Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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with H
() =
P
A
U
A
() and a function
depending only on variables
x
on sites x from a
finite set S Z
d
.
A convenient way of evaluating the expectation starts
with introduction of the modified partition function
Z
;
ðÞ¼Z
þ Z
;
¼ Z
1 þ hi
ðÞ½42
Clearly,
hi
¼
d log Z
;
ðÞ
d
¼0
½43
Thus, one may get an expression for the expectation
hi
, by forming a polymer representation of Z
,
()
and isolating terms linear in in the corresponding
cluster expansion. For the first step, in the just cited
high-temperature case with general multi-site inter-
actions, we first enlarge the original set A()ofall
polymers in (consisting of connected collections
A= (A
1
, ..., A
k
)) to W
S
() = A() [A
S
(), where
A
S
()isthesetofallcollections(A
1
, ..., A
k
)of
polymers such that each of them intersects the set S
(polymers (A
1
, ..., A
k
)are‘‘glued’’byS into a single
entity). Compatibility is defined as before by disjoint-
ness; in addition, any two collections from A
S
()are
declared to be incompatible as well as any polymer A
from A() intersecting S is considered to be incompa-
tible with any collection from A
S
(). Defining now
w
(A) = w(A)forA2A()and
w
ðAÞ ¼
Z
ðÞe
H
ðÞ
Y
x2[
A2A
1
[[A
k
A[S
ðd
x
Þ
½44
for A= (A
1
, ..., A
k
) 2A
S
(), we get Z
,
()
exactly in the form [18],
Z
;
ðÞ¼
X
IW
S
ðÞ
Y
A2I
w
ðAÞ ½45
As a result, we have
log Z
;
ðÞ¼
X
X2XðW
S
ðÞÞ
aðXÞw
X
½46
allowing easily to isolate terms linear in : namely,
the terms with multi-indices X with supp X \A
S
()
consisting of a single collection, say A
0
, that occurs
with multiplicity one, X(A
0
) = 1. Explicitly, using
X
S;A
0
ðÞ¼ X 2XðW
S
ðÞÞ : supp X \A
S
ðÞ
f
¼fA
0
g; XðA
0
Þ¼1g½47
we get
hi
¼
X
A
0
2A
S
ðÞ
X
X2X
S;A
0
ðÞ
aðXÞw
X
½48
It is easy to show that, for sufficiently small ,theseries
on the right-hand side is absolutely convergent even if
we extend A
S
()toA
S
= [
A
S
()andX
S,A
0
()to
X
S,A
0
= [
X
S,A
0
(). As a result, we have an explicit
expression for the limiting expectation hi in terms of
an absolutely convergent power series. This can be
immediately applied to show that jhihi
j decay
exponentially in distance between S and the comple-
ment of . Indeed, it suffices to find a suitable bound on
P
X
ja(X)jjwj
X
with the sum running over all clusters
X reaching from the set S to
c
. To this end one does not
need to evaluate explicitly the number of clusters of
given ‘‘diameter’’ diam(X)=
P
A
X(A)diam((A))=m
with m dist(S,
c
). The needed estimate is actually
already contained in the condition (ii) from the cluster
expansion theorem. It just suffices to choose a suitable
k and assume that is small enough to assure validity
of (40) in a stronger form,
P
A:(A)3x
jw(A)jK
(A)j
1,
yielding eventually
X
X : diamðXÞdistðS;
c
Þ
jaðXÞjjwj
X
K
distðS;
c
Þ
jSj
X
X:[
A2supp X
ðAÞ3x
jaðXÞjjwj
X
K
P
XðAÞjðAÞj
jSjK
distðS;
c
Þ
½49
Exponential decay of correlations h
1
;
2
i
=
h
1
2
i
h
1
i
h
2
i
(and the limiting h
1
;
2
i)
in distance between supports of
1
and
2
can be
established in a similar way by isolating terms
proportional to
1
2
in the cluster expansion of
log Z
,
1
;
2
(
1
;
2
) with
Z
;
1
;
2
ð
1
;
2
Þ
¼Z
1 þ
1
h
1
i
þ
2
h
2
i
þ
1
2
h
1
2
i
ðÞ½50
The resulting claim can be readily generalized to one
about the decay of the correlation h
1
;...;
k
i in
terms of the shortest tree connecting supports
S
1
,...,S
k
of the functions
1
,...,
k
.
Low-Temperature Expansions
Finally, in some models with symmetries, we can apply
cluster expansion also at low temperatures. Let us
illustrate it again in the case of Ising model. This time,
we take the partition function Z
þ
()withplus
boundary conditions. First, let us define for each
nearest-neighbor bond hx, yi its dual as the (d 1)-
dimensional closed unit hypercube orthogonal to the
segment from x to y and bisecting it at its center. For a
given configuration
, we consider the boundary of
the regions of constant spins consisting of the union
@(
) of all hypercubes that are dual to nearest-
neighbor bonds hx, yi for which
x
6¼
y
.Thecontours
corresponding to
are now defined as the connected
components of @(
). Notice that, under the fixed
boundary condition, there is a one-to-one correspon-
dence between configurations
and sets of
mutually compatible (disconnected) contours in .
Cluster Expansion 535
Observing that the number of faces in @(
) is just
the sum of the areas jj of the contours 2 ,we
get the polymer representation
Z
þ
ðÞ¼e
jEðÞj
X
exp
X
2
jj
!
½51
where the sum is over all collections of disjoint
contours in . Here E() is the set of all bonds hx, yi
with at least one endpoint x, y in .
