# Analytic Formulae for the Matrix Elements of the Transition

## Transcript Of Analytic Formulae for the Matrix Elements of the Transition

Bulg. J. Phys. 32 (2005) 124–135

Analytic Formulae for the Matrix Elements of the Transition Operators in the Symplectic Extension of the Interacting Vector Boson Model

H. G. Ganev, A. I. Georgieva, V. P. Garistov Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, BG-1784 Soﬁa, Bulgaria

Received 2 April 2005

Abstract. The tensor properties of all the generators of Sp(12,R) - the group of dynamical symmetry of the Interacting Vector Boson Model (IVBM), are given with respect to the reduction chain Sp(12,R) ⊃ U(6) ⊃ U(3) × U(2) ⊃ O(3) × (U(1) × U(1)). Matrix elements of the basic building blocks of the model are evaluated in symmetry adapted basis along the considered chain. As a result of this, the analytic form of the matrix elements of any operator in the enveloping algebra of the Sp(12,R), deﬁning a certain transition operator, can be calculated. The procedure allows further applications of the symplectic IVBM for the description of transition probabilities between nuclear collective states.

PACS number: 21.60.Fw, 03.65.Fd,02.20.Sv

1 Introduction

In the algebraic models the use of the dynamical symmetries deﬁned by a certain reduction chain of the group of dynamical symmetry, yields exact solutions for the eigenvalues and eigenfunctions of the model Hamiltonian, which is constructed from the invariant operators of the subgroups in the chain.

Something more, it is very simple and straightforward to calculate matrix elements of transition operators between the eigenstates of the Hamiltonian, as both – the basis states and the operators, can be deﬁned as tensor operators in respect to the considered dynamical symmetry. Then the calculation of matrix elements is simpliﬁed by the use of the respective generalization of the Wigner-Eckart theorem. By deﬁnition such matrix elements give the transition probabilities between the collective states attributed to the basis states of the Hamiltonian. The comparison of the experimental data with the calculated transition probabilities

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Analytic Formulae for the Matrix Elements...

is one of the best tests of the validity of the considered algebraic model. With the aim of such applications of the rotational limit of symplectic extension of IVBM, we develop in this paper a practical mathematical approach for explicit evaluation of the matrix elements of transitional operators in the model.

The algebraic IVBM was developed [1] initially for the description of the low lying bands of the well deformed even-even nuclei [2]. Recently this approach was adapted to incorporate the newly observed higher collective states, both in the ﬁrst positive and negative parity bands [3] by considering the basis states as ”yrast” states for the different values of the number of bosons N, that built them. This was achieved by extending the dynamical symmetry group U (6) to the noncompact Sp(12, R). The excellent results obtained for the energy spectrum require a further investigation of the transition probabilities in the framework of the generalized IVBM with Sp(12, R) as a group of dynamical symmetry. Thus in the present work we consider the tensor properties of the algebra generators (Section 2.) in respect to the reduction chain

Sp(12, R) ⊃ U (6) ⊃ U (3) × U (2) ⊃ O(3) × (U (1) × U (1)), (1)

and also classify the basis states (Section 3.) by the quantum numbers corresponding to the irreducible representations of its subgroups. In this way we are able to deﬁne the transition operators between the basis states and then to evaluate analytically their matrix elements (Section 4.).

2 Tensor Properties of the Generators of the Sp(12,R) Group

The basic building blocks of the IVBM [1] are the creation and annihilation oper-

ators

of

the

vector

bosons

u+m(α)

and

um(α)

(m

=

0,

±1;

α

=

±

1 2

),

which

can

be considered as components of a 6−dimensional vector, which transform ac-

cording to the fundamental U (6) irreducible representations [1, 0, 0, 0, 0, 0]6 ≡ [1]6 and its conjugated [0, 0, 0, 0, 0, −1]6 ≡ [1]∗6, respectively. These irreducible

representations become reducible along the chain of subgroups (1) deﬁning the

dynamical symmetry of the rotational limit of the model [2]. This means that

along with the quantum number characterizing the representations of U (6), the

operators are also characterized by the quantum numbers of the subgroups of

chain (1).

The only possible representation of the direct product of U (3) × U (2) belonging

to the representation [1]6 of U (6) is [1]3.[1]2, i.e., [1]6 = [1]3.[1]2. According to the reduction rules for the decomposition U (3) ⊃ O(3), the representation [1]3 of U (3) contains the representation (1)3 of the group O(3) giving the angular momentum of the bosons l = 1 with a projection m = 0, ±1. The representation [1]2 of U (2) deﬁnes the ”pseudospin” of the bosons T = 12 , whose projection is given by the corresponding representation of (U (1) × U (1)), i.e., α = ± 12 . In this way the creation and annihilation operators u+m(α) and um(α)are deﬁned

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as irreducible tensors along the chain (1), and the used phase convention and commutation relations are the following [4]:

u[[11]]63[1]2mα + = u[[11]]∗3∗6 [1]∗2mα = (−1)m+ 12 −αu[[11]]∗6∗3 [1]∗2 −m−α (2)

u[[11]]∗∗6 [1]∗mα, u[[11]]6[1] nβ = δm,nδα,β

32

32

Initially the generators of the symplectic group Sp(12, R) were written as double tensors [5] with respect to O(3) ⊃ O(2) and U (2) ⊃ U (1) reductions

ALT M T0 =

C1LmM1

n

C

T

1

T0 α1

β

u[[11]]6[1] mαu[[11]]∗∗6 [1]∗βn,

(3)

22

32

32

m,n α,β

FTLTM0 = C1LmM1nCT21 αT012 β u[[11]]63[1]2mαu[[11]]63[1]2nβ , (4)

m,n α,β

GLT M T0 =

C1LmM1

n

C

T

1

T0 α1

β

u[[11]]∗∗6 [1]∗αmu[[11]]∗∗6 [1]∗βn.

(5)

22

32

32

m,n α,β

Further they can be deﬁned as irreducible tensor operators according to the whole chain (1) of subgroups and expressed in terms of (3), (4) and (5)

A[

λ]

[

χ]6 [2T

]

TLTM = C[[11]]6[1] [1][∗1[]1∗6]∗

[λ]

[χ]6 [2T ]

C([11])3[(11])∗3 [λ(L]3)

ALTM T ,

(6)

3

2

0

3232

3

2

33 3

0

F[λ]3[[χ2]T6 ]2 TLTM0 = C[[11]]36[1]2[1][31[]16]2 [λ]3[2[χT]]62 C([11])33[(11])33[λ(L]3)3 FTLTM0 , (7)

G[

λ]

[

χ]6 [2T

]

TLTM = C[[11]]∗∗6[1]∗[1][∗1[]1∗6]∗

[λ]

[χ]6 [2T ]

C([11])∗3 [(11])∗3 [(λL]3)

GLTM T ,

(8)

3

2

0

3232

3

2

33 3

0

where, according to the lemma of Racah [6], the Clebsch-Gordan coefﬁcients

along the chain are factorized by means of isoscalar factors (IF), deﬁned for

each step of decomposition (1). It should be pointed out [4] that the U (6)−

C [χ1]6

[χ2 ]6

[λ1]3[2T1]2 [λ2]3[2T2]2

[λ]3[[χ2T]6]2 and U (3) − C([λl11)]33 [λ(l22])33[(λL]3)3 IF’s, entering in

(6), (7) and (8), are equal to ±1 and their values, are taken into account in what

follows.

