Generalized coherent states are developed for SU ( n ) systems for arbitrary n . This is done by first iteratively determining explicit representations for the SU ( n ) coherent states and then determining parametric representations useful for applications. For SU ( n ), the set of coherent states is isomorphic to a coset space SU ( n )/ SU ( n -1), and thus shows the geometrical structure of the coset space. These results provide a convenient (2 n -1)-dimensional space for the description of arbitrary SU ( n ) systems. We further obtain the metric and measure on the coset space and show some properties of the SU ( n ) coherent states.
Generalized coherent states are developed for SU ( n ) systems for arbitrary n . This is done by first iteratively determining explicit representations for the SU ( n ) coherent states and then determining parametric representations useful for applications. For SU ( n ), the set of coherent states is isomorphic to a coset space SU ( n )/ SU ( n -1), and thus shows the geometrical structure of the coset space. These results provide a convenient (2 n -1)-dimensional space for the description of arbitrary SU ( n ) systems. We further obtain the metric and measure on the coset space and show some properties of the SU ( n ) coherent states.
In this review, a general algorithm for constructing coherent states of dynamical groups for a given quantum physical system is presented. The result is that, for a given dynamical group, the coherent states are isomorphic to a coset space of group geometrical space. Thus the topological and algebraic structure of the coherent states as well as the associated dynamical system can be extensively discussed. In addition, a quantum-mechanical phase-space representation is constructed via the coherent-state theory. Several useful methods for employing the coherent states to study the physical phenomena of quantum-dynamic systems, such as the path integral, variational principle, classical limit, and thermodynamic limit of quantum mechanics, are described.
An explicit and uniform construction for coherent states (CS) for all the SU(N) groups is given and, on this basis, their properties are investigated. The CS are parametrized by the dots of a coset space, which is, in this particular case, the projective space CP N-1 and which plays the role of the phase space in the corresponding classical mechanics. The logarithm of the modulus of the CS overlap, being interpreted as assymmetric in the space, gives the Fubini-Study metric in CP N-1 . The classical limit is investigated in terms of operator symbols. h=P -1 (where P is the signature of the representation) plays the role of Planck's constant. The classical limit of the so called star commutator of the symbols generates the Poisson bracket in the corresponding phase space. The CS form an overcompleted system in the representation space and, as quantum states possess a minimum uncertainty, they minimize an invariant dispersion of the quadratic Casimir operator.
Within the generalized definition of coherent states as group orbits we study the orbit spaces and the orbit manifolds in the projective spaces constructed from linear representations. Invariant functions are suggested for arbitrary groups. The group SU (2) is studied in particular and the orbit spaces of its j = 1/2 and 1 representations completely determined. The orbits of SU (2) in CP N can be either two- or three-dimensional, the former being either isomorphic to S 2 or to RP 2 and the latter being isomorphic to quotient spaces of RP 3 . We end with a look from the same perspective to the quantum mechanical space of states in particle mechanics.
Unitary operator coherent states, as defined by Klauder (1963), Perelomov (1972) and Gilmore (1974), are considered for the harmonic series irreducible representations of U(p,q). Various properties of these states are reviewed, including their reproducing kernel and measure explicit forms, as well as the representation they lead to for the U(p,q) generators. Starting from a contraction of the su(p,q) Lie algebra, both Dyson and Holstein-Primakoff boson realisations of u(p,q) are obtained in terms of pq pairs of boson creation, annihilation operators and of the generators of a U(p)*U(q) intrinsic group. Such boson realisations are applied to determine the matrix elements of the U(p,q) generators between discrete bases classified according to the chain U(p,q) contains/implies U(p)*U(q). Finally, the isomorphism between so(4,2) and su(2,2) is employed to derive the corresponding properties of the SO(4,2) unitary operator coherent states for application in atomic physics.
The idea of construction of the nonlinear coherent states based on the hypergeometric- type operators associated to the Weyl-Heisenberg group [J:P hys:A 45(2012) 095304], are generalized to the similar states for the arbitrary Lie group SU(1, 1). By using of a discrete, unitary and irreducible representation of the Lie algebra su(1, 1) wide range of generalized nonlinear coherent states(GNCS) have been introduced, which admit a resolution of the identity through positive definite measures on the complex plane. We have shown that realization of these states for different values of the deformation pa- rameters r = 1 and 2 lead to the well-known Klauder-Perelomov and Barut-Girardello coherent states associated to the Irreps of the Lie algebra su(1, 1), respectively. It is worth to mention that, like the canonical coherent states, GNCS possess the temporal stability property. Finally, studying some statistical characters implies that they have indeed nonclassical features such as squeezing, anti-bunching effect and sub-Poissonian statistics, too.
This work can be considered as a continuation of our previous study, in which an explicit form of coherent states (cs) for all SU(N) groups was constructed by means of representations on polynomials. Here we extend that approach to any SU(l, 1) group and construct explicitly corresponding cs. The cs are parametrized by dots of a coset space, which is, in that particular case, the open complex ball CD l . This space together with the projective space CP l , which parametrizes the cs of the SU(l + 1) group, exhaust all complex spaces of constant curvature. Thus, both sets of cs provide a possibility for an explicit analysis of the quantization problem on all the spaces of constant curvature. This is why the cs of the SU(N) and SU(l, 1) groups are of importance in connection with quantization theory. The constructed cs form an overcompleted system in the representation space and, as quantum states possessing a minimum uncertainty, they minimize an invariant dispersion of the quadratic Casimir operator. The classical limit is investigated in terms of symbols of operators; the limit of the so called star commutator of the symbols generates the Poisson bracket in CD l , the latter plays the role of the phase space for the corresponding classical mechanics.