Coherent states in mathematical physics
Encyclopedia
Coherent states have been introduced in a physical context, first as quasi-classical states in quantum mechanics, then as the backbone of quantum optics and they are described in that spirit in the article
Coherent states (see also ).
However, they have generated a huge variety of generalizations, which have led to a tremendous literature in mathematical physics.
In this article, we will sketch the main directions of research on this line. For further details, we refer to several existing surveys
.
be a complex, separable Hilbert space, 
a locally compact space and 
a measure on 
. Let 
be a vector in 
,
defined for each
in 
. Assume that this set of vectors possesses the following properties:
holds in the weak sense on the Hilbert space
, i.e., for any two vectors 
in 
, the following equality holds:
A set of vectors
satisfying the two properties above is called a family of generalized coherent states.
In order to recover the previous definition (given in the article Coherent state
) of canonical or standard coherent states (CCS), it suffices to take
, the complex plane, 
and 
Sometimes the resolution of the identity condition is replaced by a weaker condition, with the vectors
simply forming a total set in 
and the functions 
, as
runs through 
, forming a reproducing kernel Hilbert space
.
The objective in both cases is to ensure that an arbitrary vector
be expressible as a linear (integral) combination of these vectors. Indeed, the resolution of the identity immediately implies that
where
.
These vectors
are square integrable, continuous functions on 
and satisfy the reproducing property
where
is the reproducing kernel, that enjoys the properties
obtained by a simple modification of their analytic structure. Let
be an infinite sequence of positive numbers (
). Define
and by convention set 
. In the same Fock space
in which the CCS were described, we now define the
related deformed or nonlinear coherent states by the expansion
The normalization factor
is chosen so that
. These generalized coherent states are overcomplete
in the Fock space and satisfy a resolution of the identity
being an open disc in the complex plane of radius 
, the radius of convergence of the series
(in the case of the CCS,
.)
The measure
is generically of the form 
(for 
), where 
is related to the
through the moment condition.
Once again, we see that for an arbitrary vector
in the Fock space, the
function
is of the form
, where 
is an analytic
function on the domain
. The reproducing kernel associated to these coherent states is
annihilation operator
by its action on the vectors 
,
and its adjoint operator
. These act on the Fock states 
as
Depending on the exact values of the quantities
, these two operators, together
with the identity
and all their commutators, could generate a wide range of algebras
including various types of deformed quantum algebra
s. 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,
where 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 (there might have more general ones).
Operators
and 
of the general type defined above
are also known as ladder operator
s . When such operators appear as generators of
representations of Lie algebras, the eigenvectors of
are usually called Barut–Girardello coherent states.
A typical example is obtained from the representations of the Lie algebra
of SU(1,1) on the Fock space
.
coherent states associated to physical Hamiltonian
s having pure point spectra.
These coherent states, known as Gazeau-Klauder coherent states, are labelled by action-angle variables.
Suppose that we are given the physical Hamiltonian
, with 
, i.e., it has the energy eigenvalues
and eigenvectors 
, which we assume to form an orthonormal basis for the
Hilbert space of states
. Let us write the eigenvalues as
by introducing a sequence of dimensionless
quantities
ordered as:
. Then, for all 
and
, the Gazeau–Klauder coherent states are defined as
where again
is a normalization factor, which turns out to be dependent on 
only.
These coherent states satisfy the temporal stability condition,
and the action identity,
While these generalized coherent states do form an overcomplete set in
, the
resolution of the identity is generally not given by an integral relation as above, but instead by an integral in Bohr's sense,
like it is in use in the theory of almost periodic functions.
Actually the construction of Gazeau–Klauder CS can be extended to vector CS and to Hamiltonians with degenerate spectra, as shown by Ali and Bagarello.
In order to see this, let us go back for a while to the case of CCS.
There, indeed, the displacement operator
is nothing but the representative in Fock space
of an element of the Heisenberg group (also called the Weyl–Heisenberg group),
whose Lie algebra
is generated by
and 
.
However, before going on with the CCS, take first the general case.
Let
be a locally compact group and suppose that it has a continuous, irreducible representation
on a Hilbert
space
by unitary operators 
. This representation is called
square integrable if there exists a non-zero vector
in 
for which the integral
converges. Here
is the left invariant Haar measure
on
.
