The Algebraic Collective Nuclear Model and SO(5)
by
Peter Shipley Turner
A thesis submitted in conformity with the requirements
for the degree of Doctor of Philosophy
Graduate Department of Physics
University of Toronto
c 2005 by Peter Shipley Turner
Copyright ii
Abstract
The Algebraic Collective Nuclear Model and SO(5)
Peter Shipley Turner
Doctor of Philosophy
Graduate Department of Physics
University of Toronto
2005
A new Bohr-Mottelson collective model is presented for the investigation of nuclear structure. It is based upon two ingredients: (i) that there exists a conceptually and computationally simple method for constructing bases of SO(5) wavefunctions that is analogous
to the construction of the well known spherical harmonics on R3 , which also provides
a means for computing SO(5) coupling coefficients in a SO(3) basis, and (ii) that there
exist generalised SU(1, 1) wavefunctions, analogous to the radial functions of R3 , appropriate for deformed nuclei that provide bases in which matrix elements of typical
collective model operators can be computed in a completely algebraic manner. Together
they make up what will be called the algebraic collective model. It is shown that the
new basis of radial wavefunctions outperforms the standard spherical basis by an order
of magnitude in the size of the basis when computing the spectrum and E2 quadrupole
transition rates of a prototypical beta-deformed nucleus. The algebraic collective model
is used to analyse a spherical to deformed collective shape phase transition of a Bohr
model nucleus, and the results are compared to the analogous transition in the interacting boson model which has attracted much attention in the community over the past
several years. Also presented is a method for the construction of all, completely generic
representations of the so(5) Lie algebra in the so(3) basis of the collective model. It is
likely the most ambitious application of vector coherent state theory to date.
iii
Dedicated to the memory of Phyllis Hughes.
iv
Acknowledgements
First and foremost, I wish to extend my sincerest thanks to my supervisor, Prof. David
Rowe. Your command of both physics and mathematics, and the artfulness with which
you use it, has set an example to which I will aspire for the rest of my days. I thank you
for that example, and it has been a privilege to call you The Boss. I am also tremendously
grateful to Prof. Joe Repka, whose knowledge is matched only by his patience, both of
which even my ceaseless ignorance could not exceed; and to Prof. Aephraim Steinberg,
for his open-mindedness in participating in a field far removed from his own, and for
giving me the opportunity to try my hand at problems in that field. Perhaps they are
not so far removed after all.
Thanks to our present and former group members from whom I’ve learned so much;
Chairul Bahri, Steve-O Bartlett, Prof. Juliana Carvalho, Santo D’Agostino, Shen Yong
Ho, Sarah Karram, Gabriela Thiamova and Wasantha Wijesundera. Thanks also go to
Prof. John Wood and Prof. Barry Sanders.
Thanks to the many people who have made my time here and elsewhere all the better; “Rav n’ Nav” Bhat, Etienne Boaknin, Barry Bruner, J.P. Bushey, Catrien DeRuyter,
Dave Dottori, Victoria Edwards, Uri Fishman, Jalani Fox, Amit Ghosh, The Grim Rapher Galea and BigAl, Zahra Hazari, Peter Henderson, the Henry family, Igor Khavkine,
Jeff Lundeen, Carrie MacTavish, Johannes Martin, Stephen McComb, Vjera Miovic,
Rob Moucha, Sas̆a Nedeljkovic, Barbara Neuhold, Jeremy O’Brien, J.P. Paglione, Roman Pahuta, Jess Phillips, Lisa Phinney, Kevin Resch, Neil Rouatt, Martin Rovers, Terri
Rudolph, Rhian Salmon, Rob Schpekkens, Mike Sutherland, Jamie Taylor, Mike Trott,
Brian Wilson, Pascal Vaudrevange, and Ian Vollrath. Thanks to Jamie Banting, Fred Beserve, Mark Brownson, Chris Gibson, Erik Johnson, Kurt Magney and the entire rugby
football club for giving me something other than schoolwork to do in Toronto. And for
the rugby. Special thanks to Yann for letting me crash at Château Robard while finishing
this thesis. Apologies to those I’ve missed.
Thanks to Marianne Khurana and Krystyna Biel for helping me to navigate graduate
school relatively unscathed. I would also like to thank the Natural Science and Engineering Research Council of Canada, the University of Toronto, and the Sumner Foundation
for their financial support.
Finally many thanks to Chris, William, Marilyn and my whole family, especially to
my parents Wendy and Brian Turner.
v
vi
Contents
Introduction
1
1 The Bohr-Mottelson Collective Model
9
1.1
The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.2
Standard Solvable Limits . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.2.1
The spherical vibrator limit . . . . . . . . . . . . . . . . . . . . .
13
1.2.2
The Wilets-Jean limit . . . . . . . . . . . . . . . . . . . . . . . .
14
The Extended Davidson Limit . . . . . . . . . . . . . . . . . . . . . . . .
15
1.3
2 The Algebraic Collective Model
17
2.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.2
SO(5) Hyperspherical Harmonics . . . . . . . . . . . . . . . . . . . . . .
19
2.2.1
Review of spherical harmonics . . . . . . . . . . . . . . . . . . . .
19
2.2.2
Definition of hyperspherical harmonics . . . . . . . . . . . . . . .
20
2.2.3
The Hilbert space L2 (S 4 )
21
2.2.4
2.2.5
. . . . . . . . . . . . . . . . . . . . . .
2
4
Constructing a basis for L (S ) . . . . . . . . . . . . . . . . . . .
23
Transformation to hyperspherical harmonics . . . . . . . . . . . .
27
2.2.6
SO(5) coupling coefficients . . . . . . . . . . . . . . . . . . . . . .
30
2.2.7
SO(5) matrix elements . . . . . . . . . . . . . . . . . . . . . . . .
31
SU(1, 1) Radial Wavefunctions . . . . . . . . . . . . . . . . . . . . . . . .
32
2.3.1
The standard spherical realisation of su(1, 1) . . . . . . . . . . . .
32
2.3.2
The extended Davidson realisation of su(1, 1) . . . . . . . . . . .
33
2.3.3
A general realisation of su(1, 1) . . . . . . . . . . . . . . . . . . .
33
2.3.4
SU(1, 1) matrix elements . . . . . . . . . . . . . . . . . . . . . . .
36
2.4
Comparison of the Spherical and Deformed Bases . . . . . . . . . . . . .
38
2.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
2.3
vii
3 A Collective Shape Phase Transition
43
3.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.2
The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.2.1
The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.2.2
E2 matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . .
47
3.2.3
The mass parameter . . . . . . . . . . . . . . . . . . . . . . . . .
48
Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.3.1
Excitation energies . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.3.2
Wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.3.3
E2 transition rates . . . . . . . . . . . . . . . . . . . . . . . . . .
49
Phases and Quasidynamical Symmetry . . . . . . . . . . . . . . . . . . .
50
3.4.1
Fitting the apparent dynamical symmetries
. . . . . . . . . . . .
50
3.4.2
The random phase approximation for α < 0.5 . . . . . . . . . . .
53
3.4.3
The adiabatic-decoupling approximation for α > 0.5 . . . . . . . .
55
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
3.3
3.4
3.5
4 Coherent State Theory
59
4.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
4.2
Scalar Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
4.2.1
Intrinsic state . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
4.2.2
Orbiter subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
4.2.3
Example: Heisenberg-Weyl group in one dimension . . . . . . . .
62
Vector Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
4.3.1
Example: free particle with spin . . . . . . . . . . . . . . . . . . .
65
Inner Products and Unitarity . . . . . . . . . . . . . . . . . . . . . . . .
66
4.3
4.4
5 Generic Representations of so(5) in a so(3) Basis
5.1
5.2
The so(5) Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
5.1.1
Irreducible representations and weight diagrams . . . . . . . . . .
70
5.1.2
Realisation of so(5) . . . . . . . . . . . . . . . . . . . . . . . . . .
72
Embedding a so(3) Subalgebra . . . . . . . . . . . . . . . . . . . . . . . .
74
5.2.1
5.3
69
The SO(5) ↓ SO(3) branching rule
. . . . . . . . . . . . . . . . .
75
Intrinsic States and Orbiter Subalgebra for so(5) . . . . . . . . . . . . . .
75
5.3.1
76
Embedding an intrinsic subalgebra . . . . . . . . . . . . . . . . .
viii
5.4
5.5
5.3.2
Realisation of the intrinsic states . . . . . . . . . . . . . . . . . .
77
5.3.3
Realisation of the so(3)-coupled states . . . . . . . . . . . . . . .
78
5.3.4
Embedding an orbiter subalgebra . . . . . . . . . . . . . . . . . .
78
The Vector Coherent State Construction for so(5) . . . . . . . . . . . . .
81
5.4.1
VCS wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . . .
81
5.4.2
Overlap coefficients of the VCS wavefunctions . . . . . . . . . . .
82
5.4.3
VCS representations . . . . . . . . . . . . . . . . . . . . . . . . .
86
5.4.4
Matrix elements of the VCS representation . . . . . . . . . . . . .
87
5.4.5
Transformation to an Hermitian basis . . . . . . . . . . . . . . . .
90
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
5.5.1
93
Tables of coefficients and matrix elements . . . . . . . . . . . . .
Conclusion
98
Bibliography
102
ix
x
List of Tables
2.1
The first five l = 6 basis polynomials and their various labels.
. . . . . .
2.2
Relationships between the general SU(1, 1) label λv and the various bases
28
of radial wavefunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
5.1
The so(5) commutation relations, with ordering [row, column]. . . . . . .
70
5.2
The so(3) content of the low dimensional so(5) irreps. Note that the values
in the spinor column are 2L. . . . . . . . . . . . . . . . . . . . . . . . . .
76
5.3
Overlap coefficients and reduced matrix elements for so(5) irrep (1, 21 ).
.
94
5.4
Overlap coefficients and reduced matrix elements for so(5) irrep (2, 12 ).
.
95
5.5
Overlap coefficients and reduced matrix elements for so(5) irrep (1, 1). . .
96
5.6
Overlap coefficients and reduced matrix elements for so(5) irrep (1,
xi
3
).
2
.
97
xii
List of Figures
1.1
Nuclear shape for various limiting values of the parameters β and γ. The
two deformed cases depicted with β > 0 have axial symmetry indicated
by the axes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
11
The low lying levels of the five dimensional harmonic oscillator Hilbert
space L2 (R5 ). Each line represents a degenerate SO(5) multiplet of states
labelled (v, n), where v is the SO(5) quantum number and n labels a
SU(1, 1) basis. The direct sum of states across a horizontal energy level
comprise a U(5) irrep labelled by N . Note that the labels are related by
N = v + 2n. Factoring out the SU(1, 1) dependence leaves a sequence of
SO(5) carrier spaces, outlined in the figure, that together span the Hilbert
space L2 (S 4 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
22
The Hilbert space L2 (S 4 ). Each line is a SO(3) multiplet labelled by the
angular momentum l. The horizontal dashed boxes contain the states
comprising a SO(5) irrep labelled by v on the vertical axis. The vertical
dashed boxes contain columns of K-band states labelled by the lowest l
value K. The pattern of K-bands repeats at intervals of ∆v = 3, and we
label each block of K-bands by t. . . . . . . . . . . . . . . . . . . . . . .
2.3
24
The action of the shift operators in the Hilbert space L2 (S 4 ), shown only
up to v = 3. Comparing with figure 2.2 should convince the reader that
with these four operations any multiplet in L2 (S 4 ) can be reached. . . . .
2.4
26
Deformed ground state wavefunctions with a = 2 sequenced as λv increases
from 5/2 to 100. The figure shows that larger λv values correspond to
states with larger mean β values, i.e., greater deformation. . . . . . . . .
xiii
35
2.5
Comparison of the eigenvalues of the ground state and the v = 1, n = 0
first excited state in the spherical and deformed bases as functions of the
number of basis states. The potential well parameters are k = 100 and
hβi = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6
39
Histogram of the ratios of the number of spherical basis states to that of
deformed states needed to achieve the ground state energy eigenvalue to
within one percent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7
40
Comparison of the low lying quadrupole intraband and interband transition rates in the spherical and deformed bases as functions of the number
of basis states. The potential well parameters are k = 100 and hβi = 5. .
3.1
The potential as a function of the control parameter α. A critical point
3.2
is reached as α passes through 0.5, whereafter there is a well about β =
p
(2α − 1)/2α. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
46
Low-lying excitation energies, as functions of α for M = 50, 100, 500 show-
ing a phase transition between the spherical vibrator (α 0.5) and soft
rotor α 0.5 that gets sharper as M increases. The curves of each rotor
band (α 0.5) are labelled in the first plot by (v1 < v2 < · · · ; ν), where
v increases with energy and ν labels the band. . . . . . . . . . . . . . . .
3.3
A sequence of M = 100 ground state wavefunctions
(α)
β 2 R00 (β)
49
for α ∈ [0, 1]
and β ∈ [0, 1.2] showing the mean deformation β and shape fluctuations
of the ground state as a function of the control parameter α. For α < 0.5,
the peak of the wavefunction changes little from that of its α = 0 limit
and its width increases slowly, while for α > 0.5 the mean deformation
increases rapidly. Note that the factor β 2 , coming from the volume element, is included so that the square of the wave function is the probability
distribution for the ground state deformation. The peaks of the wavefunctions are qualitatively similar for other values of M . However, their widths
decrease as M increases. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
50
Ratios, on the same scale, of some intraband B(E2) transition rates Rintra (v)
and interband rates Rinter (v), as defined in the text, for 0 ≤ α ≤ 2 and
M = 100. The horizontal lines are the predictions for the harmonic vi-
brator (α < 0.5) and adiabatically-decoupled rotor-vibrator (α > 0.5)
approximations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiv
51
3.5
Excitation energies, for M = 100, of a harmonic spherical vibrator and an
adiabatically-decoupled rotor-vibrator fitted to the lowest v = 0 and v = 1
excitation energies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6
5.1
5.2
52
Comparison of the diagonalised excitation energies as functions of α with
the RPA (for α < 0.5) and the ADA (for α > 0.5). . . . . . . . . . . . . .
54
Root diagram for the so(5) Lie algebra in the basis {A, S, T }. . . . . . .
70
est grade states labelled |(v, f )mi (see §5.3). . . . . . . . . . . . . . . . .
71
The weight diagrams for the fundamental irreps (0, 21 ) and (1, 0) with high1
)
2
5.3
The fundamental weight diagram (0,
5.4
The fundamental weight diagram (1, 0) in terms of boson creation operators. 73
5.5
The highest grade and weight of a general so(5) irrep. The u(2) core
in terms of boson creation operators. 72
subalgebra and grade raising/lowering operators are shown in the second
diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
77
xvi
Introduction
Nuclear collectivity
Early in the development of the theory of the nucleus there arose two very different
models. The liquid drop model of Neils Bohr [1] took a very classical view of the nucleus
as a drop of some nuclear liquid, and through the use of hydrodynamics with some
quantum corrections, the binding energies of all known nuclei were reproduced quite
well, along with a useful model for nuclear fission. This model failed in many other
respects however, and it was the shell model of Mayer and Jensen [2, 3], based upon
the successful electronic orbital model of the atom, that reproduced the so-called magic
atomic numbers, spins and parities of low lying states, as well as other properties.
These two models, each applicable in its own realm, are intuitively incompatible.
The shell model is microscopic – that is, it describes the nucleus in terms of constituent
interacting nucleons that, for most intents and purposes, are fundamental. A liquid drop
however, for most intents and purposes, has no constituent parts — at least none that
can easily be identified with the nucleons of the shell model. If from the shell model we
adopt the view of a nucleus consisting of interacting nucleons, then any properties of the
nucleus that are well described by a liquid drop-like model must be collective, in that the
nucleons are no longer independent but act as some sort of whole.
A major problem for both the liquid drop and shell models was the deviation of the
nuclear distribution from spherical symmetry, in particular the large quadrupole moments
observed in some elements. This suggested that certain nuclei are themselves deformed,
a situation that could not be satisfactorily explained by either model. The first hint of a
solution came with Rainwater’s suggestion [4] that valence nucleons outside of the closed
shells could have a polarising effect upon the core. If there were enough valence nucleons
moving coherently, the closed-shell core could itself become ellipsoidally deformed, giving
rise to the large, formerly inexplicable quadrupole moments of some nuclei.
1
2
Introduction
Aage Bohr soon after published a detailed treatise [5] along these lines, formally
expanding the surface of a nuclear distribution in second order harmonics and quantising
the resulting degrees of freedom. This was the birth of the collective model, with a
liquid drop-like core undergoing completely collective motion, to which valence nucleons
could be coupled in various ways. For this reason, it has also sometimes been called the
unified model. Together with Mottelson [6], Bohr expanded his analysis and found that
the collective model predicted vibrational and rotational excitations of the core. The
surprisingly good agreement between the spectra of these excitations and experiment
showed that many nuclei were indeed exhibiting this kind of collective behaviour. This
opened a new avenue of research in nuclear structure theory, leading to the discovery
of further collective features such as the tendency of nucleons to form pairs not unlike
electrons in a superconductor, (for an early account of nuclear collectivity, see [7]). The
tremendous impact of the theory is evidenced by Bohr, Mottelson and Rainwater being
awarded the Nobel Prize for this work in 1975.
In the fifty years that have passed, one could argue that the original Bohr-Mottelson
collective model has been eclipsed in some respects by other models, notably the interacting boson model (IBM) of Arima and Iachello [8, 9]. This thesis develops some new
approaches to the Bohr-Mottelson collective model that make it more robust and computationally useful than perhaps previously thought. Many of our results, and all of our
techniques, are also applicable beyond the Bohr model.
Group theory and algebraic models
Any microscopic model of the nucleus, such as the shell model, leads to a many-body system of interacting nucleons. Nuclei are unique in the field of many-body physics because
for all but the simplest elements there are too many particles to solve the interacting
system by brute force, and yet there are too few particles to achieve satisfactory results
using the statistical approaches of condensed matter physics. For nuclei with atomic
numbers that are nearly magic, that is nuclei with closed shells but for only a few extra
particles or holes, shell model calculations can and have been be carried out. However, as
one adds nucleons beyond closed shells the number of states to consider exponentiates,
and such calculations are not feasible. Collective models, on the other hand, do not
suffer from this restriction. Indeed, collective models are our only theoretical recourse in
certain regions of the periodic table, most especially those that are far from having closed
Introduction
3
shells. One reason for the tractability of collective models is that they are amenable to
the powerful machinery of group theory.
We will not attempt to summarise the multitude of ways that group theory and
quantum mechanics come together because we cannot do it justice here, (consider [10,
11, 12, 13, 14, 15, 16] as the tip of the proverbial iceberg). Suffice it to say that the two
are effectively inseparable; wenn Gruppentheorie eine Pest ist, dann Quantenmechanik
ebenso.‡
Group theoretical considerations were used early on in the development of the collective model, such as the prediction of the angular momentum states contained in collective
multiplets [17]. That group theory is important to the study of many-body physics in
general was being realised about the same time, see for example [18, 19]. However, it was
not until later that dynamical symmetry and the full power of the algebraic approach was
utilised. It is no coincidence that the development of dynamical symmetries and spectrum generating algebras [20, 21] ran along side that of the collective models. Indeed, it
is probably safe to say that nuclear structure physics was the driving force behind the
concept of dynamical symmetry. That these concepts have applications in all areas of
many-body physics, from molecules (see for example [22]) to condensed matter (see for
example [23]), has only begun to be recognised by researchers outside of nuclear physics
relatively recently.
We say that a model is algebraic when the Hamiltonian and other observables are
written in terms of the tensor operators associated with some useful group or groups,
usually continuous Lie groups. A Lie group G is generated by its Lie algebra g, hence
the epithet ‘algebraic’. If the Hamiltonian is invariant under the action of some group
GS , the stationary states of the model carry irreducible representations (irreps) of GS
and we say that GS is a static symmetry group of the system. All of the standard group
theoretical concepts apply: the labelling of basis states with good quantum numbers, the
ability to predict level splittings upon symmetry breaking, and selection rules for matrix
elements.
Dynamical symmetry is more general. Any group that has an irrep carried by the
Hilbert space of a model is a dynamical symmetry group for that model. Thus the action
of a dynamical group GD on any model state spans the model space, i.e., it leaves the
dynamics invariant. A spectrum generating algebra (SGA) can be defined as a Lie algebra
‡
In response to W. Pauli’s “Gruppenpest”, with all due respect.
4
Introduction
of operators such that all observables of the model are expressible as polynomials, (linear
combinations of multiple tensor products), of those operators. Now the group generated
by the SGA is a dynamical group, and any carrier space of a nontrivial irrep of the SGA is
a realisation of a subspace of the model Hilbert space. Since the dynamical group action
spans this subspace, the SGA generates the spectrum. Access to a dynamical symmetry
group is important because the SGA can be used to compute the matrix elements of the
Hamiltonian and any other observable that is a polynomial in the SGA. Exact solutions
to the model can be identified by determining the ways in which the static symmetry
group can sit inside the dynamical group. Thus the subgroup chains GD ⊃ · · · ⊃ GS
correspond to exactly solvable limits of the model, where the terms of the Hamiltonian
are scalars in the subgroups of the chain. It should be stated that the operators from
which the Hamiltonian is built may not close upon a finite algebra, so the method is not
universally applicable. However, there are generalisations of dynamical symmetry that
enable one to utilise the algebraic approach for a wider class of Hamiltonians [24, 25].
There are also exact solutions for Hamiltonians which include scalars from more than
one subgroup chain [26]. Dynamical symmetry has become a pillar of modern nuclear
structure theory (see for example [27]), and will be used throughout this thesis.
On top of the considerable predictive power of an algebraic model, there is also considerable interpretive power. At the risk of taking the microscopic interpretation too
literally, we could call a collective model phenomenological, in that it neglects substructure. However, a collective model is by no means a classical or even a semi-classical
approximation, any more than, say, a phonon is a classical object. It is merely the case
that the excitations of a collective model cannot easily be interpreted in terms of single
nucleon orbitals. If we start with an algebraic phenomenological model and can find a
SGA with observables that have a microscopic interpretation, such as nucleon creation
and annihilation operators for instance, then it remains only to construct irreps of this
SGA in order to arrive at a microscopic version of the original model. This was the route
from which the symplectic model of Rosensteel and Rowe [28, 29] emerged, where the
collective behaviour is realised as a subdynamics of the microscopic shell model. Now
phenomenological parameters can be derived rigorously and microscopic observables can
be evaluated. Thus not only does one benefit from the computational power of an algebraic model, one at the same time obtains a possible means for assigning an interpretation
to the results.
Algebraic models are good for all kinds of many-body investigations; in particular they
Introduction
5
are fertile ground for the investigation of phase transitions. When terms from different
subgroup chains occur in a Hamiltonian, the dynamical symmetries can compete to
lower the ground state energy of the system. Introducing control parameters into those
competing terms allows us to vary the symmetry and watch the spectrum evolve. We
can then discern whether the system makes a continuous, first, or second order transition
from the phase associated with one dynamical symmetry to the other. In many such
models, the system tends to ‘cling’ to its symmetry well into the parametric region
where the original symmetry is broken. This surprising phenomenon has been called
quasidynamical [30] or effective symmetry [31]. It is important since it indicates a highly
coherent mixing of states from different irreps of the limiting symmetries, and suggests
the possibility of true dynamical symmetry lurking in new regions of the parameter space.
A topic currently of much interest in and around the nuclear structure community takes
this idea to its extreme. In several recent papers [32, 33], the idea of critical point
symmetry is put forward, where at the critical point of a phase transition between two
dynamical symmetries, the spectrum is given to a high degree of accuracy by an effective
Hamiltonian with a completely new dynamical symmetry. This is important from the
perspective of the renormalisation group and scaling laws at the critical point [34].
In this thesis, we formulate the Bohr model in a new, completely algebraic way. We
call it the algebraic collective model, and we put it to work in a spherical to deformed
collective nuclear shape phase transition, where we will discuss some of these topics.
Constructing representations
Having hopefully convinced the reader of the importance of algebraic models, we must
now warn that their utility comes at a price. In order to make use of the benefits
that dynamical symmetries have to offer, we must have at our disposal representations
of the SGA in question. The states that carry these representations must reduce a
pertinent dynamical subgroup chain of the model, in the same way that the angular
momentum states |jmi reduce the subgroup chain SO(3) ⊃ SO(2) — we often say that
these states carry SO(3) irreps (labelled by j) in a SO(2) basis (indexed by m). For
many interesting subgroup chains, this reductive basis is not explicitly known and we
must therefore construct the requisite representation. More often than not, this is a
nontrivial undertaking.
In quantum mechanics, the construction of irreps is frequently carried out by a sophis-
6
Introduction
ticated process known as induction [35]. The physicist’s version of this process makes use
of coherent states. In group theoretical language, a coherent state is an element of the
orbit of some fiducial state under a group action. The rough idea is to span the carrier
space of the target representation with cleverly chosen coherent states and project out
states that both carry the irrep and reduce the desired subgroup chain. As a generalisation of this idea, vector coherent state theory [36, 37, 38] is arguably the most powerful
method available for the construction of physically utile representations.
In Bohr’s collective model, the quadrupole moments of a nuclear distribution are
the relevant degrees of freedom. Since their components span a five dimensional space,
quadrupole moments are related by orthogonal transformations in 5-space, just as vectors
in 3-space are related by rotations. SO(5), the group of special (i.e., orientation preserving) orthogonal transformations in R5 , therefore occurs in several dynamical subgroup
chains of the Bohr and most other collective models. And since the Hamiltonian for
a nucleus free of external fields must be rotationally, that is SO(3), invariant, we need
representations that reduce SO(5) ⊃ SO(3).
In this thesis, we present two new constructions of SO(5) irreps in a SO(3) basis.
The first is concerned only with the so called one-rowed irreps of SO(5). These are the
symmetric representations required by the Bohr model. This construction is completely
analogous to that of the spherical harmonics in R3 , and we call them SO(5) hyperspherical
harmonics. They make up an integral part of the algebraic collective model that we
propose. The second is a full vector coherent state construction of all generic (genuine
and spinor) irreps of the so(5) algebra in a so(3) basis. To the best of our knowledge,
it is the most demanding application of vector coherent state theory to date. These
constructions are not restricted to nuclear collectivity — they are a basis for any algebraic
model with the same embedded SO(5) ⊃ SO(3) subgroup chain.
Outline
The outline of the thesis is as follows. Chapter 1 is a review of the Bohr-Mottelson
model with an emphasis on dynamical symmetries and recent developments in constructing physically relevant bases for performing calculations. In chapter 2, we present a new
basis for the collective model. It is the culmination of two discoveries: that there exists a
conceptually and computationally simple method for constructing bases of SO(5) hyperspherical harmonics, which also provides a method for computing SO(5)⊃SO(3) coupling
Introduction
7
coefficients; and that there exist generalised SU(1, 1) wavefunctions, analogous to the radial functions of R3 , appropriate for deformed nuclei that provide bases in which matrix
elements of typical collective model operators can be computed algebraically. Together
these wavefunctions make up the basis for the algebraic collective model. The model
is used in chapter 3, where we formulate and analyse a spherical to deformed collective
shape phase transition. The results are compared to the analogous transition in the interacting boson model which has attracted much attention in the community over the
past several years. Chapter 4 is a review of coherent state theory with emphasis on its
group theoretical interpretation and its application to representation theory. It is meant
as an introduction to the concepts used in chapter 5, where we develop a method for the
construction of all, completely generic representations of the so(5) Lie algebra in a so(3)
basis.
