Proceedings TH2002 Supplement (2003) 279 – 292
c Birkhäuser Verlag, Basel, 2003
1424-0637/03/040279-14 $ 1.50+0.20/0
Proceedings TH2002
Coherent States, Induced Representations,
Geometric Quantization, and their Vector
Coherent State Extensions
D.J. Rowe
Abstract.
The theory of coherent state representations is both a theory of induced representations and of quantization. The different perspectives lead to valuable insights
and new ways of unifying and extending the two theories. The coherent state approach, and its vector coherent state generalization, have the merit that their physical content is transparent and highly practical. Indeed, they have been shown in
numerous applications to be applicable to a wide range of symmetry problems in
physics. In particular, in the language of geometric quantization, they are able to
quantize interesting physical systems with intrinsic (gauge) degrees of freedom.
1 Introduction
This presentation outlines some advances in coherent state representation theory
and shows that it provides a simple and practical way of constructing induced
representations and carrying out the prescriptions of geometric quantization. A
typical problem that motivates such developments is from nuclear physics and can
be expressed qualitatively as follows.
Given a microscopic Hamiltonian on a Hilbert space for 168 nucleons,
derive the low-energy vibrational and rotational states of the 168 Er nucleus.
The low energy-level spectrum of 168 Er is shown in Fig. 1. It is seen to comprise
a sequence of rotational bands. For example, the lowest energy states of angular
momentum L = 0, 2, 4, . . . make up the ground state band. But many other bands
are based on what are presumably vibrational excitations. This is a beautiful
problem whose solution invokes some sophisticated representation theory and the
quantization of a non-trivial model. Of course, the theory is not restricted in
application to the 168 Er nucleus or even to nuclear physics. But, it is useful to
have a specific problem in mind.
A simple model of nuclear rotations and vibrations cannot explain the 168 Er
spectrum in detail. The complete dynamics is complex and involves spin as well as
many spatial degrees of freedom. However, we are able to understand, in qualitative
microscopic terms, the rotational dynamics of the 168 Er ground and some vibra-
280
D.J. Rowe
Proceedings TH2002
Figure 1: The low-energy level spectrum of the 168 Er nucleus shown as sequences of
rotational bands. The positive parity bands are shown on the left and the negative
parity bands on the right.
tional bands by constructing a model of nuclear rotations and vibrations, whose
basic observables can be expressed in many-nucleon coordinates and momenta.
This paper presents a qualitative overall description of the underlying ideas
aimed at giving a general idea of what coherent state theory can do and the new
light it sheds on induced representation theory [1] and the theory of geometric
quantization [2, 3]. It outlines how coherent state theory and its vector coherent
extension incorporate the basic theories of induced representations, e.g., Mackey
theory [1] and Harish Chandra’s method [4] of holomorphic induction in a way
that is readily accessible to physicists and provides the explicit expression of the
results that physicists need to exploit the symmetries of their problems in real
calculations. For example, the coherent state methods don’t just give abstract
representations of Lie groups, they also give explicit algorithms for constructing
matrix elements of these representations and of their infinitesimal generators. I will
focus particularly on a recent development in which we have shown that coherent state theory reproduces the three fundamental representations of an algebraic
model given by geometric quantization. An algebraic model is essentially a model
whose basic observables can be expressed in terms of the infinitesimal generators of
a Lie group, known as a dynamical group for the model. The three representations
of a model given by geometric quantization are: the classical representations with
their Poisson bracket, the representations of prequantization, which are unitary
but reducible, and finally the irreducible unitary representations of a full quanti-
Vol. 4, 2003
Coherent States, Induced Representations, Geometric Quantization
281
zation. I will then give an illustration of how the vector coherent state methods
make fuller use of the geometrical structures available to geometric quantization
and lead to quantizations with interesting non-Abelian intrinsic gauge degrees of
freedom.
The developments reported here were done primarily in collaboration with
S.D. Bartlett, J. Repka and G. Rosensteel.
2 A classical model with symmetry
The space of states for a classical model is a smooth symplectic manifold (phase
space) P. Such a manifold admits local systems of canonical position and momentum coordinates and supports functions which satisfy Poisson bracket relationships
defined by the symplectic form for P. A state of a classical model is represented
by a point on P.
An observable for a classical model, is a smooth function on the phase space
having values at each point corresponding to the physical values of the observable
for the corresponding state of the model. Thus, a state of the model is characterized
by the values of a set of observables. We therefore consider Lie algebras of classical
observables with Lie bracket given by their Poisson bracket. A given classical model
can support many algebras of observables most of which are infinite dimensional.
