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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. J. Schwinger, ‘‘On angular momentum,’’ USAEC Report NYO-3071 共1952兲 共unpublished兲; reprinted in Quantum Theory of Angular Momentum, edited by L. C. Biedenharn and H. van Dam 共Academic, New York, 1965兲; also in A Quantum Legacy—Seminal Papers of Julian Schwinger, edited by K. A. Milton 共World Scientific, Singapore, 2000兲. 2 D. P. Arovas and A. Auerbach, Phys. Rev. B 38, 316 共1988兲; A. Auerbach and D. P. Arovas, Phys. Rev. Lett. 61, 617 共1988兲; A. Auerbach, Interacting Electrons and Quantum Magnetism 共Springer, New York, 1994兲. 3 Arvind, B. Dutta, N. Mukunda, and R. Simon, Phys. Rev. A 52, 1609 共1993兲. 4 K. Sundar, N. Mukunda, and R. Simon, J. Opt. Soc. Am. A 12, 560 共1995兲. 5 A. P. Balachandran, P. Salomonson, B. S. Skagerstam, and J. O. Winnberg, Phys. Rev. D 15, 2308 共1977兲. 6 M. V. Berry and J. M. Robbins, Proc. R. Soc. London, Ser. A 453, 1771 共1997兲. 7 V. Bargmann, Rev. Mod. Phys. 34, 829 共1962兲. 8 N. Mukunda and B. Radhakrishnan, J. Math. Phys. 15, 1320, 1332, 164, 1656 共1974兲. 9 M. Moshinsky, Rev. Mod. Phys. 34, 813 共1962兲. 10 G. W. Mackey, Group Representations in Hilbert Space 共AMS, Providence, RI, 1963兲. 11 See, for instance, J. J. de Swart, Rev. Mod. Phys. 35, 916 共1963兲. 12 N. Mukunda and L. K. Pandit, J. Math. Phys. 6, 746 共1965兲. 13 D. Sen and M. Mathur, J. Math. Phys. 42, 4181 共2001兲. 14 N. Mukunda and L. K. Pandit, Prog. Theor. Phys. 34, 46 共1965兲. 15 See, for instance, A. M. Perelomov, Usp. Fiz. Nauk 123, 23 共1977兲 关Sov. Phys. Usp. 20, 703 共1977兲兴; K. Wódkiewicz and J. H. Eberly, J. Opt. Soc. Am. B 2, 458 共1985兲. 16 V. Bargmann, Ann. Math. 48, 568 共1947兲. 17 V. Bargmann, Commun. Pure Appl. Math. 14, 187 共1961兲. 18 R. Howe, Proc. Indian Math. Soc. 97, 85 共1987兲; Trans. Am. Math. Soc. 313, 539 共1989兲, and references cited therein. 19 I. M. Gelfand and A. V. Zelevinskii, Funct. Anal. 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