The condition [24] with r() =
yields a similar
bound on the weights w() = e
jj
as in the high-
temperature expansion. To verify it, for sufficiently
large, boils down to the evaluation of number of
contours of size n that contain a fixed site.
As a result, we can employ the cluster expansion
theorem to get
log Z
þ
ðÞ¼jEðÞjþ
X
X:X2XðCðÞÞ
aðXÞw
X
½52
with an explicit formula for the limit
pðÞ¼d þ
X
X:AðXÞ30
aðXÞ
jAðXÞj
w
X
½53
Here, A(X) is the set of sites attached to contours
from supp X,
AðXÞ¼[
2supp X
AðÞ½54
with
AðÞ¼fx 2 Z
d
j such that distðx;Þ1=2g½55
As a consequence of the fact that [53] is, for large
, an absolutely convergent sum of analytic terms
a(X)w
X
= a(X)e
P
X()j j
(considered as functions
of ), the function p() is, for large , analytic in .
The fact that one can explicitly express the
difference log Z
þ
() jjp() (cf. [28]) found
numerous applications in situations where one
needs an accurate evaluation of the influence of the
boundary of the region on the partition function.
One such example is a study of microscopic
behavior of interfaces. The main idea is to use the
explicit expression in the form
Z
þ
ðÞ
¼exp pðÞjj
fg
exp
X
X:AðXÞ\
c
6¼;
aðXÞw
X
jAðXÞ\j
jAðXÞj
8
<
:
9
=
;
¼exp pðÞjj
fg
Y
X:AðXÞ\
c
6¼;
ð1 þf
X
Þ ½56
Noticing that
f
X
¼ exp aðXÞw
X
jAðXÞ\j
jAðXÞj
1
does not vanish only if A(X) \ 6¼;,wecanexpand
the product to obtain ‘‘decorations’ ’ of the boundary
@ by clusters f
X
. In the case of interface these clusters
can be incorporated into the weight of interface, while
on a fixed boundary they yield a ‘‘wall free energy.’’
The possibility of the (low-temperature) polymer
representation of the partition function in terms of
contours is based on the þ$symmetry of the
Ising model. In absence of such a symmetry, cluster
expansions can still be used, but in the framework of
Pirogov–Sinai theory (see Pirogov–Sinai Theory).
Bibliographical Notes
Cluster expansions originated from the works of Ursell,
Yvon, Mayer, and others and were first studied in terms
of formal power series. The combinatorial and enu-
meration problems considered in this framework were
summarized in Uhlenbeck and Ford (1962). For related
topics in modern language, see Bergeron et al. (1998).
The convergence results for Mayer and virial expansions
for dilute gas were first proved in the works of Penrose,
Lebowitz, Groenveld, and Ruelle (see Ruelle (1969) for
a detailed survey). General polymer models on lattice
were discussed by Gruber and Kunz (1971) (see also
Simon (1993) for discussion of high-temperature and
low-temperature cluster expansions of lattice models).
Abstract polymer models were introduced in Kotecky´
and Preiss (1986). An elegant proof of a general claim
presented by Dobrushin (1996) was further extended
and summarized by Scott and Sokal (2005). We follow
their reformulation of the Dobrushin condition. Cluster
expansions with a view on applications in quantum field
theory are reviewed in Brydges (1986).
See also: Phase Transitions in Continuous Systems;
Pirogov–Sinai Theory; Wulff Droplets.
Further Reading
Bergeron F, Labelle G, and Leroux P (1998) Combinatorial
Species and Tree-Like Structures, Coll. Encyclopaedia of
Mathematics and Its Applications, vol. 67. Cambridge, MA:
Cambridge University Press.
Brydges DC (1986) A short course on cluster expansions. In:
Osterwalder K and Stora R (eds.) Critical Phenomena, Random
Systems, Gauge Theories, pp. 129–183. Les Houches, Session
XLIII, 1984. Amsterdam/New York: Elsevier.
Dobrushin RL (1996) Estimates of semi-invariants for the Ising
model at low temperatures. In: Dobrushin RL, Minlos RA,
Shukin MA, and Vershik AM (eds.) Topics in Statistical and
Theoretical Physics, pp. 59–81. Providence, RI: American
Mathematical Society.
Gruber C and Kunz H (1971) General properties of polymer
systems. Communications Mathematical Physics 22: 133–161.
Kotecky´ R and Preiss D (1986) Cluster expansion for abstract polymer
models. Communications in Mathematical Physics 103: 491–498.
536 Cluster Expansion
Ruelle D (1969) Statistical Mechanics: Rigorous Results, The
Mathematical Physics Monograph Series. Reading, MA:
Benjamin.
Scott AD and Sokal AD (2005) The repulsive lattice gas, the
independent-set polynomial, and the Lova´sz local lemma.
Journal of Statistical Physics 118: 1151–1261.
Simon B (1993) The Statistical Mechanics of Lattice Gases,Princeton
Series in Physics, vol. 1. Princeton: Princeton University Press.
Uhlenbeck GE and Ford GW (1962) The theory of linear graphs with
applications to the theory of the virial development of the
properties of gases. In: de Boer J and Uhlenbeck GE (eds.) Studies
in Statistical Mechanics, vol. I, Amsterdam: North-Holland.