However, when we speak of a classical model with symmetry, we suppose that there
exists a minimal finite-dimensional algebra of observables having the property that
the values of its elements are sufficient to uniquely identify any point on the phase
space. I shall refer to such a finite-dimensional algebra of observables as a dynamical
algebra (also called a spectrum generating algebra) for the model.
The gradient of any smooth function on P defines a tangent vector at every
point of P, called a Hamiltonian vector field, which can be interpreted as an infinitesimal generator of displacements on P. Thus, the elements of an algebra of
observables integrate to a Lie group of transformations of P.
A finite-dimensional Lie group G of transformations of the phase space P for
a model is said to be a dynamical group for the model if it preserves the symplectic
structure of P, and hence Poisson brackets, and also has the property that if m
is any point of P then any other point m ∈ P can be reached by applying some
element g ∈ G to m; thus, we write m = g · m. In other words, a dynamical group
should be transitive on the model’s phase space. Identifying a dynamical group for
a model, has the valuable property that it puts the phase space P in one-to-one
correspondence with a factor space G/H where
H = h∈Gh·m=m .
(1)
The concepts of a dynamical algebra and a dynamical group are clearly related. In fact, as readily shown by coherent state theory, a classical representation
of the Lie algebra of a dynamical group by functions on P is, in fact, a dynamical
algebra for the model.
282
D.J. Rowe
Proceedings TH2002
The objective of quantization is to construct a Hilbert space for the model on
which its observables act as Hermitian operators. The idea of so-called canonical
quantization, proposed by Dirac, is to map the classical algebra of observables to an
irreducible unitary (Hilbert space) representation. It is now known that this cannot
be done for the infinite-dimensional algebra of all classical observables. However,
it is also recognized that an appropriate quantization is given by making the Dirac
map for a finite-dimensional subalgebra; hence we focus on a finite-dimensional
dynamical algebra of observables as defined above.
The first step towards quantization of a given classical representation starts
with the observation of the remarkable fact that the classical representation already
defines a (generally projective) unitary representation of the subgroup H, defined
by eqn. (1). This is seen in the following simple example.
An example:
Consider a phase space P which is a two sphere of radius M . A suitable
dynamical group for this phase space is then the group G = SO(3) of rotations of points on the sphere and the sphere is identified with the factor space
SO(3)/SO(2). A corresponding dynamical algebra of observables is given by the
components (Lx , Ly , Lz ) of the vector from the centre to a point on the sphere.
The Hamiltonian vector fields defined by these functions are then tangent to the
sphere and interpreted physically as classical components of angular momentum
for the model (which could be a symmetric top, for example) and mathematically
as infinitesimal generators of rotations.
In particular, Lz is an infinitesimal generator of rotations about the z axis.
But now observe that the north pole of the sphere, at which the angular momenta
have the values
Lx = Ly = 0 ,
Lz = M ,
(2)
is unmoved by a rotation about the z axis. In fact, the subgroup of SO(2) rotations
generated by Lz is the isotropy subgroup of all elements of SO(3) rotations that
leave the point (0, 0, M ) fixed. The interesting observation is that the value M
of Lz at the point (0, 0, M ) defines a unitary representation of an SO(2) rotation
through angle θ by θ → eiMθ . However, unless M is an integer, this representation
is a projective representation.
In the language of geometric quantization, we say that the classical representation of the so(3) Lie algebra, given by the classical model, is quantizable if 2M
is an integer.
In general, we say that a classical representation of a dynamical algebra
g on a phase space G/H is quantizable if the unitary representation of
H defined by the classical representation of g appears in the restriction
of some unitary representation of G to its subgroup H ⊂ G.
Vol. 4, 2003
Coherent States, Induced Representations, Geometric Quantization
283
3 The coherent state perspective
Whereas geometric quantization starts from a classical model and proceeds to
derive its quantizations, coherent state theory starts with an abstract realization
of the dynamical group G for the model and its Lie algebra g and derives the
classical and quantized representations of the model.
The coherent state perspective assumes the existence of some unspecified
(generally reducible) representation T of the dynamical group G on a Hilbert space
H and from it derives both classical and quantum representations of the model. If
one wishes to be specific, T might be thought of as the regular representation of
G. Or, in thinking of the model as a submodel of a many-particle system, it could
be a representation of G as a group of unitary transformations of a many-body
Hilbert space.
