In this paper we study the generalized coherent states (g.c.s.) in n -dimensional space. Their annihilation operators consist of linear combinations of the annihilation operators a l , and their coherent states are the product of the simple coherent states. All the known annihilation operators which have been studied as yet, and also the magnetic ones, are special cases of these operators. The new operators A k + A m form the Lie algebra U ( n ) of the Lie group U ( n ). Finally, instead of the operators a l and a l + we can use the new operators, which depend on a complex parameter.
In this paper we study the generalized coherent states (g.c.s.) in n -dimensional space. Their annihilation operators consist of linear combinations of the annihilation operators a l , and their coherent states are the product of the simple coherent states. All the known annihilation operators which have been studied as yet, and also the magnetic ones, are special cases of these operators. The new operators A k + A m form the Lie algebra U ( n ) of the Lie group U ( n ). Finally, instead of the operators a l and a l + we can use the new operators, which depend on a complex parameter.
Two new types of coherent states associated with the Cλ-extended oscillator, where Cλ is the cyclic group of order λ, are introduced. The first ones include as special cases both the Barut-Girardello and the Perelomov su(1,1) coherent states for λ = 2, as well as the annihilation-operator coherent states of the Cλ-extended oscillator spectrum generating algebra for higher λ values. The second ones, which are eigenstates of the Cλ-extended oscillator annihilation operator, extend to higher λ values the paraboson coherent states, to which they reduce for λ = 2. All these states satisfy a unity resolution relation in the Cλ-extended oscillator Fock space (or in some subspace thereof). They give rise to Bargmann representations of the latter, wherein the generators of the Cλ-extended oscillator algebra are realized as differential-operatorvalued matrices (or differential operators). The statistical and squeezing properties of the new coherent states are investigated over a wide range of parameters and some interesting nonclassical features are exhibited. 1 1
Recently, the coherent-state labeled by the real parameters (RCS) has been introduced in connec tion with the numerical path integration for many-fermion systems.!) This is a complement of the coherent-state labeled by complex vari ables (CCS) which gives a basis of the path inte gral in the generalized phase space leading to the quantum version of the time-dependent mean field. 2 ),3) The central formula of both types of coherent states is the relation of of unity. For the case of RCS, Dobacewski and Koonin 4 ) have very recently proved this relation based on the commutability of the projection operator with the algebra generating the coher ent-state. The proof is rather general, since one uses only the Lie algebraic concept of the highest weight vectors. However it presupposes a specific form of the integration measure from the outset. In this note, apart from such a Lie-alge braic approach, we construct the RCS for many fermion systems based on the geometric device of the Lie group and associated label space (the so called quotient space),S) thereby the partition of unity is proved through a natural construction of the invariant measure. We also point out a specific role of the RCS for path integration in comparison with the path integral in the represen tation of CCS. In the following we are concerned with the RCS of determinantal states.. We firstly show that the underlying group becomes an orthogonal group instead of the unitary group for the CCS. 2 ) Let (aI', ai) be annihilation operators of the ~'particle and hole states with It = 1 ~ m, i = 1 ~ n with respect to the initial determinantal state 1 ([}o> = IL~lai +10>. Consider the linear canonical transformation where T(g) is an irreducible unitary representa tion of a Lie group G and A, B, C and Dare m x m, m x n, n x m and n x n matrices. Now consider the real restriction of (1). Hence the canonical nature leads to g'gT=J<m+n) (2) or
Recently we have obtained, on the basis of a group approach to quantization, a Bargmann-Fock-like realization of the Relativistic Harmonic Oscillator as well as a generalized Bargmann transform relating fock wave functions and a set of relativistic Hermite polynomials. Nevertheless, the relativistic creation and annihilation operators satisfy typical relativistic commutation relations of the Lie product (vector-z, vector-z(sup dagger)) approximately equals Energy (an SL(2,R) algebra). Here we find higher-order polarization operators on the SL(2,R) group, providing canonical creation and annihilation operators satisfying the Lie product (vector-a, vector-a(sup dagger)) = identity vector 1, the eigenstates of which are 'true' coherent states.
A new kind of k-quantum nonlinear coherent states, i.e., the k eigenstates of the k-th power of the generalized annihilation operator of f-oscillators, are obtained and their properties are discussed. The completeness of the k states is investigated. An alternative method to construct them is proposed. It is shown that these states may form a complete Hilbert space, and all of them can be generated by a linear superposition of k Roy-type nonlinear coherent states. Physically, they can be generated by a linear superposition of the time-dependent Roy-type nonlinear coherent states at different instants.
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.
During the century of existence of the notion of coherent states, either linear or nonlinear, several schemes for their construction, theoretical or experimental, have been developed. Generally, the mathematical structure of coherent states depends on the choice of ladder operators, and consequently on the structure constants. In this paper, we propose a way to construct generalized coherent states for anharmonic oscillators that is based on a diagonal operator ordering technique (DOOT) applied to generalized hypergeometric functions, that is, on some of the most general special functions. These states are generated by the action of a pair of ladder operators, the creation and the annihilation, whose ordered normal product is equal to the dimensionless Hamiltonian of the quantum system. In addition, the action of these operators is easy to find if the expression for the dimensionless energy eigenvalues is known.