A vector
for which 
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
. Moreover,
if the group
is unimodular
, i.e., if the left and the right invariant measures
coincide, then the existence of one admissible vector implies that every vector in
is admissible. Given a square integrable representation 
and an admissible vector
, let us define the vectors
These vectors are the analogues of the canonical coherent states, written there in terms of the
representation of the Heisenberg group (however, see the section on Gilmore-Perelomov CS, below).
Next, it can be shown that the resolution of the identity
holds on
. Thus, the vectors 
constitute a family of generalized
coherent states. The functions
for all vectors
in 
are square integrable with respect to the measure
and the set of such functions, which in fact are continuous in the topology
of
, forms a closed subspace of 
. Furthermore, the mapping
is a linear isometry between 
and 
and
under this isometry the representation $U$ gets mapped to a subrepresentation of the
left regular representation
of
on 
.
of the line,
. This is the group of all 2
2 matrices of the type,
and 
being real numbers with 
. We shall also write
, with the action on 
given by 
. This group is non-unimodular, with the left invariant measure being given
by
(the right invariant measure being 
).
The affine group has a unitary irreducible representation on the Hilbert space
.
Vectors in
are measurable functions 
of the real variable
and the (unitary) operators 
of this
representation act on them as
If
is a function in 
such that its Fourier transform
satisfies the (admissibility) condition
then it can be shown to be an admissible vector, i.e.,
Thus, following the general construction outlined above, the vectors
define a family of generalized coherent states and one has the resolution of the identity
on
.
In the signal analysis literature, a vector satisfying the admissibility
condition above is called a mother wavelet and the generalized
coherent states
are called wavelets. Signals are then
identified with vectors
in 
and the function
is called the continuous wavelet transform
of the signal
.
This concept can be extended to two dimensions, the group
being replaced by the so-called similitude group of the plane, which consists of plane translations, rotations and global dilations.
The resulting 2D wavelets, and some generalizations of them,
are widely used in image processing
.
Already it cannot yield the CCS, since these are not indexed by the elements of the Heisenberg group,
but rather by points of the quotient of
the latter by its center, that quotient being precisely
. The key observation is that the center of the Heisenberg group
leaves the vacuum vector
invariant, up to a phase.
Generalizing this idea, Gilmore and Perelomov
consider a locally compact group
and a unitary irreducible
representation
of 
on the Hilbert space 
, not necessarily square integrable. Fix a vector 
in 
, of unit norm, and
denote by
the subgroup of 
consisting of all elements 
that leave it invariant up to a phase, that is,
where
is a real-valued function of 
. Let 
be the left coset space and
an arbitrary element in 
. Choosing a coset representative 
, for each coset 
, we define the vectors
The dependence of these vectors on the specific choice of the coset representative
is only through a phase. Indeed, if instead of 
, we took a different
representative
for the same coset 
, then since 
for some 
, we would have 
.
Hence, quantum mechanically, both
and
$U(g(x)')| \psi\rangle$ represent the same physical state and in particular, the projection operator
depends only on the coset. Vectors 
defined in this way are called
Gilmore–Perelomov coherent states. Since
is assumed to be irreducible, the set of all
these vectors as
runs through 
is dense in 
.
In this definition of generalized coherent states, no resolution of the identity is postulated. However,
if
carries an invariant measure, under the natural action of 
, and if the formal operator 
defined as
is bounded, then it is necessarily a multiple of the identity and a resolution of the identity is again retrieved.
Gilmore–Perelomov coherent states have been generalized to quantum groups, but for this we refer to the literature
.
compact group. On the other hand, particularly in case of failure of the Gilmore–Perelomov construction, there exist other constructions of generalized coherent
states, using group representations, which generalize the notion of square integrability
to homogeneous spaces of the group.
Briefly, in this approach one starts with a unitary
irreducible representation
and attempts to find a vector 
, a
subgroup
and a section
(see Section (fiber bundle)
)
such that
where
, 
is a bounded, positive operator with bounded inverse
and
is a quasi-invariant measure on 
. It is not assumed that
be invariant up to a phase under the action of 
and clearly, the
best situation is when
is a multiple of the identity. Although somewhat technical,
this general construction is of enormous versatility for semi-direct product groups of the
type
, where 
is a closed subgroup of 
.
Thus, it is useful for many physically important groups, such as the
Poincaré group
or the Euclidean group
, which do not have
square integrable representations in the sense of the earlier definition.
