Irreducible SU(3) Schwinger Bosons
aa r X i v : . [ m a t h - ph ] A p r Irreducible SU(3) Schwinger Bosons
Ramesh Anishetty ∗ The Institute of Mathematical Sciences,CIT-Campus, Taramani, Chennai, India
Manu Mathur † and Indrakshi Raychowdhury ‡ S. N. Bose National Centre for Basic Sciences, JD Block,Sector III, Salt Lake City, Kolkata 700098, India.
We develop simple computational techniques for constructing all possible SU(3) rep-resentations in terms of irreducible SU(3) Schwinger bosons. We show that these ir-reducible Schwinger oscillators make SU(3) representation theory as simple as SU(2).The new Schwinger oscillators satisfy certain Sp(2,R) constraints and solve the mul-tiplicity problem as well. These SU(3) techniques can be generalized to SU(N).
I. INTRODUCTION
It is well known that the simple features of SU(2) Lie algebra or angular momentumalgebra and it’s representations are lost when we study SU(3) or higher SU(N) groups.In the past, considerable work has been done in this direction [2, 3, 4, 5]. A detaileddiscussion with an extensive list of references in this context can be found in [3]. Thepurpose and motivation of this work is to address these issues and make the construction ofall SU(3) irreducible representations (irreps.) as simple and accessable as SU(2). Moreover,the techniques can be generalized to higher SU(N) groups in a straightforward manner.We use Schwinger boson representation of SU(2) [1] and SU(3) Lie algebra to illustrateour results. Infact, Schwinger boson representation of SU(2) Lie algebra has played manyimportant and diverse roles in different areas of physics due to its intrinsic simplicity. Moreexplicitly, the simplicity is because the Schwinger analysis of the SU(2) Lie algebra is in termsof it’s smallest (spin half) constituents instead of (spin one) angular momentum operatorsthemselves. These spin half operators are two quantum mechanical harmonic oscillators( j = , m = ± ) which are called Schwinger bosons. The angular momentum operators arecomposites of Schwinger bosons and belong to (higher) spin one representation. This leads ∗ e-mail: [email protected] † e-mail: [email protected] ‡ e-mail: [email protected] to enormous computational simplifications in the representation theory of SU(2) as well asit’s various coupling coefficients [1]. Besides this group theory advantages, the SchwingerSU(2) construction has also been exploited in nuclear physics [6], strongly correlated systems[7], supersymmetry and supergravity algebras [10], lattice gauge theories [11], loop quantumgravity [12] etc.In the context of SU(2) group theory, the Schwinger bosons, each carrying basic (half)unit of spin angular momentum flux, provide an explicit and simple realization of the angularmomentum algebra as well as all it’s representations [1]. In particular, the Hilbert spacecreated by the two Schwinger oscillators is isomorphic to the representation space of SU(2)group (see section II). Thus the Schwinger boson representation of SU(2) group is simple,economical as well as complete. However, all these features are lost when we generalizethe Schwinger boson construction to SU(N) with N ≥ N ≥
3. Any two states which differ by the overall presence of such an invariants willtransform in the same way under SU(N). This leads to the problem of multiplicity which inturn makes the representation theory of SU(N) ( N ≥
