Coherent states in mathematical physics
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 sketch the main directions of research on this line. For further details, we refer to several existing surveys .
- 1 A general definition
- 2 Some examples
- 3 The group-theoretical approach
- 4 Further generalization: Coherent states on coset spaces
- 5 Coherent states: a Bayesian construction for the quantization of a measure set
- 6 See also
- 7 References
A general definition
Let be a complex, separable Hilbert space, a locally compact space and a measure on . For each in , denote a vector 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) of canonical or standard coherent states (CCS), it suffices to take , the complex plane, [clarification needed] and
Sometimes the resolution of the identity condition is replaced by a weaker condition, with the vectors simply forming a total set[clarification needed] 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
These vectors are square integrable, continuous functions on and satisfy the reproducing property
where is the reproducing kernel, which satisfies the following properties
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 is 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
Barut–Girardello coherent states
By analogy with the CCS case, one can define a generalized 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 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, 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 operators . 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.
Gazeau–Klauder coherent states
A non-analytic extension of the above expression of the non-linear coherent states 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 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.
Heat kernel coherent states
Another type of coherent state arises when considering a particle whose configuration space is the group manifold of a compact Lie group K. Hall introduced coherent states in which the usual Gaussian on Euclidean space is replaced by the heat kernel on K. The parameter space for the coherent states is the "complexification" of K; e.g., if K is SU(n) then the complexification is SL(n,C). These coherent states have a resolution of the identity that leads to a Segal-Bargmann space over the complexification. Hall's results were extended to compact symmetric spaces, including spheres, by Stenzel. The heat kernel coherent states, in the case , have been applied in the theory of quantum gravity by Thiemann and his collaborators. Although there are two different Lie groups involved in the construction, the heat kernel coherent states are not of Perelomov type.
The group-theoretical approach
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 .
An example: wavelets
A typical example of the above construction is provided by the affine group 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 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
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. 
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 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 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.
Further generalization: Coherent states on coset spaces
The Perelomov construction can be used to define coherent states for any locally 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 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.
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 as 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 .
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