An algebraic representation of the particle-plus-rotor model · double coupling in which the spin degree of freedom sare strongly coupled to the rotor with SGA [R5]so(3) L. Like strong

Nuclear Physics A 636 (1998) 47169

An algebraic representationof the particle-plus-rotor model ?

Hubert de Guise 1, David J. RoweDepartment of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada

Received 18 February 1998; accepted 11 March 1998

Abstract

We investigate, using group theoretical methods, the coupling of a single particle with spin s toan axially symmetric rigid rotor by a quadrupole-quadrupole interaction. c© 1998 Elsevier ScienceB.V.

PACS: 02.20.−a; 21.60.FwKeywords: Particle-plus-rotor models; Symplectic model; Symplectic transformation.

1. Introduction

The objective of describing nuclear collective states in terms of interacting neutronsand protons was largely achieved for even nuclei when it was shown that the collec-tive model, for quadrupole vibrations and rotations, could be embedded in the shellmodel [1]. This was made possible by expressing the collective model in terms of aspectrum generating algebra (SGA), namely the symplectic algebra, which has a micro-scopic realization as a subalgebra of shell model observables. The present paper is partof a sequence of investigations aimed at a parallel expression of the collective states forodd nuclei.

By expressing the collective model in algebraic terms, one greatly simplifies appli-cations of the collective model, already at the phenomenological level. This is because

? Supported in part by N.S.E.R.C. of Canada.1 Now at Centre de Recherches Mathematiques, Universite de Montreal, C.P. 6128 Succ. Centre-Ville,

Montreal, Quebec H3C 3J7, Canada.

0375-9474/98/$19.00 c© 1998 Elsevier Science B.V. All rights reserved.PII S0375-9474(98) 00 16 9- 9

NUPHA 3933

48 H. de Guise, D.J. Rowe / Nuclear Physics A 636 (1998) 47169

one can choose basis states which reduce suitable subalgebra chains, thereby diagonal-izing a large part of the collective model Hamiltonian and facilitating the computationof matrix elements of both the Hamiltonian and other observables of the model. Then,by finding representations of the collective model algebra in the space of the shellmodel, one endows the collective model with microscopic wave functions; i.e. one givesthe collective model a microscopic interpretation. This enables one to determine whatcollective model states are compatible with the microscopic many-nucleon structure ofnuclei. It also enables one, at least in principle, to derive collective model parametersfrom corresponding shell-model observables.

In this paper, we are concerned with the particle-plus-rotor model. Various couplingschemes are possible when a rotor is coupled to a particle in a rotationally invariantmanner. We first show that these coupling schemes are defined by corresponding sub-group chains. In a subsequent paper, we will consider the coupling of a microscopicrotor, described by the symplectic model, to an extra particle.

We start with the rotationally invariant Hamiltonian

H = Hrot + hs.p. − χQ · q , (1)

where

Hrot = AR2, (2)

hs.p. =p2

2m+ 1

2mω20r

2 + a l · s+ b l · l . (3)

Hrot is the Hamiltonian for an axially symmetric rigid rotor with angular momentum R,Q is the quadrupole tensor of the rotor and q is the single-particle quadrupole tensor,with components

qν =

√16π

5r2Y2ν(θ, ϕ) . (4)

This Hamiltonian has a SGA

g = [R5]so(3)R + sp(3,R)l + su(2)s , (5)

where [R5]so(3)R is a SGA for the rotor, sp(3,R)l is a SGA for the spatial dynam-ics of the particle and su(2)s is the spin algebra of the particle. The rotor algebra[R5]so(3)R is spanned by five commuting components of the quadrupole tensor Q andthree components of the rotor angular momentum R. The algebra sp(3,R)l contains alllinear combinations of the single-particle operators pipj , xixj and xipj + pjxi, wherexi; i = 1, 2, 3 and pi; i = 1, 2, 3 are the components of the position and momentumvectors r and p of the single particle. In particular, sp(3,R)l contains p ·p and r · r andall components of the single particle angular momentum l. Sp(3,R) has a number ofimportant subalgebras, including those of the “shell model” chain

sp(3,R) ⊃ u(3) ⊃ su(3) ⊃ so(3) . (6)

H. de Guise, D.J. Rowe / Nuclear Physics A 636 (1998) 47169 49

All the terms of the Hamiltonian H of Eq. (1) are either elements of the Lie algebra gor bilinear products of elements of g. Hence, eigenstates of H belong to unirreps of g.

A particle can be coupled to a rotor in many ways. Coupling schemes useful forcomputational purposes are ones whose basis states diagonalize important parts of theHamiltonian. Such coupling schemes and the quantum numbers needed to label basisstates are naturally associated with subalgebra chains of g, of which there are many. Wemention a few of the possibilities.

1.1. Weak coupling

If the coupling constant χ is sufficiently small or if the rotor has a small intrinsicquadrupole moment, the eigenstates of H approach those of the weak coupling limit.In this limit, R, l and s are all good quantum numbers and there is no mixing ofsingle-particle states from different spherical harmonic oscillator shells. Weak couplingignores all observables but the angular momenta, so that weakly coupled states aresimply angular momentum-coupled states. Thus, weak coupling reduces the subalgebrachains

so(3)l + su(2)s ⊃ su(2)j , so(3)R + su(2)j ⊃ su(2)I , (7)

where so(3)l, su(2)s, so(3)R, su(2)j and su(2)I are the angular momentum algebrasspanned, respectively, by the components of l, s, R, j = l + s, and I = R + j. Notethat so(3) is isomorphic to su(2).

Among the multiplets of weakly coupled states, which are degenerate in the χ = 0limit, the most interesting are the so-called rotationally aligned states for which I = R+j.Because of Coriolis decoupling, these states are particularly important at high angularmomentum I and when the single particle angular momentum j is large.

1.2. Intermediate coupling

Intermediate coupling occurs when the interaction −χQ·q is small compared to energydifferences between single particle states but large compared to energy differences ofthe rotor (at sufficiently low angular momentum). We then have a variety of possiblecoupling schemes in which the particle is strongly coupled to an adiabatic rotor; thefollowing are three special cases.

1.2.1. Strong j couplingIn strong j coupling [2], the angular momentum j of the particle is a good quantum

number. This coupling scheme reduces the subalgebra chain

[R5]so(3)R + su(2)j ⊃ [R5]su(2)I ⊃ su(2)I . (8)

This coupling makes use of the fact that the direct sum Lie algebra [R5]so(3)R+su(2)j,spanned by the quadrupole moments and angular momentum operators Qν, Rk, jk, has


a subalgebra, [R5]su(2)I , spanned by the subset of operators Qν, Ik = Rk + jk. It alsomakes use of the fact that the rotor algebra [R5]so(3), like so(3), has representationscomprising states of half odd integer spins.

1.2.2. Strong l couplingIn strong l coupling [2], the primary coupling is between the rotor and the orbital

angular momentum l of the extra particle. Strong l coupling reduces the subalgebrachains[

R5]

so(3)R + so(3)l ⊃[R5]

so(3)L , (9)[R5]

so(3)L + su(2)s ⊃[R5]

su(2)I ⊃ su(2)I , (10)

where [R5]so(3)L is the algebra spanned by the operators Qν, Lk = Rk + lk. It is adouble coupling in which the spin degree of freedom s are strongly coupled to the rotorwith SGA [R5]so(3)L. Like strong j coupling, strong l coupling makes use of the factthat the rotor algebra [R5]so(3) has spinor representations.