The tensors (6), transform according to the direct product [χ]6 of the corresponding U (6) representations [1]6 and [1]∗6 [4], namely

[1]6 × [1]∗6 = [1, −1]6 + [0]6

(9)

where [1, −1]6 = [2, 1, 1, 1, 1, 0]6 and [0]6 = [1, 1, 1, 1, 1, 1]6 is the scalar U (6) representation. Further we multiply the two conjugated fundamental representations of U (3) × U (2)

[1]3[1]2 × [1]∗3[1]∗2 = ([1]3 × [1]∗3)([1]2 × [1]∗2)

= ([210]3 ⊕ [1, 1, 1]3) × ([2, 0]2 ⊕ [1, 1]2)

= [210]3[2]2 ⊕ [210]3[0]2 ⊕ [0]3[2]2 ⊕ [0]3[0]2.

(10)

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Analytic Formulae for the Matrix Elements...

Obviously, the ﬁrst three U (3) × U (2) irreducible representations (irreps) in the resulting decomposition (10) belong to the [1, −1]6 of U (6) and the last one to [0]6. Introducing the notations u+i ( 12 ) = p+i and u+i (− 12 ) = n+i , the scalar operator

A[[00]]36 [0]2

0000 = N = √12

C10m0 1−m

m

p+mp−m + n+mn−m

(11)

has the physical meaning of the total number of bosons operator N = Np+ Nn, where Np = p+mpm, Nn = n+mnm and is obviously the ﬁrst order invariant of all the unitary groups U (6), U (3) and U (2). Hence it reduces them to their respective unimodular subgroups SU (6), SU (3) and SU (2). Something more, the invariant operator (−1)N , decomposes the action space H of the Sp(12, R) generators to the even H+ with N = 0, 2, 4, ..., and odd H with N = 1, 3, 5, ..., subspaces of the boson representations of Sp(12, R) [7].

The U (3) irreps [λ]3 are shorthand notations of [n1, n2, n3]3, and are expressed in terms of Elliott’s notations [8] (λ, µ) with λ = n1 − n2, µ = n2 − n3, so in (10) we have [210]3 = (1, 1) and [0]3 = (0, 0).The corresponding values of L from the SU (3) ⊃ O(3) reduction rules are L = 1, 2 in the (1, 1) irrep and L = 0 in the (0, 0). The values of T are 1 and 0 for the U (2) irreps [2]2 and [0]2 respectively. Hence, the U (2) irreps in the direct product distinguish the equivalent U (3) irreps that appear in this reduction, and there is not degeneracy. The tensors with T = 0 correspond to the SU (3) generators

A[1−1]6 1M = √1

C 1M

[210]3[0]2 00

2 m,k 1m1k

p+mpk + n+mnk

(12)

A[[121−01]]36[0]2

2M = √1

C 2M

00 2 m,k 1m1k

p+mpk + n+mnk

(13)

representing the components of the angular LM (12) and Elliott’s quadrupole QM momentum (13) operators.

The tensors

A[[10]−3[12]]62 0101 =

32 p+mn−m ∼ T1,

m

A[[01]−3[12]]62 010−1 = − 32 n+mp−m ∼ T−1

m

A[[10]−3 [12]]62

00 = − 1 √3 (p+ p−m − n+ n−m) ∼ T0, 10 2 m m m

(14)

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H. G. Ganev, A. I. Georgieva, V. P. Garistov

correspond to the SU (2) generators, which are the components of the pseudospin operator T . And ﬁnally the tensors

A[21[01]−3[12]]62 L11M =

C1LmM1k p+m nk ,

m,k

A[21[01]−3[12]]62 L1−M1 =

C1LmM1k n+m pk ,

m,k

(15) (16)

and

A[21[01]−3[12]]62 L10M = √12 C1LmM1k(p+mpk − n+mnk),

(17)

m,k

with L = 1, 2 and M = −L, −L + 1, ..., L extend the U (3) × U (2) algebra to the U (6) one.

By analogy, the tensors (7) and (8) transform according to [4]

[1]6 × [1]6 = [2]6 + [1, 1]6,

(18)

and [1]∗6 × [1]∗6 = [−2]6 + [−1, −1]6,

respectively. But, since the basis states of the IVBM are fully symmetric, we consider only the fully symmetric U (6) representations [2]6 and [−2]6 of the operators (7) and (8). Since the two operators F and G are conjugated, i.e., (F[λ]3[[χ2]T6 ]2TLTM0 )+ = (−1)λ+µ+L−M+T −T0 G[λ]∗3[χ[2]∗6T ]∗2 TL−−TM0 , where [λ]3 = (λ, µ), we are going to present the next decompositions only for the F tensors (18). According to the decomposition rules for the fully symmetric U (6)

irreps [4], we have

[2]6 = [2]3[2]2 + [1, 1]3[0]2 = (2, 0)[2]2 + (0, 1)[0]2,

(19)

which further contain in (2, 0) L = 0, 2 with T = 1 and in (0, 1) - L = 1

with T

=

0.

Their

explicit

expressions

in

terms

of

the

creation

p

+ i

,

n+i

and

annihilation operators pi, ni at i = 0, ±1 are

F[2]3[[22]]62 L11M =

C1LmM1kp+mp+k ,

m,k

F[2]3[[22]]62 L1−M1 =

C1LmM1k n+m n+k

m,k

(20)

F[2]3[[22]]62 F[1,1[]23][60]2

LM = √1

CLM (p+ n+ − n+ p+),

10 2 m,k 1m1k m k m k

LM = √1

C LM

00 2 m,k 1m1k

p+mn+k + n+mp+k

(21)

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Analytic Formulae for the Matrix Elements...

The above operators and their conjugated ones G[λ]∗3[χ[2]∗6T ]∗2 TLTM0 change the number of bosons by two and realize the symplectic extension of the U (6) algebra. In this way we have listed all the irreducible tensor operators in respect to the reduction chain (1), that correspond to the inﬁnitesimal operators of the Sp(12, R) algebra.