8
Introduction
Chapter 1
The Bohr-Mottelson Collective
Model
The Bohr-Mottelson model [5, 6] for collective nuclear motions was motivated in part by
the observation of electric quadrupole transitions in far from closed shell nuclei whose
strengths exceed those of single particle shell model transitions by an order of magnitude
or more. Such transitions indicate a coherent activity of many single particle orbitals
simultaneously, leading to the consideration of quadrupolar deformation in the overall
shape of the nucleus. One is therefore led back to the liquid drop model of the nucleus
in attempting to describe such transitions.
The model has achieved enormous successes in its various applications, particularly
in the hands of the Frankfurt school [39, 40, 41] where it was developed into a formidable
computational tool for calculating spectra, electric transition rates and nuclear shape
distributions. Perhaps most importantly, it helped lead to the group theoretical interpretation of nuclear collectivity that has become the modus operandi of nuclear structure
physics. In particular, the quadrupole degrees of freedom and rotational invariance of
the model lead naturally to consideration of irreps of the special orthogonal Lie group
SO(5) in a SO(3) basis, the representation of which is a major theme of this thesis.
In this chapter, we review the salient features of the Bohr-Mottelson model (often
called simply the collective model). The model is over fifty years old, but there have
been several important recent developments which have rekindled interest that we also
cover here. These have led to our development of what we call the algebraic collective
model, which we discuss in detail in chapter 2.
9
10
Chapter 1. The Bohr-Mottelson Collective Model
1.1
The Model
Bohr considers the radius of a liquid drop model nucleus expanded in quadrupolar spherical harmonics
2
i
h
X
αm Ym2 (θ, φ) .
R(θ, φ) = R0 1 +
(1.1)
m=−2
R0 is the mean radius and Ym2 (θ, φ) is a second order spherical harmonic with θ and φ the
usual polar angles. The parameters αm then give the shape, restricted of course to only
quadrupolar deformation. We will adopt an equivalent set of shape parameters, given by
the quadrupole moments of the nuclear density ρ [27];
qm ∝
Z
dV r 2 Ym2 (θ, φ) ρ(r, θ, φ).
(1.2)
Now the set {qm , m = 0, ±1, ±2} characterises the shape. From the spherical harmonics
∗
they inherit the property qm
= (−1)m q−m , where ∗ indicates complex conjugation, ensur-
ing that there are five real shape parameters as befits a quadrupolar object. In spherical
∗
tensor notation we have q m = qm
.
The shape is now treated as a dynamical quantity with coordinates {qm } and con-
jugate momenta {π m }. The system is quantised in the usual way: π̂ m = −i~ ∂q∂ m , and
the Hilbert space HBM is that of all square integrable functions in five variables. Any
Hamiltonian in these coordinates describing a nucleus free of external symmetry-breaking
fields must be rotationally invariant, that is it must be a polynomial in SO(3) scalars and
solutions will have good SO(3) quantum numbers. Thus we consider only models with
SO(3) ⊃ SO(2) static symmetry.
We can make a transformation to a coordinate system that takes advantage of this rotational invariance. Rotate from the laboratory (space-fixed) coordinates q to an intrinsic
(body-fixed) frame of principal axes with coordinates q
qm =
X
n
2
q n Dnm
(Ω),
(1.3)
2
where Dnm
is a five dimensional Wigner rotation matrix and Ω represents the three Euler
angles relating the intrinsic frame to that of the laboratory. We can choose this new frame
such that q 1 = q −1 = 0 and q 2 = q −2 , leaving two manifestly SO(3) invariant parameters
{q 0 , q 2 } that characterise the shape and three parameters Ω giving the orientation. It is
11
1.1. The Model
customary to introduce variables β and γ such that
q 0 = β cos γ
1
q 2 = √ β sin γ,
2
(1.4)
and we can write the original coordinates in terms of these new variables
qm (β, γ, Ω) =
X
n
2
q n Dnm
(Ω)
β
2
2
2
= β cos γD0m
(Ω) + √ sin γ D2m
(Ω) + D−2m
(Ω) .
2
(1.5)
Since there are five independent real components in the set {qm }, they define a vector
P
β in the Euclidean space R5 . From the above definitions one finds that m q m qm =
P
2
2
m |qm | = β , and so β ∈ [0, ∞) is seen to represent the magnitude of the deformation
vector β. The other four coordinates {γ, Ω} are now seen to be the angles in a polar
representation of β. From the definition (1.4), γ is independent of the orientation and
so gives the intrinsic shape, while the orientation is given by the three Euler angles Ω.
Quadrupole deformations of a sphere yield ellipsoidal distributions, and Bohr shows [5]
that γ ∈ [0, π/3] with γ = 0 for a prolate (rugby football shaped) nucleus and γ = π/3
for an oblate (discus shaped) nucleus; see figure 1.1. For intermediate values of γ the
nucleus is triaxial; that is, it has no axis of symmetry.
β=0
β>0
γ=0
γ = π/3
Figure 1.1: Nuclear shape for various limiting values of the parameters β and γ. The
two deformed cases depicted with β > 0 have axial symmetry indicated by the axes.
The kinetic energy is proportional to the R5 Laplacian which in these coordinates is
12
Chapter 1. The Bohr-Mottelson Collective Model
given by
1
1 ∂ 4 ∂
β
− 2 Λ̂2
4
β ∂β ∂β β
1
≡ ∆2 − 2 Λ̂2 ,
β
∇2 =
(1.6)
where
3
∂
1X
L̂2k
1 ∂
sin 3γ
+
Λ̂ = −
sin 3γ ∂γ
∂γ 4
sin2 (γ − 2πk/3)
2
(1.7)
k=1
is the quadratic SO(5) Casimir operator. {L̂1 , L̂2 , L̂3 } are the SO(3) angular momentum
operators in the intrinsic frame. SO(5) acts in R5 as orthogonal transformations that
leave the magnitude β fixed. Thus SO(5) can affect only the shape coordinate γ or
the orientation Ω, and any potential independent of these coordinates must therefore be
SO(5) invariant. SO(3) acts in R3 as a physical rotation of the nuclear distribution. Thus
it affects only the orientation Ω, and since it leaves β fixed it is a subgroup of SO(5). The
embedding of SO(3) in SO(5) is specified up to conjugacy by requiring the irreducible
three dimensional SO(3) rotations of a distribution in R3 to induce an irreducible five
dimensional SO(3) action upon the quadrupole space R5 .
We will soon discuss SO(5) in further detail; for the moment it is sufficient to note
that any potential that is independent of {γ, Ω} leads to a SO(5) invariant Hamiltonian
and solutions with good SO(5) quantum number v, (note that only symmetric, or ‘one-
rowed’ SO(5) irreps are allowed for the bosonic states of the Bohr model). The eigenvalue
of Λ̂2 for a v state is v(v + 3) which can be compared to the eigenvalue l(l + 1) of the
quadratic SO(3) Casimir operator L̂2 for states of good angular momentum l. The label
v is often called the seniority in nuclear physics.
1.2
Standard Solvable Limits
There are three standard exactly solvable limits of the Bohr model which include SO(5)
in their subgroup chains. These include the spherical harmonic vibrator limit, and the
β-rigid γ-soft Wilets-Jean limit, both of which are described briefly below. The third
standard limit is the β-rigid γ-rigid (rigid rotor) limit, however it does not directly pertain
to our discussion because it does not enjoy a static SO(5) symmetry. For a more detailed
discussion of the standard limits, see reference [27].
13
1.2. Standard Solvable Limits
1.2.1
The spherical vibrator limit
Consider a nucleus with a spherical equilibrium shape that makes only small displacements in q. This is a five dimensional harmonic oscillator, since there are five independent
modes of oscillation qm . This is not to be confused with oscillations in physical 3-space
— we are in the space of quadrupole moments R5 .
The Hamiltonian is that of a simple harmonic oscillator in the quadrupole coordinates
HSV = −
X
1
~2 X m
q m qm ,
π πm + Bω 2
2B m
2
m
(1.8)
where classically π m = B q̇ m . B is a mass parameter and ω is an angular frequency,
both independent of mode m. In practice these can be determined from experiments
with nuclei that exhibit such vibrational spectra. In theory they can be computed using
hydrodynamical arguments in the liquid drop picture, although such calculations must
be corrected for various single particle shell effects such as pairing.
We can make the usual transformation to boson creation and annihilation operators
r
r
Bω
1
d†m = −
q̂m + i
π̂m
2~
2B~ω
r
r
Bω m
1
m
d = −
q̂ − i
π̂ m ,
(1.9)
2~
2B~ω
n
. These excitations are viewed as phonons each carrying angular
with [dn , d†m ] = δm
momentum 2. In practice any Bohr-Mottelson model Hamiltonian is a polynomial in
these operators, which is to say that the Heisenberg-Weyl algebra hw(5), spanned by
dm ,d†m and the identity operator, is a spectrum generating algebra of the Bohr-Mottelson
model. It is useful in practice to use the semidirect sum algebra [hw(5)]u(5) as the SGA,
since this contains the generators of the SO(3) static symmetry group as a subalgebra.
The Hamiltonian (1.8) becomes
ĤSV = ~ω
X
m
d†m dm +
5
2
,
(1.10)
and since the first term is a u(5) Casimir operator the Hamiltonian is seen to be invariant under U(5), the group of unitary transformations in five dimensions generated by
{ d†m dn }, with irrep label N for the number of quanta. In the collective model coordinates
the Hamiltonian becomes
ĤSV = −
~2 2 1
∇ + Bω 2 β 2 ,
2B
2
(1.11)
14
Chapter 1. The Bohr-Mottelson Collective Model
which is manifestly independent of γ. Thus the potential depends only upon β and the
Hamiltonian is additionally invariant under SO(5) and SO(3) transformations, so that
solutions can be labelled according to the subgroup chain
[HW(5)]U(5) ⊃ U(5) ⊃ SO(5) ⊃ SO(3) ⊃ SO(2),
N
v
τ
l
m
(1.12)
where τ is a multiplicity label distinguishing multiple states with the same v and l values.
Bases of wavefunctions that reduce this spherical vibrator subgroup chain were constructed by Chaćon, Moshinsky and Sharp [42, 43]. These wavefunctions were used by
the Frankfurt school [41, 44] in calculating solutions of the Bohr model between the
standard vibrational and rotational limits.
1.2.2
The Wilets-Jean limit
Consider now a nucleus that has a nonzero equilibrium deformation that is confined
about β0 6= 0. The potential for such a system has a deep minimum about β0 and we
again take it to be independent of {γ, Ω}. Such a potential is often called γ-unstable [45]
since the β coordinate is fixed while the γ coordinate is completely free. The situation
is analogous to a central potential in physical R3 with a sharp minimum at some r > 0.
The non-spherical equilibrium shape allows rotation as a possible mode of excitation.
However, since the potential does not fix the shape γ, it is not a rigid rotor.
If we construct the operators q̂m ∝ d†m +dm we find that their span is isomorphic to the
commutative algebra R5 . They are physically significant because the qm are proportional
to the quadrupole moments of the nucleus. One also finds that the SO(3) scalar coupling
[q̂ ⊗ q̂]00 is proportional to β·β= β 2 , and is also scalar with respect to SO(5). Thus any
potential that is polynomial in β 2 has analytic solutions that can be labelled according
to the subgroup chain
[HW(5)]U(5) ⊃ [R5 ]SO(5) ⊃ SO(5) ⊃ SO(3) ⊃ SO(2).
β0
v
τ
l
m
(1.13)
The rigidity of the beta coordinate in the Wilets-Jean limit implies that the radial
beta wavefunctions are Dirac delta functions, and are thus realisable in the Bohr model
Hilbert space only as limits of sequences of normalisable wavefunctions. This is also true
of the γ-rigid rotor limit. This reflects the fact that such rigidly defined quadrupole
moments are unphysical — a quantum mechanical collective mode is bound and should
15
1.3. The Extended Davidson Limit
allow fluctuation of the radial coordinate. Fortunately there exists another subgroup
chain which handles this limit in a more satisfactory manner, which we discuss in the
next section.
1.3
The Extended Davidson Limit
It was realised by Rohoziński et. al. [46], and later independently by Elliott et. al.
[47], that there exist exactly solvable Bohr model Hamiltonians with nonzero mean beta
deformations that need not be rigid. The situation goes back to a model proposed by
Davidson [48] for describing the rotational and vibrational states of molecules in three
dimensions. The algebraic structure of this Davidson limit, and its extension to the five
dimensional case of the collective model, was given by Rowe and Bahri [49]. It was
shown that eigenfunctions of γ-independent Hamiltonians can be classified according to
the chain
SU(1, 1) × SO(5) ⊃ U(1) × SO(3) ⊃ SO(2),
λ
v
τ
n
l
m
(1.14)
where λ labels a SU(1, 1) irreducible representation and n a basis. SU(1, 1) acts as scale
transformations on the radial coordinate β, and SO(5) acts as orthogonal transformations
in the space of quadrupole moments. The situation is analogous to that of a particle
interacting with a central potential in real 3-space, such as an electron in a hydrogen
atom where the symmetry group is SU(1, 1) × SO(3) [50]. n is the analogue of the
principal quantum number and v is the analogue of the angular momentum quantum
number.
This subgroup chain is remarkably flexible in that it can be used to describe both of
the preceding symmetry limits. For zero mean deformation one recovers the spherical
vibrator limit with λ = v + 5/2, so that v labels both the SU(1, 1) and the SO(5) irrep.
This is a manifestation of the fact that these two groups are dual, (for a detailed discussion
of group duality in nuclear physics, see [51] and references therein). For a system with
nonzero mean deformation, allowing the width of the resulting beta wavefunctions to
approach zero gives the Wilets-Jean limit.
The algebraic structure implies that the Bohr model Hilbert space of the five dimensional harmonic oscillator decomposes into the direct sum
HBM =
M
v
Hv
(1.15)
16
Chapter 1. The Bohr-Mottelson Collective Model
of subspaces labelled by seniority v, and that each subspace is a tensor product
Hv = HvSU(1,1) ⊗ HvSO(5) ,
SU(1,1)
where Hv
SO(5)
and Hv
(1.16)
are carrier spaces for SU(1, 1) and SO(5) irreps v, respectively.
This decomposition of the Bohr model Hilbert space is key to understanding the algebraic
collective model that we propose in chapter 2. The SO(5) subspaces are spanned by hyperspherical harmonic wavefunctions in the collective coordinates {γ, Ω}, while SU(1, 1)
subspaces are spanned by radial wavefunctions in the collective coordinate β.
If the β modes are sufficiently decoupled from the γ modes, so that the wavefunctions
for a fixed SU(1, 1) label n are essentially the same for all v, (or at least the vs of
interest), then a SO(5) invariant Hamiltonian can be diagonalised using only su(1, 1) as
its SGA [52]. In the next chapter we show how to algebraically evaluate radial beta
matrix elements in a deformed SU(1, 1) basis. Along with the SO(5) spherical harmonics
which we will also construct, the algebraic model based upon this SU(1, 1) × SO(5)
decomposition can be used to diagonalise an arbitrary collective model Hamiltonian.
Chapter 2
The Algebraic Collective Model
In this chapter, we develop what will be called the algebraic collective model. It is a
completely algebraic treatment of the Bohr-Mottelson collective model that provides a
much simplified technique for performing collective model calculations, geared toward
situations with SO(5) symmetry. As was shown in §1.3 the Bohr model Hilbert space
factors into the tensor product of a space of SO(5) hyperspherical harmonic wavefunctions
and a space of SU(1, 1) radial wavefunctions, analogous to the case of the regular three
dimensional harmonic oscillator. We will describe the construction of wavefunctions and
determination of matrix elements for each of these subspaces in detail. We end the chapter
with a comparison of the standard spherical basis and our new basis in calculating some
standard quantities for a prototypical collective model Hamiltonian with a variable mean
deformation.
2.1
Overview
The algebraic model adopts the SU(1, 1)×SO(5) decomposition of the Bohr model Hilbert
space. Basis states, labelled according to the subgroup chain (1.14), are of the form
|λv n v τ l mi = |λv ni ⊗ |v τ l mi.
(2.1)
The states {|λv ni, n = 0, 1, 2, · · · } are a basis for an irrep λv of SU(1, 1). We include
the subscript v on the SU(1, 1) label because in our generalised basis these labels depend
upon the seniority. The states {|v τ l mi, m = −l, −l + 1, · · · , l} are a basis for an
irrep of SO(5). The τ and l values appropriate for a state of seniority v are given
by the SO(5)↓SO(3) branching rule. Tensor products of these states therefore span a
17
18
Chapter 2. The Algebraic Collective Model
representation of the direct product group SU(1, 1)×SO(5). We can give these states
realisations in the collective coordinates {β, γ, Ω} as
|λv ni ⊗ |v τ l mi 7→ Rnλv (β) Yτvlm (γ, Ω).
(2.2)
Here, Rnλv (β) is a radial wavefunction and Yτvlm (γ, Ω) is a hyperspherical harmonic, both
of which are described in detail below.
Operators can be written in terms of products of the collective coordinates, such as
X̂ = F (β)G(γ, Ω),
(2.3)
where F and G are functions of the indicated variables and derivatives thereof. The
factorisation of the space means that matrix elements of such operators are products of
radial β matrix elements and angular γ, Ω matrix elements. Thus we can address these
two factors individually.
For the collective model, one is usually interested in low-lying energy levels and B(E2)
electric quadrupole transition rates. To compute matrix elements of a collective model
Hamiltonian in the SU(1, 1) × SO(5) basis, we typically need matrix elements of ∇2 from
equation (1.6) and of the SO(3) scalar polynomials of the quadrupole operators (see
§1.2.2)
1
(2.4)
q̂m = √ d†m + dm , m = 0, ±1, ±2 ,
2
that occur in the expansion of a rotationally-invariant Hamiltonian. The boson operators
d† were defined in equation (1.9). The Laplacian ∇2 = ∆2 − Λ̂/β 2 , defined in equation
(1.6), is a SO(5) scalar; thus, its matrix elements are given in terms of v and the radial
matrix elements of ∆2 and 1/β 2 . In order to evaluate matrix elements of the quadrupole
operators, which give the B(E2) strengths, we express them as functions of the collective
coordinates and factor them according to equation (2.3);
q̂m = β Qm (γ, Ω) ,
(2.5)
where Q, defined in equation (2.17), is proportional to a v = 1 SO(5) spherical harmonic.
Thus, matrix elements of q̂m are expressed as products of radial matrix elements and
matrix elements of Qm . Scalar operators of particular interest include, for example,
[q̂ ⊗ q̂]00 ∝ β 2 ,
and their multiples.
[q̂ ⊗ q̂ ⊗ q̂]00 ∝ β 3 cos 3γ ,
(2.6)
2.2. SO(5) Hyperspherical Harmonics
2.2
19
SO(5) Hyperspherical Harmonics
In this section we will describe a new method for constructing the SO(5) hyperspherical
harmonics that is both conceptually and computationally simpler than existing constructions, such as the one due to Chaćon, Molshinsky and Sharp [42, 43], or Rowe and Hecht
[53, 54]. It relies heavily upon the decomposition of the Hilbert space outlined in §1.3.
Here we are focusing upon the Bohr collective model, however it should be noted
that the harmonics that we construct will be applicable to any system with the same
embedded SO(5) ⊃ SO(3) symmetry. We also show how to derive SO(5) ⊃ SO(3)
coupling coefficients, (analogous to Clebsch-Gordan or Wigner coefficients), using these
harmonics. Finally we use these harmonics to evaluate some useful matrix elements in
the collective model.
2.2.1
Review of spherical harmonics
It will be helpful to keep in mind the well known spherical harmonics of R3 since we will
show that the R5 case is very much analogous. For a particle of mass m in R3 under the
influence of a central potential V (r) the Hamiltonian is
ĤR3 = −
~2 2
∇ 3 + V (r),
2m R
(2.7)
where ∇2R3 is the Laplacian on R3 . In spherical coordinates we have
∇2R3 =
1 ∂ 2∂
1 2
r
−
L̂ ,
r 2 ∂r ∂r r 2
(2.8)
where L̂2 is the quadratic SO(3) Casimir operator
L̂2 = −
∂
1 ∂2
1 ∂
sin θ
−
.
sin θ ∂θ
∂θ sin2 θ ∂φ2
(2.9)
The Hamiltonian ĤR3 is rotationally invariant, i.e., it has static SO(3) ⊃ SO(2) symmetry
and solutions are labelled with the two quantum numbers lm. The wavefunctions are
separable, ψlm (r, θ, φ) = Rl (r)Yml (θ, φ), and the functions Yml (θ, φ) which satisfy the
eigenvalue equation
L̂2 Yml (θ, φ) = l(l + 1)Yml (θ, φ)
(2.10)
are the spherical harmonics. They reduce the subgroup chain SO(3) ⊃ SO(2) and as such,
the set {Yml (θ, φ);
m = −l, · · · , l}, properly normalised, form an orthonormal basis for
20
Chapter 2. The Algebraic Collective Model
a SO(3) irrep l on the Hilbert space of square integrable functions on the 2-sphere L2 (S 2 )
with respect to the invariant measure dΩ ≡ sin θ dθ dφ.
The 2-sphere S 2 can be viewed as the subspace of R3 swept out by all SO(3) rotations
of a nonzero vector, but since the SO(2) rotations about the axis defined by that vector
add nothing, they are ‘factored out’. We say that the 2-sphere is isomorphic to the coset
space SO(3)/SO(2) of SO(3) rotations modulo SO(2).
Since the spherical harmonics are basis functions for irreps of SO(3) in a SO(2) basis,
they must be linear combinations of Wigner rotation functions. The standard definition
has
Yml (θ, φ)
=
r
2l + 1 l
D0m (Ω−1 (θ, φ)),
4π
(2.11)
where Ω(θ, φ) is an element of the coset space SO(3)/SO(2).
2.2.2
Definition of hyperspherical harmonics
Recall §1.1, where we described the Bohr-Mottelson model. If we abstract the situation
there, we find that the problem is defined by a configuration space that is isomorphic
to Euclidean R5 . Each point q ∈ R5 corresponds to a quadrupole deformation with five
spherical tensor components {qm }. The change of variables (1.5) can be viewed as a
transformation to the five hyperspherical coordinates {β, γ, Ω} in R5 , (we always lump
the three Euler angles into one symbol Ω, since the usual Euler variables (α, β, γ) would
be hopelessly confusing here). As we showed in §1.1, β is the radial coordinate that
gives the distance from the origin to the point q ∈ R5 . If we fix this length β 6= 0 and
vary the four remaining coordinates γ and Ω, we sweep out a submanifold of R5 ; it is
compact, since these four coordinates take values in finite intervals, and we call it the 4(hyper)sphere S 4 in analogy with the 2-sphere in R3 . Since the group of transformations
that sweep a 5-vector around R5 without changing its length is SO(5) by definition, and
the subgroup of SO(5) that rotates a 5-vector about its axis (and therefore adds nothing
to our submanifold) is SO(4), it must be that the coset space SO(5)/SO(4) is isomorphic
to the 4-sphere S 4 . And since S 4 is parameterised by {γ, Ω} by definition, the coset space
is also.
Now consider a Hamiltonian with a potential that depends only upon β — all of the
potentials that we considered in chapter 1 were of this form. We have
ĤR5 = −
~2 2
∇ + V (β),
2M
(2.12)
2.2. SO(5) Hyperspherical Harmonics
21
where ∇2 takes the form (1.6). The Schrödinger equation is separable, and from (2.2)
the equation governing {γ, Ω} is
Λ̂Yτvlm (γ, Ω) = v(v + 3)Yτvlm (γ, Ω),
(2.13)
where Λ̂ is the SO(5) Casimir operator (1.7). This defines the hyperspherical harmonics
Yτvlm (γ, Ω), which reduce the subgroup chain SO(5) ⊃ SO(3) ⊃ SO(2) and so form
an orthonormal basis for an irrep v of SO(5) on the Hilbert space of square integrable
functions on S 4 with respect to the SO(5) invariant measure dΓ ≡ sin 3γ dγ dΩ, where
dΩ is the standard SO(3) invariant measure on the 2-sphere and 0 ≤ γ ≤ π/3.
The differential equation (2.13) was tackled heroically by Bès [55], however only solu-
tions up to l = 6 were obtained by direct assault. The construction of solutions becomes
simpler upon recognising the fact that the hyperspherical harmonics form bases for (onerowed) irreps of SO(5), and so one can use tensor products to build larger harmonics
from smaller ones. This idea was used by Chaćon, Molshinsky and Sharp [42, 43] in their
construction of wavefunctions on the full five dimensional harmonic oscillator space, and
these were used by the Frankfurt school in their collective model calculations (see for
example [39, 40]). Here we describe a construction similar to the standard one, however
it is much simpler both to understand and to use due to the restriction of consideration
to the pertinent S 4 submanifold of R5 .
2.2.3
The Hilbert space L2 (S 4 )
Consider the full five dimensional harmonic oscillator Hilbert space, L2 (R5 ). The static
symmetry group is U(5), and so each energy level carries a one-rowed U(5) irrep labelled
by a non-negative integer N . An irrep N of U(5) contains irreps of the subgroup SO(5)
according to the branching rule
N ↓ (v = N ) ⊕ (v = N − 2) ⊕ (v = N − 4) ⊕ · · · ⊕ (v = N (mod 2)).
(2.14)
Figure 2.1 shows a schematic for L2 (R5 ) with level spacings based upon the harmonic
oscillator. Each row (U(5) irrep) has been split into SO(5) irreps v according to the
branching rule (2.14)—every line in the figure denotes a degenerate SO(5) multiplet.
Each column of states shares a single SO(5) label v, which also labels a dual SU(1, 1)
irrep since the direct sum of states in a column carries an irrep of SU(1, 1) × SO(5).
We can therefore label each multiplet by (v, n), where n indexes the SU(1, 1) basis.
22
Chapter 2. The Algebraic Collective Model
6
L2 (R5 )
N
6
(0, 3)
(1, 2)
5
4
(0, 2)
(2, 1)
(0, 1)
(5, 0)
(4, 0)
(3, 0)
(2, 0)
(6, 0)
I
@
@L2 (S 4 )
(1, 0)
1
0
(4, 1)
(3, 1)
(1, 1)
3
2
(2, 2)
(0, 0)
Figure 2.1: The low lying levels of the five dimensional harmonic oscillator Hilbert space
L2 (R5 ). Each line represents a degenerate SO(5) multiplet of states labelled (v, n), where
v is the SO(5) quantum number and n labels a SU(1, 1) basis. The direct sum of states
across a horizontal energy level comprise a U(5) irrep labelled by N . Note that the labels
are related by N = v + 2n. Factoring out the SU(1, 1) dependence leaves a sequence of
SO(5) carrier spaces, outlined in the figure, that together span the Hilbert space L2 (S 4 ).
This duality corresponds to the product structure of the L2 (R5 ) wavefunctions, in that
there exists a basis for L2 (R5 ) whose elements are a product of a β-dependent radial
function Rnλv (β) upon which SU(1, 1) acts as scale transformations, and a γ, Ω-dependent
harmonic function Yτvlm (γ, Ω) upon which SO(5) acts as 5-rotations. Thus, if we suppress
the β dependence by removing the radial functions from the Hilbert space, each column
collapses leaving one copy of each SO(5) irrep in the space of square integrable functions
of the remaining coordinates {γ, Ω}, i.e., the space of functions on the 4-sphere. Thus
we have
L2 (R5 ) ∼ L2 (R+ ) ⊗ L2 (S 4 ),
(2.15)
where L2 (R+ ) is the space of square integrable functions on the non-negative real line,
in this case functions of β.
Recall that in hyperspherical coordinates the quadrupole moments were given in (1.5)
2.2. SO(5) Hyperspherical Harmonics
23
as functions in L2 (R5 )
1
2
2
2
qm (β, γ, Ω) = β cos γD0m
(Ω) + √ β sin γ D2m
(Ω) + D−2m
(Ω) .