3) much more involved compared toSU(2) (compare representations (5) for SU(2) with (12) for SU(3)). These issues have alsobeen extensively addressed in the past [2, 3, 4, 5]. In the context of SU(3) Schwinger bosonanalysis, a systematic group theoretic procedure based on noncompact group Sp(2,R) isgiven in [5] to label the multiplicity of SU(3). In this work, exploiting this Sp(2,R) labelingin [5], we define irreducible SU(3) Schwinger bosons in terms of which construction of SU(3)representations is as simple as SU(2) (compare (5) for SU(2) with (26), instead of (12), forSU(3)). Further like in SU(2) case, the representations in terms of irreducible Schwingerbosons are multiplicity free.The plan of the paper is as follows. In section II, we start with a brief introduction toSU(2) Schwinger bosons. This section makes the presentation self contained and also allowsus to compare our SU(3) results (section IV) with the corresponding SU(2) results explicitly.In section III we discuss SU(3) Lie algebra and it’s representations in terms of Schwingerbosons. We also summarize the construction of Sp(2,R) group in [5] which commutes withSU(3). In section IV, we solve certain Sp(2,R) constraints in terms of irreducible Schwingerbosons leading to all SU(3) representations with the simplicity of SU(2). The appendixA describes the construction of SU(3) projection operators to construct the SU(3) irreps.discussed in section III. In appendix B we show that the irreducible Schwinger bosons in(1 ,
0) and (0 ,
1) representations acting on ( n, m ) SU(3) irrep. directly produce ( n + 1 , m )and ( n, m + 1) irreps. respectively. II. SU(2) SCHWINGER BOSONS
The SU (2) Lie algebra is given by a set of three angular momentum operators { ~J } ≡{ J , J , J } or equivalently by { J + , J − , J } , ( J ± ≡ J ± i J ) satisfying[ J , J ± ] = ± J ± , [ J + , J − ] = 2 J . (1)The SU (2) group has a Casimir operator given by ~J · ~J , and the different irreducible represen-tations are characterized by its eigenvalues j ( j + 1), where j is an integer or half-odd-integer.A given basis vector in representation j is labeled by the eigenvalue m of J as | j, m i .We now define a doublet of quantum mechanical oscillators or equivalently Schwinger bosons, ~a ≡ ( a , a ) and ~a † ≡ ( a † , a † ) respectively [1]. They satisfy the simpler bosonic commuta-tion relation [ a α , a † β ] = δ βα with α, β = 1 ,
2. The angular momentum operators in (1) areconstructed out of Schwinger bosons as: J a ≡ a † σ a a, (2)where σ a denote the Pauli matrices. It is easy to check that the operators in (2) satisfy the SU (2) Lie algebra with the SU(2) Casimir: ~J · ~J ≡ ~a † · ~a ~a † · ~a ! . (3)Thus the representations of SU (2) can be characterized by the eigenvalues of the totaloccupation number operator with the angular momentum satisfying, j = ( n + n )2 ≡ n n and n are the eigenvalues of a † a and a † a respectively.A general irreducible representation of SU (2) with n = n + n = 2 j can be written as, | ψ i α α ...α n = a † α a † α ...a † α n | i ≡ O α α ...α n | i . (5)Above we have defined O α α ...α n = a † α a † α ...a † α n for later convenience. The state | i denotes the vaccum state of both the Schwinger bosons, i.e., a α | i = 0 , α = 1 ,