1.2.3. Strong su(3) couplingStrong su(3) coupling is essentially the coupling scheme in which Nilsson model

states, computed with suppression of major harmonic oscillator shell mixing, are stronglycoupled to a rotor. This limit was treated algebraically in a previous paper [3]. It usesthe fact that a rotor can be regarded as an asymptotic su(3) irrep; i.e.,

[R5]so(3)R ≡ lim su(3)R . (11)

To see this, consider the su(3) operators Qsu(3)ν , Lα, where Qsu(3)

ν is an su(3)quadrupole moment and Lα is an angular momentum, and rescale the su(3) quadrupolemoments such that

Qsu(3)ν → Qν = Qsu(3)

ν /√Λ , (12)

where Λ = 4(λ2 + σ2 + λσ + 3λ + 3σ) is the eigenvalue of the second order su(3)Casimir operator when acting on any state of a representation (λ, σ). In the limit inwhich λ (and hence Λ) goes to → ∞, the commutation relations for the rescaledoperators become [4][

Qν, Qµ

]=

1Λ

[Qsu(3)ν , Qsu(3)

µ

]=

1Λ

3√

10(2 ν; 2µ|1 ν + µ

)Lν+µ → 0 as Λ→∞ ,[

Qν, Lα]

=−1√Λ

√6(2α; 1 ν|2α+ ν

)Qsu(3)α+ν

=−√

6(2α; 1 ν|2α+ ν

)Qα+ν ,[

Lν, Lµ]

=−√

2(1 ν; 1µ|1 ν + µ

)Lν+µ , (13)

i.e. they contract to the commutation relations of the rigid rotor algebra [R5]so(3).


With this contraction, we obtain a coupling scheme which reduces the subalgebrachain

[R5]so(3)R + su(3)l + su(2)s ≡ limΛ→∞

su(3)R + su(3)l + su(2)s

⊃ limΛ→∞

su(3)L + su(2)s ⊃ [R5]su(2)I ⊃ su(2)I . (14)

1.3. Strong coupling

This limit occurs when the coupling interaction χQ · q is dominant. The couplingscheme that emerges reduces the subalgebra chain

[R5]so(3)R + sp(3,R)l + su(2)s ⊃ [R5]su(2)I ⊃ su(2)I . (15)

It corresponds to a rotationally invariant version of the Nilsson model. The primaryobjective of the present paper is to give an algorithm for implementing strong couplingin a rotationally invariant manner.

2. The strong coupling scheme

2.1. Basis states for a rotor

To describe the dynamics of a rotor, we must construct a unirrep of its spectrumgenerating algebra, [R5]su(2). This is accomplished by the method of induced repre-sentations [5].

In the context of the rotor model, the inducing construction can be seen as a sys-tematic algorithm for implementing the traditional Bohr-Mottelson construction. In bothapproaches, one makes use of two frames of references: a laboratory (or space-fixed)frame and an intrinsic (or body-fixed) frame which rotates with the rotor.

The first step of the construction is to choose an irrep ρ of the R5 subalgebra. Thissubalgebra is spanned by the five quadrupole moment operators Qν , ν = ±2,±1, 0.Since R5 is Abelian, the irrep ρ is one-dimensional and corresponds to an assignmentof numerical values Qν , ν = ±2,±1, 0 to each of the quadrupole operators Qν, i.e.

ρ(Qν) = Qν . (16)

The quadrupole moments Qν are interpreted as moments relative to an intrinsic (i.e.body-fixed) frame. We assume that an intrinsic frame for the rotor is chosen such that

Qν = Q0δν,0 + Q2(δν,2 + δν,−2) . (17)

The next step in the construction of an [R5]su(2)J unirrep is to identify the intrinsicsymmetry group S of the rotor. This group is the subgroup of rotations that leave theintrinsic quadrupole moments invariant, i.e.

S = ω ∈ SU(2)|R(ω) Qν R(ω−1) = Qν , (18)


where

R(ω) Qν R(ω−1) =∑µ

QµD2µν(ω) . (19)

One sees that, if both Q0 and Q2 are non-zero, the rotor is triaxial and S is thegroup D2 of rotations through multiples of π about any of its axes, i.e. D2 comprisesΓ(±πy) = R(0,±π, 0) , Γ(±πz) = R(±π, 0, 0), and all products of these two ele-ments, with R(α, β, γ) denoting an element of SU(2) parametrized by the three Eulerangles (α, β, γ). However, if Q0 3 0 but Q2 = 0, the rotor has a symmetry axis andS is the group D∞ comprising rotations about the symmetry axis and rotations throughangle π about perpendicular axes, i.e. D∞ contains Γ(γz) = R(0, 0,±γ) , Γ(±πy) =R(0,±π, 0) and all their products. Rotors with D∞ intrinsic symmetry are often referredto as symmetric tops.

A two-dimensional irrep ΓK of D∞ with basis states ϕK, ϕK is defined, for 2K apositive integer, by the equations

ΓK(γz)ϕK = e−iγKϕK , ΓK(γz)ϕK = eiγKϕK , (20)

ΓK(πy)ϕK = ϕK , ΓK(πy)ϕK = (−1)2KϕK , K > 0 . (21)

D∞ also has one-dimensional irreps with K = 0 and basis state ϕε0 that satisfy

Γ0(γz)ϕε0 = ϕε0 , Γ0(πy)ϕε0 = ϕε0≡ εϕε0 , (22)

with ε = ±1.Thus, a symmetric top rotor has an intrinsic state ϕ0 (if K = 0), or a pair of intrinsic

states ϕK, ϕK if (K > 0), which are eigenstates of the quadrupole operators withidentical eigenvalue Q0;

Qν ϕK = δν,0Q0ϕK , Qν ϕK = δν,0Q0ϕK . (23)

Note that, since the intrinsic quadrupole moments ρ(Qν) = Qν are invariant under thesubgroup S of SU(2), the two irreps ρ of R5 and ΓK of S are compatible one with theother and give an irrep of the semi-direct product group [R5]S. It remains to induce anirrep of [R5]SU(2) from this [R5]S irrep.

Consider a general product of intrinsic and rotational wave functions of the form

ΨIηKM(Ω) = ϕK

∑N

aηIKN DINM(Ω) + ϕK∑N

aηIKNDINM(Ω) , (24)

where Ω ∈ SU(2) is the rotation which defines the orientation of the body-fixed axesrelative to the laboratory axes, and where η denotes all additional quantum numbersneeded to label the state. It is known from Mackey’s theory that, if the aηIKN coefficientsare chosen such that the ΨI

ηKM wave functions satisfy the consistency equation

ΓK(ω)ΨIηKM(Ω) = ΨI

ηKM(ωΩ) , ω ∈ S , (25)


where

ΓK(ω)[ϕKDINM(Ω)] = [ΓK(ω)ϕK] DINM(Ω) , (26)

then the wave functions ΨIηKM are a basis for an irreducible [R5]su(2)I representation.