Next, we can introduce the tensor products

T ([χ1]6[χ2]6) ω[χ]6

[λ]3[2T ]2

LM T T0

= T T [χ1]6

[λ1 ]3 [2T1 ]2

L1M1 [χ2]6 T1(T0)1 [λ2]3[2T2]2

L2 M2 T2 (T0 )2

× C[χ1]6

[χ2 ]6

[λ1 ]3 [T1 ]2 [λ2 ]3 [T2 ]2

C ω[χ]6

[λ]3[2T ]2

[λ1 ]3 [λ2 ]3 [λ]3 (L1 )3 (L2 )3 (L)3

× CML11 LM22

C L T1

T2

M (T0)1(T0)2

T T0

(22)

of two tensor operators T[λ]3[[χ2]T6]2 TLTM0 , which are as well tensors in respect to the considered reduction chain. We use (22) to obtain the tensor properties of

the operators in the enveloping algebra of Sp(12, R), containing the products

of the algebra generators. The products of second degree enter the two-body

interaction of the phenomenological Hamiltonian of this limit [4]. In this partic-

ular case we are interested in the transition operators between states differing by four bosons T[[λ4]]36[2T ]2 LMT T0 , expressed in terms of the products of two operators F[λ]3[[22]T6 ]2 TLTM0 . Making use of the decomposition (19) and the reduction rules in the chain (1), we list in Table 1 all the representations of the chain subgroups

that deﬁne the transformation properties of the resulting tensors.

[2]6 [λ1]3[2T1]2 (2, 0)[2]2 (2, 0)[2]2 (0, 1)[0]2

Table 1. Tensor products of two raising operators.

[2]6 [λ2]3[2T1]2 (2, 0)[2]2 (0, 1)[0]2 (0, 1)[0]2

[4]6 [λ]3[2T ]2 (4, 0)[4]2 (2, 1)[2]2 (0, 2)[0]2

O(3) K; L 0;0, 2, 4 1;1, 2, 3 0;0, 2

U (2) T 2 1 0

U (1) T0 0, ±1, ±2 0, ±1 0

The basis states in the H+ are also obtained as symmetrized tensor products of different degrees of the tensor operators F[λ]3[[22]T6 ]2 TLTM0 .

3 Construction of the Symplectic Basis States of the IVBM

In order to clarify the role of the tensor operators introduced in the previous section as transition operators and to simplify the calculation of their matrix elements, the basis for the Hilbert space must be symmetry adapted to the algebraic structure along the considered subgroup chain (1). It is evident from (20) and (21), that the basis states of the IVBM in the H+ (N −even) subspace of the

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H. G. Ganev, A. I. Georgieva, V. P. Garistov

boson representations of Sp(12, R) can be obtained by a consecutive application of the raising operators F[λ]3[[22]T6 ]2 TLTM0 on the boson vacuum (ground) state | 0 , annihilated by the tensor operators G[λ]3[χ[2]T6 ]2 TLTM0 | 0 = 0 and A[λ]3[χ[2]T6 ]2 TLTM0 | 0 = 0.

Thus, in general a basis for the considered dynamical symmetry of the IVBM can

be constructed by applying the multiple symmetric coupling (22) of the raising tensors F[λi][32[]26Ti]2 LTiiMT0ii with itself - [F × . . . × F ][λ]3[χ[2]T6 ]2 TLTM0 . Note that only fully symmetric tensor products [χ]6 ≡ [N ]6 are nonzero, since the raising operator commutes with itself. The possible U (3) couplings are enumerated by

the set [λ]3 = {[n1, n2, 0] ≡ (n1 − n2, n2, ); n1 ≥ n2 ≥ 0 }. In terms of the notations (λ, µ) the SU (3) content of the U (6) symmetric tensor [N ]6 is determined by λ = n1 − n2, µ = n2. The number of copies of the operator F in the symmetric tensor product [N ]6 is N/2, where N = n1 + n2 = λ + 2µ [3]. Each raising operator will increase the number of bosons N by two. Then, the

resulting inﬁnite basis is denoted by |[N ](λ, µ); KLM ; T T0 , where KLM are the quantum numbers for the non-orthonormal basis of the irrep (λ, µ).

The Sp(12, R) classiﬁcation scheme for the SU (3) boson representations, ob-

tained by applying the reduction rules [3] for the irreps in the chain (1) for even

value of the number of bosons N , is shown on Table 2. Each row (ﬁxed N ) of

the table corresponds to a given irreducible representation of the U (6) algebra.

Then the possible values for the pseudospin, given in the column next to the re-

spective value of N , are T

=

N 2

,

N 2

− 1, ... 0.

Thus when N

and T

are ﬁxed,

2T + 1 equivalent representations of the group SU (3) arise. Each of them is

distinguished by the eigenvalues of the operator T0 : −T, −T + 1, ..., T, deﬁn-

ing the columns of Table 2. The same SU (3) representations (λ, µ) arise for the

positive and negative eigenvalues of T0.

Now it is clear which of the tensor operators act as transition operators between

the basis states ordered in the classiﬁcation scheme presented in Table 2. The operators F[λ]3[[22]T6 ]2 TLTM0 with T0 = 0 (21) give the transitions between two neighboring cells (↓) from one column, while the ones with T0 = ±1(20) change the column as well (ւ). The tensors A[2,1[1]3−[01]]26 (12) and (13), which correspond to the SU (3) generators do not change the SU (3) representations (λ, µ), but can

change the angular momentum L inside it (=⇒). The SU (2) generating tensors A[0]3[1[2−]21]6 (14) change the projection T0 (→) of the pseudospin T and in this way distinguish the equivalent SU (3) irreps belonging to the different columns of

the same row of Table 2. Inside a given cell the transition between the different SU (3) irreps (⇓) is realized by the operators A[2,1[1]3−[21]]26 (15), (16) and (17), that represent the U (6) generators. The action of the tensor operators on the SU (3)

vectors inside a given cell or between the cells of Table 2. is also schematically

presented on it with corresponding arrows. In physical applications sequences

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Analytic Formulae for the Matrix Elements...

of SU (3) vectors are attributed to sequences of collective states belonging to different bands in the nuclear spectra. By means of the above analysis, the appropriate transition operators can be deﬁned as appropriate combinations of the tensor operators given in Section 2.

4 Matrix Elements of the Transition Operators in Symmetry Adapted Basis

Matrix elements of the Sp(12, R) algebra can be calculated in several ways. A direct method is to use the Sp(12, R) commutation relations [1] to derive recursion relations. Another is to start from approximate matrix element and proceed by successive approximations, adjusting the matrix elements until the commutation relations are precisely satisﬁed [9]. The third method is to make use of a vector-valued coherent-state representation theory [5], [10] to relate the matrix elements to the known matrix elements of a much simpler Weyl algebra.