In particular, the integral condition defining the operator
ensures that any vector
in 
can be written in terms of the generalized coherent states 
namely,
which is the primary aim of any kind of coherent states.
superpositions of eigenstates of some self-adjoint operator, as was the harmonic oscillator Hamiltonian for the standard CS. It is the essence of quantum mechanics that this superposition
has a probabilistic flavor. As a matter of fact, we notice that the probabilistic structure of the canonical coherent states involves two probability distributions that underlie their construction.
There are, in a sort of duality, a Poisson distribution
ruling the probability of detecting
excitations when the quantum system is in a coherent state 
,
and a gamma distribution on the set
of complex parameters, more exactly on the range 
of the square of the radial variable.
The generalization follows that duality scheme.
Let
be a set of parameters equipped with a measure 
and its associated Hilbert space 
of complex-valued functions, square integrable with respect to 
. Let us choose
in
a finite or countable orthonormal set 
:
In case of infinite countability, this set must obey the (crucial) finiteness condition:
Let
be a separable complex Hilbert space with orthonormal basis
in one-to-one correspondence with the elements of 
.
The two conditions above
imply that the family of normalized coherent states
in 
, which are defined by
resolves the identity in
:
Such a relation allows us to implement a coherent state or frame quantization of the set of parameters
by associating to a function 
that satisfies
appropriate conditions the following operator in
:
The operator
is symmetric if 
is real-valued, and it is self-adjoint (as a quadratic form) if 
is real and semi-bounded. The original 
is an upper symbol, usually non-unique, for the operator 
. It will be called a
classical observable with respect to the family
if the so-called
lower symbol of
, defined as
has mild functional properties to be made precise according to further topological properties granted to the original set
.
A last point of this construction of the space of quantum states concerns its statistical aspects.
There is indeed an interplay between two probability distributions:
(i) For almost each
, a discrete distribution,
This probability could be considered as concerning experiments performed on the system within some experimental protocol, in order to measure the spectral values of a certain self-adjoint operator
, i.e., a quantum observable, acting in 
and having the discrete spectral resolution 
.
(ii) For each
, a continuous distribution on 
,
Here, we observe a Bayesian duality typical of coherent states. There are two interpretations: the resolution of the unity verified by the coherent states
introduces a preferred prior measure on the set 
, which is the set of parameters of the discrete distribution, with this distribution itself playing the role of the likelihood function
. The associated discretely indexed continuous distributions become the related conditional posterior distribution. Hence, a probabilistic approach to experimental observations concerning
should serve as a guideline in choosing the set of the 
's.
We note that the continuous prior distribution will be relevant for the quantization whereas the discrete posterior one characterizes the measurement of the physical spectrum from which is built the coherent superposition of quantum states
.
Coherent states (see also ).
However, they have generated a huge variety of generalizations, which have led to a tremendous literature in mathematical physics.
In this article, we will sketch the main directions of research on this line. For further details, we refer to several existing surveys
.
A general definition
Let be a complex, separable Hilbert space, a locally compact space and a measure on . Let be a vector in ,defined for each
in . Assume that this set of vectors possesses the following properties:
- The mapping
is weakly continuous, i.e., for each vector 
in 
, the function 
is continuous (in the topology of 
). - The resolution of the identity
holds in the weak sense on the Hilbert space
, i.e., for any two vectors in , the following equality holds:A set of vectors
satisfying the two properties above is called a family of generalized coherent states.In order to recover the previous definition (given in the article Coherent state
Coherent state
In quantum mechanics a coherent state is a specific kind of quantum state of the quantum harmonic oscillator whose dynamics most closely resembles the oscillating behaviour of a classical harmonic oscillator...
) of canonical or standard coherent states (CCS), it suffices to take
, the complex plane, and Sometimes the resolution of the identity condition is replaced by a weaker condition, with the vectors
simply forming a total set in and the functions , as runs through , forming a reproducing kernel Hilbert spaceReproducing kernel Hilbert space
In functional analysis , a reproducing kernel Hilbert space is a Hilbert space of functions in which pointwise evaluation is a continuous linear functional. Equivalently, they are spaces that can be defined by reproducing kernels...
.