For a symmetric top rotor we have the familiar wave functions:

ΨIKM(Ω) =

√2I + 1

16π2(1 + δK0)ϕKDIKM(Ω) + (−1)I+KϕKDI−KM(Ω) , K > 0 , (27)

where the relative phase of the two terms is determined from the self-consistencycondition. They are labeled by the quantum number K, which can be taken to be non-negative since the wave function ΨI

−KM, with K > 0, differs from ΨIKM by only a

phase factor. Note that if K = 0, the irrep of the intrinsic symmetry group becomesone-dimensional with ϕ0 = εϕ0 and ε = ±1. We then regain the familiar result that aK = 0 band comprises a I = 0, 2, 4, . . . sequence of states, if ε = 1, and a I = 1, 3, 5, . . .sequence, if ε = −1.

2.2. Strongly coupled particle-plus-rotor states

We now consider wave functions for a rotor-plus-particle system in the limit in whichthe inertial parameter A in the Hamiltonian of Eq. (3) is small relative to the strength(in suitable units) of the particle-rotor interaction −χQ · q. In this limit, eigenfunctionsof H can be found which reduce the subalgebra chain (15).

When A is small, it is appropriate to replace the angular momentum of the rotor Rby the difference I − f = R and express H in the form

H = A I · I + h+ V , (28)

where

h = hs.p. − χQ · q

=p2

2m+ 1

2mω20 r

2 + a l · s+ b l · l− χ Q · q , (29)

and

V = A f · (f− 2I) . (30)

We obtain strong coupling of the particle to the rotor when A is small and the Coriolisinteraction, −2A f · I, can either be neglected or treated as a first order perturbation. Forthis reason, the Coriolis interaction is known as a decoupling interaction.

All terms in the Hamiltonian H are rotationally invariant. Moreover, h is independentof the total angular momentum. Thus, its energy levels can be determined in any frameof reference. In particular, they can be determined in the intrinsic frame of the rotor in


which the components of the quadrupole tensor Q are replaced by their intrinsic values:Qν → ρ(Qν) = Qν = δν0Q0. In this frame

h→ h = hs.p. − χQ · q = hs.p. − χQ0 (2z 2 − x2 − y2) , (31)

and h can be identified with the Nilsson model Hamiltonian

h =p2

2m+ 1

2m(ω2⊥x

2 + ω2⊥y

2 + ω2z z

2) + a l · s+ b l · l , (32)

where

ω2⊥ = ω2

0 (1 + 2α) , ω2z = ω2

0 (1− 4α) , (33)

and α is the dimensionless parameter

α =χQ0

mω20

. (34)

When h is replaced by h, it ceases to be rotationally invariant. However, it remainsinvariant under Dj

∞ ⊂ SU(2)j and has the dynamical subgroup chain

Sp(3,R)l + SU(2)s ⊃ Dj∞ ⊃ U(1)j . (35)

Eigenstates pηk,pηk of h can then be defined which reduce the subgroup chain ofEq. (35) and satisfy the equations

hpηk = εηkpηk , hpηk = εηkpηk , (36)

Γk(γz)pηk = e−iγkpηk , Γk(γz)pηk = eiγkpηk , (37)

Γk(πy)pηk = pηk , Γk(πy)pηk = (−1)2kpηk . (38)

Basis states for the combined rotor-plus-particle system, which span irreps of [R5]SU(2)I and reduce the subgroup chain

[R5]SO(3)R × Sp(3,R)l × SU(2)s ⊃ [R5]SU(2)I ⊃ SU(2)I , (39)

can now be constructed by inducing from irreps of [R5]DI∞ in the chain

[R5]DR∞ × Dl

∞ × Ds∞ ⊃ [R5]DR

∞ × Dj∞ ⊃ [R5]DI

∞ , (40)

where DR∞ ⊂ SO(3)R is the intrinsic symmetry group of the rotor and Dj

∞ ⊂ SU(2)jis the intrinsic symmetry group of the particle.

With the help of the D∞ branching rules and coupling coefficients to be found inAppendix B, we find that, if ϕ0 is an intrinsic state for a (K = 0) band of the rotorcore, we obtain strongly coupled wave functions ΨI

ηkM for the particle plus rotor inthe standard rotor model form with

ΨIηkM(Ω) =

√2I + 116π2

ϕηkDIkM(Ω) + (−1)I+kϕηkDI−kM(Ω) , k > 12 , (41)


where k = |k| is a good quantum number, and where, according to Eq. (B.13), ϕηk =ϕ0 · pηk and ϕηk = ϕ0 · pηk span a 2-dimensional representation of D∞, with

Γk(γz)ϕηk = e−iγkϕηk , Γk(γz)ϕηk = eiγkϕηk ,

Γk(πy)ϕηk = ϕηk , Γk(πy)ϕηk = (−1)2kϕηk . (42)

3. Coupling coefficients for intermediate couplings

The schemes for coupling a particle to a rotor can be expressed in terms of grouptheoretically defined coupling coefficients. There are several couplings for which the co-efficients have analytic expressions derivable by group theoretical means. These include:weak coupling and the several intermediate-coupling limits defined in Section 1. As weshow in the following sections, we can also derive analytic coupling coefficients for theasymptotic limit of strong coupling.

For weak coupling, the rotor-plus-particle states are simply SU(2)-coupled;

[ΦR × pηlj]IM =∑Mrm

(RMr; j m|I M

)ΦRMr pη ljm . (43)

Thus, the coupling coefficients (RMr; j m|I M

) are SU(2) Clebsch Gordan coeffi-

cients. For other coupling schemes, the coefficients are products of SU(2) coefficientsand reduced coupling coefficients. The reduced coupling coefficients CRlj

ηI for an ar-bitrary state |ηIM〉 are then given by the expansion

ΨηIM =∑Rlj

CRljηI [ΦR × pη lj]IM . (44)

3.1. Strong-j coupling

For strong-j coupling, we start with Eq. (41) but replace ϕηk by the D∞ productwave function

ϕηljk = pηljk · ϕ+0 , (45)

where pηljk is a single-particle state of angular momentum jk and ϕ+0 is the intrinsic state

for the core. (N.B., the group D∞ has two K = 0 irreps, Γ±0 , cf., Appendix B, where thesuperscript ± indicates that the basis states for the two representations are, respectively,even and odd under rotation through angle π about the y axis.) The correspondingcoupled wave functions are then given [6,2] by

ΨηljkIM =∑R

√2 (2R+ 1)(2I + 1)

(R 0; j k|I k

)[ΦR × pηlj]IM , k > 1

2 . (46)

Once again, we can restrict to k positive because the wave functions with k < 0 differfrom the former by only a phase factor. For rigid rotor core states of a Kπ = 0+ band,the sum extends over even values of R only. Thus, we determine that


CRljηkI =

√2 (2R+ 1)(2I + 1)

(R 0; j k|I k

). (47)

3.2. Strong-l coupling

For strong-l coupling, the possible intrinsic states are constructed as follows. We firstcouple the rotor to the orbital part of the single particle wave function. In accordancewith Eq. (B.12), the D∞ coupling decomposes as Γk⊗ Γ+

0 = Γk (k > 0) with intrinsicstates pηlk · ϕ+

0 , pηlk · ϕ+0 , where the single particle state pηlk has integral orbital

angular momentum quantum numbers lk.These intrinsic states are then coupled to states ζsks of spin angular momentum s.