Coherent States
S T Ali, Concordia University, Montreal, QC, Canada
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
Very generally, a family of coherent states is a set of
continuously labeled quantum states, with specific
mathematical and physical properties, in terms
of which arbitrary quantum states can be expressed
as linear superpositions. Since coherent states are
continuously labeled, they form overcomplete
sets of vectors in the Hilber t space of states.
Originally these states were introduced into physics
by Schro¨ dinger (1926) , as a family of quantum
states in terms of which the transition from quantum
to classical mechanics could be conveniently studied.
These states have the minimal uncertainty property,
in the sense that they saturate the Heisenberg
uncertainty relations. The name coherent state was
applied when these states were rediscovered in the
context of quantum optical radiation by Glauber,
Klauder, and Sudarshan. It was demonstrated that in
these states the correlation functions of the quantum
optical field factorize as they do in classical optics,
so that the optical field has a near-classical behavior,
with the optical beam being coherent. In this article,
we shall refer to these originally studied coh erent
states as canonical coherent states (CCS).
The canonical coherent states, apart from their
use in quantum optics, have also been found to be
extremely useful in computations in atomic and
molecular physics, in quantum statistical mechanics,
and in certain areas of mathematics and mathema-
tical physics, including harmonic analysis, symplec-
tic geometry, and quantization theory. Their wide
applicability has prompted the search for other
families of states sharing similar mathematical and
physical properties. These other families of states are
usually called generalized coherent states, even when
there is no link to optical coherence in such studies.
Some Properties of CCS
In addition to the minimal uncertainty property, the
canonical coherent states have a number of analytical
and group-theoretical properties which are taken as
starting points in looking for generalizations. We
now define the canonical coherent states mathemati-
cally and enumerate a few of these properties.
Suppose that the vectors j0i, j1i, ..., jni, ..., cor-
respond to quantum states of 0, 1, ..., n, ..., exci-
tons, respectively. The Hilbert space of these states,
in which they form an orthonormal basis, is often
known as Fock space. The canonical coherent states
are then defined in terms of this basis, for each
complex number z, by the analytic expansion:
jzi¼e
jzj
2
=2
X
1
n¼0
z
n
ffiffiffiffi
n!
p
jni½1
The states jzi are normalized to unity: hzjzi= 1.
They satisfy the formal eigenvalue equation
ajzi¼zjzi½2
where a is the annihilation operator for excitons, which
acts on the basis vectors (Fock states) jni as follows:
ajni¼
ffiffiffi
n
p
jn 1i½3
Its adjoint a
y
has the action
a
y
jni¼
ffiffiffiffiffiffiffiffiffiffiffiffi
n þ 1
p
jn þ 1i½4
and
½a; a
y
¼aa
y
a
y
a ¼ I ½5
I being the identity operator on Fock space.
Introducing the self-adjoint operators Q and P,of
position and momentum, respectively,
Q ¼
a þ a
y
ffiffiffi
2
p
; P ¼
a a
y
i
ffiffiffi
2
p
½6
it is possible to demonstrate the minimal uncertainty
property referred to above (we take h = 1):
hQihPi¼
1
2
½7
where for any observable A,
hAi¼ hzjA
2
jzihzjAjzi
2
hi
1=2
is its dispersion in the state jzi.
Coherent States 537
One can also prove the resolution of the identity,
Z
C
jzihzj
dq dp
2
¼ I ½ 8
where z = (1=
ffiffiffi
2
p
)(q ip) has been written in terms
of its real and imaginary parts (1=
ffiffiffi
2
p
)q and
(1=
ffiffiffi
2
p
)p, respectively. The above operator integral
is to be understood in the weak sense, as will be
explained later. Equation [8] incorporates the
mathematical fact that the set of vectors jzi is
overcomplete in the Hilbert space. Indeed, using [8]
any vector ji in the Hilbert space can be written as
a linear (integral) superposition of these states:
ji¼
Z
C
ðzÞjzi
dq dp
2
where is the component function, (z) = hjzi.
Thus, the coherent states jzi form a continuously
labeled total set of vectors in the Hilbert space and
since this space is separable, they are an over-
complete set.
Analytic properties of the vectors jz i emerge when
the scalar product hjzi is taken with respect to an
arbitrary vector ji in Fock space. From [1] it is
clear that
FðzÞ¼hj zi¼e
jzj
2
=2
f ðzÞ
where f is an entire analytic function in the complex
variable z. Moreover, the mapping 7!f is an
isometric embedding of the Fock space onto the
Hilbert space of analytic functions, with respect to
the norm
kf k¼
Z
C
jf ðzÞj
2
dðz; zÞ
1=2
½9
defined by the measure d(z,
z) = (1=2)e
jzj
2
dq dp.
Group-theoretical properties of the CCS can be
demonstrated by noting that
jni¼
ða
y
Þ
n
ffiffiffiffi
n!
p
j0i and aj0i¼0
using which [1] can be recast into the form
jzi¼e
jzj
2
=2
e
za
y
j0i¼UðzÞj0i
Uðz Þ¼e
za
y
za
½10
The vectors jzi and the unitary operator U(z)canbe
reexpressed in terms of the real variables q, p and the
operators Q, P as
jzi¼jq; pi¼Uðq; pÞj0i
Uðq; pÞ¼e
iðpQqPÞ
½11
The operators U(q, p) realize a (projective) unitary,
irreducible representation of the Weyl–Heisenberg
group, which is the group whose Lie algebra has the
generators Q, P, and I, obeying the commutation
relations [Q, P] = iI. The existence of the resolution
of the identity [8] is the statement of the fact that
this representation is square integrable (a notion
which will be elaborated upon in the section ‘‘Some
examples’’) which gives us the next paradigm for
building coherent states, namely by the action, on a
fixed vector, of the unitary operators of a square-
integrable representation of a locally compact
group.