3.1
Classical representations
If |0 is any state in H, the set of states
C = |g = T (g −1 )|0 | g ∈ G
(3)
is called a system of coherent states, in accord with Perelomov’s general definition
of coherent states [5]. (The use of an inverse group element in this definition is
purely for convenience; it has no significance.) The map A → A from elements of
g to functions on G, defined by
d
A(g) = g|Â|g , with  = −i T (eiAt )
,
(4)
dt
t=0
associates a function A on the factor space H\G with each element A ∈ g, where
(5)
H = h ∈ G A(hg) = A(g), ∀A ∈ g .
Moreover, this map is a classical representation with Lie bracket given by the
Poisson bracket, related to the abstract Lie bracket for g by
i
{A, B}(g) = − g|[Â, B̂]|g .
(6)
The notable fact, well known in the theory of geometric quantization, is that
the symplectic form defined on the factor space H\G by this Poisson bracket
is non-degenerate, Thus, H\G is a phase space and the map (4) is a classical
representation of the dynamical algebra.
The Lie algebra h of H is the subalgebra of g
(7)
h = A ∈ g 0|[Â, B̂]|0 = 0, ∀B ∈ g .
It follows that, for A ∈ h, the map
A → A(e) = 0|Â|0
(8)
284
D.J. Rowe
Proceedings TH2002
defines a one-dimensional representation of h (e denotes the identity element of
G). The corresponding one-dimensional representation of H
h → χ(h) = 0|T (h)|0 ,
∀h ∈ H ,
(9)
is unitary but generally projective.
3.2
The reducible unitary representations of prequantization
If a classical representation is quantizable according to the definition and provided
the restriction to H of the representation T of G actually contains the unitary
representation χ as a subrepresentation, then there exists some state |0 ∈ H
which not only has the property that 0|T (h)|0 = χ(h) but also satisfies the
stronger condition
T (h)|0 = χ(h)|0 , ∀h ∈ H .
(10)
Given such a state, there is a map from H to a space of coherent-state wave function
on G in which a state |ψ ∈ H maps to a function ψ with values
ψ(g) = 0|T (g)|ψ,
g ∈ G,
(11)
∀h ∈ H .
(12)
and for which
ψ(hg) = χ(h)ψ(g) ,
Equivalently, if ψ(Ag) is defined by
ψ(Ag) = i
d
ψ(e−iAt g)
,
dt
t=0
(13)
then the constraint condition (12) is expressed
ψ(Ag) = A(e)ψ(g) ,
∀A ∈ h .
(14)
In fact, the above construction has a natural generalization. Suppose that the
representation χ of H is contained, not as a subrepresentation of the restriction
of the representation T to the subgroup H ⊂ G, but as a component in a direct
integral decomposition of this restriction. Then there will not exist a state |0 ∈ H
having the property (12) but there will exist a functional ϕ| on a dense subspace
HD of H having the property that the coherent state wave functions
ψ(g) = ϕ|T (g)|ψ,
g ∈ G,
(15)
are well-defined for all |ψ ∈ HD and satisfy the conditions (12) and (14).
The space of such coherent-state wave functions carries a (generally reducible)
coherent-state representation of G given by
[Γ(g)ψ](g ) = ψ(g g) .
(16)
Vol. 4, 2003
Coherent States, Induced Representations, Geometric Quantization
285
This representation is known to be unitary wrt to a suitably defined inner product.
For example, if H is the Hilbert space of square integrable functions over G
(relative to the invariant measure) and T is the regular representation then, for
|ϕ a normalizable state in H, the coherent state wave functions are a subset of H.
Moreover, if |ϕ is a generic state with isotropy subgroup H ⊂ G, i.e., for which
T (h)|ϕ = χ(h)|ϕ, then the coherent-state representation Γ of G is identical to
the representation of G induced from the representation χ of the subgroup H ⊂ G.
The representation (16) also defines a corresponding unitary representation
of the Lie algbra g for which
[Γ(A)ψ](g) = i
d
ψ(ge−iAt )
,
dt
t=0
∀A ∈ g . .
(17)
This representation can be expressed in the form
[Γ(A)ψ](g) = ϕ|Â(g)T (g)|ψ = A(g)ψ(g) + i[∇A ψ](g) ,
(18)
where Â(g) = T (g)ÂT (g −1 ), A(g) = ϕ|Â(g)|ϕ is the value of the classical observable A at g, and ∇A is defined by
i[∇A ψ](g) = ϕ|Â(g)T (g)|ψ − A(g)ψ(g) .