Whenever s is a half odd integer (as we assume in this paper), i.e. s = 12 ,

32 ,

52 etc., the

possible values of ks are also half odd integer, so that it is impossible to have ks = kl.The appropriate D∞ couplings are therefore

Γks ⊗ Γk =

Γks if k = 0 ,

Γks+k ⊕ Γ|ks−k| if k>0 .(48)

Using Eqs. (B.5), (B.6) or (B.7), we find that the appropriate intrinsic states aregiven by

ϕη(lk)sK =

ζsks ·

[pηlk · ϕ+

0

]if K = k+ ks ,

ζsks ·[pηlk · ϕ+

0

]if K = ks − k > 0

ζsks ·[pηlk · ϕ+

0

]if K = k− ks > 0 ,

(49)

with ϕη(lk)sK obtained using ΓK(πy)ϕη(lk)sK = ϕη(lk)sK , along with pηl0 = (−1)l pηl0if k = 0.

The three cases can be handled simultaneously if we allow the intermediate labels kand ks to take negative, as well as positive, values. In the last case of Eq. (49), forexample, one would have ks < 0 and k > 0. This allows us to write all lab frameexpressions of strong-l wave functions in the common form

Ψη(lk)sKIM =∑RL

√2 (2R+ 1)(2I + 1)

(R 0; l k|Lk

)(Lk; s ks|I K

)×[[ΦR × pηl]× ζs]IM , K = k+ ks > 0 . (50)

It follows that the coupling coefficients for strong-l coupling are given by

CRljηkKI =

∑Lj

√2 (2R+ 1)(2L+ 1)(2j + 1)

(2I + 1)

×(R 0; l k|Lk

) (Lk; s ks|I K

)W(RlIs;Lj) , (51)

where W(RlIs;Lj) is a Racah recoupling coefficient.


3.3. Strong-su(3) coupling

Strong-su(3) coupling is based on the observation that the coupling of a rotor and ansu(3) particle is given by the Λ →∞ limit of the su(3) coupling (Λ, 0)⊗ (λ, 0) fora single particle in the shell λ. From the work of Elliott [7], it is known that only thehighest weight state of an su(3) representation is required to generate the states of arotor band. Thus, we first consider the highest weight states for the su(3) irreps in thetensor product

(Λ, 0)⊗ (λ, 0) =∑σ′

(Λ+ λ− 2σ′, σ′) . (52)

Let φnz ,nx,ny denote a single-particle (spherical harmonic oscillator) state with (nx, ny,nz) quanta associated with the (x, y, z) directions, respectively. From previous work [8]we know that the simple product φλ−σ,σ,0 · ϕ+

0 , where ϕ+0 is a rotor intrinsic state, is a

highest weight state for the su(3) representation (Λ+λ−2σ,σ) in the Λ→∞ limit. Thestate φλ−σ,σ,0 is an eigenstate of the su(3) quadrupole moment qsu(3)

0 with eigenvalue(2λ − 3σ). It has also been shown [7] that, in the Λ → ∞ limit, the representation(Λ+λ− 2σ,σ) becomes indistinguishable from a sum of axially symmetric rigid rotorreps, i.e.

limΛ→∞

(Λ+ λ− 2σ,σ) =∑kL

ΓkL , kL = σ ,σ− 2 , σ− 4 , 0 or 1 , (53)

with ε = (−1)λ when kL = 0 occurs.It follows from this that the asymptotic su(3) highest weight state φλ−σ,σ,0 ·ϕ+

0 mustbe a sum of kL > 0, D∞ states. Since ϕ+

0 carries a 1-dimensional irrep of D∞, we canuse the branching rules of Appendix B to express [3] φλ−σ,σ,0 as a sum of such states.This decomposition can be inferred if we first expand φλ−σ,σ,0 in a complete set ofangular momentum states

φλ−σ,σ,0 =λ∑

k=−λ

λ∑l=0

pλlk(ω0) 〈pλlk(ω0)|φλ−σ,σ,0〉 , (54)

and define Θλσk , Θλσk , k > 0 by

Θλσk =∑l

pλlk(ω0) 〈pλlk(ω0)|Θλ−σ,σ,0〉 ,

Θλσk = (−1)λ−k∑l

pλl,−k(ω0) 〈pλl,−k(ω0)|φλ−σ,σ,0〉 , k > 0 . (55)

(The argument ω0 of pλlk(ω0) specifies the original frequency of the shell modelpotential (cf., Eq. (29).) It can then be verified, using the elements of D∞, thatΘλσk , Θλσk span a two-dimensional representation Γk of D∞ when k > 0, whereasΘλσ0 is a basis vector for a 1-dimensional irrep Γ0 with ε = (−1)λ.


The branching rule of Eq. (53) is satisfied because the overlaps

〈pλl,−k(ω0)|φλ−σ,σ,0〉 = (−1)σ−k〈pλlk(ω0)|φλ−σ,σ,0〉 , k > 0 , (56)

which were evaluated in Ref. [8], are different from zero only when k is one of thepossible values of kL, i.e. k = σ,σ− 2 etc. Therefore,

〈pλl,−k(ω0)|φλ−σ,σ,0〉 = 〈pλlk(ω0)|φλ−σ,σ,0〉 , k > 0 . (57)

Thus, the asymptotic su(3) highest weight state φλ−σ,σ,0 · ϕ+0 is a sum of k > 0,

D∞-coupled intrinsic states

Θλσk · ϕ+0 , Θλσk · ϕ+

0 , k = σ,σ− 2, σ− 4, 0 or 1 . (58)

The overlaps 〈pλlk(ω0)|φλ−σ,σ,0〉 , k > 0, are given [8] by

〈pλlk(ω0)|φλ−σ,σ,0〉 =

√√√√ 2σ( σ12 (σ−k)

) √(λ− σ)!λ!σ!

F(λ, σlk) , k > 0 , (59)

where

F(λ, σlk) = σ! 2k

√ 1

2 (λ− l)!(l+ k)! (l− k)! 12 (λ+ l)! λ! 2l

(λ+ l+ 1)!

×

12 (σ+k)∑q=k

(−1)q

22qq!(q− k)!(l+ k− 2q)! 12 (σ+ k)− q!q− 1

2 (σ+ k) + 12 (λ− l)!

.

(60)

With the inclusion of spin, the relevant K > 0, D∞ states for the coupling of a particlein shell λ to a rigid rotor are given by

ϕλσksK =

∑l

ζsks ·[pλlk(ω0) · ϕ+

0

]· 〈pλlk(ω0)|φλ−σ,σ,0〉 if K = k+ ks∑

l


0

]· 〈pλlk(ω0)|φλ−σ,σ,0〉 if K = ks − k > 0∑

l


0

]· 〈pλlk(ω0)|φλ−σ,σ,0〉 if K = k− ks > 0 ,

(61)

where the order of the coupling reflects the fact that the asymptotic SU(3) coupling isto be done first, with the resultant coupled to spin.