However, in the present paper we use another technique for calculation of the matrix elements of the Sp(12, R) algebra, based on the fact that the representations of the SU (3) subgroup in IVBM are build with the help of the two kinds of vector bosons, which is in some sense simpler than the construction of the SU (3) representations in IBM [11] and Sp(6, R) symplectic model [12]. In the preceding sections we expressed the Sp(12, R) generators FTLTM0 , GLTM T0 , ALTM T0 and the basis states as components of irreducible tensors in respect to the reduction chain (1). Thus, for calculating their matrix elements, we have the advantage of using the Wigner-Eckart theorem in two steps. For the SU (3) → SO(3) and SU (2) → U (1) × U (1) reduction we need the standard SU (2)

NT

00

2 10

2 41

0 3 6 21 0 4 3 82 1 0 ...

Table 2. Classiﬁcation of the basis states.

T0 \ ... ± 4

A[0]3[1[−2]12]6 → (8, 0) − − − − ...

±3

(6, 0) − − −

(8, 0) (6, 1)

− − −

...

±2

F[1,1[2]3]6[0]2 ↓

(4, 0) − −

(6, 0) (4, 1)

− − (8, 0) (6, 1) (4, 2) − −

...

±1

ւ F[2]3[2[2]6]2

=⇒ A[2,[11]−3 [10]]62

(2, 0) −

A[2,[11]−3[12]]62 ⇓

(4, 0) (2, 1)

−

(6, 0)

(4, 1)

(2, 2)

−

(8, 0)

(6, 1)

(4, 2)

(2, 3)

−

...

0

(0, 0)

(2, 0) (0, 1)

(4, 0) (2, 1) (0, 2) (6, 0) (4, 1) (2, 2) (0, 3) (8, 0) (6, 1) (4, 2) (2, 3) (0, 4)

...

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H. G. Ganev, A. I. Georgieva, V. P. Garistov

Clebsch-Gordan coefﬁcient (CGC)

[N ′](λ′, µ′); K′L′M ′; T ′T0′|T[[λχ]]36[2t]2 lmtt0 |[N ] (λ, µ); KLM ; T T0 = [N ′](λ′, µ′); K′L′||T[[λχ]]36[2t]2 ||[N ](λ, µ); KL CLLM′Mlm′ CTTT′T00t′t0 .

(23)

For the calculation of the double-barred reduced matrix elements in (23) we use the form [13]

[N ′](λ′, µ′); K′L′||T[[λχ]]36[2t]2 lmtt0 ||[N ] (λ, µ); KL = [N ′](λ′, µ′)|||T[[λχ]]36[2t]2 |||[N ](λ, µ) CK(λL,µ)[λ(L]3)(3λ′,µK′)′L′

(24)

where CK(λL,µ)[λ(L]3)(3λ′,µK′)′L′ is a reduced SU (3) Clebsch-Gordan coefﬁcient (CGC), which is known analytically in many special cases [14], [15], [16], [17]

and computer codes are available to calculate them numerically [18], [19], and

[20]. Hence, for the evaluation of the matrix elements (23) of the Sp(12, R)

operators only their reduced triple-barred matrix elements are required. In order to evaluate them we ﬁrst obtain the matrix elements of the creation u+m(α) and annihilation um(α) operators (m = 0, ±1; α = ± 21 ) of the vector bosons that build them. The latter act in the Hilbert space H of the boson representation of the algebra of Sp(12, R) with a vacuum |0 deﬁned by um(α)|0 = 0. In the notations u+i ( 12 ) = p+i and u+i (− 21 ) = n+i , an orthonormal basis in H is introduced in the following way [7]:

|π, ν =

(√p+i )πi (n√+k )νk |0 ,

(25)

i,k=0,±1 πi! νk!

where π ≡ {π1, π0, π−1} run over the set of three nonnegative numbers for

which Np = πi, and the same is valid for ν ≡ {ν1, ν0, ν−1} with Nn =

i

i

νi, where Np and Nn give the number of bosons of each kind, and the total

number of bosons that build each state is N = Np + Nn. These numbers are

eigenvalues of the corresponding operators Np = p+mpm, Nn = n+mnm,

and N = Np+ Nn (11):

Np |π, ν = Np |π, ν , Nn |π, ν = Nn |π, ν .

(26)

N |π, ν = N |π, ν

(27)

As a result of the above relations the basis states |π, ν can be labelled with the quantum numbers N, Np, Nn. In the considered reduction chain (1) the labels of the SU (3) − (λ, µ) irreps are related to the numbers of the introduced n and p -vector bosons in the following way λ = Np −Nn, µ = Nn, N = λ+2µ [3]. It is simple to see that the eigenvalues in (26) Np ≡ n1 and Nn ≡ n2, where n1, n2

132

Analytic Formulae for the Matrix Elements...

deﬁned the U (3) tensor properties [λ]3 in the tensor operators (6), (7) and (8) and the symmetry adapted basis states, given in Table 2. Also the eigenvalue N in (26) corresponds to the totally symmetric U (6) irrep [N ] that deﬁnes the tensor

operators and the basis states tensor properties. Hence the basis (25) can be equivalently labelled by |[N ]; (λ, µ) with dim(λ, µ) = 21 (λ+µ+2)(λ+1)(µ+ 1) included in the normalization of the states. The action of any component of

the boson creation and annihilation operators is standardly given by

p

+ i

|[N

];

(λ

,

µ)

=

(N + 1) dim(λ, µ) |[N + 1]; (λ′, µ′) dim(λ′, µ′)

= (2µ+λ+1)(λ+µ+2)(λ+1) |[N +1]; (λ+1, µ) , (28) (λ+µ+3)(λ+2)

n+i |[N ]; (λ, µ) = (2µ+λ+1)(λ+µ+2)(λ+1)(µ+1) |[N +1]; (λ−1, µ+1) , (29) λ(λ+µ+1)(µ+2)

(2µ+λ)(λ+µ+2)(λ+1)

pi|[N ]; (λ, µ) =

λ(λ+µ+1)

|[N −1]; (λ−1, µ) , (30)

(2µ+λ)(λ+1)(µ+1)

ni|[N ]; (λ, µ) =

(λ+2)µ

|[N −1]; (λ+1, µ−1) . (31)

With the help of relations ( 28)−(31) we can evaluate the corresponding matrix

elements of the building blocks of the IVBM. They correspond to the triple-

barred reduced matrix elements in (24) as they depend only on the U (6) →

SU (3) quantum numbers. The above expressions for the action of the creation

and annihilation operators are very simple and useful, as only a single resulting

state is obtained. The Sp(12, R) generators, explicitly presented in Section 2

as

tensor

operators

in

terms

of

bilinear

products

of

p

+ i

,

n+i

,

pi

and

ni,

can

be

considered as coupled U (6) → SU (3) tensors (22) and their matrix elements

calculated using the above expressions where the state resulting from the action

of the operators taken as an intermediate state [13].