2
(2.16)
Factoring out the β dependence restricts these to functions on L2 (S 4 )
1
2
2
2
Qm (γ, Ω) = cos γD0m
(Ω) + √ sin γ D2m
(Ω) + D−2m
(Ω) .
2
(2.17)
These functions {Qm } form a basis for both the fundamental v = 1 irrep of SO(5) and
the l = 2 irrep of SO(3). They are therefore, to within normalisation, fundamental SO(5)
harmonics in a SO(3) basis. It is with these functions that we will construct all of the
SO(5) harmonics.
We can rearrange the Hilbert space L2 (S 4 ) in light of the fact that it contains one
copy of each SO(5) irrep v = 0, 1, 2, · · · . Specifically, we wish to split the SO(5) multiplets
into their constituent SO(3) multiplets. The SO(5) ↓ SO(3) branching rule for one-rowed
irreps was given in [56]; a SO(5) irrep v contains SO(3) irreps l according to
l = 2k, 2k − 2, 2k − 3, 2k − 4, · · · , k
k = v, v − 3, v − 6, · · · , kmin
(2.18)
where kmin = v (mod 3). Figure 2.2 shows a schematic for the Hilbert space based upon
the branching rule (2.18) for v up to 6. One can see that the SO(3) multiplets can be
arranged into a sequence of bands, (vertical columns in the figure), and that this band
structure repeats itself at intervals of ∆v = 3. We label each band by its lowest angular
momentum value which we call K. Each block of bands is given by a label t, the first
block being t = 0. Thus any multiplet is specified by the labels tKl.
2.2.4
Constructing a basis for L2 (S 4)
We are now prepared to construct a basis of wavefunctions for L2 (S 4 ). As mentioned in
the previous section, the functions {Qm } are the unnormalised fundamental v = 1 hyperspherical harmonics. Note that at this point we are not concerned with normalisation
or even with orthogonality — we simply wish to construct a set of functions that span
L2 (S 4 ).
Recall that the space L2 (S 4 ) is a direct sum of SO(5) carrier spaces,
L2 (S 4 ) = L2 (v = 0) ⊕ L2 (v = 1) ⊕ L2 (v = 2) ⊕ · · · .
(2.19)
24
Chapter 2. The Algebraic Collective Model
6
L2 (S 4 )
v
t=0
6
12
10
9
8
7
6
t=1
6
t=2
4
3
K=6
5
10
8
7
6
5
0
K=0
4
2
K=2
4
8
6
5
4
2
K=4
3
6
4
3
0
K=0
2
4
2
K=2
1
2
0
0
K=0
Figure 2.2: The Hilbert space L2 (S 4 ). Each line is a SO(3) multiplet labelled by the
angular momentum l. The horizontal dashed boxes contain the states comprising a
SO(5) irrep labelled by v on the vertical axis. The vertical dashed boxes contain columns
of K-band states labelled by the lowest l value K. The pattern of K-bands repeats at
intervals of ∆v = 3, and we label each block of K-bands by t.
Now, the states (vectors) of any SO(5) irrep can be constructed from tensor products
of the states carrying the fundamental irrep v = 1, (that is, the v = 1 irrep generates
the one-rowed representation ring of SO(5)), just as any SO(3) irrep is contained in a
(multiple) tensor product of the fundamental l = 1 irrep. Realising these v = 1 states as
the functions {Qm }, this means that the functions carrying any SO(5) irrep v are linear
combinations of products of Qm functions. Since these are bounded functions, and the
product of bounded elements of L2 (S 4 ) are also elements of this space, it follows that
2.2. SO(5) Hyperspherical Harmonics
25
L2 (S 4 ) is spanned by polynomials in {Qm }.
Consider the subspace of L2 (S 4 ) spanned by the highest weight SO(3) states. This
subspace looks exactly like figure 2.2 save that each multiplet is replaced by its highest
weight m = l state. From these states one can obtain all other states in L2 (S 4 ) by making
use of the SO(3) lowering operator. Therefore it suffices to construct the functions of the
SO(3) highest weight subspace, which is simplified by the fact that the product of SO(3)
highest weight functions is another highest weight function. This subspace is spanned by
polynomials of the form
ΦtKl = (Φ002 )n1 (Φ022 )n2 (Φ100 )n3 (Φ023 )n4 ,
(2.20)
where n ≡ [n1 , n2 , n3 , n4 ] are non-negative integers,
Φ002 ∝ Q2
Φ022 ∝ [Q ⊗ Q]22
Φ100 ∝ [Q ⊗ Q ⊗ Q]00
Φ023 ∝ [Q ⊗ Q ⊗ Q]33 ,
(2.21)
and the labels are related by
t = n3 ,
K = 2n2 + 2n4 ,
l = 2n1 + 2n2 + 3n4 ,
n4 = 0 or 1.
(2.22)
The square brackets indicate standard SO(3) coupling
[Q ⊗ Q]lm =
X
−2≤µ,ν≤2
(2, µ; 2, ν|lm)Qν Qµ ,
(2.23)
where (2, µ; 2, ν|lm) is a SO(3) Clebsch-Gordan coefficient.
In order to see that these polynomials do indeed span our space, note that the ground
state t = K = l = 0 corresponds to the trivial unit function n1 = n2 = n3 = n4 = 0.
To construct the highest weight function of the fundamental v = 1 irrep we need only
multiply this unit function by Q2 = Φ002 , and so on for all of the functions of a K = 0
band. Thus Φ002 is a ∆K = 0, ∆l = 2 shift operator that acts on functions in L2 (S 4 )
to yield functions with angular momentum increased by 2 and the same t,K values.
Similarly, Φ022 is a ∆K = 2, ∆l = 2 shift operator, Φ023 is a ∆K = 2, ∆l = 3 shift
operator, and Φ100 is a ∆t = 1, ∆K = 0, ∆l = 0 shift operator that yields functions in
the next block of K bands. With the help of figures 2.2 and 2.3 it is seen that these four
26
Chapter 2. The Algebraic Collective Model
6v
6
3
4
0
3
Φ023 2
1
0
2
Φ
100
Φ
022 2 6 Φ002 0
4
Figure 2.3: The action of the shift operators in the Hilbert space L2 (S 4 ), shown only
up to v = 3. Comparing with figure 2.2 should convince the reader that with these four
operations any multiplet in L2 (S 4 ) can be reached.
shift operators are enough to put the polynomials {ΦtKl } in a one-to-one correspondence
with all of the states in the highest weight subspace. Thus these polynomials constitute
the required set. Note that it is nonorthonormal with respect to the S 4 measure, so that
the labels t and K are not good quantum numbers.
From the expression for a product of Wigner functions
Dkl11 m1 (Ω)Dkl22 m2 (Ω)
=
lX
1 +l2
l=|l1 −l2 |
(l1 , k1 ; l2 , k2 |l, k1 +k2 )(l1 , m1 ; l2 , m2 |l, m1 +m2 )Dkl 1 +k2 m1 +m2 (Ω),
(2.24)
one finds that the polynomials (2.20) can be put in the form
r
√
even
2l + 1 X n
1 l
l
ΦtKl (γ, Ω) =
Fκ (γ)
Dκl (Ω) + (−1)l D−κl
(Ω) ,
4π l≥κ≥0
1 + δ0κ
(2.25)
where the generator functions Fκn are, from (2.21),
[1000]
F0
(γ) = cos γ,
[0100]
F0
(γ)
= cos 2γ,
[0010]
F0
(γ)
[0001]
F0
(γ) = 0,
[1000]
F2
(γ) = sin γ,
[0100]
F2
(γ)
= cos 3γ,
[0001]
F2
= − sin 2γ,
(γ) = sin 3γ.
(2.26)
2.2. SO(5) Hyperspherical Harmonics
27
A general function Fκn is found to take the form
Fκn (γ)
=
n1 X
n2
X
n
fijκ
(cos γ)n1 −i (sin γ)i (cos 2γ)n2 −j (sin 2γ)j (cos 3γ)n3 (sin 3γ)n4 ,
(2.27)
i=0 j=0
n
where the coefficients fijκ
are in practice calculated by computer. The normalisation in
(2.25) was chosen so that the inner product of basis functions is simply
hΦtKl |Φt0 K 0 l0 i = δll0
even Z
X
l≥κ≥0
π/3
0
0
Fκn (γ)Fκn (γ) sin γ dγ.
(2.28)
These overlaps can be computed analytically using (2.27) — any modern computer algebra package, such as MAPLE, will return exact values.
2.2.5
Transformation to hyperspherical harmonics
Now that we have a spanning set and an inner product for L2 (S 4 ), namely the set
of polynomials {Φ(n)}, we can use a recursive Gram-Schmidt procedure in order to
transform to an orthonormal basis of highest weight hyperspherical harmonics {Yτvl },
where τ is a multiplicity label. The Gram-Schmidt procedure requires that we impose
some ordering of the set of basis functions; we use the following two facts to do so.
(i) From figure 2.3 it is seen that the shift operators {Φ002 , Φ022 , Φ100 , Φ023 } have v
values 1,2,3 and 3 respectively. Thus, from (2.20) and the fact that a product
of functions with highest weights v1 and v2 has a highest weight v1 + v2 , a basis
element Φ(n) has
vmax = n1 + 2n2 + 3n3 + 3n4 .
(2.29)
It follows that a basis element Φ(n) is a linear combination of hyperspherical harmonics with v ≤ n1 +2n2 +3n3 +3n4 , and we can order basis polynomials of a given
l by their vmax values. For states with multiplicities, meaning basis polynomials
with the same l and vmax values, we order them by decreasing values of n1 .
(ii) The Q-parity is SO(5) invariant. A polynomial is said to have even(odd) Q-parity,
ΠQ = ±1, if it is of even(odd) degree in Q. From (2.21) one finds that ΠQ (n) =
(−1)vmax . Thus hyperspherical harmonics will be linear combinations of only even
or only odd polynomials which will be orthogonal to one another, and we need only
orthonormalise within the subsets of even and odd Q-parity.
28
Chapter 2. The Algebraic Collective Model
The methodology is best shown by an example; we choose l = 6 since this is the
first case with a multiplicity. Restricting our consideration to the states in figure 2.2, we
see that there are five with l = 6. Table 2.1 summarises the labels for these five states.
Thus we find that the first and third basis elements have odd Q-parity, the others even.
order
Φ(n)
t
K
l
vmax
ΠQ
1
Φ([3, 0, 0, 0]) 0
0
6
3
-1
2
Φ([2, 1, 0, 0]) 0
2
6
4
+1
3
Φ([1, 2, 0, 0]) 0
4
6
5
-1
4
Φ([3, 0, 1, 0]) 1
0
6
6
+1
5
Φ([0, 3, 0, 0]) 0
6
6
6
+1
Table 2.1: The first five l = 6 basis polynomials and their various labels.
6
6
Suppose that we’re interested in finding the multiplicity-two harmonics Y1,6
and Y2,6
,
corresponding to [3, 0, 1, 0] and [0, 3, 0, 0]. These have even parity, so we can restrict our
Gram-Schmidt procedure to the subset of even basis elements in table 2.1, i.e., functions
2, 4 and 5. Note that, due to the recursive nature of the Gram-Schmidt procedure, before
constructing a given harmonic in a set we must construct all of the preceding harmonics.
Thus, in finding harmonics 4 and 5 in this example we must also construct harmonic 2,
6
Y1,4
. The three basis elements are given by polynomials with the following γ dependent
functions [F0n , F2n , F4n , F6n ]
√
√
√
234234
650650 2
3 650650
2
cos γ cos 2γ −
cos γ sin γ sin 2γ +
sin γ cos 2γ,
Φ([2, 1, 0, 0]) ∝
5005
3003
10010
√
√
√
2 16396380
1821820
55 2
2
cos γ sin 2γ +
cos γ sin γ cos 2γ −
sin γ sin 2γ,
−
15015
110
√ 5005
√
6558552
3718 2
−
cos γ sin γ sin 2γ +
sin γ cos 2γ,
6006
286
1
(2.30)
− sin2 γ sin 2γ ,
2
2.2. SO(5) Hyperspherical Harmonics
29
√
3
3 650650
cos3 γ cos 3γ + √
Φ([3, 0, 1, 0]) ∝
cos γ sin2 γ cos 3γ,
5005
154
√
55 3
6
√ cos2 γ sin γ cos 3γ +
sin γ cos 3γ,
110
55
3
√ cos γ sin2 γ cos 3γ,
22
1 3
sin γ cos 3γ ,
2
√
3 650650
3
Φ([0, 3, 0, 0]) ∝
cos 2γ sin2 2γ,
cos3 2γ + √
5005
154
√
55 3
6
2
sin 2γ,
− √ cos 2γ sin 2γ −
110
55
3
√ cos 2γ sin2 2γ,
22
1 3
− sin 2γ .
2
(2.31)
(2.32)
The norms and matrix of overlaps for these three basis elements can now be computed
using the inner product (2.28). One finds that the squared norms are
|Φ([2, 1, 0, 0])|2 =
52
,
693
|Φ([3, 0, 1, 0])|2 =
4016
,
225225
|Φ([0, 3, 0, 0])|2 =
and that the Gram-Schmidt transformation is given by the matrix


1
0 0



 −9
1
0
.

13


330 142
−
1
251 251
1292
, (2.33)
13805
(2.34)
Thus in this example the SO(5) harmonics are given by the linear combinations
r
693
4
Φ([2, 1, 0, 0]),
Y1,6
=
52
r
225225 9
6
Φ([3, 0, 1, 0]) − Φ([2, 1, 0, 0]) ,
Y1,6
=
4016
13
r
13805
330
142
6
Φ([3, 0, 1, 0]) −
Φ([2, 1, 0, 0]) . (2.35)
Φ([0, 3, 0, 0]) +
Y2,6
=
1292
251
251
This method can be used to calculate any SO(5) ⊃ SO(3) harmonic, although due
to the recursive nature of the Gram-Schmidt process it may become computationally
intractable for large values of v and/or l.
30
Chapter 2. The Algebraic Collective Model
2.2.6
SO(5) coupling coefficients
We can use the SO(5) harmonics in a SO(3) basis to find the SO(5) ⊃ SO(3) cou-
pling coefficients. The method parallels that used by Wigner [10] to calculate the SU(2)
Clebsch-Gordan coefficients.
Consider the matrix element
hv3 τ3 l3 m3 |Yτv22l2 m2 |v1 τ1 l1 m1 i
=
Z
S4
dΓ Yτv33l3∗m3 Yτv22l2 m2 Yτv11l1 m1 ,
(2.36)
where dΓ is the measure on S 4 and we’re suppressing the argument (γ, Ω) in each harmonic here and in what follows. From the Wigner-Eckart theorem, we have
hv3 τ3 l3 m3 |Yτv22l2 m2 |v1 τ1 l1 m1 i = (v1 τ1 l1 m1 ; v2 τ2 l2 m2 |v3 τ3 l3 m3 )hv3 |||Y v2 |||v1 i,
(2.37)
where hv3 |||Y v2 |||v1 i is a SO(5)-reduced matrix element for Y v2 , viewed as an hyperspheri-
cal SO(5) tensor operator that acts simply by multiplication, and (v1 τ1 l1 m1 ; v2 τ2 l2 m2 |v3 τ3 l3 m3 )
is a SO(5) coupling coefficient. If we define the SO(3)-reduced SO(5) coupling coefficient
(v1 τ1 l1 ; v2 τ2 l2 ||v3 τ3 l3 ) by
(v1 τ1 l1 m1 ; v2 τ2 l2 m2 |v3 τ3 l3 m3 ) = (v1 τ1 l1 ; v2 τ2 l2 ||v3 τ3 l3 )(l1 m1 ; l2 m2 |l3 m3 ),
(2.38)
where (l1 m1 ; l2 m2 |l3 m3 ) is a SO(3) Clebsch-Gordan coefficient, then we can use the orthogonality of SO(3) Clebsch-Gordan coefficients
X
m1 m2
(l1 m1 ; l2 m2 |l3 m3 )(l1 m1 ; l2 m2 |l0 3 m0 3 ) = δl3 l0 3 δm3 m0 3
to arrive at the equation
Z
dΓ Yτv33l3∗m3 Yτv11l1 ⊗ Yτv22l2 l3 m3 = hv3 |||Y v2 |||v1 i(v1 τ1 l1 ; v2 τ2 l2 ||v3 τ3 l3 ).
(2.39)
(2.40)
S4
We choose the norms and phases such that the reduced matrix elements hv3 |||Y v2 |||v1 i
are positive, the coupling coefficients are real and we have the familiar normalisation
X
τ 1 l1 τ 2 l2
(v1 τ1 l1 ; v2 τ2 l2 ||v3 τ3 l3 )2 = 1.
(2.41)
This is enough to uniquely define each coupling coefficient.
In practice, the same computer algebra routine that generates the SO(5) harmonics can perform the integrals in (2.40) analytically, and produce tables of exact SO(3)reduced SO(5) coupling coefficients; see [57] for examples of several such tables.
2.2. SO(5) Hyperspherical Harmonics
31
SO(5) matrix elements
2.2.7
The availability of the SO(5) Clebsch-Gordan coefficients for the one-rowed irreps of
relevance to the collective model, means that only SO(5)-reduced matrix elements of
tensor operators are required for collective model calculations. For example, SO(3)reduced matrix elements of the SO(5) component Q of the quadrupole tensor q̂ = βQ
are given by the Wigner-Eckart theorem and equation (2.40)
√
hv 0 τ 0 l0 ||Q||vτ li = 2l0 + 1 (vτ l; 112||v 0 τ 0 l0 ) hv 0 |||Q|||vi ,
(2.42)
where hv 0 |||Q|||vi is an SO(5)-reduced matrix element and (vτ l; 112||v 0τ 0 l0 ) is a SO(3)-
reduced SO(5) Clebsch-Gordan coefficient.
The hv 0 |||Q|||vi reduced matrix elements were derived in [52] by consideration of the
multiplicity free spherical vibrational states
1
|λv = v + 52 , n = 0, v, τ = 1, l = m = 2vi = √ (d†2 )v |0i,
(2.43)
v!
where |0i is the boson vacuum and we must include the SU(1, 1) labels since q̂ acts in
SO(5)
both Hv
SU(1,1)
and Hv
. We will suppress the redundant labels λv and τ in what follows,
since they’re redundant. The fact that the maximum m = l state of a one-rowed SO(5)
irrep v is l = 2v follows from the branching rule (2.18). For these states, SO(5)-reduced
matrix elements of the quadrupole operator q̂m of equation (2.4) are given by
1
h0|(d)v+1 √12 d† + d (d† )v |0i
h0 v + 1|||q̂|||0vi = p
v! (v + 1)!
p
(v + 1)/2 .
=
(2.44)
The matrix elements of Q are given by the matrix elements q̂ = βQ, factored into radial
and harmonic products
h0 v + 1|||q̂|||0vi = h0 v + 1|β|0 vi hv + 1|||Q|||vi
The radial β matrix element is easily computed, see §2.3.4, and found to be
(2.45)
p
v + 5/2 in
the spherical vibrator basis appropriate for d† . Together with the symmetry relationship
(for our choice of phase convention)
hv|||Q|||v + 1i =
s
dim(v + 1)
hv + 1|||Q|||vi ,
dim(v)
where dim(v) is the dimension of the SO(5) irrep v, we obtain
r
r
v+1
v+3
hv + 1|||Q|||vi =
, hv|||Q|||v + 1i =
.
2v + 5
2v + 3
(2.46)
(2.47)
32
Chapter 2. The Algebraic Collective Model
2.3
SU(1, 1) Radial Wavefunctions
Having described the new approach to constructing hyperspherical harmonics on the
SO(5) Hilbert spaces, we turn our attention to the remaining SU(1, 1) spaces. We begin
with a brief review of the standard and extended su(1, 1) structure as was used in [52],
and then generalise the situation to arrive at the algebraic model basis. We then show
that the matrix elements of any integer power of β can be computed algebraically in this
basis.
2.3.1
The standard spherical realisation of su(1, 1)
The su(1, 1) Lie algebra, spanned by operators with commutation relations
[Ŝ− , Ŝ+ ] = 2Ŝ0 ,
[Ŝ0 , Ŝ± ] = ±Ŝ± ,
(2.48)
has unitary representations with orthonormal basis states {|λ ni; n = 0, 1, 2, · · · } which
satisfy the raising and lowering relations
Ŝ0 |λni = 21 (λ + 2n)|λni ,
p
Ŝ+ |λni = (λ + n)(n + 1) |λ, n + 1i ,
p
Ŝ− |λni = (λ + n − 1)n |λ, n − 1i .
(2.49)
(2.50)
(2.51)
That the sequence of λ states does not terminate for some finite value of n is a consequence
of the fact that SU(1, 1), unlike SO(5), is noncompact.
A standard realisation of the su(1, 1) operators is given by the SO(5) invariants
Ŝ+ = 21 [d† ⊗ d† ]00 ,
Ŝ− = 21 [d ⊗ d]00 ,
Ŝ0 = 41 ([d† ⊗ d]00 + [d ⊗ d† ]00 ) =
1
2
[d† ⊗ d]00 +
(2.52)
5
2
,
(2.53)
where d† is the usual spherical tensor raising operator for the five dimensional oscillator.
The states of seniority v of the collective model span a unitary representation of the
direct product group SU(1, 1) × SO(5) and satisfy the equations
Ŝ0 Rnλ Yτvlm = 21 (λ + 2n) Rnλ Yτvlm ,
p
λ
Ŝ+ Rnλ Yτvlm = (λ + n)(n + 1) Rn+1
Yτvlm ,
p
λ
Ŝ− Rnλ Yτvlm = (λ + n − 1)n Rn−1
Yτvlm ,
where Yτvlm is a hyperspherical harmonic of seniority v.
(2.54)
2.3. SU(1, 1) Radial Wavefunctions
33
In the spherical case, that is zero mean deformation, the SU(1, 1) label is related to
the seniority by λ = v + 5/2. When the above su(1, 1) operators are expanded in terms
of the quadrupole moments {qm } and the corresponding gradient operators they take the
form
1 ∇2
∇2
1
2
2
5
Ŝ± =
− 2 + (aβ) ,
(2.55)
+ (aβ) ∓ q · ∇ + 2 , Ŝ0 =
4 a2
4
a
P
where β 2 = m |qm |2 , ∇m = (−1)m ∂/∂q−m , and we’ve introduced a parameter a which
allows us to vary the width of the resulting oscillator wavefunctions. The spherical vibra-
tor Hamiltonian (1.11) is equal to 2Ŝ0 , and so these radial wavefunctions are solutions
with energy eigenvalues given by
Eλn = (λ + 2n)~ω.
2.3.2
(2.56)
The extended Davidson realisation of su(1, 1)
For deformed nuclei, there is a more useful realization that leads to beta wavefunctions of
the extended Davidson type (see §1.3). As observed by Rowe and Bahri [49], the above
SU(1,1) commutation relations remain unchanged under the substitution
∇2 → ∇ 2 −
(aβ0 )4
,
β2
(2.57)
where β0 is a deformation parameter. Under this substitution, the product wavefunctions
{Rnλ YτvLM } become eigenfunctions of a five-dimensional Davidson oscillator
~ω
∇2 a2 β04
2
ĤDav = 2~ω Ŝ0 →
− 2 + 2 + (aβ)
2
a
β
(2.58)
with energy eigenvalues that continue to be given by the harmonic oscillator expression
(2.56). However, the addition of the a2 β04 /β 2 in the SU(1, 1) operators implies that the
relationship between λ and the seniority becomes
λ=1+
2.3.3
p
(v + 3/2)2 + (aβ0 )4 .
(2.59)
A general realisation of su(1, 1)
The above SU(1,1) operators have the (sometimes) inconvenient property of acting on
both the beta and the SO(5) wavefunctions. This is because ∇2 = ∆2 − Λ̂/β 2 , where
from (1.6)
∆2 =
1 ∂ 4 ∂
β
β 4 ∂β ∂β
(2.60)
34
Chapter 2. The Algebraic Collective Model
and Λ̂ is the SO(5) Casimir operator. However, an SO(5) wavefunction Yτvlm of seniority
v is an eigenfunction of Λ̂ of eigenvalue v(v + 3). Thus, with the identities
∇2
∂
q·∇=β ,
∂β
v(v + 3)
λ v
v
2
Rn Yτ lm = Yτ lm ∆ −
Rnλ ,
β2
(2.61)
(2.62)
it follows that the beta wavefunctions of the Davidson oscillator are solutions of the radial
equation
1
∆2 v(v + 3) + (aβ0 )4
2
+ (aβ) Rnλ (β) = (λ + 2n) Rnλ (β) .
− 2 +
2
a
(aβ)2
(2.63)
Now the relationship (2.59) gives
v(v + 3) + (aβ0 )4 = λ +
1
2
λ−
5
2
,
(2.64)
and implies that the beta wavefunctions of the Davidson oscillator are solutions of the
equation
"
#
λv + 21 λv − 52
∆2
1
2
− 2 +
+ (aβ) Rnλv (β) = (λv + 2n) Rnλv (β) .
2
2
a
(aβ)
(2.65)
We take this as the definition of generalised Davidson radial wavefunctions, and replace
the original SU(1, 1) label λ with the label λv . It is useful because it implies that the basis
of deformed beta wavefunctions {Rnλv } can be defined with a variable dependence upon
the seniority of the accompanying SO(5) wavefunction. It is general since the previously
known bases are all special cases of this general label; see table 2.2.
λv
basis type
v + 5/2
spherical vibrator
p
(v + 3/2)2 + (aβ0 )4 + 1 extended Davidson
δ±
(−1)v −1
2
algebraic model
Table 2.2: Relationships between the general SU(1, 1) label λv and the various bases of
radial wavefunctions.
It follows that the operators
"
#
1
5
2
λ
+
λ
−
∆
1
∂
5
v
v
(λ )
2
2
Ŝ± v =
−
+ (aβ)2 ∓ β
+
,
4 a2
(aβ)2
∂β 2
"
#
1
5
2
λ
−
λ
+
1
∆
v
v
(λ )
2
2
Ŝ0 v =
− 2 +
+ (aβ)2 ,
4
a
(aβ)2
(2.66)
2.3. SU(1, 1) Radial Wavefunctions
35
define a unitary irrep of a general su(1, 1) algebra spanned by the radial wavefunctions
{Rnλv ; n = 0, 1, 2, · · · } for any value of λv ≥ 5/2; i.e., they satisfy the equations
(λv )
Rnλv = 12 (λv + 2n)Rnλv ,
p
(λ )
λv
,
Ŝ+ v Rnλv = (λv + n)(n + 1) Rn+1
p
(λ )
λv
Ŝ− v Rnλv = (λv + n − 1)n Rn−1
.
Ŝ0
(2.67)
It will be shown in the following section that the required matrix elements of the beta
wavefunctions can be obtained by algebraic methods so that the explicit expressions of
the wavefunctions are not needed. Nevertheless it is instructive to note that, because the
radial wavefunctions satisfy identical su(1, 1) algebraic equations to those of the spherical
vibrator, they retain the same functional form
s
2 n! a5
2
(aβ)λv −5/2 e−(aβ) /2 Lnλv −1 (a2 β 2 ),
Rnλv (β) = (−1)n
Γ(λv + n)
(2.68)
when expressed in terms of λv , where Lλnv −1 is an associated Laguerre polynomial. Figure
2.4 shows β 2 R0λv (β) as a function of λv and β.
100
80
λv
60
40
20
1
00
1
2
3
β
4
5
6
7
Figure 2.4: Deformed ground state wavefunctions with a = 2 sequenced as λv increases
from 5/2 to 100. The figure shows that larger λv values correspond to states with larger
mean β values, i.e., greater deformation.