The above range of properties, which are enjoyed
by the CCS, cannot all be expected to hold when
looking for generalizations. It then becomes neces-
sary to adopt one or other of these properties as the
starting point and to proceed from there. In so
doing, it is best first to set down a general definition
of coherent states, involving a minimal mathema-
tical structure. Motivated more by possible applica-
tions to physics, we do this in the following section.
General Definition
Let H be an abstract, separable Hilbert space over
the complexes, X a locally compact space and d a
measure on X. Let jx, ii be a family of vectors in H ,
defined for each x in X and i = 1, 2, 3, ..., N, where
N is usually a finite integer, although it could also
be infinite. We assume that this set of vectors
possesses the following properties:
1. For each i, the mapping x 7!jx, ii is weakly
continuous, that is, for each vector ji in H , the
function
i
(x) = hx, iji is continuous (in the
topology of X).
2. For each x in X, the vectors jx, ii, i = 1, 2, ..., N,
are linearly independent.
3. The resolution of the identity
X
N
i¼1
Z
X
jx; iihx; ijdðxÞ¼I
H
½12
holds in the weak sense on the Hilbert space H ,
that is, for any two vectors ji,j i in H , the
following equality holds:
X
N
i¼1
Z
X
hjx; iihx; ij idðxÞ¼hj i
A set of vectors jx, ii satisfying the above three
properties is called a family of generalized vector
coherent states. In case N = 1, the set is called a family
of generalized coherent states. Sometimes the resolu-
tion of the identity condition is replaced by a weaker
538 Coherent States
condition, with the vectors jx, ii simply forming a total
set in H and the functions F
i
(x) = hx, iji,asji runs
through H , forming a reproducing kernel Hilbert
space. Alternatively, the identity on the right-hand
side of [12] could also be replaced by a bounded,
positive operator T with bounded inverse. In this case,
the term frame is also used for the family of general-
ized coherent states. For physical applications, how-
ever, the resolution of the identity condition is always
assumed to hold, although the measure d could be of
a very general nature (possibly also singular). The
objective in all these cases is to ensure that an arbitrary
vector ji be expressible as a linear (integral)
combination of these vectors. Indeed, [12] is immedi-
ately seen to imply that
ji¼
X
N
i¼1
Z
X
i
ðxÞjx; iidðxÞ½13
where
i
(x) = hx, iji.
Associated to a family of generalized coherent
states on a Hilbert space H , there is an intrinsic
isomorphism between this space and a Hilbert space
of (in general, vector valued) continuous functions
over X. Using this isomorphism, it is always possible
to look upon coherent states as a family of
continuous functions which are square integrable
with respect to the measure d. To demonstrate this,
we note that, in view of [12], for each vector ji in
H , the vector-valued function Y(x)onx, with
components
i
(x) = hx, iji, i = 1, 2, ..., N, satisfies
the norm condition
X
N
i¼1
Z
X
j
i
ðxÞj
2
dðxÞ¼kk
2
H
This means that the set of vectors Y,asji runs
through H , is a closed subspace of the Hilbert space
L
2
C
N
(X,d)ofN-vector-valued functions on x. Let us
denote this subspace by H
K
and note that this space
is a reproducing kernel Hilbert space with a matrix-
valued kernel K(x, y) having matrix elements
Kðx; yÞ
ij
¼hx; ijy; ji; i; j ¼ 1; 2; ...; N ½14
and enjoying the properties
Kðx; yÞ
ij
¼ Kðy; xÞ
ji
; Kðx; xÞ
ii
> 0 ½15
and
X
N
‘¼1
Z
X
Kðx; zÞ
i‘
Kðz; yÞ
‘j
dðzÞ¼Kðx; yÞ
ij
½16
If e
i
, i = 1, 2, ..., N, are the vectors constituting the
canonical basis of C
N
, then for each x in X and
i = 1, 2, ..., N, the vector-valued function x
i
x
on X,
defined by x
i
x
(y) = K(y, x)e
i
, is the image in H
K
of
the generalized vector coherent state jx, ii, under the
above-mentioned isometry. The vectors x
i
x
span
the space H
K
and for an arbitrary element Y of this
Hilbert space, the reproducing property [16] of the
kernel implies the relation
Z
X
Kðx; yÞYðyÞdðyÞ¼YðxÞ½17
Conversely, given any reproducing kernel Hilbert
space, with a kernel satisfying the relations [15] and
[16], generalized coherent states can be constructed
as above in terms of this kernel. Mathematically,
therefore, generalized coherent states are just the set
of vectors naturally defined by the kernel in a
reproducing kernel Hilbert space.
Some Examples
We present in this section some of the more
commonly used types of coherent states, as illustra-
tions of the general structure given above.
