been drawn to our attention by the referee that this result of ours is a special case of more
general results available in mathematics literature.18,19兲 Further, we have shown how this ‘‘generating representation’’ for SU共3兲 can also be constructed using the theory of induced representations
and have constructively established the equivalence between the two by making use of the Bargmann representation. It is hoped that the construction presented here will have useful applications
in various branches of physics much the same way as the SU共2兲 construction has. Indeed, the work
presented here has direct relevance to SU共3兲 coherent states as will be shown in a succeeding
publication.
ACKNOWLEDGMENTS
This work was begun while N.M. was a Jawaharlal Nehru Chair Professor of the University of
Hyderabad during September 1999. N.M. gratefully acknowledges the hospitality extended to him
during his stay by the School of Physics, University of Hyderabad. We are also grateful to the
referee for bringing Refs. 18 and 19 to our attention.
APPENDIX: BOSON OPERATOR CONSTRUCTION OF SU „3…ÃSp„2,R … BASIS STATES
We give here the explicit construction of the orthonormal basis states 兩 p,q;IM Y ;m 典 for H
introduced in Eq. 共3.26兲. We deal first with the states 兩 p,q;IIY ;k 典 苸H (p,q;0) 傺H (p,q) 艚H0 having
highest SU共2兲 weight; then by repeated use of the Sp(2,R) raising operator K ⫹ ⫽aគ̂ † •bគ̂ † with
兩 p,q;IIY ;m 典 苸H (p,q;m⫺k) 傺H (p,q) ; and finally with the general state 兩 p,q;IM Y ;m 典 using the
SU共2兲 lowering operator. At each stage the normalization will be ensured.
As is well known, the boson operators â †j , b̂ †j carry the following U共2兲 quantum numbers:11
I
M
Y
â †1 ,â †2
1
2
⫾ 12
1
3
â †3
0
0
b̂ †2 ,⫺b̂ †1
1
2
⫾
b̂ †3
0
0
⫺ 32 .
1
2
⫺
共A1兲
1
3
2
3
Therefore, â ␣† b̂ ␣† ⬅â †1 b̂ †1 ⫹â †2 b̂ †2 , â †3 and b̂ †3 are SU共2兲 scalars. The I-Y multiplets present in the
SU共3兲 UIR (p,q) are listed in Eq. 共2.5兲, and are parametrized by two integers r, s. The state
兩 p,q;IIY ;k 典 苸H0 involves p factors â † and q factors b̂ † acting on the Fock vacuum 兩 0គ ,0គ 典 , and in
addition it is annihilated by K ⫺ ⫽aគ̂ •bគ̂ . We therefore start with the expression 关guided by 共A1兲兴:
(p⫺r,q⫺s) ⬍
兩 p,q;IIY ;k 典 ⫽ 共 â †1 兲 r 共 b̂ †2 兲 s
兺
n⫽0,1,...
Y 1
r⫽I⫹ ⫹ 共 p⫺q 兲 ,
2 3
C n 共 â ␣† b̂ ␣† 兲 n 共 â †3 兲 p⫺r⫺n 共 b̂ †3 兲 q⫺s⫺n 兩 0គ ,0គ 典 ,
共A2兲
Y 1
s⫽I⫺ ⫹ 共 q⫺ p 兲 .
2 3
The condition
K ⫺ 兩 p,q;IIY ;k 典 ⫽0
共A3兲
gives the recursion relation
n 共 r⫹s⫹n⫹1 兲 C n ⫽⫺ 共 p⫺r⫺n⫹1 兲共 q⫺s⫺n⫹1 兲 C n⫺1 ,
n⫽1,2,...,
共A4兲
with the solution
C n⫽
共 ⫺1 兲 n
共 p⫺r 兲 ! 共 q⫺s 兲 ! 共 r⫹s⫹1 兲 !