An important property of the above basis which results in substantial simplification is
that the beta wavefunctions {Rnλv }, defined by equations (2.66), (2.67) and (2.68), depend
36
Chapter 2. The Algebraic Collective Model
only on the value of λv . Thus λv can be chosen to be any function of the seniority v of an
accompanying SO(5) wavefunction. For example, a single v-independent value of λv = δ
would be tantamount to assigning v-dependent values to the deformation β0 given by
(aβ0 )4 = (δ + v + 1/2)(δ − v − 5/2) .
(2.69)
In fact, for reasons given below, it will be convenient to assign λv a single fixed value
δ for v even and a value δ − 1 (or δ + 1) for v odd. Thus, in the algebraic model, we
introduce a new deformation parameter δ such that
λv = δ ±
(−1)v − 1
,
2
(2.70)
as in table 2.2. Note that the undeformed value of δ is 5/2 and not zero.
2.3.4
SU(1, 1) matrix elements
Matrix elements of powers of β and of the Laplacian ∇2 are obtained algebraically as
follows. From the definition (2.66) of the su(1, 1) operators, we have the identities
(λ )
(λ )
(λ )
(aβ)2 = 2Ŝ0 v + Ŝ+ v + Ŝ− v ,
λv + 21 λv − 52
∆2
(λ )
(λ )
(λ )
= 2Ŝ0 v − Ŝ+ v − Ŝ− v .
− 2 +
a
(aβ)2
(2.71)
(2.72)
The first identity immediately gives
p
p
λv
λv
(aβ)2 Rnλv (β) = (λv + 2n) Rnλv (β) + (λv + n − 1)n Rn−1
(β) + (λv + n)(n + 1) Rn+1
(β) ,
(2.73)
2
2
2
for any value of λv . With ∇ = ∆ − Λ̂/β , the second identity implies that
λv + 21 λv − 25 − Λ̂
1 2
(λv )
(λv )
(λv )
∇ = −2Ŝ0 + Ŝ+ + Ŝ− +
.
a2
(aβ)2
(2.74)
Thus, in the evaluation of the matrix elements of ∇2 the only unknown quantities are
the matrix elements of 1/β 2 . These are readily evaluated as follows.
λv
denote the matrix element
Let fnm
Z
λv
λv
λv
(β) .
β 4 dβ Rnλv (β) (aβ)−2 Rm
fnm = fmn =
(2.75)
(Recall that the measure on the space of radial wavefunctions is β 4 dβ.) Then, from
equation (2.73), we obtain the recursion relation
p
p
λv
λv
λv
+ (λv + n − 1)n fm,n−1
(λv + 2n) fmn
+ (λv + n)(n + 1) fm,n+1
= δmn
(2.76)
2.3. SU(1, 1) Radial Wavefunctions
and the solution
λv
fnm
(−1)m−n
=
λv − 1
s
37
n! Γ(λv + m)
,
m! Γ(λv + n)
for n ≥ m.
(2.77)
The matrix elements of β are the motivation for our definition of λv in equation 2.70,
and so we switch back to the label λ to show why this is so. They are given by
(aβ) Rnλ (β) =
√
√
λ−1
λ + n − 1 Rnλ−1 (β) + n + 1 Rn+1
(β)
(2.78)
or, equivalently, by
(aβ) Rnλ (β) =
√
λ + n Rnλ+1 (β) +
√
λ+1
n Rn−1
(β).
(2.79)
These equations follow from the known identity for Kummer functions [58]
zM (n, λ + 1, z) = λM (n, λ, z) − λM (n − 1, λ, z)
(2.80)
which are related to associated Laguerre polynomials by
M (−n, λ, z) =
n! Γ(λ) λ−1
L (z).
Γ(λ + n) n
(2.81)
At first sight, it appears inconvenient that βRnλ should be expanded in terms of
wavefunctions for λ ± 1. In fact, it is not. Recall that there is an SO(5) selection rule
which states that any matrix element of a nuclear quadrupole moment is zero between
states that are both even or both odd. Thus, for example, the operator β 3 cos 3γ changes
an even seniority state into an odd seniority state and vice-versa. In order to facilitate
the calculation of matrix elements in deformed nuclei, it is therefore most convenient
to assign λv values for even and odd seniority states that differ by one, as indicated in
equation (2.70).
We can calculate the matrix elements of 1/β in a similar fashion as we did those of
1/β 2 . Equation (2.79) gives the recursion relation
1 λ
1
Rnλ−1 (β) −
Rn (β) = √
aβ
λ+n−1
r
n
1 λ
R (β)
λ + n − 1 aβ n−1
(2.82)
which has the solution
n
X
1 λ
R (β) =
(−1)n−m
aβ n
m=0
s
n! Γ(λ + m − 1) λ−1
Rm (β).
m! Γ(λ + n)
(2.83)
In any given calculation, the deformation parameter δ and the width parameter a can
be optimised to give rapid convergence, e.g., by choosing their values to minimise the
38
Chapter 2. The Algebraic Collective Model
energy expectation of a single-state variational calculation of the ground state energy.
Note that, in using a spherical vibrator basis, one has a choice of width parameter. There
is now a deformation parameter that can be further exploited to enhance the size of the
basis needed for a given level of accuracy. The wavefunction β 2 R0λv , shown for various
values of λv in figure 2.4, illustrates the fact that larger values of λv are appropriate for
nuclei with larger deformations.
2.4
Comparison of the Spherical and Deformed Bases
As a simple example, consider the SO(5) invariant collective model Hamiltonian
v(v + 3)
β2
k β4
1
−
k
+
,
Ĥv (k, hβi ; β) = − ∆2 +
2
2β 2
hβi 2 2 hβi 4
(2.84)
where k/2 is the potential well depth and hβi is a classical equilibrium deformation parameter. This is a radial (beta) Hamiltonian for states of seniority v. From §2.3.4
above, the matrix elements of Ĥv (k, hβi ; β) can be determined algebraically. We wish to
compare the performance of the spherical basis, given by λv = v + 5/2, to that of the
deformed basis given by λv = δ + ((−1)v − 1)/2 (having chosen δ − 1 for odd values of
v), in diagonalising this Hamiltonian.
We give the matrix elements of the three hermitian operators ∇2 , β 2 and β 4 explicitly,
since they will also be used in §3.2. As in §2.2.7, we designate the basis by the two labels
of interest v and n, and we need only give the upper triangular values since the matrices
are symmetric. From equations (2.62), (2.66), (2.67), (2.74) and (2.77) we find


−(λv + 2n) + (λv +1/2)(λλvv−5/2)−v(v+3)
for m = n,


−1
q

2
p
hnv|∇ |mvi
n+1
(λv + n)(n + 1) − (λv +1/2)(λλvv−5/2)−v(v+3)
for m = n + 1,
=
−1
λv +n
2
q

a


 (−1)n−m (λv +1/2)(λv −5/2)−v(v+3) m! Γ(λv +n)
for m > n + 1.
λv −1
n! Γ(λv +m)
(2.85)
From equation (2.73) we find


λv + 2n
for m = n,

 p
a2 hnv|β 2 |mvi =
(λv + n)(n + 1) for m = n + 1,


 0
for m > n + 1.
The matrix elements of β 4 are therefore given by
X
hnv|β 4 |mvi =
hnv|β 2 |jvihjv|β 2 |mvi.
j
(2.86)
(2.87)
39
2.4. Comparison of the Spherical and Deformed Bases
v = 0, n = 0, k = 100, hβi = 5
−36
−38
−40
eigenvalue
eigenvalue
−38
spherical basis
deformed basis
−42
−44
−40
spherical basis
deformed basis
−42
−44
−46
−48
v = 1, n = 0, k = 100, hβi = 5
−36
−46
0
10
20
30
40
number of basis states
50
−48
0
10
20
30
number of basis states
40
50
Figure 2.5: Comparison of the eigenvalues of the ground state and the v = 1, n = 0 first
excited state in the spherical and deformed bases as functions of the number of basis
states. The potential well parameters are k = 100 and hβi = 5.
We optimise each basis over the adjustable parameter(s) by fitting the v = 0, n = 0
ground state energy via a variational procedure. The ground state energy is in general
h00|Ĥ0 (k, hβi )|00i =
4δ + 5 2
kδ −2 kδ(δ + 1) −4
a −
a +
a .
8(δ − 1)
hβi 2
2hβi 4
(2.88)
For the ground state in the spherical basis, δ = 5/2 and a is the only parameter to
be optimised. For the deformed basis, we optimise both a and δ. The optimisation is
performed simply by minimising equation (2.88) with respect to a and/or δ.
Armed with bases so optimised, we can diagonalise Ĥv in an increasingly large Hilbert
space and watch how the eigenvalues converge. Figure 2.5 shows the energies of the
ground and first excited states computed as functions of the number of basis states in
both the spherical and deformed bases. It can be seen that, in both cases, a single
deformed basis state already gives remarkably accurate results.
We can quantify the improvement by taking the ratio of the number of optimised
spherical basis states to the number of optimised deformed basis states needed to achieve
the eigenvalue to some accuracy. Figure 2.6 is a histogram of such ratios for k ranging
from 100 to 1000 and hβi from 1 to 10. The accuracy is set to one percent. It is clear
that the deformed basis outperforms the spherical by large factors that increase with
deformation and well depth.
We can also calculate B(E2) transition rates for this model algebraically. Figure 2.7
compares the intraband transition
B(E2 : n = 0, v = 1 → n = 0, v = 0) ∝ h01|||q̂|||00i
(2.89)
40
Chapter 2. The Algebraic Collective Model
40
ratio
30
20
10
0
10
de
1000
8
for
ma
800
6
tio
nh 4
βi
600
2
200
th
ep
ll d
400
k
we
Figure 2.6: Histogram of the ratios of the number of spherical basis states to that of
deformed states needed to achieve the ground state energy eigenvalue to within one
percent.
41
4.8
1.6
4.6
1.4
B(E2:0, 1 → 0, 0)
k = 100, hβi = 5
4.4
4.2
B(E2:1, 0 → 0, 1)
k = 100, hβi = 5
1.2
spherical basis
deformed basis
4
transition rate
transition rate
2.5. Summary
3.8
1
spherical basis
deformed basis
0.8
0.6
0.4
3.6
0.2
3.4
0
10
20
30
40
number of basis states
50
0
10
20
30
40
number of basis states
50
Figure 2.7: Comparison of the low lying quadrupole intraband and interband transition
rates in the spherical and deformed bases as functions of the number of basis states. The
potential well parameters are k = 100 and hβi = 5.
and the interband transition
B(E2 : n = 1, v = 0 → n = 0, v = 1) ∝ h10|||q̂|||01i
(2.90)
in the two bases as a function of the number of states in the diagonalisation. The
calculations make use of the radial matrix elements of β and the SO(5) reduced matrix
elements of §2.2.7 — we will give more details for such calculations in §3.2.2.
2.5
Summary
This completes our description of the algebraic collective model. In short, it takes the
SU(1, 1)×SO(5) structure that is admitted by the collective model and develops new
bases for both the radial and harmonic sectors of the Bohr model Hilbert space. These
bases are completely algebraic, in that coupling coefficients and matrix elements can be
computed algebraically.
The algebraic model gives considerably simplified methods for collective model calculations, especially those for deformed nuclei. We have confirmed by illustrative calculations that the use of our generalised beta wavefunctions reduces the number of basis
states needed for collective model calculations of deformed nuclei by an order of magnitude. When combined with the fact that in order to to compute the coupled rotationalvibrational spectrum of a deformed nucleus one typically needs ten to twenty SO(5)
harmonics for each angular momentum state and each radial wavefunction, the gain is
enormous. The convergence of the eigenvalues can also be altered by optimising the
42
Chapter 2. The Algebraic Collective Model
parameters a and δ differently. For instance, one could minimise the energy of the head
state of a different band, or even fit the basis states to available experimental data.
We have given expressions for the harmonic SO(5)-reduced matrix elements of Q and
the radial matrix elements of ∇2 , β, β 2 , 1/β and 1/β 2 . The derivation of SO(5) coupling
coefficients enable one to obtain the other matrix elements by combinations of these or by
similar methods. Together these allow one to calculate algebraically the matrix elements
of any collective model potential that is built from SO(5) tensors and has a Laurent series
expansion in the β variable.
Perhaps more important than the substantial practical benefit that it affords, the
algebraic model serves to make the Bohr model a much more robust tool for modelling
nuclear structure, and gives it the potential for exploring regions inaccessible or unsuited
for other models. We will have more to say along these lines in the conclusions.
Chapter 3
A Collective Shape Phase Transition
In this chapter we continue our investigation of deformation in the Bohr collective model
by studying a spherical vibrator to soft rotor shape phase transition. The methods introduced in developing the algebraic model of chapter 2 make this investigation straightforward. The potential used is that of a corresponding transition in the interacting boson
model (IBM) [59]. We begin with a brief overview of phase transitions and dynamical
symmetry, followed by the formulation of the model and the results. It is shown that
the phases are well approximated by the standard Bohr model limits with a second order
transition between them, and that the results closely parallel those of the IBM with the
collective model mass parameter set equal to twice the boson number. We discuss these
results in the following section with regards to quasidynamical symmetry , concluding
with a brief summary.
3.1
Overview
Phase transitions arise in algebraic models as systems adopt different dynamical symmetry limits. By adjusting a control parameter, interactions corresponding to different
subgroup chains are strengthened or weakened. Therefore in general the Hamiltonian of
a system exhibiting a transition is of the form
Ĥ(α) = (1 − α)Ĥ1 + αĤ2 ,
(3.1)
where α is the tunable parameter. The two non-commuting Hamiltonians Ĥ1 and Ĥ2
possess different dynamical symmetries, each corresponding to a phase of the system.
In the majority of cases the transition between the two is smooth, in that observable
43
44
Chapter 3. A Collective Shape Phase Transition
quantities go smoothly into one another as α is varied. If however there is a narrow range
of α values over which these properties change drastically, the transition is classified
as first or second order depending on whether that drastic change is discontinuous or
continuous, respectively. For finite systems such as nuclei, this narrow range of α usually
becomes singular in the infinite limit and defines some critical α value. This is an
example of what is often referred to loosely as a quantum phase transition, since it occurs
at zero temperature and is therefore assumed to be driven by quantum and not thermal
fluctuations.
Here we concern ourselves with such a transition in the collective model. The two
dynamical symmetry limits are those of the spherical vibrator and the gamma-soft rotor.
The algebraic collective model developed in chapter 2 lends itself incredibly well to this
situation since the basis can be optimised for each value of the transition parameter α
as it goes from the spherical phase to the deformed phase. Since the phase transition is
SO(5) invariant, the Hamiltonian can be evaluated algebraically for all values of α using
only the SU(1, 1) machinery developed in §2.3.
Our analysis parallels that of a previous investigation [59] of the U(5) to O(6) transition in the interacting boson model. The IBM has U(6) dynamical symmetry, and
calculations are carried out in the space of u(6) representations labelled by the boson
number N . As the size of an irrep, i.e., the number of bosons, becomes large, the U(5)
and O(6) symmetry limits of the IBM become more and more akin to the spherical vibrator and soft-rotor limits of the Bohr model, a process known as contraction, (the situation
is analogous to that of the P and Q distribution formalism of quantum optics [60]). The
potential giving rise to IBM deformation stems from an O(6) pairing interaction which
can be written in terms of the Bohr coordinates (1.4) using coherent state theory [59, 61].
The result is a SO(5) invariant collective model potential that goes like
V2 = N (−β 2 + β 4 ).
(3.2)
We will show that a collective mass parameter M in the Bohr model plays the rôle that
N does in the IBM. It was also shown in [59] that the persistence of apparent dynamical
symmetries could be explained with limiting approximations in the two phases of the
IBM. We make similar approximations here for the Bohr transition and show that they
too give the small and large α limits quite well.
45
3.2. The Model
3.2
The Model
3.2.1
The Hamiltonian
The spherical vibrator Hamiltonian is, from equation (1.11),
Ĥ1 = −
~2 2 1
∇ + Bω 2 β 2 .
2B
2
(3.3)
The gamma soft (though not Wilets-Jean) Hamiltonian is, from equation (3.2),
Ĥ2 = −
~2 2 1
∇ + Bω 2 − β 2 + β 4 /b2 ,
2B
2
(3.4)
where we’ve included a unit of quadrupole length b. From equation 3.1, the phase transition Hamiltonian is therefore
h
i
~2 2 1
2
2
4 2
Ĥ(α) = − ∇ + Bω (1 − 2α)β + αβ /b .
2B
2
(3.5)
where ∇2 is the R5 Laplacian of equation (1.6), B is a collective mass parameter, and ω
p
is a vibrational angular frequency. It will be convenient to set b = ~/B0 ω equal to 1
and write B = M B0 , so that the mass parameter M is dimensionless, and to set the unit
of energy ~ω = 1. The Hamiltonian then becomes
Ĥ(α) =
i
1 h ∇2
−
+ (1 − 2α)M β 2 + αM β 4 .
2
M
(3.6)
Comparing this to equation (3.2), it is seen that the potential energy component of
Ĥ2 becomes identical to that of the IBM model when M = 2N . The phase transition
potential
M
(1 − 2α)β 2 + αβ 4
(3.7)
2
exhibits a critical point at α = 0.5 at which the coefficient of the β 2 term vanishes. For
p
α ≤ 0.5 the minimum is at β = 0, while for α > 0.5 it is at β = (2α − 1)/2α. Figure
Vα (β) =
3.1 shows a plot of this potential as a function of the transition parameter α, which
clearly shows the emergence of a potential well for α > 0.5. This well serves to stabilise
a nonzero mean value of β and so gives rise to deformation.
The SU(1, 1)×SO(5) basis (2.1) of the algebraic model is particularly well suited to
this situation. As in our simple model of §2.4, the SO(5) invariance means we need only
label states with v and n. The width parameter a and the deformation parameter δ can
be optimised for each value of α, and so as the system transits from a spherical phase
46
Chapter 3. A Collective Shape Phase Transition
V
1
0
−1
1
0
0.5
α
β
1
1.5
2
Figure 3.1: The potential as a function of the control parameter α. A
pcritical point is
reached as α passes through 0.5, whereafter there is a well about β = (2α − 1)/2α.
to a deformed phase the basis can be adjusted accordingly. The matrix elements of the
various terms of the phase transition Hamiltonian (3.6), they being ∇2 , β 2 and β 4 , were
all computed algebraically in §2.4. The ground state energy is given by
h00|Ĥ(α)|00i =
4δ + 5
M
M
a2 + (1 − 2α)δa−2 + αδ(δ + 1)a−4 ,
8M (δ − 1)
2
2
(3.8)
and by minimising this expression numerically for a given M , we find optimal a and δ
values as functions of α. We will denote the bases so optimised as {|nviα }.
The Hamiltonian hnv|Ĥ(α)|mviα is therefore readily diagonalised for any value of α
to give eigenstates in the expanded form
|νviα =
X
n
(vα)
Dνn
|nviα ,
(3.9)
where D (vα) is the diagonalising transformation for a given α and ν now indexes the
(α)
energy eigenbasis. These states have product wavefunctions given by Rνv (β) Yτvlm (γ, Ω)
47
3.2. The Model
with
(α)
Rνv
(β) =
X
(vα) λv
Rn (β),
Dνn
(3.10)
n
where radial and harmonic wavefunctions are given by the algebraic collective model
of chapter 2. Note that since the Hamiltonian is symmetric D (vα) is orthogonal, i.e.,
D −1 = D T .
3.2.2
E2 matrix elements
As in §2.1 we assume that the electric quadrupole operator is proportional to the mass
quadrupole operator,
where d†m and dm
Z
(3.11)
Q̂m = Z q̂m = √ (d†m + dm ),
2
= (−1)m dm are raising and lowering operators for the Hamiltonian
Ĥ(α = 0) and Z is an effective charge which will be used to normalise the transition
rates. SO(5)-reduced E2 matrix elements are then determined algebraically as in §2.2.7,
hnv|||Q̂|||mv 0 i = Zhnv|β|mv 0 ihv|||Q|||v 0i.
(3.12)
From equations (2.78), (2.79) and (2.47) we have the following nonzero matrix elements
for any value of α
r
v+3
hνv|β|µ v + 1iα ,
2v + 3
r
v
hνv|β|µ v − 1iα ,
hνv|||Q̂|||µ v − 1iα = Z
2v + 3
where the beta matrix elements in the energy eigenbasis are
X
(vα) (v 0 α)
hνv|β|µv 0 iα =
Dnν
Dµm hnv|β|mv 0 i,
hνv|||Q̂|||µ v + 1iα = Z
(3.13)
(3.14)
nm
and
p
λv + n,
√
a hnv|β|n − 1 v + 1i =
n,
p
a hnv|β|n v − 1i =
λv + n − 1,
√
a hnv|β|n + 1 v − 1i =
n + 1.
a hnv|β|n v + 1i =
Reduced E2 transition rates are then given for each value of α by
v+3
|hνv|β|µ v + 1iα |2 ,
B(E2; νv → µ v + 1)α = Z 2
2v + 3
v
2
B(E2; νv → µ v − 1)α = Z
|hνv|β|µ v − 1iα |2 .
2v + 3
(3.15)
(3.16)
48
Chapter 3. A Collective Shape Phase Transition
3.2.3
The mass parameter
Typical values of the mass parameter appropriate for nuclear structure physics are obtained by considering ratios of E2 transition rates for decay of the first excited state in
deformed nuclei to those in spherical nuclei. For the spherical limit, λv = v + 5/2 and the
v+5/2
radial wavefunctions Rn
are seen to be exact solutions to the Hamiltonian Ĥ(α = 0)
with a2 = M . The expressions in §3.2.2 give
B(E2; 01 → 00)α=0 =
Z2
Z2
=
.
2a2
2M
(3.17)
With the approximation that for large values of α the mean deformation is equal to the
p
minimum of the phase transition potential, h0v|β|0 v − 1iα ≈ (2α − 1)/2α ≈ 1, we
have
Z2
.
5
(3.18)
2M
B(E2; 01 → 00)∞
=
.
B(E2; 01 → 00)0
5
(3.19)
B(E2; 01 → 00)α=∞ =
The ratio is then predicted to be given by
A comparison of values of this ratio for well-deformed and spherical nuclei in the heavy
elements then suggests that M should take values in the range 10 − 200 [62]. We present
results for M = 100 in most cases.
3.3
3.3.1
Numerical Results
Excitation energies
The choice of where to truncate the Hilbert space and still obtain accurate results in
a numerical diagonalisation of the Hamiltonian is dictated by the value asssigned to
the mass parameter M , the maximum value of α, and the number of eigenstates under
consideration. For M = 100, we find that limiting the basis to states of n ≤ 10 gives
accurate results for the range of α considered. For larger values of M we find it necessary
to take values of n up to a maximum of the order of M/10.
Figure 3.2 is a sequence of plots of the low-lying excitation energies for the model
with increasing mass M . They show that as M is increased, the transition region shrinks
producing a sharp critical point at α = 0.5 as M → ∞.
49
3.3. Numerical Results
Excitation energy
M = 50
M = 100
M = 500
4
(0; 2) 4
4
3
3
3
2
2
1
1
2
(0, 1, 2; 1)
1
(1, 2, 3, 4; 0)
0
0.5
1
α
1.5
20
0.5
1
α
1.5
2 0
0.5
1
α
2
1.5
Figure 3.2: Low-lying excitation energies, as functions of α for M = 50, 100, 500 showing
a phase transition between the spherical vibrator (α 0.5) and soft rotor α 0.5 that
gets sharper as M increases. The curves of each rotor band (α 0.5) are labelled in the
first plot by (v1 < v2 < · · · ; ν), where v increases with energy and ν labels the band.
3.3.2
Wavefunctions
(α)
The ground state radial wavefunctions β 2 R00 (β) for the M = 100 ground state of the
Hamiltonian Ĥ(α), defined by equation (3.10) and multiplied by β 2 so that they become
normalised relative to the measure dβ instead of the more usual β 4 dβ measure, are
shown in Fig. 3.3 as functions of α. The figure indicates that, for α < 0.5, the mean
value of β remains close to its value for the α = 0 spherical oscillator while, for α > 0.5,
it corresponds to that of a nonspherical nucleus with a nonzero mean deformation.
3.3.3
E2 transition rates
Equation (3.16) is used to calculate reduced E2 transition rates as functions of α, where
the transformations D (vα) are given by the numerical diagonalisation above. Figure 3.4
shows ratios of the interband transitions
Rinter (v) = B(E2; 1, v → 0, v + 1)α /B(E2; 1, 0 → 0, 1)α ,
(3.20)
and the intraband transitions
Rintra (v) = B(E2; 0, v → 0, v − 1)α /B(E2; 0, 1 → 0, 0)α ,
(3.21)
50
Chapter 3. A Collective Shape Phase Transition
M = 100
1
0.8
α
0.6
0.4
2
1
0
0.2
0.4
0.6
β
0.8
1
1.2
(α)
Figure 3.3: A sequence of M = 100 ground state wavefunctions β 2 R00 (β) for α ∈ [0, 1]
and β ∈ [0, 1.2] showing the mean deformation β and shape fluctuations of the ground
state as a function of the control parameter α. For α < 0.5, the peak of the wavefunction
changes little from that of its α = 0 limit and its width increases slowly, while for α > 0.5
the mean deformation increases rapidly. Note that the factor β 2 , coming from the volume
element, is included so that the square of the wave function is the probability distribution
for the ground state deformation. The peaks of the wavefunctions are qualitatively similar
for other values of M . However, their widths decrease as M increases.
for several low-lying v.
3.4
3.4.1
Phases and Quasidynamical Symmetry
Fitting the apparent dynamical symmetries
An apparent persistence of the dynamical symmetries associated with the small and
large α limits is very evident in the above results. At α = 0, the Hamiltonian is U(5)invariant and is that of an isotropic five dimensional harmonic vibrator. In the asymptotic
limit of large α, the solutions approach those of an adiabatically-decoupled soft-gamma
rotor-vibrator with a SU(1, 1)×[R5 ]SO(5) dynamical symmetry group. The latter can
be regarded as a contraction of the IBM dynamical symmetry group SU(1, 1)×O(6),
for which gamma-soft rotations are described by an O(6) dynamical group and beta-
51
3.4. Phases and Quasidynamical Symmetry
v=4
4
M = 100
Rintra(v)
3.5
v=3
3
diagonalisation
vibrator
rotor−vibrator
2.5
v=2
2
Rinter(v)
1.5
1
v=1
v=2
v=3
0.5
0
0.5
1
α
1.5
2
Figure 3.4: Ratios, on the same scale, of some intraband B(E2) transition rates Rintra (v)
and interband rates Rinter (v), as defined in the text, for 0 ≤ α ≤ 2 and M = 100. The
horizontal lines are the predictions for the harmonic vibrator (α < 0.5) and adiabaticallydecoupled rotor-vibrator (α > 0.5) approximations.
vibrational excitations by an SU(1, 1) dynamical group.
For a harmonic spherical vibrator with U(5) symmetry, the excitation energies should
all be integer multiples of the lowest excitation energy, as they are at α = 0. Thus we fit
the spectrum
Eνv = (v + 2ν)ωvβ (α),
(3.22)
where ωvβ (α) is determined by the first excitation energy for each α. The extent to which
the excitation energies satisfy this relationship are shown for α < 0.5 in figure 3.5.
Similarly, for a deformed state of the model with an adiabatically-decoupled softgamma rotor-vibrator spectrum, the energy levels should be given by a two-parameter
expression of the form
Eνv (α) = νωβ (α) + A(α)v(v + 3) .