A large class of generalizations of the canonical
coherent states [1] is obtained by a simple modifica-
tion of their analytic structure. Let x
1
x
2
x
n
be an infinite sequence of positive numbers
(x
1
6¼ 0). Define x
n
!=x
1
x
2
x
n
and by convention
set x
0
!=1. In the same Fock space in which the CCS
were described, we now define the related deformed
or nonlinear coherent states via the analytic
expansion
jzi¼Nðjzj
2
Þ
1=2
X
1
n¼0
z
n
ffiffiffiffiffiffiffi
x
n
!
p
jni½18
The normalization factor N(jzj
2
) is chosen so that
hz jzi= 1. These generalized coherent states are
overcomplete in the Fock space and satisfy a
resolution of the identity of the type
Z
D
jzihzjNðjzj
2
Þdðz; zÞ¼I ½19
D being an open disk in the complex plane of radius
L, the radius of convergence of the series
P
1
n = 0
(z
n
=
ffiffiffiffiffiffiffi
x
n
!
p
). (In the case of the CCS, L = 1.)
The measure d is generically of the form d d(r)
(for z = re
i
), where d is related to the x
n
! through
the moment condition
x
n
!
2
¼
Z
L
0
r
2n
dðrÞ; n ¼ 0; 1; 2; ... ½20
This means that once the quantities x
n
! are specified,
the measure d is to be determined by solving the
Coherent States 539
moment problem [20], which of course may not
always have a solution. This puts a constraint on the
type of sequences {x
n
} which may be used in the
construction.
Once again, we see that for an arbitrary vector ji
in the Fock space, the function F(z) = h jzi, of the
complex variable z, is of the form F(z) =
N(jzj
2
)
1=2
f (z), where f is an analytic function on
the domain D. The reproducing kernel associated to
these coherent states is
Kð
z; z
0
Þ¼hzjz
0
i
¼Nðjzj
2
ÞNðjz
0
j
2
Þ
hi
1=2
X
1
n¼0
ðzz
0
Þ
n
x
n
!
½21
By analogy with [2], one can define a generalized
annihilation operator A by its action on the vectors jzi,
Ajzi¼zjzi½22
and its adjoint operator A
y
. These act on the Fock
states jni as follows:
Ajni¼
ffiffiffiffiffi
x
n
p
jn 1i
A
y
jni¼
ffiffiffiffiffiffiffiffiffiffi
x
nþ1
p
jn þ 1i
½23
Depending on the exact values of the quantities x
n
,
these two operators, together with the identity I and
all their commutators, could generate a wide range
of algebras including various deformed quantum
algebras. The term nonlinear, as often applied to
these generalized coherent states, comes again from
quantum optics, where many such families of states
are used in studying the interaction between the
radiation field and atoms, and the strength of the
interaction itself depends on the frequency of
radiation. Of course, these coherent states will not
in general have either the group-theoretical or the
minimal uncertainty properties of the CCS.
The following is an example of generalized
coherent states of the above type, built over the
unit disk, D= {z 2 C jjzj < 1}: on the Fock space,
we define the states
jzi¼ð1 r
2
Þ
X
1
n¼0
ð2Þ
n
n!
1=2
z
n
jni r ¼jzj½24
where = 1, 3=2, 2, 5=2, ...,and
ðaÞ
m
¼
ða þ mÞ
ðaÞ
¼ aða þ 1Þða þ 2Þða þ m 1Þ
Comparing [24] with [18] we see that x
n
= n=(2 þ
n 1) so that lim
n !1
x
n
= 1. Thus, the infinite sum
is convergent for any z lying in the unit disk. These
generalized coherent states arise from representa-
tions of the group SU(1, 1) belonging to the discrete
series, each irreducible representation being labeled
by a specific value of the index . The associated
Hilbert space of functions, analytic on the unit disk,
is a subspace of L
2
(D,d
), with
d
ðz; zÞ¼ð2 1Þ
ð1 r
2
Þ
22
r dr d
z ¼ re
i
which can be obtained by solving the moment
problem [20]. The resolution of the identity satisfied
by these states is
2 1
Z
D
jzihzj
r dr d
ð1 r
2
Þ
2
¼ I ½25
The associated generalized creation and annihilation
operators are
Ajni¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n
2 þ n 1
r
jn 1i
A
y
jni¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n þ 1
2 þ n
r
jn þ 1i
½26
so that, clearly, [A, A
y
] 6¼ I.
Operators A and A
y
of the general type defined in
[23] are also known as ladder operators. When such
operators appear as generators of representations of
Lie algebras, their eigenvectors (see [22]) are usually
called Barut–Girardello coherent states. As an example,
the representation of the Lie algebra of SU(1,1) on the
Fock space is generated by the three operators K
þ
, K
,
and K
3
, which satisfy the commutation relations
½K
3
; K
¼K
; ½K
; K
þ
¼2K
3
½27
They act on the vectors jni as follows:
K
jni¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
nð2 þ n 1Þ
p
jn 1i
K
þ
¼ K
y
K
3
jni¼ð þ nÞjni
½28
Thus, K
j0i= 0 and
jni¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n!ð2Þ
n
p
K
n
þ
j0i
The Barut–Girardello coherent states jzi are now
defined as the formal eigenvectors of the ladder
operator K
:
K
jzi¼zjzi; z 2 C ½29
They have the analytic form
jzi¼
jzj
21
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
I
21
ð2jzjÞ
p
X
1
n¼0
z
n
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n!ð2 þ n 1Þ!
p
jni½30
540 Coherent States
where I
(x) is the order- modified Bessel function
of the first kind. These coherent states satisfy the
resolution of the identity,
2
Z
C
jzihzjK
21
ð2rÞI
21
ð2rÞr dr d ¼ I
z ¼ re
i
½31
where again, K
(x) is the order- modified Bessel
function of the second kind.