2. The statesin (5) are completely symmetric in all the n = 2 j number of α indices. Graphically, therepresentation (5) is given by Young tableau in Fig. 1.As mentioned in the introduction, the aim of the present work is to define irreducibleSU(3) Schwinger bosons in terms of which all SU(3) representations retain the simplicityof (5). However, before going into technical details, we briefly summarize the essentialpreliminary ideas of SU(3) Lie algebra and it’s representations in terms of SU(3) Schwingerbosons. α α α n . . . n = 2 j boxes FIG. 1: The SU(2) representation in terms of Young Tableau. Each Schwinger boson a † α in (5)creates a Young tableau box. III. SU(3) SCHWINGER BOSONS
The rank of the SU (3) group is two. Therefore, to cover all SU(3) irreducible representa-tions we need two independent harmonic oscillator triplets belonging to the two fundamentalrepresentations 3 and 3 ∗ . Let’s denote them by { a † α } and { b † α } respectively with α = 1 , , SU (3) group are written as [8]: Q a = a † λ a a − b † ˜ λ a b (6)In (6), a = 1 , , ..., λ a are the Gell Mann matrices for the triplet (3) representation, − ˜ λ a are the corresponding matrices for the anti-triplet 3 ∗ representation where ˜ λ denotes thetranspose of λ . Throughout the paper, the upper and lower indices are in the conjugate 3(i.e, transforming as a † α ) and 3 ∗ (i.e, transforming as b † α ) representations respectively. Theoperators Q a satisfy the SU (3) algebra amongst themselves, i.e, [ Q a , Q b ] = if abc Q c where f abc are the SU(3) structure constants [8]. The defining relations (6) also imply:[ Q a , ( a † ) α ] = 12 ( a † ) β ( λ a ) αβ , [ Q a , b † α ] = −
12 ( λ a ) βα b † β . (7)Thus the operators ( a † ) α and ( b † ) α transform according to 3 and 3 ∗ representations. Thecorresponding Young tableau are given by one box and two vertical boxes respectively. As Q a , (a = 1 , , ..,
8) in (6) involve both creation and annihilation operators of a and b types,it is clear that: [ Q a , N a ] = 0 , [ Q a , N b ] = 0 . (8)The SU(3) Casimirs are given by the total occupation numbers of a and b type oscillators: N a = a † · a, N b = b † · b. (9)We represent their eigenvalues by n and m respectively and the SU(3) vacuum ( n = 0 , m =0) by the state | i . A. Symmetric vs. Mixed representations
A general SU(3) irreducible representation is characterized by ( n, m ). Note that n andm are the number of single and double boxes in a Young tableau diagram. At this stage, itis convenient to define the most basic SU(3) tensor operator: O α α ...α n β β ...β m ≡ ( a † ) α ( a † ) α ... ( a † ) α n ( b † ) β ( b † ) β ... ( b † ) β m , (10)with O α α ...α n ≡ ( a † ) α ( a † ) α ... ( a † ) α n and O β β ...β m ≡ ( b † ) β ( b † ) β ... ( b † ) β m . A general irre-ducible ( n, m ) representation of SU (3), denoted by | ψ i α α ...α n β β ...β m , satisfies the following threeconditions [13]: C1: symmetry in all upper ( α ) indices. C2: symmetry in all lower ( β ) indices. C3: tracelessness in any of it’s upper ( α ) and lower ( β ) indices.Let us first consider the simpler pure irreps. of type ( n,
0) and (0 , m ) respectively: | ψ i α α ...α n = O α α ...α n | i ; | ψ i β β ...β m = O β β ...β n | i (11)These pure representations satisfy C1 and C2 as the Schwinger boson creation operatorscommute amongst themselves and the condition C3 is redundant. The ( n,
0) and (0 , m )Young tableau are given by: β m . . .α β β α . . . α n m boxes ∈ ∗ n boxes ∈ FIG. 2: The two symmetric representations ( n,
