However, they have generated a huge variety of generalizations, which have led to a tremendous amount of literature in mathematical physics.
satisfying the two properties above is called a family of generalized coherent states.
We present in this section some of the more commonly used types of coherent states, as illustrations of the general structure given above.
A large class of generalizations of the CCS is obtained by a simple modification of their analytic structure.
In the same Fock space in which the CCS were described, we now define the related deformed or nonlinear coherent states by the expansion
These generalized coherent states are overcomplete in the Fock space and satisfy a resolution of the identity
By analogy with the CCS case, one can define a generalized annihilation operator
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.
[5] A typical example is obtained from the representations of the Lie algebra of SU(1,1) on the Fock space.
[7] 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.[8] 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.
[11] Although there are two different Lie groups involved in the construction, the heat kernel coherent states are not of Perelomov type.
Gilmore and Perelomov, independently, realized that the construction of coherent states may sometimes be viewed as a group theoretical problem.
be a locally compact group and suppose that it has a continuous, irreducible representation
This representation is called square integrable if there exists a non-zero vector
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).
The affine group has a unitary irreducible representation on the Hilbert space
In the signal analysis literature, a vector satisfying the admissibility condition above is called a mother wavelet and the generalized coherent states
The key observation is that the center of the Heisenberg group leaves the vacuum vector
Generalizing this idea, Gilmore and Perelomov[12][13][14][15] consider a locally compact group
Gilmore–Perelomov coherent states have been generalized to quantum groups, but for this we refer to the literature.
[21][22][23][24][25][26] 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.
Although somewhat technical, this general construction is of enormous versatility for semi-direct product groups of the type
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.
As a matter of fact, we notice that the probabilistic structure of the canonical coherent states involves two probability distributions that underlie their construction.
In case of infinite countability, this set must obey the (crucial) finiteness condition:
Such a relation allows us to implement a coherent state or frame quantization of the set of parameters
A last point of this construction of the space of quantum states concerns its statistical aspects.