The intrinsic state ϕλσksK is seen to be a sum of strong-l states. Thus, we obtain thelab frame wave functions

ΨλσksKIM =∑l

Ψλ(lk)sKIM 〈pλlk(ω0)|φλ−σ,σ,0〉 , K > 0 , (62)


where we follow the rules detailed in Section 3.2 to determine the signs of k andks required to evaluate 〈pλlk(ω0)|φλ−σ,σ,0〉 and the Clebsch1Gordan coefficients thatenter in the construction of strong-l states. It follows that the coupling coefficients forstrong-SU(3) coupling are given by

CRljλσksKI =

∑L

√2 (2R+ 1)(2L+ 1)(2j + 1)

2I + 1(R0; lk|Lk)(Lk; sks|IK)

×W(RlIs;Lj)〈pλlk(ω0)|φλ−σ,σ,0〉 . (63)

4. The asymptotic Nilsson basis

The eigenfunctions of the full Nilsson Hamiltonian, Eq. (32), are generally only par-tially defined by a dynamical subgroup chain. An exception occurs when the parametersa and b of the Hamiltonian are put equal to zero or when the term

a l · s+ b l · l (64)

in the Hamiltonian is treated as a first order perturbation. The Nilsson model wavefunctions then reduce the subgroup chains

Sp(3,R)l ⊃ U(1)× U(2)l ⊃ Dl∞ , SU(2)s ⊃ Ds

∞ , Dl∞ × Ds

∞ ⊃ Dj∞ , (65)

and there are no missing quantum numbers. The basis that results is the well-knownasymptotic Nilsson basis.

4.1. The Hamiltonian

When using the asymptotic basis, it is convenient to express the rotor-plus-particleHamiltonian of Eq. (1) in the form

H = Hrot. + h0 + V , (66)

with

Hrot. =A I · I ,

h0 =p2

2m+ 1

2mω20r

2 − χ Q · q , (67)

and

V = a l · s+ b l · l+ A (f · f− 2I · f) . (68)

The components Hrot and h0 are then diagonal in the asymptotic basis.


4.2. The asymptotic basis as a deformed spherical basis

The Hamiltonian Hrot. + h0 commutes with the total orbital angular momentum L =R + l operator for the system; this makes it possible to construct eigenstates of goodtotal orbital angular momentum.

One starts by diagonalizing h0 in the intrinsic frame of the rotor. In this frame,the quadrupole moments of the rotor are replaced by their intrinsic values; i.e. Qν →ρ(Qν) = Qν . We suppose that the intrinsic quadrupole moments are those of an axiallysymmetric Kπ

R = 0+ rotor. Thus, Qν = δν0Q0 and

h0 → hcyl. =p2

2m+ 1

2mω20r

2 − χQ0 (2z 2 − x2 − y2) , (69)

In terms of the familiar creation and destruction operators

a†j =

√mω0

2~

(xj −

ipjmω0

), aj =

√mω0

2~

(xj +

ipjmω0

), (70)

this gives

hcyl. = ~ω0[(1− 2α)(a†z az + 1

2 )− α (a†z a†z + az az)

]+~ω0

[(1 + α)(a†xax + 1

2 ) + 12α (a†xa

†x + axax)

]+~ω0

[(1 + α)(a†yay + 1

2 ) + 12α (a†ya

†y + ayay)

]. (71)

The Hamiltonian hcyl. is an element of the single particle sp(3,R)l algebra (cf. Ap-pendix A). Moreover, hcyl. is diagonalized by an Sp(3,R)l group transformation of thespherical basis. Since hcyl. is a sum of three simple harmonic oscillator Hamiltonians,the transformation O which diagonalizes hcyl. is carried out by eliminating the shellmixing terms for each of the three components separately. The desired transformationis of the form

O = Sz (ε)Sy(δ)Sx(δ) , (72)

where

Sj(β) = e−12β(a†j a

†j −ajaj) . (73)

O is an asymmetric scale transformation in which the parameters ε and δ are fixed bythe requirement that the mixing of different single particle shells is suppressed in thetransformed Hamiltonian Ohcyl.O−1. One also notes that O is D∞ invariant; thus, itpreserves the D∞ invariance of hcyl.

We determine that

Sj(β) a†j Sj(−β) = a†j coshβ + aj sinhβ ,

Sj(β) aj Sj(−β) = aj coshβ + a†j sinhβ . (74)

Thus, under the action of O, which maps an operator X 7→ O XO−1,


a†z a†z 7→ a†z a

†z cosh2 ε+ az az sinh2 ε+ 2(a†z az + 1

2 ) cosh ε sinh ε ,

az az 7→ az az cosh2 ε+ a†z a†z sinh2 ε+ 2(a†z az + 1

2 ) cosh ε sinh ε ,

(a†z az + 12 ) 7→ (a†z az + 1

2 )(cosh2 ε+ sinh2 ε) + (a†z a†z + az az) cosh ε sinh ε . (75)

Similar formulae hold for the x and y operators with the replacement ε→ δ.To evaluate the parameters ε and δ, we use Eq. (71) to obtain O hcyl.O−1 explicitly.

Requiring that the shell mixing terms vanish then leads to the equations for ε and δ:

12 (1− 2α) sinh 2ε− α cosh 2ε = 0 , 1

2 (1 + α) sinh 2δ+ 12α cosh 2δ = 0 . (76)

These are easily solved to give

eε =1

[1− 4α]1/4=√ω0

ωzand eδ =

1[1 + 2α]1/4

=√ω0

ω⊥. (77)

The transformed Hamiltonian is diagonal in the basis of eigenstates of a harmonicoscillator with frequency ω0,

Ohcyl.O−1 = ~ω0

[√1− 4α(a†z az + 1

2 ) +√

1 + 2α(a†xax + a†yay + 1)]. (78)

Thus, as a result of the transformation, O hcyl.O−1 has become u(1) + u(2) invariant,where u(1) is the Lie algebra spanned by a†z az and u(2) is spanned by a†i aj, i, j =x, y.Ohcyl.O−1 can be related to the strong-su(3) problem if we rewrite

O hcyl.O−1 = ~ω013 (√

1− 4α+ 2√

1 + 2α) (a†z az + a†xax + a†yay + 32 )

+ 13 (√

1− 4α−√

1 + 2α) qsu(3)0 , (79)

where

qsu(3)0 = ~ω0 (2a†z az − a†xax − a†yay) (80)

is the ν = 0 component of the su(3) quadrupole tensor. Thus, in the intrinsic frame, thetransformed Hamiltonian is the strong-su(3) Hamiltonian of Ref. [3] plus a diagonal,spherically symmetric term.

4.3. Asymptotic strong-coupling coefficients

4.3.1. Intrinsic statesWe have shown in Section 3.3 that to generate strong-SU(3) particle-plus-core states,

it is sufficient to combine the intrinsic state of the rotor with a single particle U(3)state φp−σ,σ,0 which is an eigenstate of qsu(3)

0 . Since, as pointed out earlier, O hcyl.O−1

is basically a strong-su(3) Hamiltonian, the particle-plus-core states for hcyl. are tobe inferred from its eigenstates O−1φp−σ,σ,0. Thus, as in Eq. (55), we define theasymptotic intrinsic states ξpσk , ξpσk for the particle by


ξpσk =∑λl

pλlk(ω0) 〈pλlk(ω0)| O−1 |φp−σ,σ,0〉 ,

ξpσk = (−1)p−k∑λl

pλl,−k(ω0) 〈pλl,−k(ω0)| O−1 |φp−σ,σ,0〉 ,

k = σ,σ− 2, . . . , 1 or 0 , (81)

where pλlk(ω0) is an eigenstate of a spherical harmonic oscillator of frequency ω0.When combined with spin wave function and the wave function of the rotor, the

strongly coupled intrinsic states are given by the immediate generalization of Eq. (61).For instance, with K = k+ ks, we find

ϕpσksK = ζsks · [ξpσk · ϕ+0 ] =

∑λl


0 ] 〈pλlk(ω0)| O−1 |φp−σ,σ,0〉 .