The objective in both cases is to ensure that an arbitrary vector
be expressible as a linear (integral) combination of these vectors. Indeed, the resolution of the identity immediately implies thatwhere
.These vectors
are square integrable, continuous functions on and satisfy the reproducing propertywhere
is the reproducing kernel, that enjoys the propertiesSome examples
We present in this section some of the more commonly used types of coherent states, as illustrations of the general structure given above.Nonlinear coherent states
A large class of generalizations of the CCS isobtained by a simple modification of their analytic structure. Let
be an infinite sequence of positive numbers (
). Define and by convention set . In the same Fock spaceFock space
The Fock space is an algebraic system used in quantum mechanics to describe quantum states with a variable or unknown number of particles. It is named after V. A...
in which the CCS were described, we now define the
related deformed or nonlinear coherent states by the expansion
The normalization factor
is chosen so that. These generalized coherent states are overcompletein the Fock space and satisfy a resolution of the identity
being an open disc in the complex plane of radius , the radius of convergence of the series(in the case of the CCS,
.)The measure
is generically of the form (for ), where is related to the
through the moment condition.Once again, we see that for an arbitrary vector
in the Fock space, thefunction
is of the form
, where is an analyticAnalytic
Generally speaking, analytic refers to the "having the ability to analyze" or "division into elements or principles."It can also have the following meanings:-Natural sciences:* In chemistry:...
function on the domain
. The reproducing kernel associated to these coherent states isBarut–Girardello coherent states
By analogy with the CCS case, one can define a generalizedannihilation operator
by its action on the vectors ,and its adjoint operator
. These act on the Fock states asDepending on the exact values of the quantities
, these two operators, togetherwith the identity
and all their commutators, could generate a wide range of algebrasincluding various types of deformed quantum algebra
Quantum algebra
Quantum algebra is one of the top-level mathematics categories used by the arXiv.Subjects include:*Quantum groups*Skein theories*Operadic algebra*Diagrammatic algebra*Quantum field theory-External links:**R. Jagannathan,...
s. 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,
where 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 (there might have more general ones).
Operators
and of the general type defined aboveare also known as ladder operator
Ladder operator
In linear algebra , a raising or lowering operator is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising operator is sometimes called the creation operator, and the lowering operator the annihilation operator...
s . When such operators appear as generators of
representations of Lie algebras, the eigenvectors of
are usually called Barut–Girardello coherent states.
A typical example is obtained from the representations of the Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
of SU(1,1) on the Fock space
Fock space
The Fock space is an algebraic system used in quantum mechanics to describe quantum states with a variable or unknown number of particles. It is named after V. A...
.
Gazeau–Klauder coherent states
A non-analytic extension of the above expression of the non-linear coherent states is often used to define generalizedcoherent states associated to physical Hamiltonian
Hamiltonian
Hamiltonian may refer toIn mathematics :* Hamiltonian system* Hamiltonian path, in graph theory** Hamiltonian cycle, a special case of a Hamiltonian path* Hamiltonian group, in group theory* Hamiltonian...
s having pure point spectra.
These coherent states, known as Gazeau-Klauder coherent states, are labelled by action-angle variables.
Suppose that we are given the physical Hamiltonian
, with , i.e., it has the energy eigenvalues and eigenvectors , which we assume to form an orthonormal basis for theHilbert space of states
. Let us write the eigenvalues as by introducing a sequence of dimensionlessquantities
ordered as:. Then, for all and, the Gazeau–Klauder coherent states are defined aswhere again
is a normalization factor, which turns out to be dependent on only.These coherent states satisfy the temporal stability condition,
and the action identity,
While these generalized coherent states do form an overcomplete set in
, theresolution of the identity is generally not given by an integral relation as above, but instead by an integral in Bohr's sense,
like it is in use in the theory of almost periodic functions.
Actually the construction of Gazeau–Klauder CS can be extended to vector CS and to Hamiltonians with degenerate spectra, as shown by Ali and Bagarello.
The group-theoretical approach
Gilmore and Perelomov, independently, realized that the construction of coherent states may sometimes be viewed as a group theoretical problem.In order to see this, let us go back for a while to the case of CCS.
There, indeed, the displacement operator
is nothing but the representative in Fock space
Fock space
The Fock space is an algebraic system used in quantum mechanics to describe quantum states with a variable or unknown number of particles. It is named after V. A...
of an element of the Heisenberg group (also called the Weyl–Heisenberg group),
whose Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
is generated by
and .However, before going on with the CCS, take first the general case.