5 Conclusions

In the present paper we investigate the tensor properties of the algebra generators of Sp(12, R) with respect to the reduction chain (1). Sp(12, R) is the group

133

Analytic Formulae for the Matrix Elements of the Transition Operators in the Symplectic Extension of the Interacting Vector Boson Model

H. G. Ganev, A. I. Georgieva, V. P. Garistov Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, BG-1784 Soﬁa, Bulgaria

Received 2 April 2005

Abstract. The tensor properties of all the generators of Sp(12,R) - the group of dynamical symmetry of the Interacting Vector Boson Model (IVBM), are given with respect to the reduction chain Sp(12,R) ⊃ U(6) ⊃ U(3) × U(2) ⊃ O(3) × (U(1) × U(1)). Matrix elements of the basic building blocks of the model are evaluated in symmetry adapted basis along the considered chain. As a result of this, the analytic form of the matrix elements of any operator in the enveloping algebra of the Sp(12,R), deﬁning a certain transition operator, can be calculated. The procedure allows further applications of the symplectic IVBM for the description of transition probabilities between nuclear collective states.

PACS number: 21.60.Fw, 03.65.Fd,02.20.Sv

1 Introduction

In the algebraic models the use of the dynamical symmetries deﬁned by a certain reduction chain of the group of dynamical symmetry, yields exact solutions for the eigenvalues and eigenfunctions of the model Hamiltonian, which is constructed from the invariant operators of the subgroups in the chain.

Something more, it is very simple and straightforward to calculate matrix elements of transition operators between the eigenstates of the Hamiltonian, as both – the basis states and the operators, can be deﬁned as tensor operators in respect to the considered dynamical symmetry. Then the calculation of matrix elements is simpliﬁed by the use of the respective generalization of the Wigner-Eckart theorem. By deﬁnition such matrix elements give the transition probabilities between the collective states attributed to the basis states of the Hamiltonian. The comparison of the experimental data with the calculated transition probabilities

124

1310–0157 c 2005 Heron Press Ltd.

Analytic Formulae for the Matrix Elements...

is one of the best tests of the validity of the considered algebraic model. With the aim of such applications of the rotational limit of symplectic extension of IVBM, we develop in this paper a practical mathematical approach for explicit evaluation of the matrix elements of transitional operators in the model.

The algebraic IVBM was developed [1] initially for the description of the low lying bands of the well deformed even-even nuclei [2]. Recently this approach was adapted to incorporate the newly observed higher collective states, both in the ﬁrst positive and negative parity bands [3] by considering the basis states as ”yrast” states for the different values of the number of bosons N, that built them. This was achieved by extending the dynamical symmetry group U (6) to the noncompact Sp(12, R). The excellent results obtained for the energy spectrum require a further investigation of the transition probabilities in the framework of the generalized IVBM with Sp(12, R) as a group of dynamical symmetry. Thus in the present work we consider the tensor properties of the algebra generators (Section 2.) in respect to the reduction chain

Sp(12, R) ⊃ U (6) ⊃ U (3) × U (2) ⊃ O(3) × (U (1) × U (1)), (1)

and also classify the basis states (Section 3.) by the quantum numbers corresponding to the irreducible representations of its subgroups. In this way we are able to deﬁne the transition operators between the basis states and then to evaluate analytically their matrix elements (Section 4.).

2 Tensor Properties of the Generators of the Sp(12,R) Group

The basic building blocks of the IVBM [1] are the creation and annihilation oper-

ators

of

the

vector

bosons

u+m(α)

and

um(α)

(m

=

0,

±1;

α

=

±

1 2

),

which

can

be considered as components of a 6−dimensional vector, which transform ac-

cording to the fundamental U (6) irreducible representations [1, 0, 0, 0, 0, 0]6 ≡ [1]6 and its conjugated [0, 0, 0, 0, 0, −1]6 ≡ [1]∗6, respectively. These irreducible

representations become reducible along the chain of subgroups (1) deﬁning the

dynamical symmetry of the rotational limit of the model [2]. This means that

along with the quantum number characterizing the representations of U (6), the

operators are also characterized by the quantum numbers of the subgroups of

chain (1).

The only possible representation of the direct product of U (3) × U (2) belonging

to the representation [1]6 of U (6) is [1]3.[1]2, i.e., [1]6 = [1]3.[1]2. According to the reduction rules for the decomposition U (3) ⊃ O(3), the representation [1]3 of U (3) contains the representation (1)3 of the group O(3) giving the angular momentum of the bosons l = 1 with a projection m = 0, ±1. The representation [1]2 of U (2) deﬁnes the ”pseudospin” of the bosons T = 12 , whose projection is given by the corresponding representation of (U (1) × U (1)), i.e., α = ± 12 . In this way the creation and annihilation operators u+m(α) and um(α)are deﬁned

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H. G. Ganev, A. I. Georgieva, V. P. Garistov

as irreducible tensors along the chain (1), and the used phase convention and commutation relations are the following [4]:

u[[11]]63[1]2mα + = u[[11]]∗3∗6 [1]∗2mα = (−1)m+ 12 −αu[[11]]∗6∗3 [1]∗2 −m−α (2)

u[[11]]∗∗6 [1]∗mα, u[[11]]6[1] nβ = δm,nδα,β

32

32

Initially the generators of the symplectic group Sp(12, R) were written as double tensors [5] with respect to O(3) ⊃ O(2) and U (2) ⊃ U (1) reductions

ALT M T0 =

C1LmM1

n

C

T

1

T0 α1

β

u[[11]]6[1] mαu[[11]]∗∗6 [1]∗βn,

(3)

22

32

32

m,n α,β

FTLTM0 = C1LmM1nCT21 αT012 β u[[11]]63[1]2mαu[[11]]63[1]2nβ , (4)

m,n α,β

GLT M T0 =

C1LmM1

n

C

T

1

T0 α1

β

u[[11]]∗∗6 [1]∗αmu[[11]]∗∗6 [1]∗βn.

(5)

22

32

32

m,n α,β

Further they can be deﬁned as irreducible tensor operators according to the whole chain (1) of subgroups and expressed in terms of (3), (4) and (5)

A[

λ]

[

χ]6 [2T

]

TLTM = C[[11]]6[1] [1][∗1[]1∗6]∗

[λ]

[χ]6 [2T ]

C([11])3[(11])∗3 [λ(L]3)

ALTM T ,

(6)

3

2

0

3232

3

2

33 3

0

F[λ]3[[χ2]T6 ]2 TLTM0 = C[[11]]36[1]2[1][31[]16]2 [λ]3[2[χT]]62 C([11])33[(11])33[λ(L]3)3 FTLTM0 , (7)

G[

λ]

[

χ]6 [2T

]

TLTM = C[[11]]∗∗6[1]∗[1][∗1[]1∗6]∗

[λ]

[χ]6 [2T ]

C([11])∗3 [(11])∗3 [(λL]3)

GLTM T ,

(8)

3

2

0

3232

3

2

33 3

0

where, according to the lemma of Racah [6], the Clebsch-Gordan coefﬁcients

along the chain are factorized by means of isoscalar factors (IF), deﬁned for

each step of decomposition (1). It should be pointed out [4] that the U (6)−

C [χ1]6

[χ2 ]6

[λ1]3[2T1]2 [λ2]3[2T2]2

[λ]3[[χ2T]6]2 and U (3) − C([λl11)]33 [λ(l22])33[(λL]3)3 IF’s, entering in

(6), (7) and (8), are equal to ±1 and their values, are taken into account in what

follows.