(82)

Hence, a lab frame wave function with this intrinsic state is given by

ΨpσksKIM =∑λl

Ψλ(lk)sKIM 〈pλlk(ω0)| O−1 |φp−σ,σ,0〉 , (83)

and the corresponding coupling coefficients are

CRljλσksKI =

∑L

√2 (2R+ 1)(2L+ 1)(2j + 1)

2I + 1(R0; lk|Lk)(Lk; sks|IK)

×W(RlIs;Lj)〈pλlk(ω0)| O−1 |φp−σ,σ,0〉 . (84)

4.3.2. Overlap integralsFirst, we claim that

(−1)l+k〈pλl,−k(ω0)| O−1 |φp−σ,σ,0〉 = (−1)p−σ〈pλlk(ω0)| O−1 |φp−σ,σ,0〉 , (85)

k> 0 .

For this, we note that

(−1)l−k〈pλlk(ω0)| O−1 |φp−σ,σ,0〉 = 〈pλl,−k(ω0)|Ry(π)O−1 |φp−σ,σ,0〉= 〈pλl,−k(ω0)| O−1 Ry(π)|φp−σ,σ,0〉 , (86)

since O−1, being D∞-invariant, commutes with Ry(π) ∈ D∞. If we now write

Ry(π)|φp−σ,σ,0〉 =Ry(π)(a†z)p−σ√(p − σ)!

(a†y)σ

√σ!|φ000〉

=

[Ry(π)

(a†z)p−σ√(p − σ)!

R−1y (π)

](a†y)

σ

√σ!

Ry(π)|φ000〉 , (87)

then, since

Ry(π) a†y R−1y (π) = a†y , (88)


and a†z transforms as the m = 0 component of an l = 1 tensor operator, we find

Ry(π)(a†z)p−σR−1y (π) = (−1)p−σ(a†z)p−σ , (89)

which completes the proof of the claim.As a result of the claim, we need only evaluate 〈pλlk(ω0)| O−1 |φp−σ,σ,0〉 for k > 0.

To this end, it is convenient to rewrite O as a z -scaling transformation, Sz (ε−δ), timesa rotationally invariant volume change V(δ), i.e.

O = Sz (ε− δ)V(δ) , (90)

where

V(δ) = e−12 δ∑

i(a†i a

†i −aiai) and Sz(ε− δ) = e−

12 (ε−δ) (a†z a

†z−az az ) . (91)

Note that the transformations V(δ) and Sz(ε− δ) commute with one another. Thus weobtain

〈pλlk(ω0)|O−1|φp−σ,σ,0〉 = 〈pλlk(ω0)|V−1(δ)S−1z (ε− δ)|φp−σ,σ,0〉 ,

=∑q

〈pλlk(ω0)|V−1(δ) |φq−σ,σ,0〉

×〈φq−σ,σ,0|S−1z (ε− δ)|φp−σ,σ,0〉 ,

=∑q

〈pλlk(ω0)|V−1(δ) |pqlk(ω0)〉〈pqlk(ω0)|φq−σ,σ,0〉

×〈φq−σ,σ,0|S−1z (ε− δ)|φp−σ,σ,0〉 , (92)

where we have used Eq. (59) and the fact that V commutes with the components of l.The matrix elements 〈φq−σ,σ,0|Sz(ε − δ)|φp−σ,σ,0〉 can be evaluated by observing

that the three binomials

J+ = 12 a†z a†z , J0 = 1

4 (a†z az + az a†z) and J− = 1

2 az az (93)

are the generators of an su(1,1) subalgebra.Following Ui [9], we have

〈J M ′|e−14 β (a†a† − aa)|J M〉 = (−1)M

′−MdJM ′M(β) , (94)

where dJM ′M(β) is an su(1,1) d-function. Depending on whether p − σ is even orodd, we are dealing with the unirrep J = 1

4 or 34 of su(1,1) and we find that

Aq,p,σ(ε− δ)≡ 〈φq−σ,σ,0|S−1z (ε− δ)|φp−σ,σ,0〉,

= 〈φq−σ,σ,0|e12 (ε−δ) (a†z a

†z−az az )|φp−σ,σ,0〉,

=

(−1)M′z−Mz d

1/4M ′

z Mz(2(δ− ε)) if p − σ is even,

(−1)M′z−Mz d

3/4M ′

z Mz(2(δ− ε)) if p − σ is odd,

(95)


where

M ′z = 1

2 (q− σ+ 12 ), Mz = 1

2 (p − σ+ 12 ) , eδ−ε =

[1− 4α1 + 2α

]1/4

=√ωz

ω⊥. (96)

Ui [9] has also given the analytic expression for the su(1,1) d-function:

dJM ′M(β) =

√Γ(M ′ + J )Γ(M ′ −J + 1)Γ(M+ J )Γ(M−J + 1)

1Γ(M ′ −M+ 1)

×(cosh 12β)−2J+M−M ′

(sinh 12β)M

′−M

×2F1(−M+ J ,M ′ + J ;M ′ −M+ 1; tanh2 12β) , (M ′ >M) ,

= (−1)M′−MdJMM ′(β) , (M >M ′) ,

(97)

where 2F1(a, b; c; z) is a hypergeometric function.The matrix elements 〈pλlk(ω0)| V(δ)−1 |pqlk(ω0)〉 are also related to su(1,1) d-

functions. This time, however, the relevant su(1,1) subalgebra is spanned by the rota-tionally invariant operators

J+ = 12

∑i=x,y,z

a†i a†i , J0 = 1

4

∑i=x,y,z

(a†i ai + aia

†i

), J− = 1

2

∑i=x,y,z

aiai . (98)

To discover which representation is appropriate, consider the state pllk(ω0) of angularmomentum l with λ = l quanta. Since J− commutes with l and J− removes two quantafrom any state, J−pllk(ω0) is proportional to a state with l − 2 quanta and angularmomentum l. But there is no such state. Thus, J−pllk(ω0) = 0, implying that pllk(ω0)is a lowest weight state for a representation of su(1,1) labeled by the eigenvalueJ = 1

2 (l+ 32 ) of J0. Furthermore, since J+ and J0 both commute with l, states sharing

common lk labels belong to the same su(1,1) unirrep. The su(1,1) weight M of thestate pλlk(ω0) is just the eigenvalue of J0, i.e. M = 1

2 (λ + 32 ), so that we have the

correspondence pλlk(ω0)↔ |J M k〉 with k distinguishing multiple occurrences of thesu(1,1) unirrep J in our Hilbert space of harmonic oscillator states.