Let
be a locally compact group and suppose that it has a continuous, irreducible representation on a Hilbertspace
by unitary operators . This representation is calledsquare integrable if there exists a non-zero vector
in for which the integralconverges. Here
is the left invariant Haar measureHaar measure
In mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....
on
.A vector
for which is said to beadmissible, and it can be shown that the existence of one such vector
guarantees the existence of an entire dense set of such vectors in
. Moreover,if the group
is unimodularUnimodular
In mathematics, unimodular may refer to any of the following:* unimodular lattice* unimodular matrix* unimodular polynomial matrix* unimodular form* unimodular group...
, i.e., if the left and the right invariant measures
coincide, then the existence of one admissible vector implies that every vector in
is admissible. Given a square integrable representation and an admissible vector, let us define the vectorsThese vectors are the analogues of the canonical coherent states, written there in terms of the
representation of the Heisenberg group (however, see the section on Gilmore-Perelomov CS, below).
Next, it can be shown that the resolution of the identity
holds on
. Thus, the vectors constitute a family of generalizedcoherent states. The functions
for all vectors in are square integrable with respect to the measure and the set of such functions, which in fact are continuous in the topologyof
, forms a closed subspace of . Furthermore, the mapping is a linear isometry between and andunder this isometry the representation $U$ gets mapped to a subrepresentation of the
left regular representation
Regular representation
In mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation....
of
on .An example: wavelets
A typical example of the above construction is provided by the affine groupAffine group
In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.It is a Lie group if K is the real or complex field or quaternions....
of the line,
. This is the group of all 22 matrices of the type, and being real numbers with . We shall also write, with the action on given by . This group is non-unimodular, with the left invariant measure being givenby
(the right invariant measure being ).The affine group has a unitary irreducible representation on the Hilbert space
.Vectors in
are measurable functions of the real variable
and the (unitary) operators of thisrepresentation act on them as
If
is a function in such that its Fourier transformFourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
satisfies the (admissibility) conditionthen it can be shown to be an admissible vector, i.e.,
Thus, following the general construction outlined above, the vectors
define a family of generalized coherent states and one has the resolution of the identity
on
.In the signal analysis literature, a vector satisfying the admissibility
condition above is called a mother wavelet and the generalized
coherent states
are called wavelets. Signals are thenidentified with vectors
in and the functionis called the continuous wavelet transform
Continuous wavelet transform
A continuous wavelet transform is used to divide a continuous-time function into wavelets. Unlike Fourier transform, the continuous wavelet transform possesses the ability to construct a time-frequency representation of a signal that offers very good time and frequency localization...
of the signal
.This concept can be extended to two dimensions, the group
being replaced by the so-called similitude group of the plane, which consists of plane translations, rotations and global dilations.
The resulting 2D wavelets, and some generalizations of them,
are widely used in image processing
Image processing
In electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as a photograph or video frame; the output of image processing may be either an image or, a set of characteristics or parameters related to the image...
.
Gilmore–Perelomov coherent states
The construction of coherent states using group representations described above is not sufficient.Already it cannot yield the CCS, since these are not indexed by the elements of the Heisenberg group,
but rather by points of the quotient of
the latter by its center, that quotient being precisely
. The key observation is that the center of the Heisenberg groupleaves the vacuum vector
invariant, up to a phase.Generalizing this idea, Gilmore and Perelomov
consider a locally compact group
and a unitary irreduciblerepresentation
of on the Hilbert space , not necessarily square integrable. Fix a vector in , of unit norm, anddenote by
the subgroup of consisting of all elements that leave it invariant up to a phase, that is,where
is a real-valued function of . Let be the left coset space and an arbitrary element in . Choosing a coset representative , for each coset , we define the vectorsThe dependence of these vectors on the specific choice of the coset representative
is only through a phase. Indeed, if instead of , we took a differentrepresentative
for the same coset , then since for some , we would have .Hence, quantum mechanically, both
and$U(g(x)')| \psi\rangle$ represent the same physical state and in particular, the projection operator
depends only on the coset. Vectors defined in this way are calledGilmore–Perelomov coherent states. Since
is assumed to be irreducible, the set of allthese vectors as
runs through is dense in .In this definition of generalized coherent states, no resolution of the identity is postulated. However,
if
carries an invariant measure, under the natural action of , and if the formal operator defined asis bounded, then it is necessarily a multiple of the identity and a resolution of the identity is again retrieved.