(3.23)
The spectrum of states given by this expression are shown for α > 0.5 in Fig. 3.5 with
A(α) and ωβ (α) adjusted for each alpha separately. The figure shows that, except for a
relatively narrow transition region about α = 0.5, the computed excitation-energies are
52
Chapter 3. A Collective Shape Phase Transition
Excitation energy
4
M = 100
diagonalisation
vibrator
rotor−vibrator
3
2
1
0
0.2
0.4
0.6
α
0.8
1
1.2
1.4
Figure 3.5: Excitation energies, for M = 100, of a harmonic spherical vibrator and an
adiabatically-decoupled rotor-vibrator fitted to the lowest v = 0 and v = 1 excitation
energies.
described well by models with these dynamical symmetries. The ratios of reduced E2transition rates are constant in these two limits. As shown in figure 3.4, these rates also
clearly exhibit the same apparent persistence of the dynamical symmetries associated
with the two limits.
Fitting the parameters to the spectra of the vibrator and rotor-vibrator limits cannot
identify a true dynamical symmetry. What we would like is to derive the parameters
from approximations to the model that have precise dynamical symmetries, and determine whether or not the resulting spectra reproduce those of the original model to some
acceptable level of accuracy. If the agreement is good we say that the model exhibits
quasidynamical symmetry (see [63] for a recent review). One could be surprised to see
spherical vibrator symmetry persisting for α > 0 and soft-rotor symmetry persisting for
α > 0.5. This is because at these intermediate α values, the limiting symmetry of the
Hamiltonian is badly broken and eigenstates become highly mixed combinations of states
with limiting symmetry. The situation can be understood in terms of embedded representations, which make precise the tendency for some coherent superpositions of states
from different irreps of a Lie algebra to behave as though they belonged to an unmixed
53
3.4. Phases and Quasidynamical Symmetry
‘average’ irrep; see references [64, 65] for more details. The preceeding results indicate
that the model is indeed exhibiting quasidynamical symmetry in the two phases, and we
wish to nail this down in the next two sections.
3.4.2
The random phase approximation for α < 0.5
We first consider the nature of the persistent U(5) symmetry in the α < 0.5 domain. The
Hamiltonian Ĥ(0) at α = 0 is that of a five dimensional harmonic oscillator expressible
in the form
1 ∇2
Ĥ(0) =
−
+ M β2 =
2
M
1
2
X
m
d†m dm + dm d†m .
(3.24)
It has a U(5) symmetry group. More generally, for α < 0.5, Ĥ(α) can be approximated
by the five dimensional harmonic oscillator Hamiltonian obtained by neglecting terms in
β of degree higher than quadratic. This is equivalent to a random phase approximation
(RPA) [7]
1 ∇2
+ M (1 − 2α)β 2 ,
−
2
M
This approximate Hamiltonian is brought to diagonal form
Ĥ(α) ≈ Ĥ RPA (α) =
Ĥ RPA (α) =
1
2
√
1 − 2α
X
†
†
Dm
D m + D m Dm
m
(3.25)
(3.26)
by a transformation to new boson operators
†
= xd†m − ydm ,
d†m → Dm
(3.27)
dm → Dm = xdm − yd†m ,
with
x=
1
2
√
1
√ + ω ,
ω
y=
1
2
√
1
√ − ω ,
ω
ω=
√
1 − 2α.
(3.28)
Thus, the excitation energies are given in the RPA by integer multiples of ω. The E2
operators are given by
Z
Z
†
Q̂m = √
(d†m + dm ) = √
(Dm
+ Dm ),
2M
2M ω
(3.29)
√
which means that E2 transition rates are enhanced by the factor 1/ 1 − 2α. However,
ratios of E2 transition rates remain constant.
Figure 3.6 shows the RPA excitation energies in comparison with the precisely computed values. It is seen that the RPA provides a good approximation for smaller values
54
Chapter 3. A Collective Shape Phase Transition
of α but starts to break down as α approaches the critical value 0.5. RPA ratios for
E2 transition rates were shown already for α < 0.5 by the horizontal lines in Fig. 3.4.
Absolute RPA B(E2) transition rates will, of course, diverge as α approaches 0.5 due to
√
the 1/ 1 − 2α factor. This is a familiar result of an approaching phase transition (cf.
Thouless’ theorem [66]). For larger values of the mass M it is observed that the RPA
holds up for larger values of α.
M = 100
diagonalisation
RPA
ADA
Excitation energy
4
3
2
1
0
0.2
0.4
0.6
α
0.8
1
1.2
1.4
Figure 3.6: Comparison of the diagonalised excitation energies as functions of α with the
RPA (for α < 0.5) and the ADA (for α > 0.5).
To the extent that the RPA provides an accurate description of the spectral properties
of the mixed-symmetry Hamiltonian Ĥ(α), it follows that the properties so described
are given by a harmonic vibrational model with a U(5) symmetry. However, the U(5)
symmetry group of the RPA changes continuously as α changes since the D-bosons,
which define the U(5) group, change. Thus, the success of the RPA indicates that while
the states of the Hamiltonian Ĥ(α) continue to be well described by a model with a
U(5) symmetry for a range of values of α, the U(5) symmetry group changes with α.
The relationships between the U(5)α symmetry groups for different values of α are given
by SU(1, 1) conjugation. As was noted in reference [59], the RPA transformation of the
boson operators is a SU(1, 1) transformation. This implies that, as a D-boson vacuum
state, the ground state of the RPA Hamiltonian is given, for each value of α by an
3.4. Phases and Quasidynamical Symmetry
55
SU(1, 1) transformation of the α = 0 vacuum state. Thus, the RPA ground state for
α 6= 0 is a squeezed (in fact dilated) SU(1, 1) coherent state relative to the α = 0 vacuum
state. This can be seen explicitly by observing that the ground-state wavefunction for
the RPA Hamiltonian (3.25) contains a width parameter that increases linearly with
√
1/ ω = (1 − 2α)−1/4 . Thus, in the RPA the width of the ground-state wavefunction
diverges as α approaches 0.5 and the frequency ω collapses to zero. Of course, the RPA
breaks down as a valid approximation before this happens.
3.4.3
The adiabatic-decoupling approximation for α > 0.5
At large values of α and relatively low values of the seniority v, the model acquires a
deformed equilibrium shape with β exhibiting small vibrational fluctuations about some
value β0 . It is also known that, if the vibrational fluctuations in the value of β about β0 are
small, then there is an effective decoupling of the high-frequency beta-vibrational degrees
of freedom and the relatively slow SO(5) rotational degrees of freedom [52]. This is the
Wilets-Jean approximation, (see §1.2.2), which is approached rapidly with increasing α.
The parameters of such an adiabatic approximation are derived by recalling from §1.1
that the Laplacian of the Bohr model is a sum of two terms
∇2 = ∇ 2 −
Λ̂
,
β2
(3.30)
where ∇2 acts only on the β wavefunction and Λ̂ is the SO(5) Casimir operator. An
adiabatic-decoupling approximation (ADA) for the Hamiltonian Ĥ(α) is then given (see
[52]) by
Ĥ(α) ≈ Ĥ ADA =
i
1 h ∇2
1
Λ̂.
−
+ (1 − α)M β 2 + αM (−β 2 + β 4 ) +
2
M
2M β02
(3.31)
The energy levels are given in this approximation by
Eνv = Eν0 +
1
v(v + 3),
2M β02
(3.32)
where {Eν0 } are the energy levels of the Hamiltonian Ĥ(α) in seniority v = 0 states. The
corresponding energy-level spectra are compared with the numerically computed spectra
in figure 3.6.
Reduced E2 transition rates are given in the ADA by equation (3.16) with the vindependent β matrix elements
hνv|β|ν, v − 1iα = β0 (α),
hνv|β|ν − 1, v + 1iα = hν0|β|ν − 1, 1iα .
(3.33)
56
Chapter 3. A Collective Shape Phase Transition
In this approximation, the value of β0 should be very close to the minimum of the
potential (3.7) and therefore given by β02 ≈ (2α − 1)/2α. However, for computing the
ratios (3.20) and (3.21), the β matrix elements are not needed. The ratios of reduced E2
transition rates between rotational states of the ground-state multiplet are given in the
ADA by
Rintra (v) =
5v
B(E2; 0v → 0, v − 1)
=
,
B(E2; 0, 1 → 0, 0)
2v + 3
(3.34)
B(E2; 1v → 0, v + 1)
v+3
=
,
B(E2; 1, 0 → 0, 1)
2v + 3
(3.35)
and the ratios between β-vibrational states and those of the ground-state multiplet are
given by
Rinter (v) =
They are independent of α and given by the horizontal lines shown for α > 0.5 in figure
3.4.
It is again found, both for the energy-level spectrum and E2 transition rates that the
ADA works increasingly well as the mass M increases and that its domain of validity
extends more and more closely to the lower α = 0.5 limit with increasing M . These
results signify that when the ADA provides an accurate approximation to the solutions
of the original model Hamiltonian Ĥ(α), then indeed, to that level of accuracy, the model
has an SU(1, 1)×[R5 ]SO(5) quasidynamical symmetry. Thus, the adiabatic-decoupling
approximation shows that there is a single SU(1, 1)×[R5 ]SO(5) dynamical group in the
rotor-vibrator domain, corresponding to the beta and SO(5)-rotational degrees of freedom as defined for the model. However, unlike the RPA in which the symmetry group
U(5) evolved with changing α, in the ADA it is the irrep which evolves with α. From
the subgroup chain (1.13), the irrep of [R5 ]SO(5) appropriate for a given value of α is
defined by the value of β0 and, as we have seen, it has an α-dependent value close to
p
(2α − 1)/2α. Thus, the SU(1, 1)×[R5 ]SO(5) quasidynamical group remains fixed but
its irreps evolve with changing α.
3.5
Summary
This completes our analysis of the spherical vibrator to deformed gamma-soft rotor phase
transition in the Bohr collective model. The analysis parallels that of reference [59], in
which the U(5) to O(6) transition of the interacting boson model was considered. The
two models are related by a group contraction — that is, in the limit of large boson
number N , the U(5) and O(6) limits of the IBM approach those of the U(5) (spherical
3.5. Summary
57
vibrator) and [R5 ]SO(5) (deformed soft rotor) limits of the collective model. Thus, a
phase transition between these two limits in the IBM is expected to share many features
with that of the collective transition. The IBM potential, mapped into the Bohr model
Hilbert space using coherent states [59], is shown to correspond directly to a Bohr model
with the boson number N replaced by half the collective mass parameter M . Thus the
large N limit of the IBM, where the transition becomes sharp, is seen to correspond to
a Bohr model with large M . Figure 3.2 shows that the collective model transition also
becomes sharp in this limit. The model also predicts [67] that the energy levels at the
critical point should scale as M −1/3 which, given the correspondence M = 2N , agrees
very closely with the N −0.3 scaling result of the IBM.
By analysing the apparent persistent symmetries for α > 0 and α > 0.5 using simplified models it is seen that both phases of the transition exhibit quasidynamical symmetry.
For M = 100, in the vibrator phase, a random phase approximation shows that the spectrum and B(E2) transitions are reproduced by a different U(5)α dynamical symmetry
group for each α . 0.2. In the rotor phase, an adiabatically-decoupled approximation
shows that the properties are reproduced by different Wilets-Jean irreps, labelled by the
(α)
deformation β0 , of a fixed SU(1, 1) × [R5 ]SO(5) dynamical symmetry group for each
α & 0.8.
The algebraic collective model is used to perform the calculations. Since the model is
SO(5) invariant, su(1, 1) is the spectrum generating algebra and only the radial SU(1, 1)
basis is required is computing the Hamiltonian matrix. This basis is optimised for each
value of the transition parameter α by minimising the ground state energy. Better agreement can be achieved by minimising the energy of the lowest state in each v band separately. The basis remains orthonormal since v is a good quantum number, however in a
model without SO(5) as a static symmetry this would not be the case.
58
Chapter 3. A Collective Shape Phase Transition
Chapter 4
Coherent State Theory
In this chapter, we review the key features of coherent state theory in order that the
construction of the generic so(5) irreps in chapter 5 might be more transparent. We give
several examples and introduce some new terminology which will be used throughout.
The reader familiar with the group theoretical approach to coherent states and their use
in representation theory may safely skip to chapter 5.
4.1
Background
Coherent states were introduced by Schrödinger [68] in his analysis of the harmonic
oscillator in quantum mechanics, and were made popular in quantum optics by Glauber
[69] and Sudarshan [70]. It was also at this time that Klauder began the generalisation [71]
of the the harmonic oscillator coherent states to those of Lie groups other than HW(1), a
process that culminated in the ‘generalised’ coherent state theory of Perelomov [72] and
Gilmore [73]. By now there are several books and review articles that cover both the
physical and mathematical aspects of coherent states in great detail [74, 75, 76, 77].
Our interest in the theory of coherent states here is driven by the considerable advantage it gives us in the construction of representations of groups and algebras. There have
been many developments along these lines, the most powerful, from a physical perspective, being vector coherent state theory [36, 37, 38, 78]. In chapter 5 we will be concerned
ultimately with vector coherent state representations of so(5) for physical applications,
and the following overview will introduce only the concepts necessary for an understanding of this problem. It should be stated that the general theory is capable of handling a
much broader range of situations [79].
59
60
Chapter 4. Coherent State Theory
Roughly speaking, given a Lie group (algebra) the idea of vector coherent state theory
is to find suitable well understood subgroups (subalgebras) and formally suppress the
details of their representation theory, leaving a simplified problem to solve. We begin
with the definition of scalar coherent states and representations, leading naturally into
that of vector coherent states. In doing so we introduce some new terminology which
will be used throughout this thesis.
4.2
Scalar Coherent States
The coherent states with which physicists are most familiar are probably those of the
simple harmonic oscillator, {|zi; z ∈ C}. These have been associated with quantal states
of the electromagnetic field since the work of Glauber [69], where he showed that there
were three equivalent ways of defining them:
(i) |zi is an eigenstate of the simple harmonic oscillator annihilation operator â;
â|zi = z|zi.
(4.1)
(ii) |zi is a minimum uncertainty state, (setting constants to unity);
1
∆q̂ ∆p̂ = ,
2
where
√
(4.2)
q
√
2q̂ = â + ↠, i 2p̂ = â − ↠and ∆X̂ = hz|X̂ 2 |zi − hz|X̂|zi2 .
(iii) |zi is a displaced vacuum;
|zi = D(z)|0i,
(4.3)
where D(z) = exp(z↠− z ∗ â).
It was realised that definition (iii) had a group theoretical interpretation, namely that
the displacement operator defines a representation of the dynamical group hw(1) for the
simple harmonic oscillator. Thus the problem of generalising the notion of coherent states
to other physical situations comes to defining coherent states for other dynamical groups,
and (iii) is the adopted definition. Note that definitions (i) and (ii) do not in general
lead to the same coherent states as (iii), if they lead to physically useful coherent states
at all [76].
These observations allow for the definition of generalised scalar coherent states. Consider a representation T of a Lie group G carried by an inner product space V. The states
61
4.2. Scalar Coherent States
{T † (g)|φi; g ∈ G} span some subspace Uφ ⊆ V for any fixed state |φi ∈ V. If G has an
invariant measure, we can map the space Uφ into a space of square integrable functions
on the group
Uφ → L2 (G);
|ψi 7→ Ψ(g) = hφ|T (g)|ψi.
(4.4)
A representation Γ of G on L2 (G) is then given by
0 Γ(g )Ψ (g) = hφ|T (g)T (g 0)|ψi = Ψ(gg 0 ),
g, g 0 ∈ G,
(4.5)
since Γ(g)Γ(g 0 ) = Γ(gg 0 ) for g, g 0 ∈ G. Moreover there is a representation of the Lie
algebra g of G given by
Γ(X)Ψ (g) = hφ|T (g)T (X)|ψi,
X ∈ g,
g ∈ G,
(4.6)
since Γ(X), Γ(Y ) = Γ([X, Y ]) for X, Y ∈ g. The states |gi ≡ T † (g)|φi are called scalar
coherent states, the functions Ψ(g) are coherent state wavefunctions and the representation Γ is called a coherent state representation.
So far the coherent state construction has little or no obvious advantage over the
original representation T on V. However we can immediately identify some advantageous
properties.
4.2.1
Intrinsic state
The choice of the state |φi has important consequences for the resulting coherent states.
We call this the intrinsic state vector. If the intrinsic vector |φi lies in a G-invariant
subspace of V, then Uφ will be an invariant subspace of V and the coherent state rep-
resentation Γ will be isomorphic to a subrepresentation of T . Iterating this idea, if |φi
lies in an irreducible subspace, then Γ will be an irrep. This provides one advantage
over the original representation T , which may have been reducible. Other choices such
as requiring |φi to be an eigenstate of an unperturbed Hamiltonian, or requiring |φi to
be a highest weight state of an SGA lead to other potentially desirable properties. A
good choice of intrinsic vector therefore enables one to make a fairly arbitrary choice of
original representation. One can for example use a well known representation in order to
induce a new representation with specific properties of interest.
62
Chapter 4. Coherent State Theory
4.2.2
Orbiter subgroup
In the above construction, the use of the entire group G guarantees that the states
{T † (g)|φi; g ∈ G} span the carrier space Uφ . However, the entire group G is often not
necessary. The greatest advantage afforded by the coherent state construction is gained
by identifying a subset S ⊂ G such that the states {T † (s)|φi; s ∈ S} also span the
carrier space Uφ . One such subset S is easily identified, since we can ‘factor out’ the
group elements that leave the intrinsic state invariant, up to phase, under the action of
T . These elements form a group, called the isotropy subgroup Hφ ⊂ G of the intrinsic
state |φi,
Hφ = {h ∈ G | T † (h)|φi = eiχ(h) |φi},
(4.7)
and our subset is given by a set of representatives of the coset space
S = Hφ \G.
(4.8)
T † (g)|φi = T † (hs)|φi = T † (s)T † (h)|φi = eiχ(h) T † (s)|φi.
(4.9)
One now has
The phase factor eiχ(h) gives a one dimensional representation of the isotropy subgroup
Hφ , and the coherent state representation of G resulting from this construction is said
to be induced from this subgroup representation. Since global phases are irrelevant for
the resulting quantal states, the restricted set {T † (s)|φi} are often called the coherent
states for G with respect to the intrinsic state |φi. They are again guaranteed to span
the space Uφ .
Often, we wish to use a subset of G that is also a group; any subgroup Hω ⊂ G
such that the basis {T † (h)|φi; h ∈ Hω } spans Uφ can be used to construct coherent state
wavefunctions. One generally seeks a subgroup that has well understood properties and
is of physical interest. We call Hω an orbiter group, since the orbit of |φi under the
action of Hω must span the irrep space, and we call its Lie algebra the orbiter subalgebra.
One now has scalar coherent state wavefunctions in L2 (Hω ) given by
Ψ(h) = hφ|T (h)|ψi.
4.2.3
(4.10)
Example: Heisenberg-Weyl group in one dimension
Consider the complexified Heisenberg-Weyl group HW(1) with Lie algebra hw(1) spanned
by {a, a† , I} with the usual commutation relations. This is a spectrum generating algebra
63
4.2. Scalar Coherent States
for the one dimensional harmonic oscillator Ĥ = a† a + 21 I. Take the representation
T carried by the Hilbert space H of the one dimensional harmonic oscillator which is
spanned by the Fock states {|ni; n = 0, 1, · · · , ∞}, where
√
n|n − 1i,
T (a)|ni =
√
T (a† )|ni =
n + 1|n + 1i,
T (I)|ni = |ni.
(4.11)
An arbitrary element of HW(1) is given by g(z, ϕ) = eza
T (g(z, ϕ))|ni = ezT (a
† )−z ∗ T (a)+iϕT (I)
|ni,
† −z ∗ a+iϕI
and the states
z ∈ C, ϕ ∈ R,
(4.12)
span H for any fixed |ni. One could use this basis to define coherent states, but it is
unwieldy.
By choosing the intrinsic state vector to be the harmonic oscillator ground state |0i,
we find that the isotropy subgroup U(1) of |0i is generated by I. If we factor this phase
out, choosing g(z, 0) as coset representatives, we arrive at Glauber’s coherent states
T (g(z, 0))|0i = ezT (a
† )−z ∗ T (a)
|0i.
(4.13)
In fact, we arrive at the same results using the orbiter subgroup generated by a† ; that
is the states
†
T (h(z))|0i = ezT (a ) |0i,
(4.14)
span H. Any state |ψi in H therefore has a scalar coherent state wavefunction
Ψ(w) = h0|ew
∗ T (a)
|ψi,
w ∈ C.
(4.15)
The coherent state representation of HW(1) on this space is given by
†
∗
Γ(g(z, ϕ))Ψ (w) = h0|ew∗T (a) ezT (a )−z T (a)+iϕT (I) |ψi
= eiϕ+w
∗ z− 1 |z|2
2
h0|e(w
∗ −z ∗ )T (a)
|ψi
∝ Ψ(w − z).
(4.16)
The coherent state representation of hw(1) on this space is given by
∂
∂
h0|ezT (a) |ψi =
Ψ(z),
Γ(a)Ψ (z) = h0|ezT (a) T (a)|ψi =
∂z
∂ z
Γ(a† )Ψ (z) = h0|ezT (a) T (a† )|ψ = h0| T (a† ) + zT (I) ezT (a) |ψi = zΨ(z),
Γ(I)Ψ (z) = Ψ(z).
(4.17)
This is known as the Bargmann-Segal representation [27].
64
Chapter 4. Coherent State Theory
4.3
Vector Coherent States
Although the scalar coherent state construction described above can be applied to many
important Lie groups, the condition that the states {T † (h)|φi} span the carrier space of
the coherent state representation is highly restrictive. One way to apply the coherent
state method to a larger class of physically interesting subgroup chains is to allow for
more than one intrinsic state vector. This yields the extremely powerful vector coherent
state (VCS) method.
Much of the method is the same as that for scalar coherent states. Consider a representation T of a Lie group G on a vector space H. Instead of just one, choose a set of
intrinsic states {|φj i}, which span a so-called intrinsic space Hι ⊂ H. The condition is
that the union of the orbits of the intrinsic states under the action of a suitably chosen
orbiter subgroup Hω ⊆ G spans an invariant subspace of the representation T . At this
point, the ordered set of coherent state wavefunctions
Ψj (h) = hφj |T (h)|ψi,
h ∈ Hω ,
(4.18)
completely determines the state |ψi ∈ H. The question is how to choose intrinsic states
and an orbiter group that best reflects the physics of a situation, while remaining mathematically tractable. VCS theory accomplishes this by choosing the basis of the intrinsic
space to carry an irrep of a useful subgroup Hι ⊆ G, call it the intrinsic subgroup, which
still respects the spanning condition under the action of an orbiter subgroup Hω ⊆ G;
that is, starting from a basis {|φj i} of the intrinsic space, the action of Hω generates the
full carrier space for an irrep of G, (in this case Hι and Hω are sometimes called complementary). This is the essential difference between scalar and vector coherent states —
the former utilises a one dimensional irrep on the intrinsic space, the latter a (finite) multidimensional irrep. The representation theory of Hι should be well understood, and we
call its algebra the intrinsic subalgebra. Let Hι be a finite dimensional carrier space for a
unitary irrep of Hι with basis wavefunctions {Φj } such that there is a natural embedding
Hι → H ι ,
|φj i 7→ Φj .
(4.19)
An arbitrary state |ψi ∈ H then has a wavefunction that takes values in Hι , called a
vector coherent state;
Ψ(h) =
X
j
Φj hφj |T (h)|ψi,
h ∈ Hω .
(4.20)
65
4.3. Vector Coherent States
The coherent state representations of the group G and the algebra g are then defined in
the same way as that for scalar coherent states;
X
Γ(g)Ψ (h) =
Φj hφj |T (h)T (g)|ψi,
j
4.3.1
X
Γ(X)Ψ (h) =
Φj hφj |T (h)T (X)|ψi,
j
g ∈ G,
X ∈ g.
(4.21)
Example: free particle with spin
Consider a spin s particle in three dimensions. This situation can be modelled algebraically by the semidirect product group [HW(3)]SU(2), with semidirect sum Lie algebra [hw(3)]su(2) spanned by the operators {xi , pi , I, Ji ; i = 1, 2, 3}. The commutation
relations for this algebra are
[xi , pj ] = i~δij I,
[Ji , Jj ] = Jk ,
[xi , xj ] = 0,
j
[pi , pj ] = 0,
k
[Ji , x ] = ix , [Ji , pj ] = ipk (ijk) cyclic.
(4.22)
Let the intrinsic subspace of spin states for the particle at the origin be spanned by
{|smi}. Map these intrinsic states to a Hilbert space of spin wavefunctions Φsm , which
carries a convenient representation S of the intrinsic spin subalgebra su(2) spanned by
{Ji ; i = 1, 2, 3}.
The entire Hilbert space for the particle is then spanned by the orbit of the intrin-
sic states under the action of the orbiter subgroup R3 , which is generated by {pi ; i =
1, 2, 3} ⊂ [hw(3)]su(2). Denote a hermitian representation of [hw(3)]su(2) on this space
ˆ Jˆi }. Then a unitary representation of an element of the orbiter subgroup is
by {x̂i , p̂i , I,
i
given by the translation operator T (r) = e ~
P
i
r i p̂i
; r i ∈ R, and an arbitrary state of the
particle |ψi has the vector coherent state wavefunction
X
X
i P i
Ψ(r) =
Φsm hsm|T (r)|ψi =
Φsm hsm|e ~ i r p̂i |ψi.
m
(4.23)
m
With the identities (where X̂ is the representation of an arbitrary element of [hw(3)]su(2))
hsm|T (r)X̂|ψi = hsm|T (r)X̂T (−r)T (r)|ψi
iX j
= hsm| X̂ +
r [p̂j , X̂] + · · · T (r)|ψi,
~ j
i
hsm|x̂i |ψi = 0,
m
i P
∂
j
p̂ |ψi = −i~ i hsm|e ~ j r p̂j |ψi,
X ∂r
Φsm hsm|Jˆi =
S(Ji )Φsm hsm|,
P
hsm|e ~
X
j
r j p̂j i
m
(4.24)
66
Chapter 4. Coherent State Theory
one finds that the coherent state representation of [hw(3)]su(2) is given by
X
i P
j
Γ(xi )Ψ (r) =
Φsm hsm|e ~ j r p̂j x̂i |ψi = r i Ψ(r),
m
X
i P
∂
j
Γ(pi )Ψ (r) =
Φsm hsm|e ~ j r p̂j p̂i |ψi = −i~ i Ψ(r),
∂r
m
h
X
i P
∂ i
∂
j
Γ(Ji )Ψ (r) =
Φsm hsm|e ~ j r p̂j Jˆi |ψi = S(Ji ) − i r j k − r k j Ψ(r), (ijk) cyclic,
∂r
∂r
m
Γ(I)Ψ (r) = Ψ(r)
(4.25)
This is the Schrödinger representation of a particle with intrinsic spin [27].
The vector coherent state construction effectively incorporates all of the details of the
intrinsic state space, in this case the spin space spanned by {|smi}, via the assumption
that the wavefunctions Φsm and representation S of the intrinsic subalgebra are well
understood. The focus is then upon the representation theory of the rest of the model,
in this case the free kinematics. This illustrates the facility with which VCS theory can
create algebraic models by starting with a well known representation and build up extra
structure by embedding the well known situation in a larger system, viewing the former
as intrinsic. A representation on the larger system is then induced from that of the well
known intrinsic system by the methods detailed above.
4.4
Inner Products and Unitarity
The construction of coherent states provides us with a basis for the carrier space of a
representation, however this basis does not necessarily come equipped with a natural inner
product. In quantum mechanics we normally require the representations of operators
corresponding to transformations of a physical system to be unitary, since this conserves
probabilities. This amounts to defining a unique inner product on the Hilbert space of the
coherent state representation, with respect to which the carrier states are orthonormal.
