sh(2/2) Superalgebra Eigenstates and Generalized Supercoherent and Supersqueezed States

Abstract

The superalgebra eigenstates (SAES) concept is introduced and then applied to find the SAES associated to the sh(2/2) superalgebra, also known as Heisenberg-Weyl Lie superalgebra. This implies to solve a Grassmannian eigenvalue superequation. Thus, the sh(2/2) SAES contain the class of supercoherent states associated to the supersymmetric harmonic oscillator and also a class of supersqueezed states associated to the osp(2/2) + ⊃ sh(2/2) superalgebra, where osp(2/2) denotes the orthosymplectic Lie superalgebra generated by the set of operators formed from the quadratic products of the Heisenberg-Weyl Lie superalgebra generators. The properties of these states are investigated and compared with those of the states obtained by applying the group-theoretical technics. Moreover, new classes of generalized supercoherent and supersqueezed states are also obtained. As an application, the superHermitian and η-pseudo-superHermitian Hamiltonians without a defined Grassmann parity and isospectral to the harmonic oscillator are constructed. Their eigenstates and associated supercoherent states are calculated.

Coherent and squeezed states of quantum Heisenberg algebras

“…On the other hand, we have found new classes of deformed squeezed states, parametrized by a real paragrassmann number, i.e., a number z such that z k 0 = 0, for some k 0 ∈ N. These states can be normalized, even if z is considered as a complex paragrassmann number. In this last case, when k 0 = 2, we can should interpret z as an odd complex Grassmann number and compare this new classes of deformed squeezed states with the ones associated to the η-superpseudo-Hermitian Hamiltonians [15].…”

“…We will see that these states will be new deformations of the standard coherent and squeezed states of the harmonic oscillator system and we will recover them in the limit when the deformation parameters go to zero. The approach of AES also gives us the possibility to construct, starting from a deformed algebra, some Hamiltonians, of physical systems to which these deformed coherent and squeezed states are associated, similarly as for algebras and superalgebras [14,15].…”

The oscillator model for the Lie superalgebra $\mathfrak {sh}(2|2)$sh(2|2) and Charlier polynomials

“…The algebra eigenstates associated to a real Lie algebra have been introduced by Brif [9]; they can be considered as generalizations of coherent or squeezed states. In [2], some superalgebra eigenstates of the Heisenberg-Weyl superalgebra sh(2|2) have been computed, and properties of these states were investigated. In the current section, we shall determine the eigenstates (and eigenvalues) of a general self-adjoint odd element of the real Lie superalgebra S. Given the conjugations (12), the most general self-adjoint odd element, up to an overall factor, is…”

“…With the purpose of describing an oscillator model, we study the spectrum and the formal eigenvectors of a general self-adjoint odd elementq of S in section 3. This is done in a broader context, since there has been some interest in the general construction of so-called superalgebra eigenstates [2]. The spectral analysis of this operatorq leads to a Jacobi matrix, and yields the infinite discrete spectrum S = {.…”

Gradings, Braidings, Representations, Paraparticles: Some Open Problems

“…One can find a host of applications, in either problems of physics or mathematics. For instance: in [18][19][20] we have constructions of coherent states in models described by this algebra; in [16,21] it is applied in the Jaynes-Cummings model; and in [22] in a variant of this model. In [23][24][25][26][27][28][29][30] this algebra is used for studying problems of the representation theory of Lie algebras, Lie superalgebras and their deformations.…”