0) and (0 , m ) respectively. Like in SU(2) case, eachsingle (double) box corresponds to a † α ( b † α ). Therefore, as far as symmetric representations of SU(3) are concerned, each single (double)Young tableau box represents a Schwinger boson creation operator a † ∈ (cid:16) b † ∈ ∗ (cid:17) . Thisconstruction is simple and retain the simplicity of SU(2). However, this simplicity is lostwhen we consider mixed representations ( n = 0 , m = 0). A ( n, m ) representation of SU(3)has to satisfy C3 in addition to C1 and C2. The states in ( n, m ) irrep. are given by [4]: | ψ i α ,α ,...α n β ,β ,...,β m ≡ h O α α ...α n β β ...β m + L n X l =1 m X k =1 δ α l β k O α α ..α l − α l ..α n β β ..β k − β k ...β m + L n X ( l ,l ) m X ( k ,k ) δ α l α l β k β k O α ..α l − α l ..α l − α l ..α n β ..β k − β k ..β k − β k ..β m + L n X ( l ,l l ) m X ( k ,k k ) δ α l α l α l β k β k β k O α ..α l − α l ..α l − α l ..α l − α l ..α n β ..β k − β k ..β k − β k ..β k − β k ...β m + .. + L q n X l ..l q =1 m X k ..k q =1 δ α l α l ..α lq β k β k ..β kq O α α ..α l − α l ..α l − α l ..α lq − α lq +1 ..α n β β ..β k − β k ..β k − β k ..β kq − β kq +1 ...β m i | i (12)where q = min( n, m ) , δ α α ..α r β β ...β r ≡ δ α β δ α β . . . δ α r β r and all the sums in (12) are over differentindices, i.e, l = l ... = l q and k = k ... = k q . The coefficients L r are given by [9]: L r ≡ ( − r ( a † · b † ) r ( n + m + 1)( n + m )( n + m − ... ( n + m + 2 − r ) , (13)The coefficients in (13) are chosen to satisfy the condition C3: X i l ,j k =1 δ α l β k | ψ i α ,α ,...α n β ,β ,...,β m = 0 , for all l = 1 , ...n, and k = 1 , ...m . (14)The projection operators implementing (14) in the Hilbert space of Schwinger bosons areconstructed in appendix A. The Young tableau for the ( n, m ) irrep. (12) is shown in Fig. 3. β m . . . α β β α . . . α n m boxes ∈ ∗ n boxes ∈ FIG. 3: A mixed ( n, m ) representation Young tableau. Like in SU(2) case, each single (double)box corresponds to the irreducible Schwinger boson A † α ( B † α ) in (26). It is clear that the tracelessness condition C3 makes the mixed irreps. in (12) much moreinvolved and complicated compared to the symmetric ones (11). As a result, like in SU (2)case, a chain of n number of a † and m number of b † operators acting on the vacuum doesnot serve the purpose for SU(3). In the next section, we make the construction of all SU(3)representation as simple as SU(2). In other words, we restore 1 − a · b and a † · b † which are SU(3) invariant operators. Another relatedissue is the problem of multiplicity [5] arising due to the above invariants. Given a state | ψ i α α ...α n β β ...β m in (12), we consider the following tower of states:( a † · b † ) ρ | ψ i α α ...α n β β ...β m ρ = 0 , , .... ∞ . (15)All the infinite states in (15) transform like ( n, m ) irrep. as they differ by different powersof the SU(3) invariant operators. In [5] it is shown that these infinite number of SU(3)degenerate states can be uniquely labeled by the quantum numbers of the group Sp(2,R)which commutes with SU(3). More explicitly, if we define the following SU(3) invariantoperators, k + ≡ a † · b † , k − ≡ a · b, k ≡
12 ( N a + N b + 3) , then it is easy to check that they satisfy Sp(2,R) or SU(1,1) algebra:[ k − , k + ] = 2 k , [ k , k + ] = k + , [ k , k − ] = − k − . (16)It is shown in [5] that the states in (15) are in 1 − D + k , labeled by | k, m ′ i with k = ( n + m + 3)(= , , , .... ) and m ′ = k + ρ .As k − | k, m ′ = k i = 0, the states in (12) are all annihilated by k − , i.e: k − | ψ i α ,α ,...α n β ,β ,...,β m = a · b | ψ i α ,α ,...α n β ,β ,...,β m ≡ . (17)Note that the symmetric states in (11) are trivially annihilated by a · b as they contain either a † s or b † s only. The mixed states are also annihilated by k − . As an example, let us considerthe simplest mixed state | ψ i αβ ∈ k − | ψ i αβ = ( a γ b γ ) (cid:18) a † α b † β − δ αβ a † · b † (cid:19) | i = 0 . Infact, the tracelessness condition (14) and the k − annihilation condition (17) are exactlyequivalent (see Appendix A). IV. THE IRREDUCIBLE SU(3) SCHWINGER BOSONS