Although the scalar and vector coherent states outlined above guarantee that the coherent
states span the Hilbert space, it is often necessary to transform this basis to a unitary
one. A useful method for doing so is the so-called K-matrix procedure [36, 79, 80].
The idea is straightforward; given a coherent state representation Γ̂ for a group G
and some basis of carrier states {ψ̃i }, assume that the matrices Γ̃
X
ψ̃j Γ̃ji (g), g ∈ G,
Γ̂(g)ψ̃i =
j
(4.26)
67
4.4. Inner Products and Unitarity
are not unitary. Introduce a change of basis to the states ψi given by the K-matrix
ψi = K ψ̃i =
X
ψ̃j Kji ,
(4.27)
ψj Γji (g)
(4.28)
j
such that the matrices Γ
Γ̂(g)ψi =
X
j
are unitary. Thus we have
Γ(g) = K −1 Γ̃(g)K.
(4.29)
The K matrix is not unique since any K 0 = U K with U unitary accomplishes the
same thing. This allows us to choose bases with useful features, such as the reduction
of a subgroup chain G ⊃ H, by choosing an appropriate K. This is made slightly more
complicated when there are multiplicities in the reduction of irreps of G down to irreps
of H. In this case there is a missing label in the subgroup chain, and we introduce some
nondegenerate operator whose eigenvalues can serve as that label (see equation (5.55)
and discussion thereabout). The procedure is undertaken in §5.4.5, where we perform it
for the subalgebra chain so(5) ⊃ so(3).
68
Chapter 4. Coherent State Theory
Chapter 5
Generic Representations of so(5) in a
so(3) Basis
In the previous two chapters we have been concerned only with completely symmetric (one-rowed) representations of SO(5), since the excitations of the collective model
are bosonic. In the interest of completeness, and in order to break ground for future
researchers, here we will construct all of the representations of the generating algebra
so(5) in the same so(3) basis as the symmetric collective model representations. It is
probably the most ambitious application of vector coherent state theory to date.
Our construction is a generalisation of the basis for the one-rowed so(5) irreducible
representations of Rowe and Hecht [53, 54], and complements existing bases such as
that of the Gel’fand chain SO(5) ⊃ SO(4) ⊃ SO(3) [81] and Hecht’s basis for the chain
SO(5) ⊃ SO(4) ∼ SU(2) × SU(2) [82].
We begin with a brief overview of the so(5) Lie algebra and its representation theory.
Unfortunately the generic irreps of SO(5) do not share the simplified structure of the
one-rowed irreps, so that for instance we do not a priori have an inner product inherited
from the S 4 subspace. We therefore use coordinates different from {β, γ, Ω} and must
appeal to the full power of vector coherent state theory.
5.1
The so(5) Lie Algebra
The special orthogonal Lie algebra so(5) is defined as the linear span of infinitesimal
generators of the Lie group SO(5), the connected (unit determinant) orthogonal transformations in five dimensions. There are ten such linearly independent generators, (SO(5)
69
70
Chapter 5. Generic Representations of so(5) in a so(3) Basis
has ten ‘Euler angles’), so so(5) is a ten dimensional vector space over the real numbers
equipped with a Lie product; the commutator. It is semisimple and isomorphic to the
symplectic group sp(2). The root diagram of so(5), or more precisely its complex extension, is given in figure 5.1.
T+
I
@
@
A++
6
@
@
A−+ S− @
@
@
S 0 , T0
@
@
?
A−−
@
S+
- A+−
@
R
@ T−
Figure 5.1: Root diagram for the so(5) Lie algebra in the basis {A, S, T }.
We choose a basis for so(5) wherein there are two mutually commuting su(2) subalgebras {S+ , S0 , S− } and {T+ , T0 , T− }, which together span a SO(4) subalgebra. The
commutation table for so(5) in this basis is given in table 5.1.
A++
A+−
A−−
A−+
S+
S0
− 21 A++
− 21 A+−
A++
0
S+
S0 + T 0
T+
0
A+−
−S+
0
T−
S0 − T 0
0
A−−
−S0 − T0
−T−
−T+
T0 − S 0
S0
1
2 A++
1
2 A+−
S−
A−+
−A−−
A−+
S+
0
T+
0
T0
1
2 A++
T−
A+−
0
A++
− 21 A+−
0
0
S−
−A+−
− 21 A−−
0
−A−+
− 21 A−−
0
−S−
0
A++
− 21 A−+
A+−
−A++
0
S+
S−
−A−+
A−−
T+
T0
T−
0
− 21 A++
1
2 A+−
−A+−
1
2 A−−
− 12 A−+
0
−A++
1
2 A−−
1
2 A−+
0
0
0
−S+
2S0
0
−S−
0
A−+
0
0
0
0
0
0
0
0
−T+
2T0
S−
0
0
0
0
1
2 A−+
0
0
0
T+
−A−−
0
0
0
−2T0
0
A−−
0
−2S0
0
0
0
T−
−T−
Table 5.1: The so(5) commutation relations, with ordering [row, column].
5.1.1
Irreducible representations and weight diagrams
so(5) is a rank two algebra, (it has two dimensional Cartan subalgebras), so each of its
irreps is characterised by a two dimensional lattice called a weight diagram. At each
lattice point in a weight diagram, there is one or more weights representing a linearly
0
71
5.1. The so(5) Lie Algebra
independent state in the carrier space of the corresponding irrep; thus the dimension of
an irrep is given by the number of weights in its weight diagram. The weight diagram for
the adjoint irrep, carried by the algebra itself as a vector space, is called the root diagram
and has one weight for each linearly independent element of the algebra; ten for so(5).
The entire representation ring of so(5) is generated by two fundamental irreps of so(5);
one five dimensional and the other four dimensional. (The latter actually corresponds to
an irrep of the double covering of SO(5), SP IN (5). We call such irreps spinor in analogy
to the so(3) case.)
We can classify an arbitrary irrep of so(5), like any semisimple Lie algebra, by a
highest weight which we label (v, f ). From the origin of a weight diagram, (the centre
of symmetry), v gives the number of lattice sites to move ‘north’ (the A++ direction of
figure 5.1) and f the number of lattice diagonals to move ‘northeast’ (the S+ direction)
before arriving at the highest weight. With this labelling scheme the fundamental irreps
are (1, 0) (dimension 5, diamond), and (0, 21 ) (dimension 4, square). Since the tensor
product of highest weight vectors is another highest weight vector, with weight equal to
the sum of the original weights, this labelling of the fundamental irreps implies that v is
an integer and f is a half-integer. The dimension of an arbitrary irrep (v, f ) is given by
the Weyl formula, in terms of these labels,
1
dim(v, f ) = (2v + 2f + 3)(v + 2f + 2)(v + 1)(2f + 1).
6
(5.1)
We denote an abstract Hilbert space that carries the irrep (v, f ) by H (v,f ) .
|(0,
1 −1
) i
2 2
r
r|(0,
1 1
) i
2 2
r
@
r
r
r|(1, 0)0i
@
@
@r
r
@
@r
Figure 5.2: The weight diagrams for the fundamental irreps (0, 21 ) and (1, 0) with highest
grade states labelled |(v, f )mi (see §5.3).
The highest weight state and indeed all of the weights bordering an irrep’s weight
diagram are multiplicity free. We call the horizontal (‘east-west’) string of weights containing the highest weight the highest grade of a so(5) irrep, which will play a key rôle
in our construction.
72
Chapter 5. Generic Representations of so(5) in a so(3) Basis
The weight diagrams of the fundamental irreps are given in figure 5.2. Due to the
fact that the fundamental weight diagrams are diamond and square shaped, it follows
from the nature of the tensor product that so(5) irreps generally have octagonal weight
diagrams, (taking the square and diamond as octagons with appropriate sides of zero
length).
5.1.2
Realisation of so(5)
In order to facilitate our construction we adopt boson realisations [83] for our algebra,
also known as Weil realisations [84]. Since the fundamental irreps of so(5) are four and
five dimensional, we consider the subgroup chains U(4) ⊃ SO(5) and U(5) ⊃ SO(5),
where U(4) and U(5) are the symmetry groups of the four and five dimensional harmonic
oscillators respectively.
The u(4) realisation
†
Let {ζm
; m = ± 32 , ± 21 } be creation operators with boson commutation relations
†
[ζ n , ζm
] = δnm I,
(5.2)
2n
† n
where ζ n = (ζn† )† and I is the identity operator. Then {Z2m
≡ ζm
ζ } is a basis for the
complex extension of u(4). Let these creation operators act upon the ground state boson
vacuum |0i and create the weights in the fundamental so(5) irrep (0, 21 ) as shown in figure
5.3.
ζ †1 |0i r
2
ζ †−3 |0i r
2
ζ †3 |0i
r
2
ζ †−1 |0i
r
2
Figure 5.3: The fundamental weight diagram (0, 21 ) in terms of boson creation operators.
The following so(5) ⊂ u(4) embedding yields the correct so(5) commutation relations
73
5.1. The so(5) Lie Algebra
in agreement with table 5.1;
1
3
√1 (Z −3 − Z −1 )
A−− = √12 (Z−3
− Z−1
)
1
3
2
−3
−1
1
1
1
3
A−+ = √2 (Z1 + Z−3 )
A+− = √2 (Z3 + Z−1 )
1
T+ = Z1−1
T− = Z−1
3
S+ = −Z3−3
S− = −Z−3
−1
)
T0 = 21 (Z11 − Z−1
−3
1
S0 = 2 (Z33 − Z−3 ).
A++ =
(5.3)
The u(5) realisation
Let {ην† ; ν = ±2, ±1, 0} be harmonic oscillator creation operators with boson commutation relations
[η ν , 絆 ] = δνµ I,
(5.4)
where η ν = (ην† )† and I is the identity operator. Then {Eµν ≡ 絆 η ν } is a basis for the
complex extension of u(5). Let these creation operators act upon the ground state boson
vacuum |0i and create the weights in the fundamental so(5) irrep (1, 0) as shown in figure
5.4.
η2† |0i
r
@
†
η−1
|0ir
@ r
η0† |0i@@r
@
†
@rη1 |0i
†
η−2
|0i
Figure 5.4: The fundamental weight diagram (1, 0) in terms of boson creation operators.
The following so(5) ⊂ u(5) embedding yields the correct so(5) commutation relations
in agreement with table 5.1;
A++ = E20 − E0−2
0
A−− = E02 − E−2
−2
T+ = E21 + E−1
−1
T− = E12 + E−2
A+− = E10 + E0−1
S+ = E2−1 + E1−2
0
A−+ = E01 + E−1
2
1
S− = E−1
+ E−2
−1
−2
− E−2
)
T0 = 21 (E22 − E11 + E−1
−1
−2
− E−2
).
S0 = 21 (E22 + E11 − E−1
(5.5)
74
Chapter 5. Generic Representations of so(5) in a so(3) Basis
Although our labelling scheme suggests that these creation operators have good an-
gular momentum quantum numbers, we must emphasise that we have not yet identified a
so(3) ⊂ so(5) embedding, and therefore such an identification is at this point premature.
5.2
Embedding a so(3) Subalgebra
As stated at the outset, we wish to construct a representation of so(5) that reduces a
physically relevant so(3) subalgebra, which corresponds with the SO(3) transformations
of the quadrupole moments in a nuclear collective model. In order that the embedding in
the generic case coincides with that of the symmetric case used in the previous chapters,
we consider a so(3) embedding in so(5) such that the fundamental so(5) irreps (1, 0) and
(0, 21 ) are simultaneously irreducible in so(3).
In fact, we need only impose the condition that (0, 21 ) be irreducible with respect to
so(3). Thus (0, 21 ) has so(3) label L = 32 , since the dimension of a so(3) irrep is 2L + 1.
Note that the the Lie algebra so(3) is isomorphic to su(2) so that half odd integer labels
are well defined. By decomposing the tensor product (0, 21 ) ⊗ (0, 21 ) in both so(5) and in
so(3), one finds that the five dimensional fundamental so(5) irrep (1, 0) has L = 2 and
that the ten dimensional adjoint irrep (0, 1) reduces to a direct sum of L = 1 and L = 3.
Since so(5) itself transforms as the adjoint irrep, we deduce that it is spanned by a set
of L = 1 angular momentum, or dipole, operators {L̂k ; k = 0, ±1} and a set of L = 3
octupole operators {Ôν ; ν = 0, ±1, ±2, ±3}.
In this dipole/octupole basis, the so(5) commutation relations, written in coupled
form, are
√
L̂, L̂ 1k = 2 L̂k
√
Ô, Ô 1k = 2 7 L̂k
√
L̂, Ô 3ν = 2 3 Ôν
√
Ô, Ô 3ν = − 6 Ôν .
(5.6)
The so(3) subalgebra spanned by {L̂k } will play the rôle of angular momentum in our
construction. Note that the embedding is not yet uniquely defined—there is still freedom
in specifying actual L = 1 and L = 3 combinations of root vectors in the {A, S, T }
basis, or equivalently, in specifying the Cartan subalgebra in this dipole/octupole basis.
We must ensure compatibility with our vector coherent state construction below before
making such a specification.
5.3. Intrinsic States and Orbiter Subalgebra for so(5)
5.2.1
75
The SO(5) ↓ SO(3) branching rule
This choice of so(3) ⊂ so(5) fixes the SO(5) ↓ SO(3) branching rule, in that we can now
determine which so(3) irreps occur in a given so(5) irrep (v, f ). Note that multiple copies
of a so(3) irrep can occur in a so(5) irrep, which we index with a multiplicity label τ .
The so(3) irrep of maximum L in a given (v, f ) is always multiplicity free.
Explicit branching rules for the one-rowed irreps (v, 0) are given in [56]:
L = 2K, 2K − 2, 2K − 3, 2K − 4, · · · , K
K = v, v − 3, v − 6, · · · , Kmin ;
(5.7)
where Kmin = v (mod 3). For irreps (0, f ) the branching rules are given in [85]:
L = 3K, 3K − 2, 3K − 3, 3K − 4, · · · , K
K = 2f, 2f − 2, 2f − 4, · · · , Kmin ;
(5.8)
where Kmin = 2f (mod 2).
There is no known explicit algorithm for finding the so(3) content of a generic so(5)
irrep. However, there exist computer routines which do so. One such routine is based
upon the idea of coupling the grade lowering operators (see figure 5.5) successively to
each grade of a so(5) weight diagram and projecting out angular momenta [86]. Another
method is given by Schur function techniques [87]. Table 5.2 gives the branching rules
for several small irreps.
5.3
Intrinsic States and Orbiter Subalgebra for so(5)
We now apply the VCS method to so(5). In light of the coherent state construction
overview of chapter 4, it is hopefully clear that the ‘key rôles’ to which we alluded in
sections §5.1.1 and §5.2 are that the highest grade states will span our intrinsic subspace
and that the so(3) subalgebra in (5.6) will be our orbiter subalgebra. In this section
we give the intrinsic subalgebra embedding and a realisation of the intrinsic states. We
also give a realisation of the so(3)-coupled basis states, i.e., the states which reduce the
SO(5) ⊃ SO(3) subgroup chain. We then give the necessary conditions for the orbits
of the intrinsic states to span an arbitrary so(5) irrep. Finally, we show that specifying
the relationship between the so(3)-coupled basis and the intrinsic basis is equivalent to
embedding a so(3) orbiter, and we choose a relationship with which to carry out the VCS
construction.
76
Chapter 5. Generic Representations of so(5) in a so(3) Basis
Genuine irreps
Spinor irreps
(v, f ) L
(v, f )
2L
(1, 0)
2
(0, 1/2) 3
(0, 1)
1, 3
(1, 1/2) 1, 5, 7
(2, 0)
2, 4
(0, 3/2) 3, 5, 9
(1, 1)
1, 2, 3, 4, 5
(1, 3/2) 1, 3, 5, 7, 7, 9, 11, 13
(0, 2)
0, 2, 3, 4, 6
(2, 1/2) 3, 5, 7, 9, 11
(3, 0)
0, 3, 4, 6
(0, 5/2) 3, 5, 7, 9, 11, 15
(2, 1)
1, 2, 3, 3, 4, 5, 5, 6, 7
(1, 5/2) 1, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 17, 19
(1, 2)
1, 2, 2, 3, 4, 4, 5, 5, 6, 7, 8 (2, 3/2) 1, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 17
(0, 3)
1, 3, 3, 4, 5, 6, 7, 9
(3, 1/2) 3, 5, 7, 9, 9, 11, 13, 15
(4, 0)
2, 4, 5, 6, 8
(0, 7/2) 3, 5, 7, 9, 9, 11, 13, 15, 17, 21
Table 5.2: The so(3) content of the low dimensional so(5) irreps. Note that the values
in the spinor column are 2L.
5.3.1
Embedding an intrinsic subalgebra
We choose the highest grade states of a so(5) irrep as the intrinsic space for the VCS
construction. The reasons for this are threefold. First, it will be shown that the orbit of
the highest grade spans the carrier space of the so(5) irrep under the action of a SO(3)
orbiter subgroup which we describe below. Second, there is a natural embedding of u(2)
in so(5) that respects the grade structure of the weight diagrams, and that the highest
grade is always a multiplicity free irrep of this u(2). Thus, u(2) is an intrinsic (core)
subalgebra. Third, the representation theory of u(2) is well understood and will make
the VCS construction straightforward.
The u(2) ⊂ so(5) embedding is spanned by the ‘horizontal’ root vectors in figure
5.1. From table 5.1, we have the standard u(2) commutation relations for the linear
combinations
√
√
u(2) ∼ span{ 2A−+ , S0 − T0 , 2A+− , S0 + T0 }.
(5.9)
A so(5) irrep uniquely identifies a corresponding u(2) irrep carried by the highest grade,
and a highest weight state of the former is simultaneously a highest weight of the latter.
Thus, a so(5) label (v, f ) is also a u(2) label, which we can view as a u(1) label v and a
su(2) label f , since u(2) ∼ u(1) ⊕ su(2). This justifies our use of half integer values for
f . With this labelling scheme a highest grade state in H(v,f ) is denoted |(v, f )mi, where
77
5.3. Intrinsic States and Orbiter Subalgebra for so(5)
m ∈ {−f, −f + 1, · · · , f } is a usual so(2) ⊂ su(2) label. The highest weight state for
(v, f ) is therefore |(v, f )f i in this notation.
highest weight
highest grade r
r r
r r r
r
r
r
r
r
r
r
r
r
r
r
r
T+
@
@
|(vf )f i
r
@
r r
@
r r @r
@
@
S−
A++
@
@
@
@
S+ grade raising
u(2) core
@
A−−
@
@
@
T− grade lowering
Figure 5.5: The highest grade and weight of a general so(5) irrep. The u(2) core subalgebra and grade raising/lowering operators are shown in the second diagram.
Each so(5) irrep weight diagram stratifies into horizontal grades with respect to this
u(2) core. The grade raising operators {T+ , A++ , S+ } and grade lowering operators
{S− , A−− , T− } shown in figure 5.5 take states in a given grade to states in a higher
or lower grade respectively.
5.3.2
Realisation of the intrinsic states
Figure 5.2 shows that the fundamental so(5) irrep (1, 0) has one highest grade basis
state denoted |(1, 0)0i, and that (0, 21 ) has two highest grade basis states |(0, 12 ) 21 i and
|(0, 21 ) −1
i. From our realisation in §5.1.2, we have
2
η2† |0i = |(1, 0)0i
ζ †3 |0i = |(0, 12 ) 21 i
2
†
ζ 1 |0i = |(0, 12 ) −1
i.
2
2
(5.10)
The represention theory of su(2) is very well studied, which makes this intrinsic space
easy to work with. In particular, a basis for an arbitrary highest grade u(2) irrep (v, f )
is known to be given by
(η2† )v (ζ †3 )f +m (ζ †1 )f −m
2
2
|0i.
|(v, f )mi = p
v!(f + m)!(f − m)!
(5.11)
78
Chapter 5. Generic Representations of so(5) in a so(3) Basis
5.3.3
Realisation of the so(3)-coupled states
From §5.2, we have already fixed the so(3) content of a so(5) irrep. We denote the so(3)coupled basis states for an irrep (v, f ) by |(v, f )τ LM i, where the values of τ and L for
a fixed (v, f ) are given by the branching rule. M ∈ {−L, −L + 1, · · · , L} is a standard
so(2) ⊂ so(3) label. These states reduce the SO(5) ⊃ SO(3) subgroup chain. We can
realise the so(3)-coupled basis states for the fundamental irreps in terms of harmonic
oscillator boson creation operators, much as we have already done for the weight states
in §5.1.2. In §5.3.4, these so(3)-coupled boson operators will be linearly related to the
weight space boson operators of §5.1.2.
Let {p†n ; n = ± 23 , ± 21 } and {d†ν ; ν = ±2, ±1, 0} be boson creation operators for a four
and a five dimensional harmonic oscillator respectively, with the commutation relations
[pn , p†m ] = δnm I
[dν , d†µ ] = δνµ I,
(5.12)
where pn = (p†n )† , dν = (d†ν )† and I is the identity operator. Since they are to create states
with good angular momentum, these operators are the components of an L = 2 spherical
tensor d† and an L =
3
2
spherical tensor p† . Hence we have the cotensor relations
3
pn = (−1)n+ 2 p−n
dν = (−1)ν+2 d−ν .
(5.13)
The so(3)-coupled basis states for the fundamental so(5) irreps are now
p†n |0i = |(0, 12 )1, 32 , ni
d†ν |0i = |(1, 0)1, 2, νi.
(5.14)
A conveniently normalised state of maximum angular momentum in an arbitrary so(5)
irrep is given by
(d†2 )v (p†3 )2f
2
|0i,
(5.15)
|(v, f )1, Lmax , M = Lmax i = p
v!(2f )!
where τ = 1 since this state is always multiplicity free. This is the generalisation of
equation (2.43).
5.3.4
Embedding an orbiter subalgebra
We are now in a position to finalise our embedding of a so(3) orbiter subalgebra. To do
so, we must express the so(3) operators {L̂k } from (5.6) in terms of the {A, S, T } basis
79
5.3. Intrinsic States and Orbiter Subalgebra for so(5)
of so(5) such that the intrinsic highest grade states of an so(5) irrep span H (v,f ) under
the action of the SO(3) orbiter subgroup generated by {L̂k }.
Theorem 1. (Rowe and Hecht) For a so(3) orbiter subalgebra that has no component
lying in the u(2) intrinsic subalgebra, the states
{R(Ω)|(v, f )mi;
m = −f, · · · , f ;
Ω ∈ SO(3)}
span the so(5) carrier space H(v,f ) .
Proof:
For a highest weight state of a semisimple Lie algebra irrep, all other weights can
be generated by repeated application of a set of lowering operators. In our {A, S, T }
so(5) basis these lowering operators are {A−+ , S− , A−− , T− }. Operating upon the highest
weight state repeatedly with A−+ ∈ u(2) generates the highest grade states. It follows
that in order to span the space of an arbitrary so(5) irrep by a group acting upon the
highest grade states, the span of the generators of that group must include the span of
{S− , A−− , T− }.
Let {L̂i ; i = 1, 2, 3} be a hermitian basis for a so(3) ⊂ so(5) subalgebra, where
+
−
0
[L̂i , L̂j ] = L̂k , (ijk) cyclic. Write L̂i = L̂−
i + L̂i + L̂i , where L̂i is composed of grade
lowering operators, L̂0i is composed of u(2) core operators, and L̂+
i is composed of grade
raising operators. By hermiticity, if L̂i has a L̂+
i component it must also have a corresponding L̂−
i component. Thus, if no L̂i lies in the 0 grade u(2) core, then each L̂i must
have at least one nonzero L̂−
i component. By linear independence it must be that the
span of {L̂−
i } equals the span of {S− , A−− , T− }.
QED
This argument also shows that the one-rowed so(5) irreps (v, 0) can be spanned by
scalar coherent states with a SO(3) orbiter. This construction was performed in [54].
With the spherical tensor operators p† and d† , we can construct so(3) operators with
the L = 1 coupling
√
L̂k = − 5[p† ⊗ p]1k
and
√
10[d† ⊗ d]1k ,
(5.16)
√
√
where the factors − 5 and 10 ensure the correct commutation relations. From (5.3)
and (5.5), we have so(5) ⊂ u(4) in terms of ζ and so(5) ⊂ u(5) in terms of η. Therefore,
by giving the p and d operators in terms of the ζ and η operators respectively and using
(5.16), one arrives at a so(3) ⊂ so(5) embedding.
80
Chapter 5. Generic Representations of so(5) in a so(3) Basis
If one were to make the ‘obvious’ correspondence p†n = ζn† and d†ν = ην† one finds
that the orbiter subalgebra has L̂0 = 3S0 + T0 ∈ u(2), which violates the conditions of
theorem 1. That is, if the highest grade states have good orbiter angular momentum,
their SO(3) orbits do not span an arbitrary so(5) irrep space.
π
π
We choose instead the relationship p†n = e− 4 (S+ −S− ) ζn† e 4 (S+ −S− ) , which gives
p†3 =
2
†
†
√1 (ζ 3
2
2
+ ζ †−3 )
p†1 = ζ †1
2
2
†
p −1 = ζ −1
2
2
†
p −3 =
2
2
†
√1 (ζ −3
2
2
− ζ †3 ).
2
(5.17)
The same transformation applied to the d operators yields
d†2 =
d†−1
=
†
√1 (η
2 2
†
− η−1
)
†
√1 (η
2 −1
+
d†1 =
†
√1 (η
2 1
d†0 = η0†
η2† )
d†−2
=
†
− η−2
)
†
√1 (η
2 −2
+
(5.18)
η1† ).
From (5.16), (5.3) and (5.5) we have the so(3) ⊂ so(5) embedding
3
L̂0 = T0 − (S+ + S− )
2√
L̂± = 2T± + 3(A±∓ + A∓∓ ),
(5.19)
√
where L± = ∓ 2L±1 is the usual so(3) raising operator. It is clear that no Lk lies
entirely within the u(2) core, and theorem 1 is satisfied. This so(3) generates our orbiter
subgroup.
We can also at this point construct the octupole operators, (5.6), by coupling the
spherical tensors to L = 3, which completes a dipole/octupole basis of §5.2 for so(5);
Ôµ =
√
5[p† ⊗ p]3µ
and
√
10[d† ⊗ d]3µ .
(5.20)
From (5.16), (5.3) and (5.5) we find
1
Ô0 = 3T0 + (S+ + S− )
√ 2
Ô±1 = ∓( 3T± − A±∓ − A∓∓ )
r
5
(A±± − A∓± )
Ô±2 =
2
√
5
Ô±3 =
(∓2S0 − S+ + S− ).
2
(5.21)
5.4. The Vector Coherent State Construction for so(5)
5.4
81
The Vector Coherent State Construction for so(5)
With the intrinsic states and orbiter subalgebra fixed, we can now define VCS wavefunctions and construct representations. The methodology follows along the lines set out in
§4.2.
5.4.1
VCS wavefunctions
First we identify the subspace of highest grade (HG) states with an intrinsic Hilbert
space of wavefunctions for a u(2) irrep by a map
(v,f )
HHG → Hι(v,f ) ;
(v,f )
|(v, f )mi 7→ Φm
.
(5.22)
These Φ functions can be realised in many standard ways; we use the Bargmann realisation below.
Armed with theorem 1 and the intrinsic and orbiter subalgebras from §5.3, we are now
in a position to map an entire so(5) carrier space H(v,f ) into a space of square integrable
vector valued functions on SO(3);
H(v,f ) → L2(v,f ) (SO(3));
|ψi 7→ Ψ.