The tensors (6), transform according to the direct product [χ]6 of the corresponding U (6) representations [1]6 and [1]∗6 [4], namely

[1]6 × [1]∗6 = [1, −1]6 + [0]6

(9)

where [1, −1]6 = [2, 1, 1, 1, 1, 0]6 and [0]6 = [1, 1, 1, 1, 1, 1]6 is the scalar U (6) representation. Further we multiply the two conjugated fundamental representations of U (3) × U (2)

[1]3[1]2 × [1]∗3[1]∗2 = ([1]3 × [1]∗3)([1]2 × [1]∗2)

= ([210]3 ⊕ [1, 1, 1]3) × ([2, 0]2 ⊕ [1, 1]2)

= [210]3[2]2 ⊕ [210]3[0]2 ⊕ [0]3[2]2 ⊕ [0]3[0]2.

(10)

126

Analytic Formulae for the Matrix Elements...

Obviously, the ﬁrst three U (3) × U (2) irreducible representations (irreps) in the resulting decomposition (10) belong to the [1, −1]6 of U (6) and the last one to [0]6. Introducing the notations u+i ( 12 ) = p+i and u+i (− 12 ) = n+i , the scalar operator

A[[00]]36 [0]2

0000 = N = √12

C10m0 1−m

m

p+mp−m + n+mn−m

(11)

has the physical meaning of the total number of bosons operator N = Np+ Nn, where Np = p+mpm, Nn = n+mnm and is obviously the ﬁrst order invariant of all the unitary groups U (6), U (3) and U (2). Hence it reduces them to their respective unimodular subgroups SU (6), SU (3) and SU (2). Something more, the invariant operator (−1)N , decomposes the action space H of the Sp(12, R) generators to the even H+ with N = 0, 2, 4, ..., and odd H with N = 1, 3, 5, ..., subspaces of the boson representations of Sp(12, R) [7].

The U (3) irreps [λ]3 are shorthand notations of [n1, n2, n3]3, and are expressed in terms of Elliott’s notations [8] (λ, µ) with λ = n1 − n2, µ = n2 − n3, so in (10) we have [210]3 = (1, 1) and [0]3 = (0, 0).The corresponding values of L from the SU (3) ⊃ O(3) reduction rules are L = 1, 2 in the (1, 1) irrep and L = 0 in the (0, 0). The values of T are 1 and 0 for the U (2) irreps [2]2 and [0]2 respectively. Hence, the U (2) irreps in the direct product distinguish the equivalent U (3) irreps that appear in this reduction, and there is not degeneracy. The tensors with T = 0 correspond to the SU (3) generators

A[1−1]6 1M = √1

C 1M

[210]3[0]2 00

2 m,k 1m1k

p+mpk + n+mnk

(12)

A[[121−01]]36[0]2

2M = √1

C 2M

00 2 m,k 1m1k

p+mpk + n+mnk

(13)

representing the components of the angular LM (12) and Elliott’s quadrupole QM momentum (13) operators.

The tensors

A[[10]−3[12]]62 0101 =

32 p+mn−m ∼ T1,

m

A[[01]−3[12]]62 010−1 = − 32 n+mp−m ∼ T−1

m

A[[10]−3 [12]]62

00 = − 1 √3 (p+ p−m − n+ n−m) ∼ T0, 10 2 m m m

(14)

127

H. G. Ganev, A. I. Georgieva, V. P. Garistov

correspond to the SU (2) generators, which are the components of the pseudospin operator T . And ﬁnally the tensors

A[21[01]−3[12]]62 L11M =

C1LmM1k p+m nk ,

m,k

A[21[01]−3[12]]62 L1−M1 =

C1LmM1k n+m pk ,

m,k

(15) (16)

and

A[21[01]−3[12]]62 L10M = √12 C1LmM1k(p+mpk − n+mnk),

(17)

m,k

with L = 1, 2 and M = −L, −L + 1, ..., L extend the U (3) × U (2) algebra to the U (6) one.

By analogy, the tensors (7) and (8) transform according to [4]

[1]6 × [1]6 = [2]6 + [1, 1]6,

(18)

and [1]∗6 × [1]∗6 = [−2]6 + [−1, −1]6,

respectively. But, since the basis states of the IVBM are fully symmetric, we consider only the fully symmetric U (6) representations [2]6 and [−2]6 of the operators (7) and (8). Since the two operators F and G are conjugated, i.e., (F[λ]3[[χ2]T6 ]2TLTM0 )+ = (−1)λ+µ+L−M+T −T0 G[λ]∗3[χ[2]∗6T ]∗2 TL−−TM0 , where [λ]3 = (λ, µ), we are going to present the next decompositions only for the F tensors (18). According to the decomposition rules for the fully symmetric U (6)

irreps [4], we have

[2]6 = [2]3[2]2 + [1, 1]3[0]2 = (2, 0)[2]2 + (0, 1)[0]2,

(19)

which further contain in (2, 0) L = 0, 2 with T = 1 and in (0, 1) - L = 1

with T

=

0.

Their

explicit

expressions

in

terms

of

the

creation

p

+ i

,

n+i

and

annihilation operators pi, ni at i = 0, ±1 are

F[2]3[[22]]62 L11M =

C1LmM1kp+mp+k ,

m,k

F[2]3[[22]]62 L1−M1 =

C1LmM1k n+m n+k

m,k

(20)

F[2]3[[22]]62 F[1,1[]23][60]2

LM = √1

CLM (p+ n+ − n+ p+),

10 2 m,k 1m1k m k m k

LM = √1

C LM

00 2 m,k 1m1k

p+mn+k + n+mp+k

(21)

128

Analytic Formulae for the Matrix Elements...

The above operators and their conjugated ones G[λ]∗3[χ[2]∗6T ]∗2 TLTM0 change the number of bosons by two and realize the symplectic extension of the U (6) algebra. In this way we have listed all the irreducible tensor operators in respect to the reduction chain (1), that correspond to the inﬁnitesimal operators of the Sp(12, R) algebra.