(19)
Thus, if {Aα } is a basis for the Lie algebra g and we make the expansion
Â(g) =
Aα (g)Âα ,
(20)
α
it follows that
i[∇A ψ](g) =
Aα (g) i∂α − θα ψ(g) ,
(21)
i
d
ψ(e− Aα t g)t=0 ,
dt
(22)
α
where
i∂α ψ(g) = i
and
θα = ϕ|Âα |ϕ = Aα (e) .
(23)
As shown explicitly in ref. [6], ∇A is the covariant derivative for the function A
and the coherent state representation
Γ(A) = A + i∇A ,
(24)
is identical to the corresponding representation of prequantization.
Note, however, that the coherent state construction also gives representations on other spaces. Moreover, by choosing the functional ϕ| to have special
properties, it is possible to induce irreducible coherent state representations.
286
3.3
D.J. Rowe
Proceedings TH2002
The irreducible unitary representations of a full quantization
For a full quantization, the coherent state representation should be irreducible as
well as unitary. It will be irreducible if ϕ| can be chosen such that only the component of a state |ψ ∈ HD belonging to an irrep gives a non-vanishing contribution
to the wave function
ψ(g) = ϕ|T (g)|ψ , g ∈ G .
(25)
Such a functional can often be defined, for example, by extending the condition
(14) to a suitable subalgebra p in the chain h ⊂ p ⊂ gc , where gc is the complex
extension of g. Let χ̃ denote a one-dimensional irrep of p ⊂ gc for which
χ̃(A) = A(e) ,
∀A ∈ h ,
(26)
and let ϕ| be a functional on a dense subspace HD ⊆ H such that
ϕ|XT (g)|ψ = χ̃(X)ψ(g) ,
∀X ∈ p .
(27)
This condition includes the conditions expressed by eqn. (14) and, for a suitable
choice of p it may be sufficient to ensure that the coherent state representation is
irreducible.
For example, if g is semisimple and h is a Cartan subalgebra, |ϕ could be a
highest or a lowest weight state for an irrep. An appropriate choice of p would then
be the parabolic subalgebra of gc containing h and a set of raising (or lowering)
operators. A suitable subalgebra p ⊂ gc is known in the language of geometric
quantization as a polarization.
3.4
An example
Let g be the Heisenberg-Weyl algebra; it is spanned by the position x and momentum p observables plus the identity 1 with Poisson brackets
{x, p} = 1 ,
{1, x} = {1, p} = 0 .
(28)
Let x → x̂, p → p̂, 1 → Iˆ denote a generic representation of this algebra (supposing
we did not already have a realization of this representation) such that
[x̂, p̂] = iIˆ ,
ˆ x̂] = [I,
ˆ p̂] = 0 .
[I,
(29)
For any normalized state |0 in the representation space for which 0|x̂|0 =
0|p̂|0 = 0, define the coherent states
i
i
{|x, p = e− xp̂ e px̂ |0 ; x, p ∈ R} .
(30)
A classical representation of the HW algebra is then given by
x̂
→ X (x, p) = x, p|x̂|x, p = x ,
p̂ → P(x, p) = x, p|p̂|x, p = p ,
Iˆ → I(x, p) = 1 ,
(31)
Vol. 4, 2003
Coherent States, Induced Representations, Geometric Quantization
287
with Poisson brackets given, for example, by
i
{x, p} = − x, p|[x̂, p̂]|x, p .
(32)
A state |ψ in the (unspecified) representation space has a coherent state wave
function
ψ(x, p) = x, p|ψ ,
(33)
and prequantization gives the representation of the HW algebra
Γ(x) = x + i
∂
,
∂p
Γ(p) = −i
∂
,
∂x
Γ(x) = 1 ,
(34)
as linear operators on these wave functions. This representation is reducible. To
get an irreducible representation, choose a functional ϕ| on the dense subspace
of continuous wave functions in the representation space which is an eigenstate of
the operator x̂ with zero eigenvalue. The coherent state wave functions
i
i
ψ(x, p) = ϕ|e− px̂ e xp̂ |ψ
(35)
then reduce to the subset of x-independent functions and the coherent state representation reduces to
Γ(x) = x ,
Γ(p) = −i
∂
,
∂x
Γ(x) = 1 .
(36)
This representation is now irreducible.
For the above prequantization, the isotropy subalgebra h is the subalgebra
of g spanned by the identity element 1. For the full quantization, h is extended
to the polarization p spanned by 1 and x. Since p is in fact a subalgebra of g it
is said to be a real polarization. The Bargmann-Segal quantization is the complex
polarization spanned by 1 and x + ip.