Gilmore–Perelomov coherent states have been generalized to quantum groups, but for this we refer to the literature
.
Further generalization: Coherent states on coset spaces
The Perelomov construction can be used to define coherent states for any locallycompact group. On the other hand, particularly in case of failure of the Gilmore–Perelomov construction, there exist other constructions of generalized coherent
states, using group representations, which generalize the notion of square integrability
to homogeneous spaces of the group.
Briefly, in this approach one starts with a unitary
irreducible representation
and attempts to find a vector , asubgroup
and a sectionSection
Section may refer to:* Section * Section * Archaeological section* Histological section, a thin slice of tissue used for microscopic examination* Section, an instrumental group within an orchestra...
(see Section (fiber bundle)
Section (fiber bundle)
In the mathematical field of topology, a section of a fiber bundle π is a continuous right inverse of the function π...
)
such thatwhere
, is a bounded, positive operator with bounded inverseand
is a quasi-invariant measure on . It is not assumed that be invariant up to a phase under the action of and clearly, thebest situation is when
is a multiple of the identity. Although somewhat technical,this general construction is of enormous versatility for semi-direct product groups of the
type
, where is a closed subgroup of .Thus, it is useful for many physically important groups, such as the
Poincaré group
Poincaré group
In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...
or the Euclidean group
Euclidean group
In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space...
, which do not have
square integrable representations in the sense of the earlier definition.
In particular, the integral condition defining the operator
ensures that any vector in can be written in terms of the generalized coherent states namely,which is the primary aim of any kind of coherent states.
Coherent states: a Bayesian construction for the quantization of a measure set
We now depart from the standard situation and present a general method of construction of coherent states, starting from a few observations on the structure of these objects assuperpositions of eigenstates of some self-adjoint operator, as was the harmonic oscillator Hamiltonian for the standard CS. It is the essence of quantum mechanics that this superposition
has a probabilistic flavor. As a matter of fact, we notice that the probabilistic structure of the canonical coherent states involves two probability distributions that underlie their construction.
There are, in a sort of duality, a Poisson distribution
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...
ruling the probability of detecting
excitations when the quantum system is in a coherent state ,and a gamma distribution on the set
of complex parameters, more exactly on the range of the square of the radial variable.The generalization follows that duality scheme.
Let
be a set of parameters equipped with a measure and its associated Hilbert space of complex-valued functions, square integrable with respect to . Let us choosein
a finite or countable orthonormal set :In case of infinite countability, this set must obey the (crucial) finiteness condition:
Let
be a separable complex Hilbert space with orthonormal basis in one-to-one correspondence with the elements of .The two conditions above
imply that the family of normalized coherent states
in , which are defined byresolves the identity in
:Such a relation allows us to implement a coherent state or frame quantization of the set of parameters
by associating to a function that satisfiesappropriate conditions the following operator in
:The operator
is symmetric if is real-valued, and it is self-adjoint (as a quadratic form) if is real and semi-bounded. The original is an upper symbol, usually non-unique, for the operator . It will be called aclassical observable with respect to the family
if the so-calledlower symbol of
, defined ashas mild functional properties to be made precise according to further topological properties granted to the original set
.A last point of this construction of the space of quantum states concerns its statistical aspects.
There is indeed an interplay between two probability distributions:
(i) For almost each
, a discrete distribution,This probability could be considered as concerning experiments performed on the system within some experimental protocol, in order to measure the spectral values of a certain self-adjoint operator
, i.e., a quantum observable, acting in and having the discrete spectral resolution .(ii) For each
, a continuous distribution on ,Here, we observe a Bayesian duality typical of coherent states. There are two interpretations: the resolution of the unity verified by the coherent states
introduces a preferred prior measure on the set , which is the set of parameters of the discrete distribution, with this distribution itself playing the role of the likelihood functionLikelihood function
In statistics, a likelihood function is a function of the parameters of a statistical model, defined as follows: the likelihood of a set of parameter values given some observed outcomes is equal to the probability of those observed outcomes given those parameter values...
. The associated discretely indexed continuous distributions become the related conditional posterior distribution. Hence, a probabilistic approach to experimental observations concerning
should serve as a guideline in choosing the set of the 's.We note that the continuous prior distribution will be relevant for the quantization whereas the discrete posterior one characterizes the measurement of the physical spectrum from which is built the coherent superposition of quantum states
.