C ,
n! 共 p⫺r⫺n 兲 ! 共 q⫺s⫺n 兲 ! 共 r⫹s⫹n⫹1 兲 ! 0
n⫽1,2,... .
共A5兲
Using this in Eq. 共A2兲, and after some algebra, the normalized state is found to be
兩 p,q;IIY ;k 典 ⫽NpqIY
共 â †1 兲 r 共 b̂ †2 兲 s
r!
s!
(p⫺r,q⫺s) ⬍
共 â ␣† b̂ ␣† 兲 n 共 â †3 兲 p⫺r⫺n 共 b̂ †3 兲 q⫺s⫺n
共 ⫺1 兲 n
兩 0គ ,0គ 典 苸H (p,q;0) ,
n!
共 p⫺r⫺n 兲 ! 共 q⫺s⫺n 兲 !
n⫽0,1,... 共 r⫹s⫹n⫹1 兲 !
共A6兲
NpqIY ⫽ 兵 r!s! 共 r⫹s⫹1 兲 ! 共 p⫺r 兲 ! 共 q⫺s 兲 ! 共 p⫹s⫹1 兲 ! 共 q⫹r⫹1 兲 !/ 共 p⫹q⫹1 兲 ! 其 1/2.
⫻
兺
From Eq. 共3.27a兲 we know that vectors in H (p,q;m⫺k) for m⬎k are obtained from vectors in
m⫺k
by applying K ⫹
. Further, the normalization is controlled by Eq. 共3.25b兲. We thus
H
obtain
(p,q;0)
兩 p,q;IIY ;m 典 ⫽ 兵 共 2k⫺1 兲 !/ 共 m⫺k 兲 ! 共 m⫹k⫺1 兲 ! 其 1/2共 aគ̂ † •bគ̂ † 兲 m⫺k 兩 p,q;IIY ;k 典 苸H (p,q;m⫺k) .
共A7兲
The last step is to reach a general value M ⭐I for the SU共2兲 magnetic quantum number. For
this we apply the SU共2兲 lowering operator J ⫺ ⫽â †2 â 1 ⫺b̂ †1 b̂ 2 (I⫺M ) times to the state 共A7兲, keeping track of normalization. This leads to the result
兩 p,q;IM Y ;m 典 ⫽ 兵 共 I⫹M 兲 !/2I! 共 I⫺M 兲 ! 其 1/2共 â †2 â 1 ⫺b̂ †1 b̂ 2 兲 I⫺M 兩 p,q;IIY ;m 典 .
共A8兲
If we combine Eqs. 共A6兲 to 共A8兲 we get the complete expression
兩 p,q;IM Y ;m 典 ⫽NpqIY 兵 共 2k⫺1 兲 ! 共 I⫹M 兲 ! 共 I⫺M 兲 !/ 共 m⫺k 兲 ! 共 m⫹k⫺1 兲 !2I! 其 1/2
I⫺M (p⫺r,q⫺s) ⬍
⫻ 共 aគ̂ •bគ̂ 兲
†
⫻
† m⫺k
兺
L⫽0
兺
n⫽0
共 ⫺1 兲 n⫹I⫺M ⫺L 共 â ␣ b̂ ␣ 兲 n
•
n!
共 r⫹s⫹n⫹1 兲 !
†
†
共 â †3 兲 p⫺r⫺n 共 b̂ †3 兲 q⫺s⫺n 共 â †1 兲 r⫺L 共 â †2 兲 L 共 b̂ †2 兲 s⫺I⫹M ⫹L
共 p⫺r⫺n 兲 ! 共 q⫺s⫺n 兲 ! 共 r⫺L 兲 !
L!
共 b̂ †1 兲 I⫺M ⫺L
共 s⫺I⫹M ⫹L 兲 ! 共 I⫺M ⫺L 兲 !
兩 0គ ,0គ 典 .
共A9兲
We thus have explicit expressions for all the normalized basis states 兩 p,q;IM Y ;m 典 of H.
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1