A nonanalytic extension of the expression [18] is
often used to define generalized coherent states
associated to physical Hamiltonians having pure
point spectra. These coherent states, known as
Gazeau–Klauder coherent states, are labeled by
action–angle variables. Suppose that we are given
the physical Hamiltonian H =
P
1
n = 0
E
n
jnihnj, with
E
0
= 0, that is, it has the energy eigenvalues E
n
and
eigenvectors jni, which we assume to form an
orthonormal basis for the Hilbert space of states H .
Let us write the eigenvalues as E
n
= !
n
by introdu-
cing a sequence of dimensionless quantities {
n
}
ordered as: 0 =
0
<
1
<
2
< . Then, for all J 0
and 2 R, the Gazeau–Klauder coherent states are
defined as
jJ;i¼NðJÞ
1=2
X
1
k¼0
J
n=2
e
i
n
ffiffiffiffiffiffi
n
!
p
jni½32
where again N is a normalization factor, which
turns out to be dependent on J only. These coherent
states satisfy the temporal stability condition
e
iHt
jJ;i¼jJ;þ !t i½33
and the action identity
hJ;jHjJ;i
H
¼ !J ½34
While these generalized coherent states do form an
overcomplete set in H , the resolution of the identity
is generally not given by an integral relation of the
type [12].
For the second set of examples of generalized
coherent states, we take the group-theoretical structure
of the CCS as the point of departure. Let G be a
locally compact group and suppose that it has a
continuous, irreducible representation on a Hilbert
space H by unitary operators U(g), g 2 G.This
representation is called square integrable if there exists
a nonzero vector j i in H for which the integral
cð Þ¼
Z
G
jh jUðgÞ ij
2
dðgÞ½35
converges. Here d is a Haar measure of G, which
for definiteness, we take to be the left-invariant
measure. (The value of the above integral is
independent of whether the left- or the right-invariant
measure is used, so we could just as well have used
the right-invariant measure.) A vector j i,satisfying
[35], is said to be admissible, and it can be shown
that the existence of one such vector guarantees the
existence of an entire dense set of such vectors in H .
Moreover, if the group G is unimodular, that is, if the
left- and the right-invariant measures coincide, then
the existence of one admissible vector implies that
every vector in H is admissible. Given a square-
integrable representation and an admissible vector
j i, let us define the vectors
jgi¼
1
ffiffiffiffiffiffiffiffiffi
cð Þ
p
UðgÞj i½36
for all g in the group G. These vectors are to be seen
as the analogs of the canonical coherent states [11],
written there in terms of the representation of the
Weyl–Heisenberg group. Next, it can be shown that
the resolution of the identity
Z
G
jgihgjdðgÞ¼I
H
½37
holds on H . Thus, the vectors jgi constitute a family
of generalized coherent states. The functions
F(g) = hgji for all vectors ji in H are square
integrable with respect to the measure d and the
set of such functions, which in fact are continuous in
the topology of G, forms a closed subspace of
L
2
(G,d). Furthermore, the mapping 7!F is a
linear isometry between H and L
2
(G,d) and under
this isometry the representation U gets mapped to a
subrepresentation of the left regular representation
of G on L
2
(G,d).
A typical example of the above construction is
provided by the affine group, G
Aff
. This is the group
of all 2 2 matrices of the type
g ¼
ab
01
½38
a and b being real numbers with a 6¼ 0. We shall
also write g = (b, a). This group is nonunimodular,
with the left-invariant measure being given by
d(b, a) = (1=a
2
)db da. (The right-invariant measure
is (1=a)db da.) The affine group has a unitary
irreducible representation on the Hilbert space
L
2
(R, dx). Vectors in L
2
(R,dx) are measurable
functions (x) of the real variable x and the
(unitary) operators U(b, a) of this representation
act on them in the manner
ðUðb; aÞÞðxÞ¼
1
ffiffiffiffiffiffi
jaj
p
x b
a
½39
Coherent States 541
If is a function in L
2
(R, dx) such that its Fourier
transform
b
satisfies the condition
Z
R
j
b
ðkÞj
2
jkj
dk < 1½40
then it can be shown to be an admissible vector, that is,
cð Þ¼
Z
G
Aff
jh jUðb; aÞ ij
2
db da
a
2
< 1
Thus, following the general construction outlined
above, the vectors
jb; ai¼
1
ffiffiffiffiffiffiffiffiffi
cð Þ
p
Uðb; aÞ ; ðb; aÞ2G
Aff
½41
define a family of generalized coherent states and
one has the resolution of the identity
Z
G
Aff
jb; aih b; aj
db da
a
2
¼ I ½42
on L
2
(R,dx).
In the signal-analysis literature a vector satisfying
the admissibility condition [40] is called a mother
wavelet and the generalized coherent states [41] are
called wavelets. Signals are then identified with
vectors ji in L
2
(R,dx) and the function
Fðb; aÞ¼hb; aji½43
is called the continuous wavelet transform of the
signal .
There exist alternative ways of constructing
generalized coherent states using group representa-
tions. For example, the Perelomov method is based
on the observation that the vector j0i, appearing in
the construction of the canonical coherent states in
[10] and [11] using the representation of the Weyl–
Heisenberg group, is invariant up to a phase, under
the action of its center. Consequently, the coherent
states jzi, as written in [10], are labeled, not by
elements of the group itself, but only by the points in
the quotient space of the group by its (central) phase
subgroup. Generally, let G be a locally compact
group and U a unitary irreducible representation of
it on the Hilbert space H . We do not assume U to be
square integrable. We fix a vector j i in H , of unit
norm and denote by H the subgroup of G consisting
of all elements h for which
UðhÞj i¼e
i!ðhÞ
j i½44
where ! is a real-valued function of h. Let X = G=H
be the left-coset space and x an arbitrary element in X.