Next, we can introduce the tensor products

T ([χ1]6[χ2]6) ω[χ]6

[λ]3[2T ]2

LM T T0

= T T [χ1]6

[λ1 ]3 [2T1 ]2

L1M1 [χ2]6 T1(T0)1 [λ2]3[2T2]2

L2 M2 T2 (T0 )2

× C[χ1]6

[χ2 ]6

[λ1 ]3 [T1 ]2 [λ2 ]3 [T2 ]2

C ω[χ]6

[λ]3[2T ]2

[λ1 ]3 [λ2 ]3 [λ]3 (L1 )3 (L2 )3 (L)3

× CML11 LM22

C L T1

T2

M (T0)1(T0)2

T T0

(22)

of two tensor operators T[λ]3[[χ2]T6]2 TLTM0 , which are as well tensors in respect to the considered reduction chain. We use (22) to obtain the tensor properties of

the operators in the enveloping algebra of Sp(12, R), containing the products

of the algebra generators. The products of second degree enter the two-body

interaction of the phenomenological Hamiltonian of this limit [4]. In this partic-

ular case we are interested in the transition operators between states differing by four bosons T[[λ4]]36[2T ]2 LMT T0 , expressed in terms of the products of two operators F[λ]3[[22]T6 ]2 TLTM0 . Making use of the decomposition (19) and the reduction rules in the chain (1), we list in Table 1 all the representations of the chain subgroups

that deﬁne the transformation properties of the resulting tensors.

[2]6 [λ1]3[2T1]2 (2, 0)[2]2 (2, 0)[2]2 (0, 1)[0]2

Table 1. Tensor products of two raising operators.

[2]6 [λ2]3[2T1]2 (2, 0)[2]2 (0, 1)[0]2 (0, 1)[0]2

[4]6 [λ]3[2T ]2 (4, 0)[4]2 (2, 1)[2]2 (0, 2)[0]2

O(3) K; L 0;0, 2, 4 1;1, 2, 3 0;0, 2

U (2) T 2 1 0

U (1) T0 0, ±1, ±2 0, ±1 0

The basis states in the H+ are also obtained as symmetrized tensor products of different degrees of the tensor operators F[λ]3[[22]T6 ]2 TLTM0 .

3 Construction of the Symplectic Basis States of the IVBM

In order to clarify the role of the tensor operators introduced in the previous section as transition operators and to simplify the calculation of their matrix elements, the basis for the Hilbert space must be symmetry adapted to the algebraic structure along the considered subgroup chain (1). It is evident from (20) and (21), that the basis states of the IVBM in the H+ (N −even) subspace of the

129

H. G. Ganev, A. I. Georgieva, V. P. Garistov

boson representations of Sp(12, R) can be obtained by a consecutive application of the raising operators F[λ]3[[22]T6 ]2 TLTM0 on the boson vacuum (ground) state | 0 , annihilated by the tensor operators G[λ]3[χ[2]T6 ]2 TLTM0 | 0 = 0 and A[λ]3[χ[2]T6 ]2 TLTM0 | 0 = 0.

Thus, in general a basis for the considered dynamical symmetry of the IVBM can

be constructed by applying the multiple symmetric coupling (22) of the raising tensors F[λi][32[]26Ti]2 LTiiMT0ii with itself - [F × . . . × F ][λ]3[χ[2]T6 ]2 TLTM0 . Note that only fully symmetric tensor products [χ]6 ≡ [N ]6 are nonzero, since the raising operator commutes with itself. The possible U (3) couplings are enumerated by

the set [λ]3 = {[n1, n2, 0] ≡ (n1 − n2, n2, ); n1 ≥ n2 ≥ 0 }. In terms of the notations (λ, µ) the SU (3) content of the U (6) symmetric tensor [N ]6 is determined by λ = n1 − n2, µ = n2. The number of copies of the operator F in the symmetric tensor product [N ]6 is N/2, where N = n1 + n2 = λ + 2µ [3]. Each raising operator will increase the number of bosons N by two. Then, the

resulting inﬁnite basis is denoted by |[N ](λ, µ); KLM ; T T0 , where KLM are the quantum numbers for the non-orthonormal basis of the irrep (λ, µ).

The Sp(12, R) classiﬁcation scheme for the SU (3) boson representations, ob-

tained by applying the reduction rules [3] for the irreps in the chain (1) for even

value of the number of bosons N , is shown on Table 2. Each row (ﬁxed N ) of

the table corresponds to a given irreducible representation of the U (6) algebra.

Then the possible values for the pseudospin, given in the column next to the re-

spective value of N , are T

=

N 2

,

N 2

− 1, ... 0.

Thus when N

and T

are ﬁxed,

2T + 1 equivalent representations of the group SU (3) arise. Each of them is

distinguished by the eigenvalues of the operator T0 : −T, −T + 1, ..., T, deﬁn-

ing the columns of Table 2. The same SU (3) representations (λ, µ) arise for the

positive and negative eigenvalues of T0.

Now it is clear which of the tensor operators act as transition operators between

the basis states ordered in the classiﬁcation scheme presented in Table 2. The operators F[λ]3[[22]T6 ]2 TLTM0 with T0 = 0 (21) give the transitions between two neighboring cells (↓) from one column, while the ones with T0 = ±1(20) change the column as well (ւ). The tensors A[2,1[1]3−[01]]26 (12) and (13), which correspond to the SU (3) generators do not change the SU (3) representations (λ, µ), but can

change the angular momentum L inside it (=⇒). The SU (2) generating tensors A[0]3[1[2−]21]6 (14) change the projection T0 (→) of the pseudospin T and in this way distinguish the equivalent SU (3) irreps belonging to the different columns of

the same row of Table 2. Inside a given cell the transition between the different SU (3) irreps (⇓) is realized by the operators A[2,1[1]3−[21]]26 (15), (16) and (17), that represent the U (6) generators. The action of the tensor operators on the SU (3)

vectors inside a given cell or between the cells of Table 2. is also schematically

presented on it with corresponding arrows. In physical applications sequences

130

Analytic Formulae for the Matrix Elements...

of SU (3) vectors are attributed to sequences of collective states belonging to different bands in the nuclear spectra. By means of the above analysis, the appropriate transition operators can be deﬁned as appropriate combinations of the tensor operators given in Section 2.

4 Matrix Elements of the Transition Operators in Symmetry Adapted Basis

Matrix elements of the Sp(12, R) algebra can be calculated in several ways. A direct method is to use the Sp(12, R) commutation relations [1] to derive recursion relations. Another is to start from approximate matrix element and proceed by successive approximations, adjusting the matrix elements until the commutation relations are precisely satisﬁed [9]. The third method is to make use of a vector-valued coherent-state representation theory [5], [10] to relate the matrix elements to the known matrix elements of a much simpler Weyl algebra.