(5.23)
A vector coherent state wavefunction Ψ is given explicitly by
Ψ(Ω) =
X
m
(v,f )
Φm
h(v, f )m|R(Ω)|ψi,
Ω ∈ SO(3).
(5.24)
For an so(3)-coupled basis state defined in §5.3.3, the VCS wavefunctions have the
form
(v,f )
Ψτ LM (Ω) =
X
m
=
X
m,K
≡
X
(v,f )
Φm
h(v, f )m|R(Ω)|(v, f )τ LM i
(v,f )
L
Φm
h(v, f )m|(v, f )τ LKiDKM
(Ω)
(v,f )
(v,f )
L
Φm
amK (τ L)DKM
(Ω),
(5.25)
m,K
where we have used the SO(3) transformation properties of |(v, f )τ LM i expressed in
L
terms of Wigner functions DKM
(Ω). These VCS wavefunctions are basis vectors for
carrier spaces of coherent state so(5) irreps that reduce so(3). We describe these repre-
sentations in §5.4.3.
82
Chapter 5. Generic Representations of so(5) in a so(3) Basis
5.4.2
Overlap coefficients of the VCS wavefunctions
Since the Φ and D bases are known, it follows that the VCS wavefunctions are completely
(v,f )
determined by the overlaps amK (τ L) ≡ h(v, f )m|(v, f )τ LKi, which can be viewed as
expansion coefficients in a ΦD basis. Using our realisations of the intrinsic and so(3)coupled basis states from §5.3, we can explicitly construct these overlaps for the funda-
mental so(5) irreps.
Fundamental (0, 12 ) wavefunctions
From (5.25), (5.10) and (5.14) the VCS wavefunctions for (0, 12 ) are
(0, 1 )
(0, 12 )
Ψ1, 32,n (Ω) = Φ 1
2
2
(0, 1 )
|R(Ω)|(0, 12 )1, 23 , ni
h(0, 12 ) 12 |R(Ω)|(0, 12 )1, 32 , ni + Φ −1 2 h(0, 12 ) −1
2
2
(0, 12 )
3
2
(0, 12 )
1
2
+ Φ −1 h0|ζ R(Ω)p†n |0i
2
X
X
3
3
1 3
−3
1
(0, 1 )
(0, 21 )
2
2
(Ω) + Φ −1 2 h0|p 2
(Ω)
= Φ 1 h0| √ (p 2 − p 2 )
p†m |0iDmn
p†m |0iDmn
2
2
2
m
m
3
3
(0, 1 ) 1
(0, 12 ) 32
2
(5.26)
= Φ 1 2 √ D 32 n (Ω) − D −3
(Ω)
+
Φ
D 1 n (Ω).
−1
2
2
2 n
2
2 2
= Φ1
2
h0|ζ
R(Ω)p†n |0i
Comparing this result to (5.25) we find that the nonzero overlaps for (0, 21 ) are
1
(0, 1 )
a 1 , 32 (1, 32 ) = √ ,
2 2
2
1
1
(0, 2 )
(1, 32 ) = − √ ,
a 1 , −3
2 2
2
(0, 1 )
a −1 2, 1 (1, 32 ) = 1.
2
(5.27)
2
Fundamental (1, 0) wavefunctions
From (5.25), (5.10) and (5.14) the VCS wavefunctions for (1, 0) are
(1,0)
(1,0)
Ψ1,2,ν (Ω) = Φ0
(1,0)
h(1, 0)0|R(Ω)|(1, 0)1, 2, νi
h0|η 2 R(Ω)d†ν |0i
X
1
(1,0)
2
= Φ0 h0| √ (d2 + d−1 )
d†µ |0iDµν
(Ω)
2
µ
1
(1,0)
2
2
= Φ0 √ D2ν
(Ω) + D−1ν
(Ω) .
2
= Φ0
(5.28)
83
5.4. The Vector Coherent State Construction for so(5)
Comparing this result to (5.25) we find that the nonzero overlaps for (1, 0) are
1
(1,0)
a0,2 (1, 2) = √ ,
2
1
(1,0)
a0,−1 (1, 2) = √ .
2
(5.29)
Generic (v, f ) wavefunctions
A wavefunction in an arbitrary so(5) irrep can be constructed similarly. Before going
(v,f )
further, we will make the u(2) intrinsic wavefunctions Φm
Let the Hilbert space
(v,f )
Hι
explicit.
of intrinsic state wavefunctions Φ(v,f ) be the space of real
polynomials in {x1 , x2 , y}. In the Bargmann realisation we have
(v,f )
Φm
(x, y)
xf1 +m xf2 −m y v
|0i
=p
(f + m)!(f − m)!v!
(5.30)
for the intrinsic space wavefunctions. If we define the operator Y † ≡ x∗1 ζ †3 + x∗2 ζ †1 + y ∗ η2† ,
2
2
we find that for |ψi ∈ H(v,f )
h0|eY R(Ω)|ψi =
X
m
(v,f )
Φm
h(v, f )m|R(Ω)|ψi = Ψ(Ω),
(5.31)
where |0i is the boson vacuum state in H(v,f ) .
Following the technique used above for the fundamental wavefunctions, we can con-
struct an arbitrary so(3)-coupled basis state as follows.
†
†
|(v, f )τ LM i = N p† ⊗ · · · ⊗ p† ⊗ d
⊗
·
·
·
⊗
d
{z
} τ LM |0i,
|
{z
} |
2f
(5.32)
v
where N is a normalisation factor and τ labels inequivalent LM -couplings. A generic
wavefunction now looks like
(v,f )
Ψτ LM = N h0|eY R(Ω) p† ⊗ · · · ⊗ p† ⊗ d† ⊗ · · · ⊗ d† τ LM |0i
X
L
= N
h0|eY p† ⊗ · · · ⊗ p† ⊗ d† ⊗ · · · ⊗ d† τ LK |0iDKM
(Ω).
(5.33)
K
In theory we could now calculate both N and the above matrix element for any (v, f )
using equations (5.17), (5.18) and the commutation relations (5.12) in order to arrive at
the overlap coefficients. However, this is an incredibly tedious task for even modest v
or f . Not only that, but for situations in which more than one inequivalent coupling is
84
Chapter 5. Generic Representations of so(5) in a so(3) Basis
possible, i.e., τ > 1, one must be sure that the different couplings chosen do not yield
linearly dependent wavefunctions.
The following theorem shows that overlaps for basis wavefunctions in any irrep can
in fact be constructed by coupling the wavefunctions, as opposed to coupling the boson
operators. This ensures that an arbitrary so(5) basis wavefunction can be built from
those of equations (5.26) and (5.28), simplifying the construction considerably.
We know that an arbitrary irrep (v, f ) is contained in a tensor product of multiple
(1, 0) and (0, 21 ). Since the highest grade uniquely identifies a so(5) irrep (v, f ), it is
the intrinsic wavefunctions Φ(v,f ) that uniquely identify the basis wavefunctions for that
irrep. The u(2) formula
s
(v1 ,f1 ) (v2 ,f2 )
Φm
Φ m2
=
1
(f + m)!(f − m)!v!
(v,f )
Φm
,
(f1 + m1 )!(f1 − m1 )!(f2 + m2 )!(f2 − m2 )!v1 !v2 !
(5.34)
where v = v1 + v2 , f = f1 + f2 and m = m1 + m2 , suggests the following generalisation
of a theorem of Rowe and Hecht [54].
Theorem 2. The product of VCS wavefunctions Ψ(v1 ,f1 ) (Ω)Ψ(v2 ,f2 ) (Ω) is a VCS wavefunction in the irrep (v1 + v2 , f1 + f2 ).
Proof:
†
†
Let ZM
create a state |(v1 , f1 )τ1 L1 M1 i ∈ H(v1 ,f1 ) and ZM
a state |(v2 , f2 )τ2 L2 M2 i ∈
1
2
†
H(v2 ,f2 ) . Then ZM
is of degree 2fi in the p† operators and of degree vi in the d† operators
i
for i = 1, 2. We have
(v ,f )
(v ,f )
†
†
2
1
|0i
|0ih0|eY R(Ω)ZM
(Ω) = h0|eY R(Ω)ZM
(Ω)Ψτ22L2 M
Ψτ11L1 M
2
1
2
1
X
X
†
†
L2
L1
(Ω)
|0iDK
h0|eY ZK
(Ω)
|0iDK
h0|eY ZK
=
2 M2
2
1 M1
1
K2
K1
=
X
K1 K2
†
e−Y
h0|eY ZK
1
†
L2
L1
(Ω),
(Ω)DK
e−Y |0iDK
|0ih0|eY ZK
2 M2
1 M1
2
since |0i = e−Y eY |0i = e−Y |0i. Focusing on the matrix elements, we have
X
1
† −Y
†
†
†
h0|eY ZK
e
|0i
=
h0|ZK
+ [Y, ZK
] + [Y, [Y, ZK
]] + · · · |0i,
i = 1, 2.
i
i
i
i
2!
n
Since Y is an annihilation operator, only the (vi + 2fi )th term contributes to the matrix
element. That is, the operator
W Ki ≡
1
†
[Y, · · · , [Y, ZK
] · · · ],
(vi + 2fi )! |
{z i
}
vi +2fi
i = 1, 2
5.4. The Vector Coherent State Construction for so(5)
85
is scalar and we can write
(v ,f )
(v ,f )
1
2
Ψτ11L1 M
(Ω)Ψτ22L2 M
(Ω) =
1
2
X
K1 K2
=
X
K1 K2
=
X
K1 K2
=
X
K1 K2
L1
L2
h0|WK1 |0ih0|WK2 |0iDK
(Ω)DK
(Ω)
1 M1
2 M2
L1
L2
h0|WK1 WK2 |0iDK
(Ω)DK
(Ω)
1 M1
2 M2
†
†
L1
L2
h0|eY ZK
ZK
e−Y |0iDK
(Ω)DK
(Ω)
1
2
1 M1
2 M2
†
†
L1
L2
h0|eY ZK
ZK
|0iDK
(Ω)DK
(Ω),
2
1 M1
2 M2
1
†
†
since ZK
ZK
is of degree 2(f1 + f2 ) in p† and of degree v1 + v2 in d† , and e−Y |0i = |0i.
1
2
Finally, from the properties of the Wigner function we have (suppressing the argument
Ω)
(v ,f )
(v ,f )
1
2
Ψτ11L1 M
Ψτ22L2 M
=
1
2
X
LK1 K2
=
X
LKK1
=
X
LK
=
X
0 L
τL
†
†
L
h0|eY ZK
ZK
|0i(L1 , K1 ; L2 , K2 |L, K)(L1 , M1 ; L2 , M2 |L, M )DKM
1
2
(L1 , M1 ; L2 , M2 |L, M )h0|eY
†
†
L
|0iDKM
ZK−K
×(L1 , K1 ; L2 , K − K1 |L, K)ZK
1
1
†
L
(L1 , M1 ; L2 , M2 |L, M )h0|eY ZK
|0iDKM
(v +v ,f1 +f2 )
cτL0 (L1 , M1 ; L2 , M2 |L, M )Ψτ 0 1LM 2
,
where M1 + M2 = M , L = |L1 − L2 |, · · · , L1 + L2 and cτL0 is a multiplicity coefficient.
This is a sum of wavefunctions in the irrep (v1 + v2 , f1 + f2 ).
QED.
Theorem 2 ensures that, starting from bases for the two fundamental irreps, we can
construct bases for all irreps. As can be seen in the proof of the theorem, in order to find
bases of wavefunctions that reduce so(3) we need only to take so(3) coupled products of
wavefunctions. Note that for a given (v, f ), there are many combinations v1 + v2 = v,
f1 + f2 = f that can be used to find wavefunctions. When the multiplicity τ for a
particular wavefunction in irrep (v, f ) is one, these many combinations are proportional
to one another and as such are identical once normalised. If τ > 1, then the different
combinations yield wavefunctions which span a τ -dimensional L subspace of the irrep
(v, f ). We use a Gram-Schmidt procedure to find a basis for each degenerate subspace
that is orthonormal with respect to the inner product
X (v,f )
(v,f )
bmK (τ L) bmK (τ 0 L) = δτ τ 0 .
mK
(5.35)
86
Chapter 5. Generic Representations of so(5) in a so(3) Basis
However, this inner product does not in general yield a unitary SO(5) action. We use Ψ̃
to denote wavefunctions that are orthonormal with respect to the inner product (5.35),
and b to denote their overlap coefficients. We reserve the notation Ψ with overlaps a for
wavefunctions orthonormal with respect to the unitary so(5) inner product which we will
determine by K matrix methods in §5.4.5. Using the symbol to indicate a coupling of
two wavefunctions with the same argument, (in this case Ω), we have
(v,f )
(v ,f )
(v ,f )
Ψ̃τ LM = [Ψ̃τ11L1 1 Ψ̃τ22L2 2 ]LM
X (v1 ,f1 ) (v1 ,f1 )
L1
=
Φm1 bm1 K1 (τ1 L1 )DK
1
m 1 K1
=
X
(v,f )
(v,f )
L
Φm
bmK (τ L)DKM
,
X
(v ,f )
L2
(v2 ,f2 )
Φm
bm22 K22 (τ2 L2 )DK
2
2
m 2 K2
LM
(5.36)
mK
where v = v1 + v2 , f = f1 + f2 , and
(v,f )
bmK (τ L) =
X
m 1 K1
(v ,f )
(v ,f )
2 2
(f1 , m1 ; f2 , m−m1 |f m) (L1 , K1 ; L2 , K−K1 |LK) bm11 K11 (τ1 L1 ) bm−m
(τ2 L2 )
1 K−K1
(5.37)
are the unnormalised overlap coefficients of the new wavefunction. Equation (5.37) and
the Gram-Schmidt inner product (5.35) give us an iterative procedure with which we can
generate all b-coefficients.
(v,f )
We now have an algorithm for constructing a basis of VCS wavefunctions Ψ̃τ LM for an
arbitrary so(5) irrep (v, f ) which is orthonormal with respect to (5.35) and reduces the
so(3) subalgebra. These wavefunctions are completely determined by the real coefficients
(v,f )
bmK (τ L).
5.4.3
VCS representations
As was shown in §4.2, a coherent state representation Γ of the so(5) operators is defined
by the action
(v,f )
[Γ(X̂µ` )Ψτ LM ](Ω) =
X
m
=
X
m
=
X
m,K
=
(v,f )
Φm
h(v, f )m|R(Ω)X̂µ` |(v, f )τ LM i
(v,f )
Φm
h(v, f )m|R(Ω)X̂µ` R(Ω−1 )R(Ω)|(v, f )τ LM i
L
(v,f )
(Ω)
h(v, f )m|R(Ω)X̂µ` R(Ω−1 )|(v, f )τ LKiDKM
Φm
X
m,K,ν
(v,f )
`
L
Φm
h(v, f )m|X̂ν` |(v, f )τ LKiDνµ
(Ω)DKM
(Ω),
(5.38)
87
5.4. The Vector Coherent State Construction for so(5)
where X̂ ` is a so(5) operator in a spherical tensor basis, such as the dipole/octupole
basis. We can write (5.38) in the coupled form
(v,f )
[Γ(X̂ ` ) ⊗ Ψτ L ]L0 M 0 (Ω) =
=
X
0
m,K,K 0 ,ν
X
m,K,K 0 ,ν
(v,f )
L
Φm
h(v, f )m|X̂ν` |(v, f )τ LKi(L, K; `, ν|L0 , K 0 )DK
0 M 0 (Ω)
(v,f )
Φm
h(v, f )m|
X
(v,f )
Φm
amN (τ 0 L0 )h(v, f )τ 0 L0 N |X̂ν` |(v, f )τ LKi
0
L
×(L, K; `, ν|L0 , K 0 )DK
0 M 0 (Ω)
(v,f )
(v,f )
Φm
amN (τ 0 L0 )
m,K 0 ,τ 0 ,N
=
X
τ0
0
(v,f )
m,K,K 0 ,ν,τ 0 ,N
=
τ 0N
|(v, f )τ 0 L0 N ih(v, f )τ 0 L0 N | X̂ν`
L
×|(v, f )τ LKi(L, K; `, ν|L0 , K 0 )DK
0 M 0 (Ω)
X
=
X
(v,f )
Ψτ 0 L0 M 0 (Ω)
h(v, f )τ 0 L0 ||X̂ ` ||(v, f )τ Li
L0
√
δ
N K 0 DK 0 M 0 (Ω)
2L0 + 1
h(v, f )τ 0 L0 ||X̂ ` ||(v, f )τ Li
√
,
2L0 + 1
(5.39)
where we have used the Wigner-Eckart theorem to introduce a SU(2) reduced matrix element, (this result is true of any so(3)-coupled basis). The VCS representation of the so(5)
operators is therefore given by the reduced matrix elements h(v, f )τ 0 L0 ||X̂ ` ||(v, f )τ Li in
an orthonormal (i.e., unitary) so(5) basis {Ψ}.
5.4.4
Matrix elements of the VCS representation
From (5.38) or (5.39) we can see that the VCS action of the so(5) operators is determined
by the matrix elements h(v, f )m|X̂ν` |(v, f )τ LKi. Evaluation of these matrix elements is
accomplished using the dipole/octupole basis for X̂ ∈ so(5). We make the following
observations, which are all a direct consequence of the VCS construction:
(i) {L̂+ , L̂0 , L̂− } (cf. equation 5.19) span so(3) and act naturally upon |(v, f )τ LKi;
√
√
(ii) { 2A+− , S0 − T0 , 2A−+ , S0 + T0 } span the u(2) core and act naturally upon
h(v, f )m|;
(iii) {S− , A−− , T− } are adjoint to the grade raising operators and therefore annihilate
h(v, f )m|.
88
Chapter 5. Generic Representations of so(5) in a so(3) Basis
These observations and (5.19), (5.21) allow us to find the matrix elements of every root
vector in so(5) in terms of operators with known actions;
p
√
h(v, f )m| 2A+− |(v, f )τ LKi =
(f + m)(f − m + 1)h(v, f ) m − 1|(v, f )τ LKi
h(v, f )m|(S0 − T0 )|(v, f )τ LKi = mh(v, f )m|(v, f )τ LKi
p
√
h(v, f )m| 2A−+ |(v, f )τ LKi =
(f − m)(f + m + 1)h(v, f ) m + 1|(v, f )τ LKi
h(v, f )m|(S0 + T0 )|(v, f )τ LKi = (v + f )h(v, f )m|(v, f )τ LKi,
(5.40)
for the u(2) intrinsic algebra and
2
h(v, f )m|S+ |(v, f )τ LKi = h(v, f )m| (T0 − L̂0 )|(v, f )τ LKi
3
1 √
h(v, f )m|T+ |(v, f )τ LKi = h(v, f )m| (− 3A+− + L̂+ )|(v, f )τ LKi
2
1
h(v, f )m|A++ |(v, f )τ LKi = h(v, f )m|(−A−+ + √ L̂− )|(v, f )τ LKi
3
h(v, f )m|S− |(v, f )τ LKi = 0
h(v, f )m|T− |(v, f )τ LKi = 0
h(v, f )m|A−− |(v, f )τ LKi = 0,
(5.41)
for the remaining so(5) operators. Thus, the expressions for the matrix elements of the
dipole operator are
h(v, f )m|L̂+ |(v, f )τ LKi =
p
(v,f )
(L − K)(L + K + 1) am K+1 (τ L),
(v,f )
h(v, f )m|L̂0 |(v, f )τ LKi = K amK (τ L),
p
(v,f )
(L + K)(L − K + 1) am K−1 (τ L),
h(v, f )m|L̂− |(v, f )τ LKi =
(5.42)
89
5.4. The Vector Coherent State Construction for so(5)
and of the octupole operator are
5
(v,f )
(2v + 2f + m − K) amK (τ L),
3
√ p
(v,f )
− 5 (f − m)(f + m + 1) am+1 K (τ L),
r
5p
(v,f )
(L + K)(L − K + 1) am K−1 (τ L),
+
6
5 p
(v,f )
√
(f + m)(f − m + 1) am−1 K (τ L),
2 2
√
3p
(v,f )
(L − K)(L + K + 1) am K+1 (τ L),
−
2
1
(v,f )
(5v + 5f − 5m − K) amK (τ L),
3
1 p
(v,f )
(L + K)(L − K + 1) am K−1 (τ L),
−√
3
√
5p
(v,f )
(f + m)(f − m + 1) am−1 K (τ L),
−
2
√
5
(v,f )
(v + f + 2m + K) amK (τ L).
(5.43)
3
h(v, f )m|Ô3 |(v, f )τ LKi = −
h(v, f )m|Ô2 |(v, f )τ LKi =
h(v, f )m|Ô1 |(v, f )τ LKi =
h(v, f )m|Ô0 |(v, f )τ LKi =
h(v, f )m|Ô−1 |(v, f )τ LKi =
h(v, f )m|Ô−2 |(v, f )τ LKi =
h(v, f )m|Ô−3 |(v, f )τ LKi =
√
Now we can write the action of so(5) (in the dipole/octupole basis) on the non-unitary
VCS basis wavefunctions. The action of the dipole operators is the so(3) standard
(v,f )
(v,f )
Γ(L̂0 )Ψ̃τ LM = M Ψ̃τ LM ,
(v,f )
Γ(L̂± )Ψ̃τ LM =
p
(v,f )
(L ∓ M )(L ± M + 1)Ψ̃τ L M ±1 .
(5.44)
The action of the octupole operators is more complicated. Using (5.39) we write a coupled
octupole action, in the non-unitary basis,
(v,f )
[Γ(Ô) ⊗ Ψ̃τ L ]L0 M 0 ≡
X
mKm0 K 0
(v,f )
(v,f )
(v,f )
0
L
bmK (τ L) MmKL,m0 K 0 L0 Φm0 DK
0M 0 ,
(5.45)
90
Chapter 5. Generic Representations of so(5) in a so(3) Basis
where we have defined the matrix M(v,f ) with nonzero entries
(v,f )
MmKL,mKL0 =
(v,f )
MmKL,m+1 K+1 L0 =
(v,f )
MmKL,m−1 K+2 L0 =
(v,f )
MmKL,m K+3 L0 =
(v,f )
MmKL,m+1 K−2 L0 =
(v,f )
MmKL,m K−3 L0 =
1
(5v + 5f − 5m − K)(L, K; 3, 0|L0 , K)
3 √
3p
−
(L + K)(L − K + 1)(L, K − 1; 3, 1|L0 , K)
2
1 p
(L − K)(L + K + 1)(L, K + 1; 3, −1|L0 , K)
−√
3
5 p
√
(f − m)(f + m + 1)(L, K; 3, 1|L0 , K + 1)
2 2
√ p
− 5 (f + m)(f − m + 1)(L, K; 3, 2|L0 , K + 2)
r
5p
(L − K)(L + K + 1)(L, K + 1; 3, 2|L0 , K + 3)
6
√
5
−
(2v + 2f + m − K)(L, K; 3, 3|L0 , K + 3)
√ 3
5p
(f − m)(f + m + 1)(L, K; 3, −2|L0 , K − 2)
−
2
√
5
(v + f + 2m + K)(L, K; 3, −3|L0 , K − 3).
(5.46)
3
Equation (5.39) tells us how M(v,f ) is related to the reduced matrix elements. However,
those matrices are not yet unitary. In the next section we will handle this.
5.4.5
Transformation to an Hermitian basis
Although we have expressions for the so(5) basis vectors and the action of our so(5)
operators, they are not yet unitary—recall that (5.35) is not a unitary so(5) inner product.
The condition for a unitary representation of a group is an hermitian representation of
its algebra. In terms of reduced matrix elements in an so(3) basis, with λ serving for any
other necessary labels, this condition is
`
hλ0 L0 M 0 |X̂µ` |λLM i∗ = h(λLM |(−1)µ X̂−µ
|λ0 L0 M 0 i
(5.47)
0
0
0
0
(LM ; `µ|L M ) 0 0 `
∗
µ (L M ; ` − µ|LM )
√
√
hλ
L
||
X̂
||λLi
=
(−1)
hλL||X̂ ` ||λ0 L0 i,
0
2L + 1
2L + 1
and so
`
0
0
hλL||X̂ ||λ L i = (−1)
µ
r
2L + 1 (LM ; `µ|L0 M 0 )
hλ0 L0 ||X̂ ` ||λLi∗ .
2L0 + 1 (L0 M 0 ; ` − µ|LM )
(5.48)
The prefactor simplifies and one finds that the hermiticity condition is, in terms of our
so(5) labels,
0
h(v, f )τ L||X̂||(v, f )τ 0 L0 i = (−1)L−L h(v, f )τ 0 L0 ||X̂||(v, f )τ Li∗ .
(5.49)
91
5.4. The Vector Coherent State Construction for so(5)
The reduced matrices of the dipole operator L̂ are diagonal since L̂ defines our so(3)
basis, and as such they satisfy (5.49). We therefore focus upon the octupole operator. In
the non-unitary Ψ̃ basis, we can equate (5.45) and (5.39) which gives
X
mKm0 K 0
(v,f )
(v,f )
bmK (τ L) MmKL,m0 K 0 L0
(v,f )
Φ m0
L0
DK
0M 0
X
=
(v,f )
(v,f )
Ψ̃τ 0 L0 M 0 (Ω)
τ0
X
=
Õ 0 0
√ τ L ,τ L
2L0 + 1
(v,f )
(v,f )
bm0 K 0 (τ 0 L0 ) Φm0
τ 0 m0 K 0
(5.50)
(v,f )
Õτ 0 L0 ,τ L
L0
,
DK 0 M 0 √ 0
2L + 1
where Õ (v,f ) denotes the reduced matrix for the octupole operator in the non-unitary
basis. Equating the coefficients of ΦD yields
X
(v,f )
(v,f )
bmK (τ L)MmKL,m0 K 0 L0
=
X
τ0
mK
(v,f )
Õτ 0 L0 ,τ L
(v,f )
0 0
bm0 K 0 (τ L ) √ 0
2L + 1
,
(5.51)
and (5.35) allows us to isolate the non-unitary reduced matrix elements
(v,f )
Õτ 0 L0 ,τ L =
X (v,f )
√
(v,f )
(v,f )
2L0 + 1
bmK (τ L)MmKL,m0 K 0 L0 bm0 K 0 (τ 0 L0 ).
(5.52)
mKm0 K 0
We write the transformation from the non-unitary basis (b(v,f ) coefficients) to the
unitary basis (a(v,f ) coefficients) as a K matrix
(v,f )
Ψτ LM =
X
(v,f )
(v,f )
Ψ̃τ 0 LM Kτ 0 τ (L).
(5.53)
τ0
This basis change transforms Õ (v,f ) into the desired reduced unitary matrix
h(v, f )τ 0 L0 ||Ô||(v, f )τ Li =
X
σσ 0
(v,f )
(v,f )
[K (v,f ) (L)]−1
τ 0 σ 0 Oσ 0 L0 ,σL Kστ (L).
(5.54)
The construction of the transformations K (v,f ) takes place in two steps; orthogonalisation and normalisation.
Orthogonalisation
States of different L and/or M are already orthogonal since the preliminary inner product
(v,f )
(5.35) respects so(3) unitarity. We may therefore focus upon L-blocks Õτ L,τ 0 L , which can
be done formally by considering matrix elements of the hermitian so(3) scalar operator
[L̂ ⊗ Ô ⊗ L̂ ⊗ L̂]00 + [L̂ ⊗ L̂ ⊗ Ô ⊗ L̂]00 ,
(5.55)
92
Chapter 5. Generic Representations of so(5) in a so(3) Basis
since they are proportional to those of Ô within any L-block. If we diagonalise (5.55) in
each L-block with a transformation K(v,f ) , then (5.49) will be satisfied after normalisation.