We have already seen that all symmetric SU(3) representations (11) retain the simplicityof SU(2). In this next section we define irreducible SU(3) Schwinger bosons in terms ofwhich: • all mixed representations also remain as simple as SU(2) (compare (12) with (26) ofthis section). • the representations are multiplicity free (see eqn. (29) and (30)).With the above motivation in mind, we need to define new Schwinger bosons A † α and B † α which satisfy the following properties:(i) A † and B † increase N a and N b by 1 with A † ∈ B † ∈ ∗ ,(ii) they commute amongst themselves to maintain the symmetry properties C1 and C2,(iii)the tracelessness property C3 is obtained by demanding the equivalent (appendix A)Sp(2,R) constraint. More explicitly: k − (cid:16) A † α | ψ i α α ...α n β β ...β m (cid:17) = [ k − , A † α ] | ψ i α α ...α n β β ...β m = 0 . (18)The most general form of A † α consistent with (i) and (iii) is: A † α = a † α + L( N a , N b ) k + b α (19)Now, L( N a , N b ) is fixed by: k − A † α | ψ i α α ...α n β β ...β m = (1 + ( n + m + 3)L( n + 1 , m + 1)) b α | ψ i α α ...α n β β ...β m = 0 (20)In (22) we have made use of (16) and (17). This fixesL( n, m ) = − n + m + 1) (21)and we get: A † α = a † α − N a + N b + 1 k + b α , A α = a α − b † α k − N a + N b + 1 (22)Similarly, B † α = b † α − N a + N b + 1 k + a α , B α = b α − a † α k − N a + N b + 1 (23)It is easy to check that the irreducible Schwinger boson creation operators commute amongstthemselves: h A † α , A † β i = 0 , h B † α , B † β i = 0 , h A † α , B † β i = 0 . (24)The other commutation relations acting on the SU(3) irreps. are: h A α , A † β i | ψ i α α ...α n β β ...β m = (cid:18) δ βα − N a + N b + 2 B † α B β (cid:19) | ψ i α α ...α n β β ...β m , h A α , B † β i | ψ i α α ...α n β β ...β m = − N a + N b + 2 B † α A β | ψ i α α ...α n β β ...β m , (25) h B α , B † β i | ψ i α α ...α n β β ...β m = (cid:18) δ αβ − N a + N b + 2 A † α A β (cid:19) | ψ i α α ...α n β β ...β m . A. SU(3) Representations
Hence a general ( n, m ) irreducible representation of SU (3) can be written in terms ofthese irreducible Schwinger bosons as: | Ψ i α α ...α n β β ...β m ≡ O α α ...α n β β ...β m | i = A † α A † α . . . A † α n B † β B † β . . . B † β m | i (26)The simple construction (26) is equivalent to (12). To see the equivalence, it is instructive tofirst consider some simple examples. The two simplest fundamental representations ((1 , , | Ψ i α = A † α | i = (cid:18) a † α − N a + N b + 1 k + b α (cid:19) | i = a † α | i ≡ | ψ i α | Ψ i β = B † β | i = (cid:18) b † β − N a + N b + 1 k + a β (cid:19) | i = b † β | i ≡ | ψ i β . (27)The equations (27) also demonstrate that all the symmetric (i.e ( n, , (0 , m )) representationsin terms of irreducible Schwinger boson are exactly same as before. This is of course trivial.The simplest mixed (1 ,