However, in the present paper we use another technique for calculation of the matrix elements of the Sp(12, R) algebra, based on the fact that the representations of the SU (3) subgroup in IVBM are build with the help of the two kinds of vector bosons, which is in some sense simpler than the construction of the SU (3) representations in IBM [11] and Sp(6, R) symplectic model [12]. In the preceding sections we expressed the Sp(12, R) generators FTLTM0 , GLTM T0 , ALTM T0 and the basis states as components of irreducible tensors in respect to the reduction chain (1). Thus, for calculating their matrix elements, we have the advantage of using the Wigner-Eckart theorem in two steps. For the SU (3) → SO(3) and SU (2) → U (1) × U (1) reduction we need the standard SU (2)

NT

00

2 10

2 41

0 3 6 21 0 4 3 82 1 0 ...

Table 2. Classiﬁcation of the basis states.

T0 \ ... ± 4

A[0]3[1[−2]12]6 → (8, 0) − − − − ...

±3

(6, 0) − − −

(8, 0) (6, 1)

− − −

...

±2

F[1,1[2]3]6[0]2 ↓

(4, 0) − −

(6, 0) (4, 1)

− − (8, 0) (6, 1) (4, 2) − −

...

±1

ւ F[2]3[2[2]6]2

=⇒ A[2,[11]−3 [10]]62

(2, 0) −

A[2,[11]−3[12]]62 ⇓

(4, 0) (2, 1)

−

(6, 0)

(4, 1)

(2, 2)

−

(8, 0)

(6, 1)

(4, 2)

(2, 3)

−

...

0

(0, 0)

(2, 0) (0, 1)

(4, 0) (2, 1) (0, 2) (6, 0) (4, 1) (2, 2) (0, 3) (8, 0) (6, 1) (4, 2) (2, 3) (0, 4)

...

131

H. G. Ganev, A. I. Georgieva, V. P. Garistov

Clebsch-Gordan coefﬁcient (CGC)

[N ′](λ′, µ′); K′L′M ′; T ′T0′|T[[λχ]]36[2t]2 lmtt0 |[N ] (λ, µ); KLM ; T T0 = [N ′](λ′, µ′); K′L′||T[[λχ]]36[2t]2 ||[N ](λ, µ); KL CLLM′Mlm′ CTTT′T00t′t0 .

(23)

For the calculation of the double-barred reduced matrix elements in (23) we use the form [13]

[N ′](λ′, µ′); K′L′||T[[λχ]]36[2t]2 lmtt0 ||[N ] (λ, µ); KL = [N ′](λ′, µ′)|||T[[λχ]]36[2t]2 |||[N ](λ, µ) CK(λL,µ)[λ(L]3)(3λ′,µK′)′L′

(24)

where CK(λL,µ)[λ(L]3)(3λ′,µK′)′L′ is a reduced SU (3) Clebsch-Gordan coefﬁcient (CGC), which is known analytically in many special cases [14], [15], [16], [17]

and computer codes are available to calculate them numerically [18], [19], and

[20]. Hence, for the evaluation of the matrix elements (23) of the Sp(12, R)

operators only their reduced triple-barred matrix elements are required. In order to evaluate them we ﬁrst obtain the matrix elements of the creation u+m(α) and annihilation um(α) operators (m = 0, ±1; α = ± 21 ) of the vector bosons that build them. The latter act in the Hilbert space H of the boson representation of the algebra of Sp(12, R) with a vacuum |0 deﬁned by um(α)|0 = 0. In the notations u+i ( 12 ) = p+i and u+i (− 21 ) = n+i , an orthonormal basis in H is introduced in the following way [7]:

|π, ν =

(√p+i )πi (n√+k )νk |0 ,

(25)

i,k=0,±1 πi! νk!

where π ≡ {π1, π0, π−1} run over the set of three nonnegative numbers for

which Np = πi, and the same is valid for ν ≡ {ν1, ν0, ν−1} with Nn =

i

i

νi, where Np and Nn give the number of bosons of each kind, and the total

number of bosons that build each state is N = Np + Nn. These numbers are

eigenvalues of the corresponding operators Np = p+mpm, Nn = n+mnm,

and N = Np+ Nn (11):

Np |π, ν = Np |π, ν , Nn |π, ν = Nn |π, ν .

(26)

N |π, ν = N |π, ν

(27)

As a result of the above relations the basis states |π, ν can be labelled with the quantum numbers N, Np, Nn. In the considered reduction chain (1) the labels of the SU (3) − (λ, µ) irreps are related to the numbers of the introduced n and p -vector bosons in the following way λ = Np −Nn, µ = Nn, N = λ+2µ [3]. It is simple to see that the eigenvalues in (26) Np ≡ n1 and Nn ≡ n2, where n1, n2

132

Analytic Formulae for the Matrix Elements...

deﬁned the U (3) tensor properties [λ]3 in the tensor operators (6), (7) and (8) and the symmetry adapted basis states, given in Table 2. Also the eigenvalue N in (26) corresponds to the totally symmetric U (6) irrep [N ] that deﬁnes the tensor

operators and the basis states tensor properties. Hence the basis (25) can be equivalently labelled by |[N ]; (λ, µ) with dim(λ, µ) = 21 (λ+µ+2)(λ+1)(µ+ 1) included in the normalization of the states. The action of any component of

the boson creation and annihilation operators is standardly given by

p

+ i

|[N

];

(λ

,

µ)

=

(N + 1) dim(λ, µ) |[N + 1]; (λ′, µ′) dim(λ′, µ′)

= (2µ+λ+1)(λ+µ+2)(λ+1) |[N +1]; (λ+1, µ) , (28) (λ+µ+3)(λ+2)

n+i |[N ]; (λ, µ) = (2µ+λ+1)(λ+µ+2)(λ+1)(µ+1) |[N +1]; (λ−1, µ+1) , (29) λ(λ+µ+1)(µ+2)

(2µ+λ)(λ+µ+2)(λ+1)

pi|[N ]; (λ, µ) =

λ(λ+µ+1)

|[N −1]; (λ−1, µ) , (30)

(2µ+λ)(λ+1)(µ+1)

ni|[N ]; (λ, µ) =

(λ+2)µ

|[N −1]; (λ+1, µ−1) . (31)

With the help of relations ( 28)−(31) we can evaluate the corresponding matrix

elements of the building blocks of the IVBM. They correspond to the triple-

barred reduced matrix elements in (24) as they depend only on the U (6) →

SU (3) quantum numbers. The above expressions for the action of the creation

and annihilation operators are very simple and useful, as only a single resulting

state is obtained. The Sp(12, R) generators, explicitly presented in Section 2

as

tensor

operators

in

terms

of

bilinear

products

of

p

+ i

,

n+i

,

pi

and

ni,

can

be

considered as coupled U (6) → SU (3) tensors (22) and their matrix elements

calculated using the above expressions where the state resulting from the action

of the operators taken as an intermediate state [13].

5 Conclusions

In the present paper we investigate the tensor properties of the algebra generators of Sp(12, R) with respect to the reduction chain (1). Sp(12, R) is the group

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