4 A microscopic model of nuclear rotations and vibrations
The above methods can be applied to a non-trivial microscopic model of the
rotational-vibrational dynamics of a nucleus proposed by Weaver, Biedenharn,
and Cusson [7] and known as the CM(3) model, where the acronym stands for
Collective Motion in 3 dimensions. The quantization of this model was first given
by Rosensteel [8].
The set of observables for a model of rotations and vibrations naturally includes six Cartesian quadrupole moments which, for a many-particle nucleus, have
a natural microscopic expression
xni xnj = Qji , i, j = 1, 2, 3 ,
(37)
Qij =
n
288
D.J. Rowe
Proceedings TH2002
where (xn1 , xn2 , xn3 ) are Cartesian coordinates for the n’th particle. These observables define the deformation and orientation of an ellipsoidally shaped nucleus.
They span a real Abelian Lie algebra which we denote by R6 . We also need vibrational and angular (rotational) momentum observables of the nucleus which can
be interpreted as infinitesimal generators of shape change. A natural choice is the
nine moment-of-momentum observables
xni pnj , i, j = 1, 2, 3 ,
(38)
Xij =
n
which are infinitesimal generators of a general linear group GL(3, R) and span a
gl(3, R) Lie algebra.
The span of these combined sets of observables {Qij , Xij } is the semi-direct
sum Lie algebra [R6 ]gl(3, R) which we take as the dynamical algebra for our model.
An element of the corresponding dynamical group is conveniently expressed in the
form
i
(39)
(B, g) ≡ e− nij Bij Qij g ,
where B is a real symmetric matrix and g ∈ GL(3, R).
A question that arises is: why are there nine momentum observables {Xij } in
this model but only six deformation observables {Qij }? In fact, this is a common
situation and is automatically looked after in the construction when the elements
of the isotropy subgroup H are factored out to give a phase space H\G. A physical
interpretation of how this might be achieved in the present example is given by
considering a standard factorization of a GL(3, R) group element as a product of
a rotation, a diagonal matrix, and another rotation:
g = Ω dΩ .
(40)
The result of applying such a sequence of transformations to a spherical nuclear
density distribution is illustrated in fig. 2. It is seen that the first rotation Ω
produces an intrinsic rotational motion but no observable effect on the quadrupole
moments of the nucleus. We refer to such unobservable rotations as intrinsic vortex
rotations. If H is set equal to SO(3) and identified with the subgroup of vortex
rotations, the factor space H\G becomes a 12-dimensional phase space with six
local coordinates and six local momentum in the neighbourhood of any point on
its surface.
The above microscopic expressions of the model observables, gives the dynamical group of the model an immediate unitary representation on the space of
many-particle wave functions; we denote this representation by
i
(B, g) → e− nij
Bij Q̂ij
T (g) ,
(41)
where
T (g) is the unitary representation of an element g ∈ GL(3, R) and Q̂ij =
n x̂ni x̂nj . However, this representation is highly reducible and cannot be considered a quantization of the model. A quantization is obtained, for example, by
Vol. 4, 2003
Coherent States, Induced Representations, Geometric Quantization
289
Figure 2: Application of the general linear transformation g = Ω dΩ to a spherical
nucleus. The first rotation produces an intrinsic rotational motion but no observable effect on the quadrupole moments of the nucleus.
choosing a functional |ϕ that is a zero-eigenstate of the quadrupole moments and
has angular momentum zero. The coherent state wave functions
i
ψ(g) ≡ ϕ|e− nij
Bij Q̂ij
T (g)|ψ = ϕ|T (dΩ)|ψ
(42)
are then seen to be independent of both B and Ω ; i.e., they satisfy
ψ(Ωg) = ψ(g) .
(43)
The coherent state representation of the quadrupole operators on these wave functions is given by
[Γ(Qij )ψ](g) = (g̃g)ij ψ(g) ,
(44)
and the general linear transformations by
[Γ(g)ψ](g ) = ψ(g g) .
(45)
This coherent state representation, is equivalent to a Mackey induced representation and is known to be irreducible. Thus, it is a quantization of the CM(3)
model.
5 Intrinsic degrees of freedom in a vector coherent state extension
From a geometrical perspective, one can think of the manifold SO(3)\GL(3, R)
as the configuration space for the above model. One can also regard the coherent
state wave functions as sections of a complex line bundle over SO(3)\GL(3, R).