Choosing a coset representative g(x) 2 G , for each
coset x, we define the vectors
jxi¼Uðgð xÞÞj i½45
in H . The dependence of these vectors on the specific
choice of the coset representative g(x), is only
through a phase. Thus, if instead of g(x) we took a
different representative g(x)
0
2 G for the same coset
x, then since g(x)
0
= g(x)h for some h 2 H, in view of
[44] we would have U(g(x)
0
)j i= e
i!(h)
jxi. Hence,
quantum mechanically, both j xi and U(g(x)
0
)j i
represent the same physical state and in particular,
the projection operator jxihxj depends only on the
coset. Vectors jxi, defined in this manner, are called
Gilmore–Perelomov coherent states. Since U is
assumed to be irreducible, the set of all these vectors
as x runs through G=H is dense in H . In this
definition of generalized coherent states, no resolu-
tion of the identity is postulated. However, if X
carries an invariant measure, under the natural
action of G, and if the formal operator B defined as
B ¼
Z
X
jxihxjdð xÞ
is bounded, then it is necessarily a multiple of the
identity and a resolution of the identity is again
retrieved.
The Perelomov construction can be used to define
coherent states for any locally compact group. On
the other hand, there exist other constructions of
generalized coherent states, using group representa-
tions, which generalize the notion of square integr-
ability to homogeneous spaces of the group. Briefly,
in this approach one starts with a unitary irreducible
representation U and attempts to find a vector j i,a
subgroup H and a section : G=H ! G such that
Z
G=H
jxihxjdðxÞ¼T ½46
where jxi= U((x))j i, T is a bounded, positive
operator with bounded inverse and d is a quasi-
invariant measure on X = G=H. It is not assumed
that j i be invariant up to a phase under the action
of H and clearly, the best situation is when T is a
multiple of the identity. Although somewhat techni-
cal, this general construction is of enormous
versatility for semidirect product groups of the type
R
n
o K, where K is a closed subgroup of GL(n, R).
Thus, it is useful for many physically important
groups, such as the Poincare´ or the Euclidean group,
which do not have square-integrable representations
in the sense of the earlier definition (see eqn [35]).
The integral condition [46] ensures that any vector
ji in H can be written in terms of the jxi. Indeed, it
542 Coherent States
is easy to see that one has the integral representation
of a vector,
ji¼
Z
X
ðxÞjxidðxÞ
ðxÞ¼hxjT
1
i
in terms of the generalized coherent states.
The canonical coherent states satisfy the minimal
uncertainty relation [7]. It is possible to build
families of coherent states by generalizing from this
condition. To do this, one typically starts with two
self-adjoint generators in the Lie algebra of a
particular group representation and then looks for
appropriate eigenvectors of a complex combination
of these two generators. For two self-adjoint
operators B and C on a Hilbert space H , satisfying
the commutation relation [B, C] = iD and any
normalized vector in H , one can prove the
Heisenberg uncertainty relation
ðBÞ
2
ðCÞ
2
hDi
2
4
½47
where hXi= hjXi and ð XÞ
2
= hX
2
ihXi
2
, for
any operator X on H . More generally, one can prove
the Schro¨ dinger–Robertson uncertainty relation
ðBÞ
2
ðCÞ
2
1
4
hDi
2
þhFi
2
hi
½48
where hFi= hBC þ CBi2hBihCi measures the
correlation between B and C in the state .
If hFi= 0, the above relation reduces to the
Heisenberg uncertainty relation. On the other
hand, if hDi= 0, the Heisenberg uncertainty rela-
tions become redundant. Suppose now that B and
C are two self-adjoint elements of the Lie algebra in
the unitary irreducible representation of a Lie group
and we look for states ji which minimize the
uncertainty relation [48], that is, for which
the equality holds. It turns out that such states
can be found by considering the linear combination
B þ iC, for a fixed complex number , and solving
the formal eigenvalue equation
B þ iC½jz;i¼zjz;i
with z ¼hBiþihCi
½49
Solutions to this equation for which jj= 1 are
called squeezed states, since in this case B 6¼
C.
Generally, the states jz, i are known as intelligent
states. As an example, for the operators Q and P in
[6], for which one has
ðQÞ
2
ðPÞ
2
1
4
1 þhFi
2
hi
taking the combination Q þ iP, one obtains the
minimal uncertainty states,
jz;i¼Nðz;Þ
1=2
e
wða
y
Þ
2
=2
e
ðz=
ffiffi
2
p
Þð1þwÞa
y
j0i½50
N(z, ) being a normalization constant and
w = (1 )=(1 þ ). The case = 1 does not lead
to any solutions, while = 1 gives the canonical
coherent states [10]. For real 6¼ 1 the above states
are the well-known squeezed states of quantum
optics.
Our final example is that of a family of vector
coherent states, which will be obtained essentially
by replacing the complex variable z in [18] by a
matrix variable. We choose the domain =C
22
(all 2 2 complex matrices), equipped with the
measure
dðZ ; Z
y
Þ¼
e
tr½ZZ
y
4
Y
2
k;j¼1
dx
kj
^ dy
kj
where Z is an element of and z
kj
= x
kj
þ iy
kj
are its
entries. One can then prove the matrix orthogon-
ality relation
Z
Z
k
Z
y‘
dðZ ; Z
y
Þ
¼
1
2
Z
tr½Z
k
Z
y‘
dðZ ; Z
y
ÞI
2
¼ bðkÞI
2
; k;‘¼ 0; 1; 2; ...; 1½51
I
2
being the 2 2 identity matrix and
bðkÞ¼
ðk þ 3Þ!