However, if there are irreps (v, f ) where one or more L-blocks have repeated eigenvalues,
then this operator cannot always resolve the degeneracy. Indeed, there are such irreps.
Now, we are guaranteed that a suitable transformation K (v,f ) exists, since we know that
every representation of a compact Lie algebra is equivalent to an hermitian representation.
We find that diagonalising the operator
[L̂ ⊗ Ô ⊗ Ô ⊗ L̂]00
(5.56)
provides suitable diagonalising transformations K(v,f ) for all irreps that we’ve calculated.
Whether or not there is a single operator that when diagonalised will provide a suitable
transformation for every irrep is an open question. We write, following (4.29),
X
(v,f )
(v,f )
(v,f )
[K(v,f ) (L0 )]−1
Oτ 0 L0 ,τ L =
τ 0 σ 0 Õσ 0 L0 ,σL Kστ (L)
(5.57)
σσ 0
for the orthogonalised matrix.
Normalisation
Once the orthogonalising transformation is made using K(v,f ) , it remains only to nor(v,f )
malise. This is accomplished with scaling factors kτ L
(v,f )
(v,f )
∈ R such that
(v,f )
Kτ τ 0 (L) = Kτ τ 0 (L) kτ 0 L .
From (5.49), (5.54) and (5.58) we find
(v,f ) 2
kτ L
(v,f )
k τ 0 L0
(5.58)
(v,f )
= (−1)
L−L0
Oτ L,τ 0 L0
(v,f )
Oτ 0 L0 ,τ L
,
which leads to the hermitian reduced matrix elements
v
u
(v,f )
u
Oτ L,τ 0 L0 (v,f )
0
0 0
t
L−L
(−1)
Oτ 0 L0 ,τ L
h(v, f )τ L ||Ô||(v, f )τ Li =
(v,f )
Oτ 0 L0 ,τ L
v
u (v,f ) u Oτ L,τ 0 L0 (v,f )
= t (v,f ) Oτ 0 L0 ,τ L .
O 0 0 (5.59)
(5.60)
τ L ,τ L
We now have an algorithm for finding the matrix elements of all so(5) operators in
any irrep (v, f ) and for transforming the basis of VCS wavefunctions to one in which
their reduced matrices are hermitian.
93
5.5. Summary
5.5
Summary
This completes our vector coherent state construction of the generic irreducible representations (v, f ) of so(5) in a so(3) basis. We first realise the states of the five dimensional
fundamental (1, 0) irrep in terms of u(5) bosons (cf. §1.2.1), and those of the four dimensional fundamental (0, 21 ) irrep in terms of u(4) bosons. The physical requirement
dictated by the rotational behaviour of collective model quadrupole moments specifies
the so(3) ⊂ so(5) branching rule, namely that the (0, 12 ) irrep must have angular momentum L = 32 . It follows that the (1, 0) irrep has L = 2, as required by the collective
model, and that the so(5) algebra is spanned by three L = 1 and seven L = 3 operators.
Since we wish the resulting representation to reduce the SO(5) ⊃ SO(3) chain, the
vector coherent state procedure set out in chapter 4 adopts a SO(3) orbiter group generated by the L = 1 so(5) operators. We show that the highest grade states of a (v, f ) irrep
provide a suitable complementary u(2) intrinsic space, and find an so(3)-coupled basis
such that any (v, f ) carrier space is spanned by the orbits of these intrinsic states under
this orbiter action. Thus we have SO(3) coherent states that take U(2) vector values.
Both of these groups and their algebras have well known representations.
Using these realisations, we compute the coherent state wavefunctions of the two
fundamental so(5) irreps and prove that wavefunctions of any other irrep are products
thereof. This provides a basis of wavefunctions for any (v, f ). Since there can be multiple
occurrences of so(3) irreps in so(5), we introduce a provisional inner product on the
space of wavefunctions and use the Gram-Schmidt procedure to arrive at a provisional
orthonormal basis. It is provisional because it does not lead to unitary so(5) irreps.
From the known actions of u(2) and so(3) on the intrinsic and so(3)-coupled bases, we
compute the matrix elements of the so(5) operators in the provisional basis. This gives
us non-unitary irreps of so(5). We perform a two-stage K-matrix orthonormalisation of
these matrices in order to find the necessary unitarising transformations. Applying these
transformations to the provisional basis, we finally arrive at a unitary so(5)⊃so(3) basis.
The results for the one-rowed irreps (v, 0) agree with those of reference [54]. We give
some of these values in the following section.
5.5.1
Tables of coefficients and matrix elements
(v,f )
The tables below give the overlap coefficients amK (τ L) for the unitary basis of coherent state wavefunctions (5.25) and unitary SO(3)-reduced so(5) matrix elements
94
Chapter 5. Generic Representations of so(5) in a so(3) Basis
h(v, f )τ L||L̂||(v, f )τ 0 L0 i, h(v, f )τ L||Ô||(v, f )τ 0 L0 i for the first four generic (v 6= 0, f 6= 0)
so(5) irreps from table 5.2. Since the diagonalisation of (5.56) must be done numerically,
the values are given in floating point form with six decimal places. The K values for the
coefficients are such that all wavefunctions in an irrep share the same K − m index, and
we use this fact in the tabulation. For those L values that have multiplicities, we index
them writing Lτ .
Overlaps
Reduced matrix elements
m\K − m
3
0
−3
1/2
0
0.474341
0
-1/2
0
-0.387298
0
1/2
0
0.517549
0.327326
-1/2
-0.534522
0.422577
0
1/2
0.499999
0.084515
-0.377964
-1/2
0.462910
0.414039
0
L
1/2
5/2
7/2
L̂
Ô
1/2
5/2
7/2
1.224744
0
0
0
6.123724
5.999999
0
7.245688
0
6.123724
2.535462
11.338934
0
0
11.224972
−5.999999
−11.338934
3.070597
Table 5.3: Overlap coefficients and reduced matrix elements for so(5) irrep (1, 12 ).
Reduced matrix elements
m\K − m
5
2
−1
−4
1/2
0
0
0.439155
0
-1/2
0
0.414039
0.207019
0
1/2
0
0.365148
-0.204124
0.241522
-1/2
0
-0.158113
-0.316227
0
1/2
0
0.365148
-0.204124
0.241522
-1/2
0
0.516397
-0.129099
0
1/2
0
0.217597
0.255883
0.174077
-1/2
-0.426401
0.023262
0.232621
0
1/2
0.353553
0.082572
0
-0.190692
-1/2
0.261116
0.190692
0.190692
0
L
3/2
5/2
7/2
9/2
11/2
L̂
Ô
3/2
5/2
7/2
9/2
11/2
3.872983
0
0
0
0
4.225771
9.486832
10.954451
0
0
7.245688
0
−6.866065
−9.486832
11.409582
0
0
10.954451
1.380131
0
0
6.866065
0
0
−1.380131
8.391463
9.486832
11.224972
0
0
−11.258858
−8.164965
11.775681
15.732132
0
8.391463
8.164965
12.465753
11.742179
0
0
0
0
20.712315
0
−9.486832
11.775681
−11.742179
8.628704
0
5.5. Summary
Overlaps
Table 5.4: Overlap coefficients and reduced matrix elements for so(5) irrep (2, 12 ).
95
Reduced matrix elements
4
1
−2
−5
1
0
0
0.380319
0
0
0
-0.439155
0
0
-1
0
-0.358568
0
0
1
0
0.462910
-0.231455
0
0
0
0.267261
0.267261
0
-1
0
-0.377964
0
0
1
0
0.306186
0.387298
0
0
0
0.111803
0.353553
0
-1
0.499999
0.223606
0
0
1
0
0.258774
-0.146385
-0.273861
0
-0.316227
0.464834
-0.059761
0
-1
-0.387298
0.327326
0
0
1
0.353553
0
-0.073192
-0.223606
0
0.387298
0.084515
-0.223606
0
-1
0.316227
0.267261
0
0
L
1
2
3
4
5
L̂
Ô
1
2
3
4
5
2.449489
0
0
0
0
0
9.258201
−1.224744
9.315885
0
0
5.477225
0
0
0
1.463850
9.710083
−9.258201 −3.585685 −8.660254
0
0
9.165151
0
0
−1.224744
8.660254
9.721111
9.082951
9.082951
0
0
0
13.416407
0
−9.315885
1.463850
−9.082951
−5.946187
12.377975
0
0
0
0
18.165902
0
−9.710083
9.082951
−12.377975
0
Table 5.5: Overlap coefficients and reduced matrix elements for so(5) irrep (1, 1).
Chapter 5. Generic Representations of so(5) in a so(3) Basis
m\K − m
96
Overlaps
Overlaps
m\K − m
5
3/2
0
1/2
0
-1/2
0
-3/2
0
3/2
0
1/2
0
-1/2
0
-3/2
0
3/2
0
1/2
0
-1/2
0
-3/2
0
3/2
0
1/2
0
-1/2
0
-3/2
0.208236
3/2
0
1/2
0
-1/2
0
-3/1 -0.724373
3/2
0
1/2
0
-1/2
0.455096
-3/2
0.643602
3/2
0
1/2
-0.316452
-1/2 -0.467431
-3/2 -0.443444
3/2
0.403399
1/2
0.474678
-1/2
0.474678
-3/2
0.350573
2
0
0
0
0.431252
0
0
0.441902
-0.360811
0
-0.528173
-0.771447
-0.578585
-0.666136
-0.525280
-0.216048
0.463617
0.008606
-0.229439
0.169828
0.298976
0.394124
0.371584
0.198620
0.344020
0.226294
0.480398
0.618721
0.382507
-0.023853
0
0.116857
0.286242
−1
0.396130
0.560213
0
0
0.662853
0
-0.441902
0
0.177154
0.417558
0.192861
0
0.063536
0.059902
-0.432096
0
0.559438
0.527444
0.339656
0
0.105334
0.347585
0.198620
0
-0.140542
-0.265008
-0.149346
0
-0.029214
-0.123946
-0.233715
0
Reduced matrix elements
−4
0
0
0
0
0
0
0
0
-0.560213
0
0
0
-0.324072
0.223850
0
0
0.254742
0.309605
0
0
-0.394124
-0.371584
0
0
-0.078390
-0.021335
0
0
0.075431
0.202403
0
0
−7
L
1/2
0
0
1.224744
1/2
0
0
0
0
0
0
3/2
0
0
0
0
0
0
5/2
−8.964214
0
0
0
0
0
7/21
7.598895
0
0
0
0
0
7/22
5.949760
0
0
0
0
0
9/2
0
0
0
0.335648
0
0
11/2
0
0
0
-0.223765
0
0
13/2
0
0
0
L̂
Ô
3/2
5/2
7/21
7/22
9/2
11/2
13/2
0
0
0
−8.964214
0
−7.598895
0
−5.949760
0
0
0
0
0
0
3.872983
−4.225771
0
9.486832
0
−12.794126
0
4.038603
0
6.866065
0
0
0
0
0
−9.486832
7.245688
−7.606388
0
−6.510670
0
−2.843828
0
−5.163977
0
12.928374
0
0
0
−12.794126
0
6.510670
11.224972
−0.634212
0
−10.096484
0
3.705020
0
−1.797956
0
−13.815832
0
4.038603
0
2.843828
0
−10.096484
11.224972
6.403215
0
17.274085
0
8.534437
0
0.178497
0
−6.866065
0
−5.163977
0
−3.705020
0
−17.274085
15.732132
0.530457
0
8.616404
0
12.924606
0
0
0
−12.928374
0
−1.797956
0
8.534437
0
−8.616404
20.712315
−11.283690
0
13.274930
0
0
0
0
0
13.815832
0
−0.178497
0
12.924606
0
−13.274930
26.124700
−2.884731
Table 5.6: Overlap coefficients and reduced matrix elements for so(5) irrep (1, 32 ).
97
98
Conclusion
Conclusion
We have presented a new algebraic collective model and illustrated its utility in several
nuclear structure calculations. The purpose of our calculations was not to find agreement
with any particular experiment, but to refine the Bohr model so that it can be brought
to bear on a wider class of topics currently of interest. The results from our investigation
of the spherical to deformed shape phase transition indicate that the collective model has
more to contribute to nuclear structure physics than might have been thought.
In the introduction to this thesis, we stated that the Bohr-Mottelson model may have
been ‘eclipsed’ in some respects by newer collective models, particularly the interacting
boson model. By this we mean that the IBM has become the de facto standard for
phenomenological collective model calculations. This is due in part to the fact that the
IBM enjoys a straightforward algebraic structure, in that an entire IBM calculation is
carried out in the space of one u(6) irrep N — U(6) is a dynamical group for the IBM,
and since it is compact the carrier space of an irrep N is finite dimensional. Recall that
the dynamical group of the Bohr model is HW(5), a noncompact group with necessarily
infinite dimensional unitary irreps, and so the IBM has an advantage in some situations.
This advantage is gained by introducing a new type of boson into the Bohr picture; thus
in the IBM there is a spin-0 s boson in addition to Bohr’s spin-2 d boson. In order to
keep the spectrum generating algebra compact, any IBM operator that includes a factor
d† (d) is paired with a factor s(s† ), leaving the total number of bosons N fixed. This is
a purely mathematical construct, and although there have been interpretations of the
IBM s and d bosons as spin 0 and 2 coupled valence nucleon pairs [88], one can still
argue that the IBM does not share the straightforward physical interpretation of the
Bohr model based solely upon the spin-2 phononic excitations of a quadrupole oscillator
and the hyperspherical coordinates {β, γ, Ω} that describe ellipsoidally deformed nuclear
distributions.
What the algebraic model developed in this thesis shows is that the original Bohr
99
100
Conclusion
model has a robust algebraic structure that is equally, if not more, straightforward than
the IBM, and that it is certainly more than simply ‘geometric’, as the Bohr model has
often been viewed. One therefore need not necessarily appeal to less intuitive models in
order to do phenomenological calculations. Despite the enormous gain in understanding
of the way nuclear collective motions emerge from microscopic models in terms of interacting nucleons, it is always important to have such phenomenological results before
embarking on complex and less intuitive calculations. Phenomenology gives the gross
nuclear structure, dictated by symmetry, that is essentially model-independent. Indeed,
random matrix calculations [89] show that the vibrational and rotational behaviour of
nuclei is a completely general phenomenon. The merit of the algebraic model will of
course be judged by how many researchers adopt it in performing these analyses, and we
hope to have made a strong case for it here.
For transparency, all of the Hamiltonians that we’ve considered in this thesis have
been SO(5) invariant. This is by no means a necessity, since the hyperspherical harmonics
and their coupling coefficients enable the algebraic collective model to handle SO(5)
tensor operators in general. Currently this basis is being used at Toronto to tackle such
problems.
The techniques used in the algebraic collective model are quite general. The algebraic methods apply to central force problem in three-dimensional space, such as the
vibrational-rotational motions of diatomic molecules, and to models with Euclidean coordinate spaces of any dimension. Thus the method for constructing hyperspherical harmonics should extend to any SO(2n + 1) (n ∈ Z+ ) group. A natural next step would be
the construction of a Bohr-type model with collective octupolar l = 3 deformations. This
would involve the construction of a SO(7)⊃SO(3) basis, which could be accomplished in
much the same way as we have done here for SO(5)⊃SO(3).
The SO(5) hyperspherical harmonics themselves are a basis for any algebraic model
with the same embedded SO(3) symmetry. One such situation outside of nuclear physics
is in the analysis of the Jahn-Teller effect in octahedral crystals [90], where nearly degenerate three-fold and two-fold normal modes of crystal vibration lead to the consideration
of one-rowed SO(5) irreps in a rotational SO(3) basis. An intriguing feature of this system is that methods have been developed for treating modes with different frequencies
of oscillation. It would be interesting to see where a Bohr model with nondegenerate
quadrupolar modes would lead.
The generic so(5) irreps that we have developed could be applied to a collective model
101
Conclusion
that includes more than one type of particle, for instance something that can be identified
with an L = 3/2 boson. Iachello’s dynamical supersymmetry [91] is one such model,
however it deals only with representations that have f = 0 or f =
1
2
in our notation.
He shows that including a spin 3/2 particle in the IBM leads to consideration, as we
have mentioned, of the double covering groups of the O(6) ⊃ O(5) ⊃ O(3) chain, namely
SPIN(6)⊃SPIN(5)⊃SPIN(3). The extra terms that occur in the Casimir operators of
these groups give rise to spectra that closely resemble that of odd-A nuclei such as
If one were to seriously pursue such a model, the f =
1
2
191
78 Ir.
irreps of so(5) would be needed.
A more direct use of our generic so(5) irreps would occur in a model where neutron and
proton collectivity were treated separately. Here one would have overall wavefunctions
that were products of two one-rowed Bohr model states, one for the proton density and
the other for that of the neutrons. Since the tensor product of two one-rowed so(5) irreps
contains components of the type (v, f ), the generic so(5) irreps would arise naturally in
this scheme.
It is clear that the vector coherent state construction of generic so(5) irreps is computationally intensive compared to the construction of the hyperspherical harmonics. This
is because we cannot restrict the U(5) Hilbert space under consideration to the hypersphere S 4 . Since a generic so(5) irrep is a tensor product of a one-rowed (v, 0) irrep and
a (0, f ) irrep, and that the latter are realised in a U(4) Hilbert space, it is conceivable
that the procedure could be simplified by identifying an advantageous restriction to a
subspace of U(4). However, it is difficult to see how such a restriction would behave, since
it would not benefit from the helpful analogy with spherical harmonics, although there
exist constructions for spinor harmonics [92] that might be of use in such an endeavour.
102
Conclusion
Bibliography
[1] N. Bohr, Nature 137 (1936), 344.
[2] M. G. Mayer, Phys. Rev. 75 (1949), 1969.
[3] O. Haxel, J. H. D. Jensen and H. E. Suess, Phys. Rev. 75 (1949), 1766.
[4] J. Rainwater, Phys. Rev. 79 (1950), 432.
[5] A. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 26 No.14 (1952).
[6] A. Bohr and B. R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27 No.16 (1953).
[7] D. J. Rowe, “Nuclear Collective Motion: Models and Theory,” Methuen, London,
1970.
[8] A. Arima and F. Iachello, Ann. Phys. 99 (1976), 253.
[9] F. Iachello and A. Arima, “The Interacting Boson Model,” Cambridge University
Press, Cambridge, 1987.
[10] E. P. Wigner, “Gruppentheorie und Ihre Anwendung auf die Quantenmechanik der
Atomspektren,” F. Vieweg und Sohn, Braunschweig, 1931. English translation by
J. J. Griffin, Academic Press, New York, 1959.
[11] H. Weyl, “Gruppentheorie und Quantenmechanik,” republished in english by Dover,
New York, 1950.
[12] V. Heine, “Group Theory in Quantum Mechanics,” republished by Dover, New York,
1993.
[13] M. Hammermesh, “Group Theory,” Addison-Wesley, Reading, 1964.
103
104
Bibliography
[14] D. J. Rowe, “Practical Group Theory with Applications,” 3rd Edition, University
of Toronto Press, Toronto, 1997.
[15] D. H. Sattinger and O. L. Weaver, “Lie Groups and Algebras with Applications to
Physics, Geometry, and Mechanics,” Springer-Verlag, New York, 1993.
[16] M. Nakahara, “Geometry, Topology and Physics,” IOP Publishing, Bristol, 1998.
[17] G. Rakavy, Nuc. Phys. 4 (1957), 289.
[18] G. Racah, “Group Theory and Spectroscopy,” Institute for Advanced Study, Princeton, 1951.
[19] M. Moshinsky, “Group Theory and the Many-Body Problem”, Gordon and Breach,
New York, 1968.
[20] S. Goshen and H. J. Lipkin, Ann. Phys. (NY) 6 (1959), 301.
[21] A. O. Barut, A. Bohm and Y. Ne’eman, “Dynamical Groups and Spectrum Generating Algebras” (2 vols.), World Scientific, 1988.
[22] F. Iachello and R. D. Levine, “Algebraic Theory of Molecules,” Oxford University
Press, Oxford, 1995.
[23] M. Guidry, L.-A. Wu, Y. Sun and C.-L. Wu, Phys. Rev. B 63 (2001), 134516.
[24] A. Leviatan, Phys. Rev. Lett. 77 (1996), 818.
[25] P. Van Isacker, Phys. Rev. Lett. 83 (1999), 4269.
[26] D. Kusnezov, Phys. Rev. Lett. 79 (1997), 537.
[27] D. J. Rowe, Prog. Part. Nucl. Phys. 37 (1996), 265.
[28] G. Rosensteel and D. J. Rowe, Phys. Rev. Lett. 38 (1977), 10.
[29] D. J. Rowe, Rep. Prog. Phys. 48 (1985), 1419.
[30] D. J. Rowe, P. Rochford and J. Repka, J. Math. Phys. 29 (1988), 572.
[31] P. O. Hess, A. Algora, M. Hunyadi and J. Cseh, Eur. Phys. J. A 15 (2002), 449.
[32] F. Iachello, Phys. Rev. Lett. 85 (2000), 3580.
Bibliography
105
[33] F. Iachello, Phys. Rev. Lett. 87 (2001), 052502.
[34] N. Goldenfeld, “Lectures on Phase Transitions and the Renormalization Group,”
Frontiers of Physics 85, Perseus, Reading, 1992.
[35] G. W. Mackey, “Induced Representations of Groups and Quantum Mechanics,” Benjamin, New York, 1968.
[36] D. J. Rowe, J. Math. Phys. 25 (1984), 2662.
[37] J. Deenen and C. Quesne, J. Math. Phys. 25 (1984), 1638; 2354.
[38] D. J. Rowe, G. Rosensteel and R. Glimore, J. Math. Phys. 26 (1985), 2787.
[39] P. O. Hess, M. Seiwert, J. Maruhn and W. Greiner, Z.Physik A 296 (1980), 147.
[40] P. O. Hess, J. Maruhn and W. Greiner, J. Phys. G 7 (1981), 737.
[41] D. Troltenier, J. A. Maruhn and P. O. Hess, Numerical Application of the Geometric
Collective Model in “Computational Nuclear Physics 1”, eds. K. Langanke, J. A.
Maruhn, and S. E. Koonin.
[42] E. Chaćon, M. Moshinsky and R. T. Sharp, J. Math. Phys. 17 (1976), 668.
[43] E. Chaćon and M. Moshinsky, J. Math. Phys. 18 (1977), 870.
[44] J. M. Eisenberg and W. Greiner, “Nuclear Models,” 3rd edition, North Holland,
1987.
[45] L. Wilets and M. Jean, Phys. Rev. 102 (1955), 788.
[46] S.G. Rohoziński, J. Srebrny and H. Horbaczewska, Z. Physik 268 (1974), 401.
[47] J. P. Elliott, J. A. Evans and P. Park, Phys. Lett. 169 B (1986), 309.
[48] P. M. Davidson, Proc. R. Soc. 135(1932), 459.
[49] D. J. Rowe and C. Bahri, J. Phys. A: Math. Gen. 31 (1998), 4947.
[50] J. Čı́žek, J. Paldus, Int. J. Quantum Chem. 12 (1977), 875.
[51] S. D’Agostino, Ph.D. Thesis, University of Toronto (2005).
106
Bibliography
[52] D. J. Rowe, Nuc. Phys. A 735 (2004), 372.
[53] D. J. Rowe, J. Math. Phys. 35 (1994), 3163.
[54] D. J. Rowe and K. T. Hecht, J. Math. Phys. 36 (1995), 4711.
[55] D. R. Bès, Nuc. Phys. 10 (1959), 373.
[56] S. A. Williams and D. L. Pursey, J. Math. Phys. 9 (1968), 1230.
[57] D. J. Rowe, P.S. Turner and J. Repka, J. Math. Phys. 45 (2004), 2761.
[58] Handbook of Mathematical Functions, eds. M. Abramowitz and I. A. Stegun, National Bureau of Standards AMS-55, Washington D.C., 1964.
[59] D. J. Rowe, Nucl. Phys. (2004), in press.
[60] P. Maystre and M. Sargent III, “Elements of Quantum Optics,” Springer-Verlag,
New York, 1991.
[61] A. E. L. Dieperink, O. Scholten and F. Iachello, Phys. Rev. Lett. 44 (1980), 1747.
[62] S. Raman, C. H. Malarkey, W. T. Milner, C. W. Nestor, Jr. and P. H. Stelson, At.
Data and Nuc. Data Tables 36 (1987), 1.
[63] D. J. Rowe, Embedded representations and quasidynammical symmetry, in “Computational and Group Theoretical Methods in Nuclear Physics,” Proc. Symp. in
Honour of J. P. Draayer’s 60th Birthday. Eds. O. Castanos, J. Escher, J. Hirsch, S.
Pittel and G. Stoicheva, World Scientific, Singapore, 2004.
[64] D. J. Rowe, P. Rochford and J. Repka, J. Math. Phys. 29 (1988), 572.
[65] P. Rochford and D. J. Rowe, Phys. Lett. B 210 (1988), 5.
[66] D. J. Thouless, Nucl. Phys. 21 (1960),225.
[67] D. J. Rowe, P. S. Turner and G. Rosensteel, Phys. Rev. Lett. 93 (2004), 232502.
[68] E. Schrödinger, Naturwiss 14 (1927), 644.
[69] R. Y. Glauber, Phys. Rev. 130 (1963), 2529; 131 (1963), 2766.
[70] E. C. G. Sudarshan, Phys. Rev. Lett. 10 (1963), 277.
Bibliography
107
[71] J. R. Klauder, J. Math. Phys. 4 (1963), 1055 and 1058.
[72] A. M. Perelomov, Comm. Math. Phys. 26 (1972), 222.
[73] R. Gilmore, J. Math. Phys. 15 (1974), 2090.
[74] J. R. Klauder and B-S. Skagerstam, “Coherent States; Applications in Physics and
Mathematical Physics,” World Scientific, Singapore, 1985.
[75] A. Perelomov, “Generalized Coherent States and Their Applications,” SpringerVerlag, Berlin, 1986.
[76] W. Zhang, D. H. Feng and R. Gilmore, Rev. Mod. Phys. 62 (1990), 867.
[77] S. T. Ali, J.-P. Antoine and J.-P. Gazeau, from “Coherent States, Wavelets and
Their Generalizations: A Mathematical Overview” Springer-Verlag, 1999.
[78] K. T. Hecht, “The Vector Coherent State Method and its Application to Problems
of Higher Symmetry,” Lecture Notes in Physics Vol. 290, Springer, 1987.
[79] D. J. Rowe and J. Repka, J. Math. Phys. 32 (1991), 2614.
[80] D. J. Rowe, J. Math. Phys. 36 (1994), 1520.
[81] I. M. Gel’fand and M. L. Tseitlin, Doklady Akad. Nauk SSSR 71 (1950), 825; 1017.
[82] K. T. Hecht, Nuc. Phys. 63 (1965) 177.
[83] A. Klein and E. R. Marshalek, Rev. Mod. Phys. 63 (1991), 375.
[84] M. Kashiwara and M. Vergne, Invent. Math. 44 (1978), 1.
[85] T. M. Corrigan, F. G. Margetan and S. A. Williams, Phys. Rev. C 14 (1976), 2279.
[86] C. Bahri, private communication.
[87] S. D’Agostino, private communication.
[88] T. Otsuka, A. Arima and F. Iachello, Nucl. Phys. 309 (1978), 1.
[89] R. Bijker and A. Frank, Nuc. Phys. News 11 (2001), 15.
[90] B. R. Judd, Can. J. Phys. 52 (1974), 999.
108
[91] F. Iachello, Phys. Rev. Lett. 44 (1980), 772.
[92] K. D. Rothe, Phys. Rev. 170 (1968), 1548.
Bibliography