1) states are: | Ψ i αβ ≡ A † α B † β | i = (cid:18) a † α − N a + N b + 1 k + b α (cid:19) (cid:18) b † β − N a + N b + 1 k + a β (cid:19) | i = a † α b † β − δ αβ a † · b † ) ! | i ≡ | ψ i αβ . (28)The equivalence between (12) and (26) is explicitly established using the method of inductionin Appendix B. Infact, it is easy to see that the states (26) satisfy all the three conditions:C1, C2 and C3 mentioned in section III. The symmetry properties C1 and C2 of (26) followfrom the commutation relations (24). The tracelessness property C3 of all the mixed states( n, m ) with n, m = 1 , . . . ∞ in (26) also follows (infact obvious) from the tracelessness ofthe octet state (28). To see this, we consider: | Ψ i γα ...α n γβ ...β m = A † · B † | Ψ i α ...α n β ...β m = A † α . . . A † α n B † β . . . B † β m | Ψ i γγ = 0 (29)In (29), we have used the fact that all the A † s and B † s commute amongst themselves (24)and the octet state (28) is traceless. Therefore, the SU(3) ( n, m ) representation states in(26) are exactly same as the states in (12): | Ψ i α α ...α n β β ...β m ≡ | ψ i α α ...α n β β ...β m . We further note that: A · B | Ψ i α α ...α n β β ...β m = a · b | Ψ i α α ...α n β β ...β m = 0 . (30)Hence, the present construction in terms of irreducible Schwinger bosons also solves themultiplicity problem. The eqns. (29) and (30) show that it is no longer possible to construct0the infinite tower (15) in terms of irreducible Schwinger bosons. We again emphasize thateach Young tableau single (double) box ∈ ∗ ) representation in Figure 3 corresponds tothe irreducible Schwinger boson creation operator A † α ( B † α ). This SU(3) representationfeature is again like SU(2) representations. V. SUMMARY & DISCUSSION
We conclude that the irreducible Schwinger bosons make the SU(3) representation theoryas simple as SU(2). By constructing irreducible Schwinger bosons commuting with k − , weare able to produce all the SU(3) irreps. with the ease of SU(2). Infact, the irreducibleSchwinger bosons play important physical role in lattice gauge theories. In SU(2) latticegauge theories the SU(2) Schwinger bosons create the electric or angular momentum fluxesalong the links [11]. In SU(3) lattice gauge theory, the corresponding role is played byirreducible SU(3) Schwinger bosons [15].We now briefly discuss the extension of the ideas in this paper to the SU(N) group. TheSU(N) generalization of the SU(2) Schwinger mapping was done in [14] in the context ofSU(N) coherent states. The SU(N) group has rank ( N − Q a = N − X r =1 Q a [ r ] = N − X r =1 a † [ r ] λ a [ r ]2 a [ r ] . (31)In (31), the index r (= 1 , , ..., ( N − N −
1) fundamental representationsof SU(N) group. The invariance group is now bigger than Sp(2,R) and can again be used todefine irreducible SU(N) Schwinger bosons. This will again lead to a simplified representa-tion theory of SU(N). The work in this direction is in progress and will be reported elsewhere.
Acknowledgment:
The authors would like to thank H. S. Sharatchandra for useful discussions.
APPENDIX A: SU(3) PROJECTION OPERATORS
We now construct projection operators P ( n, m ) such that, P ( n,m ) O α α ...α n β β ...β m | i = | ψ i α α ...α n β β ...β m (A1)where, O α α ...α n β β ...β m is defined in (10). It is clear that the state | ψ i α α ...α n β β ...β m will transform in thesame way as O α α ...α n β β ...β m | i . Hence the projection operator can contain only SU (3) invariant1operators. Thus the most general form of P ( n,m ) is given by: P ( n,m ) ≡ ∞ X r =0 l r ( n, m )( k + ) r ( k − ) r = q X r =0 l r ( n, m )( k + ) r ( k − ) r (A2)where, q = min( n, m ).Applying k − on (A1) and equating it to zero, we get the recurrence relation:( n + m + 2 − r )( r − l r ( n, m ) = − l r − ( n, m ) (A3)Choosing the overall normalization l = 1, the solution of (A3) is l r ( n, m ) = ( − r r !