2ðk þ 1Þðk þ 2Þ
k ¼ 1; 2; 3; ...; bð0Þ¼1
½52
Consider the Hilbert space
~
H = L
2
C
2
(,d) of square
integrable, two-component vector-valued functions
on and in it consider the vectors jY
i
k
i, i = 1, 2,
k = 0, 1, 2, ...,1 , defined by the C
2
-valued
functions,
Y
i
k
ðZ
y
Þ¼
1
ffiffiffiffiffiffiffiffiffiffi
bðkÞ
p
Z
yk
i
½53
where the vectors
i
, i = 1, 2, form an orthonormal
basis of C
2
. By virtue of [51], the vectors jY
i
k
i
constitute an orthonormal set in
~
H , that is,
hY
i
k
jY
j
‘
i
~
H
¼
k‘
ij
Denote by H
K
the Hilbert subspace of
~
H generated
by this set of vectors. This can be shown to be a
reproducing kernel Hilbert space of analytic
Coherent States 543
functions in the variable Z
y
, with the matrix valued
kernel K : 7! C
22
:
KðZ
0y
; Z Þ¼
X
2
i¼1
X
1
k¼0
Y
i
k
ðZ
0y
ÞY
i
k
ðZ
y
Þ
y
¼
X
2
i¼1
X
1
k¼0
Z
0yk
Z
k
bðkÞ
½54
Vector coherent states in H
K
are then naturally
associated to this kernel and are given by
jZ ; ii¼
X
2
j¼1
X
1
k¼0
jy
Z
k
i
ffiffiffiffiffiffiffiffiffiffi
bðkÞ
p
jY
j
k
i
that is; jZ ; iiðZ
0y
Þ¼KðZ
0y
; Z Þ
i
½55
for i = 1, 2 and all Z in . They satisfy the resolution
of the identity
X
2
i¼1
Z
jZ ; iihZ ; ijdðZ ; Z
y
Þ¼I
H
K
½56
The expression for the jZ , ii in [55], involving the
sum, should be compared to [18], of which it is a
direct analog.
Some Applications of Coherent States
Generalized coherent states have many applications
in physics, signal analysis, and mathematics, of
which we mention a few here. As an example of
an application of deformed coherent states, we take
x
n
¼
q
n
q
n
q q
1
1=2
; q > 0 ½57
in the definition of these states in [18]. It is then easy
to see that the operators A and A
y
, defined in [23],
satisfy the q-deformed commutation relation
AA
y
qA
y
A ¼ q
N
½58
where N is the usual number operator, which acts
on the Fock states as Njni= njni. Clearly, in the
limit as q !1, these q-deformed coherent states go
over to the canonical coherent states, with the
operators A and A
y
becoming the usual creation
and annihilation operators a and a
y
, respectively.
The operators A and A
y
and the commutation
relation [58] describe a system of q-deformed
oscillators, which have been used to describe, for
example, the vibrations of polyatomic molecules.
The potential energy between the atoms of such
a molecule has anharmonic terms, leading to
a deformation of the usual oscillator algebra,
generated by the operators a and a
y
.
As already mentioned, generalized coherent states
are widely used in signal analysis. The wavelet
transform F(b, a) = hb, aji, introduced in [43],isa
time–frequency transform, in which the parameter b
is identified with time and 1/a with frequency.
Wavelet transforms are used extensively to analyze,
encode, and reconstruct signals arising in many
different branches of physics, engineering, seismo-
graphy, electronic data processing, etc. Similarly, the
canonical coherent states, as written in [11], give
rise to the transform F(q, p) = hq, p ji. Again, if q is
interpreted as time and p as frequency, then this is
just the windowed Fourier transform, also used
extensively in signal processing. More general
wavelets, from higher-dimensional affine groups,
are used to analyze higher-dimensional signals,
while wavelet like transforms from other groups
have been used to study signals exhibiting different
geometries. In particular, wavelet transforms from
spherical geometries have been applied to the study
of brain signals and to astrophysical data.
Our final example is taken from quantization
theory. A quantization technique is a method for
performing the transition from a given classical
mechanical system to its quantum counterpart.
Many methods have been developed to accomplish
this and the use of coherent states is one of them.
Suppose that we are given a family of coherent
states jxi in a Hilbert space H , where the set X from
which x is taken is a classical phase space. This
means that X is a symplectic manifold with an
associated 2-form !, which defines a Poisson
bracket on the set of observables of the classical
system, which are real-valued functions on X. There
is a natural measure d!, defined on X by the 2-form
!. Let us assume that the coherent states jxi satisfy a
resolution on the identity with respect to this
measure:
Z
X
jxihxjd!ðxÞ¼I
H
In this case, the coherent states may be used to
quantize the observables of the classical system in
the following way: let f be a real-valued function on
X, representing a classical observable and suppose
that the formal operator
b
f ¼
Z
X
f ðxÞjxihx jd!ðxÞ½59
is well defined as a self-adjoint operator on H . Then
we may take the operator
b
f to be the quantized
observable corresponding to the classical observable
f. Suppose that we have two such operators,
b
f and
b
g,
544 Coherent States