Thus, we have

〈pλlk(ω0)|V−1(δ)|pqlk(ω0)〉 = (−1)M′−MdJM ′M(−2δ) , (99)

where

J = 12 (l+ 3

2 ) , M ′ = 12 (λ+ 3

2 ) , M = 12 (q+ 3

2 ) , e−δ = [1 + 2α]1/4 ,

(100)

and the d-function is given in Eq. (97). Note that the overlap

〈pλlk(ω0)|V−1(δ)|pqlk(ω0)〉

does not depend on the “multiplicity” label k.


The final expression for 〈pλlk(ω0)| O−1 |φp−σ,σ,0〉 is therefore

〈pλlk(ω0)| O−1 |φp−σ,σ,0〉 =∑q

(−1)M′−MdJM ′M(−2δ) 〈pqlk(ω0)|φq−σ,σ,0〉

×Aq,p,σ(ε− δ) , (101)

where 〈pλlk(ω0)|φq−σ,σ,0〉 is given by Eq. (59).

Note: The phases of the d-function (−1)M′−MdJM ′M(−2δ) for the rotationally in-

variant operators of Eq. (98) and of the overlap 〈pqlk(ω0)|φq−σ,σ,0〉 are consistent withthe su(3) states defined by Sharp et al. [11]. If one uses the harmonic oscillator states of[12] or [13], then the change 〈pqlk(ω0)|φq−σ,σ,0〉 → (−1)(q−l)/2〈pqlk(ω0)|φq−σ,σ,0〉should be made.

Coming back to the claim made in Eq. (85), we see that the phases are such that

〈pλl,−k(ω0)| O−1 |φp−σ,σ,0〉 = 〈pλlk(ω0)| O−1|φp−σ,σ,0〉 , k > 0 , (102)

since, if p is even (resp. odd), only even (resp. odd) values of λ will occur, which inturns implies that only even (resp. odd) values of l will occur, while non-zero overlaps〈pλlk(ω0)|φq−σ,σ,0〉 only occur when k − σ is even. This completes the calculation ofall overlaps required for the construction of the strong symplectic wave functions.

4.4. Matrix elements in the asymptotic Nilsson basis

The computation of matrix elements of a l · s + b l · l and A R2

is straightforwardnow that we have explicit expressions for the asymptotic Nilsson basis in terms ofstates of good angular momentum with coefficients given by Eq. (101). The matrixelements simplify further when the operator under consideration commutes with the

volume transformation V(δ), as is the case for a l · s + b l · l and A R2. For then

O = Sz (ε− δ)V(δ) can be replaced by just Sz(ε− δ) and, from Eq. (83), we obtain,for example,

〈Ψp ′σ ′k ′sK ′I ′M ′ |a l · s+ b l · l |ΨpσksKIM〉

=∑λl

〈Ψλ(lk ′)sK ′I ′M ′ |a l · s+ b l · l |Ψλ(lk)sKIM〉

×[〈pλlk ′(ω0)|Sz(ε− δ)|φp ′−σ ′,σ ′,0〉∗ 〈pλlk(ω0)|Sz(ε− δ)|φp−σ,σ,0〉

]×δII ′δKK ′δMM ′ , (103)

with

〈Ψλ(lk ′)sKIM| l · s |Ψλ(lk)sKIM〉

=∑j

(s k ′s ; l k

′|j K) (s ks; l k|j K

)12 [j(j + 1)− l(l+ 1)− s(s+ 1)] . (104)

The su(2) Clebsch1Gordan coefficients in the above matrix element arise from therelation



0

]=∑j

(s ks; l k|j K

)ϕλljk · ϕ+

0 (105)

used convert strong-l states into a sum of strong-j states.

Matrix elements of R2

= I · I − d · d− 2 I · d are evaluated in a similar way with

〈Ψλ(lk ′)sK ′IM|I · d|Ψλ(lk)sKIM〉 =∑j

(s k ′s ; l k

′|j K) (s ks; l k|j K

)×〈ΨλljK ′IM| I · d |ΨλljKIM〉 , (106)

where 〈ΨλljK ′IM| I · d |ΨλljKIM〉 is a familiar Coriolis matrix element.

5. Conclusion

The strategy of this paper, that of expressing the rotationally invariant Hamiltonian

H = Hrot + hs.p. − χ Q · q

of Eq. (1) in terms of generators of the dynamical algebra g of Eq. (5), has allowed usto go beyond the standard methods of the Nilsson model. In particular, by identifyingsp(3,R)l as the dynamical algebra of the deformed as well as the spherical harmonicoscillator, it has become possible to diagonalize exactly the Hamiltonian A I · I + h0,where

h0 =p2

2m+ 1

2mω0r2 − χ Q · q ,

by a group transformation O. As a result, we have been able to express asymptoticNilsson model states as sums of angular momentum-coupled rotor-plus-particle states

ΨpσksKIM =∑λRlj

CRljλσksKI[ΦR × pλ lj]IM ,

with group theoretically defined coupling coefficients CRljλσksKI, given explicitly by

Eqs. (84) and (101).In arriving at this result, we were able to solve the problem of labeling angular

momentum states IM by using the index K, which labels irreps of the intrinsic symmetrysubgroup [R5]DI

∞ ⊂ G of the coupled rotor-plus-particle system. Multiple occurrencesof the irreps K are distinguished by considering the D∞ couplings of the subgroups inthe chains

[R5]SO(3)R⊃ [R5]DR∞ ,

SU(2)s⊃Ds∞ ,

Sp(3,R)l ⊃ U(1)× U(2)⊃U(1)× Dl∞ , (107)

which completely specify the construction of the various K irreps through the labels(pσks).


The problem of handling infinitely many states with given angular momentum labelslk, in the expansion of deformed single-particle wave functions, was solved by the useof the non-compact group SU(1,1) and its associated d-functions.

Although the formalism presented here was derived for the case of a single extranucleon coupled to an axially symmetric core, many extensions can be consideredwithout too much difficulty. The mathematical framework pertinent to these more generalcases has been discussed elsewhere [8].

Appendix A. Review of the symplectic algebra

We assume throughout this paper that we are dealing with harmonic series represen-tations of the non-compact symplectic algebra sp(3,R), which is spanned by the 21operators

Aij =A∑a=1

a†iaa†ja , Cij = 1

2

A∑a=1

(a†iaaja + ajaa†ia) , Bij =

A∑a=1

aiaaja , (A.1)

where a labels the particle number, A is the total number of particles, i, j = x, y, z labelthe directions of space and the operators a†ja and aja are defined in the usual way by

a†ja =

√mω0

2~

(xja −

ipjamω0

), aja =

√mω0

2~

(xja +

ipjamω0

), (A.2)

The harmonic series are unitary representations carried by subspaces of many-particleharmonic oscillator states. They are characterized by lowest but not highest weight states.

The Cartan subalgebra of sp(3,R) coincides with that of the u(3) subalgebra spannedby the nine Cij operators. Thus, sp(3,R) irreps are labeled by three integers, which canbe chosen to be N(Λ,µ). N is the eigenvalue of C11 +C22 +C33, which is equal to thenumber of quanta plus the zero point energy for A nucleons, 3A/2. (Λ,µ) are the usualsu(3) labels. The symplectic operators Aij and Bij are u(3) tensors of the type 2(2, 0)and −2(0, 2) respectively, and all the states of an sp(3,R) representation N(Λ,µ) canbe generated by laddering up with the Aij operators from the states of the lowest u(3)subspace, labeled by N(Λ,µ).