( n + m + 1) . . . ( n + m + 2 − r ) = ( − r r ! ( n + m + 1 − r )!( n + m + 1)! , (A4)leading to: P ( n,m ) = 1( n + m + 1)! ∞ X r =0 ( − r r ! ( n + m + 1 − r )!( k + ) r ( k − ) r (A5)The action of the projection operator on the state O α α ...α n β β ...β m | i leads to (12). Infact theprojection operator in (A5) is idempotent, i.e, it satisfies: P ( n,m ) | ψ i α α ...α n β β ...β m = | ψ i α α ...α n β β ...β m (A6)The above property is obvious as k − annihilates the states | ψ i α α ...α n β β ...β m and therefore only theidentity (i.e, r = 0 term) in (A5) contributes. APPENDIX B: ACTION OF IRREDUCIBLE SCHWINGER BOSONS
In this appendix we show that the states in (26) are same as the SU(3) irreps. in (12).These results are obvious for all the symmetric representations as shown in section IV A Wehave also seen this equivalence for the octet (1 ,
1) representation. We now use the methodof induction for the general case. Let us assume the equivalence for ( n, m ) representation: | Ψ i α α ...α n β β ...β m ≡ | ψ i α α ...α n β β ...β m . (B1)We now need to prove: | Ψ i α α ...α n +1 β β ...β m +1 = | ψ i αα α ...α n +1 ββ β ...β m +1 (B2)2Let us first consider the case n → n + 1 and m → m . The l.h.s. of (B2) is: | Ψ i αα α ...α n β β ...β m ≡ A † α | Ψ i α α ...α n β β ...β m = A † α | ψ i α α ...α n β β ...β m = (cid:18) a † α − N a + N b + 1 k + b α (cid:19) ∞ X r =1 l r ( n, m ) k r + k r − ! O α α ...α n β β ...β m | i = (cid:16) O αα α ...α n β β ...β m | {z } T − k + b α ( n + m + 2) O α α ...α n β β ...β m | {z } T + ∞ X r =0 l r ( n, m ) k r + a † α k r − O α α ...α n β β ...β m | {z } T − n + m + 2) ∞ X r =0 l r ( n, m ) k + b α k r + k r − O α α ...α n β β ...β m | {z } T (cid:17) | i (B3)We use: [ a † α , k r − ] = − rk r − − b α , [ b α , k r + ] = rk r − a † α , to write the third term ( T ) and the fourth term ( − T ) in (B3) as: T = ∞ X r =1 l r ( n, m ) k r − k r + O αα α ...α n β β ...β m | {z } T − ∞ X r =1 rl r ( n, m ) k r + k r − − b α O α α ...α n β β ...β m | {z } T T = ∞ X r =1 rl r ( n, m ) n + m + 2 k r + k r − a † α O α α ...α n β β ...β m | {z } T − ∞ X r =1 r l r ( n, m ) n + m + 2 k r + k r − − b α O α α ...α n β β ...β m | {z } T + ∞ X r =1 l r ( n, m ) n + m + 2 k r +1+ k r − b α O αα α ...α n β β ...β m | {z } T (B4)The defining eqn. (A4) implies: l r ( n + 1 , m ) ≡ n + m + 2 − rn + m + 2 l r ( n, m ) . (B5)Using (B5), (A5) and (A1), we get:( T + T − T ) | i = | ψ i αα ...α n β β ...β m . (B6)We now need to show: − T + T − T − T = 0 . Using (B5) and − l r ( n, m ) n + m + 2 = ( r + 1) l r +1 ( n + 1 , m ) , l ( n + 1 , m ) = − n + m + 2 , T − T = − ∞ X r =1 rl r ( n + 1 , m ) k r + k r − − b α O α α ...α n β β ...β m (B7) − T − T = ∞ X r =1 ( r + 1) l r +1 ( n + 1 , m ) k r +1+ k r − + l ( n + 1 , m ) k + ! b α O α α ...α n β β ...β m = ∞ X r =1 rl r ( n + 1 , m ) k r + k r − − b α O α α ...α n β β ...β m ≡ T − T (B8)Similarly, we can prove the case: m → m + 1 and n → n . Thus we have also explicitlyproved that the simple SU(3) Schwinger boson states (26) (which are exact SU(3) analoguesof the SU(2) construction (5)) are indeeed the SU(3) irrep. states (12). [1] J. Schwinger U.S Atomic Energy Commission Report NYO-3071, 1952 or D. Mattis, TheTheory of Magnetism (Harper and Row, 1982).[2] M. Moshinsky, Rev. Mod. Phys.
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