This bundle is, in fact, a line bundle that is associated to the principle GL(3, R) →
SO(3)\GL(3, R) bundle by the identity representation of the subgroup SO(3) ⊂
GL(3, R).
290
D.J. Rowe
Proceedings TH2002
Figure 3: Coherent state wave functions for a zero vorticity representation of the
rotor-vibrator model are sections of a complex line bundle associated to the principle GL(3, R) → SO(3)\GL(3, R) bundle by the identity representation of SO(3).
The latter perspective suggests a more general construction with non-zero
vorticity in which the wave functions are sections of a vector bundle associated to
the principle GL(3, R) → SO(3)\GL(3, R) bundle by a non-trivial representation
of SO(3). Such a generalization sounds complicated but, in fact, it is a natural
extension of a coherent state representation to a vector coherent state representation.
Such an irrep is readily constructed as follows. Let ρ̂ denote an SO(3) irrep
of angular momentum v on a (2v + 1)-dimensional Hilbert space with orthonormal
basis {ξvν ; ν = −v, . . . , +ν}. Thus, we suppose that
ρ̂(Ω)ξvν =
µ
v
ξvµ Dµν
(Ω) ,
Ω ∈ SO(3) .
(46)
Now, instead of a single functional |ϕ, we consider a set of functionals {|vν; ν =
−v, . . . , +v} which are all zero eigenstates of the quadrupole moments and transform under rotations as states of angular momentum v and z-component ν, i.e.,
T (Ω)|vν =
µ
v
|vµ Dµν
(Ω) ,
Ω ∈ SO(3) ⊂ GL(3, R) .
(47)
Vector-valued coherent state wave functions are then defined by
Ψ(g) ≡
ν
i
ξvν vν|e− nij
Bij Q̂ij
T (g)|ψ =
ν
ξvν ψν (g) ,
(48)
Vol. 4, 2003
Coherent States, Induced Representations, Geometric Quantization
291
Figure 4: Vector coherent state wave functions for a non-zero vorticity representation of the rotor-vibrator model are sections of a vector bundle associated to
the principle GL(3, R) → SO(3)\GL(3, R) bundle by a multidimensional irrep of
SO(3).
where
ψν (g) = vν|T (g)|ψ .
(49)
Thus, VCS wave functions are seen to have intrinsic (vortex spin) degrees of freedom. They continue to be independent of B. However, instead of eqn. (43), they
satisfy the identity
Ψ(Ωg) =
ρ̂(Ω)ξvν ψν (g) = ρ̂(Ω)Ψ(g) .
(50)
ν
VCS wave functions are, in fact, sections of the vector bundle associated to the
principle GL(3, R) → SO(3)\GL(3, R) bundle by the irrep ρ̂ of SO(3). Moreover,
as shown in ref. [6], VCS representations can be expressed in the language of
geometrical quantization. However, to use them, it is not necessary to understand
this geometrical interpretation. As a result of their simplicity, they have been
widely used by physicists to construct the explicit matrices of a wide range of
irreducible Lie algebra and Lie group representations and in the quantization of
algebraic models.
292
D.J. Rowe
Proceedings TH2002
References
[1] Mackey G W 1952 Ann. of Math. 55 101; 1968 Induced Representation of
Groups and Quantum Mechanics (New York: Benjamin); 1978 Unitary Group
Representations in Physics, Probability and Number Theory (Reading, MA:
Benjamin)
[2] Kostant B 1970 On Certain Unitary Representations which Arise from a
Quantization Theory, in Group Representations in Mathematics and Physics,
Lecture Notes in Physics, Vol. 6 (Berlin: Springer); Quantization and Unitary
Representations, in Letures in Modern Analysis and Applications III, Lecture
Notes in Mathematics, Vol. 170 (Berlin, Springer)
[3] Souriau J–M 1966 Comm. Math. Phys. 1 374; 1970 Structure des systèmes
dynamiques (Paris: Dunod)
[4] Harish-Chandra 1955 Amer. J. Math. 77 743; 1956 78 1; 1956 78 564
[5] Perelomov A 1986 Generalized Coherent States and Their Applications
(Berlin: Springer-Verlag)
[6] Bartlett S D, Rowe D J and Repka J 2002 J. Phys. A: Math. Gen. 35 5599
[7] Weaver L, Biedenharn L C and Cusson R Y 1973 Ann. Phys. (N.Y.) 77 250
[8] Rosensteel G and Rowe D J 1976 Ann. Phys. (N.Y.) 96 1
D.J. Rowe
Department of Physics, University of Toronto
Toronto, Ontario M5S 1A7
Canada