There are two one-particle harmonic series irrep; they are distinguished by parity. Fora single particle representation, the sums in (A.1) contain only a single term (i.e. A=1).Single particle operators will be denoted by lower case letters, so that we have

aij = a†i a†j , cij = 1

2 (a†i aj + aja†i ) , bij = aiaj . (A.3)

Single particle lowest weight states, i.e. states annihilated by bij , belong to the su(3)irrep (0, 0) or (1, 0). Hence, single particle irreps of sp(3,R) are labeled by 3

2 (0, 0)(positive parity) or 5

2 (1, 0) (negative parity). These representations contain, respec-tively, the u(3) shells (2n+ 3

2 )(2n, 0) and (2n+ 52 )(2n+ 1, 0) with n = 0, 1, 2, . . .


Appendix B. Branching rules and basis states for D∞ coupling

A generic irrep ΓK (K > 0) of D∞ is two-dimensional with basis states ϕK , ϕKthat satisfy Eqs. (20) and (21). D∞ also has one-dimensional irreps with K = 0 andbasis state ϕε0 that satisfy

Γ(γz)ϕε0 = ϕε0 , Γ(πy)ϕε0 = ϕε0≡ εϕε0 , (B.1)

where ε is either +1 or −1.To find the Clebsch1Gordan series

ΓK1 ⊗ ΓK2 =∑K

cKΓK , K > 0 , (B.2)

where cK is the multiplicity of the irrep ΓK in the decomposition of the above product,we first determine the possible values of the K quantum number in the space spannedby the products of basis states of ΓK1 and ΓK2 .

If K1 > 0 and K2 > 0, so that ΓK1 and ΓK2 are both two-dimensional, the representa-tion space for ΓK1 ⊗ ΓK2 is spanned by the 4 states ϕK1 ·ϕK2 , ϕK1 ·ϕK2

, ϕK1·ϕK2 , ϕK2

·ϕK1. Acting with Γ(γz) on these states, we find, using Eq. (20), that

Γ(γz)[ϕK1 · ϕK2

]=[Γ(γz)ϕK1

]·[Γ(γz)ϕK2

]= e−i(K1+K2)γ ϕK1 · ϕK2 ,

Γ(γz)[ϕK1 · ϕK2

]= e−i(K1−K2)γ ϕK1 · ϕK2

,

Γ(γz)[ϕK1· ϕK2

]= ei(K1−K2)γ ϕK1

· ϕK2 , (B.3)

Γ(γz)[ϕK1· ϕK2

]= ei(K1+K2)γ ϕK1

· ϕK2.

Thus, if K1 3 K2, the representation ΓK1+K2 will occur once in (B.2), along with onecopy of the representation Γ|K1−K2 | , i.e.

ΓK1 ⊗ ΓK2 = ΓK1+K2 ⊕ Γ|K1−K2| , K1 3 K2 , K1, K2 > 0 . (B.4)

Basis states for ΓK1+K2 are given simply by

ϕK = ϕK1 · ϕK2 , ϕK = ϕK1· ϕK2 , K = K1 + K2, K1, K2 > 0 . (B.5)

For the irrep Γ|K1−K2|, we choose the basis states

ϕK = ϕK1 · ϕK2, ϕK = ϕK1

· ϕK2 for K = K1 − K2 > 0, K1 > K2 > 0 , (B.6)

and

ϕK = ϕK1· ϕK2 , ϕK = ϕK1 · ϕK2

for K = K2 − K1 > 0, K2 > K1 > 0 . (B.7)

If K1 = K2, the representation ΓK with K = 2K1 occurs once, and there are two Γ0

irreps. The latter can be distinguished by the Γ(πy) ∈ D∞ operator. Thus, if we set

ϕε0 =1√2

[ϕK1 · ϕK1

+ ε · ϕK1· ϕK1

], K1 = K2 > 0 , (B.8)


it can be verified that Γ(πy)ϕε0 = εϕε0 = ϕε0

in accordance with Eq. (B.1). Thus, wefind

ΓK1 ⊗ ΓK1 = Γ2K1 ⊕ Γ+0 ⊕ Γ−0 , K1 = K2 > 0 , (B.9)

with basis states given by Eq. (B.5) with K1 = K2 for Γ2K1 and by Eq. (B.8) for Γε0.

Suppose now that K1 > 0 but K2 = 0. The resulting two-dimensional space ΓK1 ⊗ Γε0

is spanned by ϕK1 · ϕε0 , ϕK1· ϕε0. It follows from the action of Γ(γz) on these states

that the only possible value of K is K = K1, i.e.

ΓK1 ⊗ Γε0 = ΓK1 , K1 > 0 , K2 = 0 , (B.10)

with basis states

ϕK1 · ϕε0 , ϕK1· ϕε

0 , K1 > 0 , K2 = 0 , (B.11)

where the phase of the second basis state has been adjusted so that Eq. (21) is verified.If K1 = 0 but K2 > 0, then

Γε0 ⊗ ΓK2 = ΓK2 , (B.12)

with basis states

ϕε0 · ϕK2 , ϕε0· ϕK2 , K2 > 0 , K1 = 0 . (B.13)

Finally, if K1 = K2 = 0, we have

Γε10 ⊗ Γε2

0 = Γε0 , ε = ε1 · ε2 , K1 = K2 = 0 , (B.14)

with basis state

ϕε0 = ϕε10 · ϕ

ε20 . (B.15)

References

[1] G. Rosensteel and D.J. Rowe, Phys. Rev. Lett. 38 (1977) 10; Ann. Phys. (N.Y.) 126 (1980) 343;D.J. Rowe, Rep. Prog. Phys. 48 (1985) 1419.

[2] Hubert de Guise, David J. Rowe and Ryoji Okamoto, Nucl. Phys. A 575 (1994) 46.[3] Hubert de Guise and David J. Rowe, Nucl. Phys. A 589 (1995) 58.[4] J. Carvalho, Ph.D. Thesis, Toronto (1984);

R. Le Blanc, J. Carvalho and D.J. Rowe, Phys. Lett. B 140 (1984) 155.[5] G.W. Mackey, Induced Representations of Groups and Quantum Mechanics (Benjamin, New York,

1968).[6] D.M. Brink et al., J. Phys. G: Nucl. Phys. 13 (1987) 629.[7] J.P. Elliott, Proc. Roy. Soc. A 245 (1958) 562;

D.J. Rowe, R. Le Blanc and J. Repka, J.Phys. A: Math. Gen. 22 (1989) L309.[8] Hubert de Guise and David J. Rowe, J. Math. Phys. 36 (1995) 6991.[9] H. Ui, Prog. Theor. Phys. 44 (1970) p. 703 .

[10] H. Ui, Prog. Theor. Phys. 44 (1970) p. 689.[11] R.T. Sharp, H.C. von Bayer and S.C. Pieper, Nucl. Phys. A127 (1969) p. 513.[12] J-P. Blaizot and G. Ripka, Quantum theory of finite systems (MIT Press, Cambridge, Mass., 1986).[13] P.M. Morse, and H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953) vols.

I and II.

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