Remarks on the q-quantization

Supersymmetry of Parafermions

“…In Section 3 we consider particular examples of finitedimensional supersymmetric systems related to various deformation schemes of the singlemode oscillator algebra. This includes the usual parafermions [1], the generalized deformed parafermions with internal Z 2 structure [10], the finite-dimensional q-deformed oscillator [13,14,15] and the q-deformed parafermions [16,17]. We find also that the appropriate linear combination of the Hamiltonians H n and H a in the case of usual parafermions gives rise to nonlinear supersymmetries characterized, like in the case of parabosons, by the presence of arbitrary (fixed) number of singlet states.…”

“…1. The parafermions [1], the generalized deformed parafermions with internal Z 2 structure [10], the finite-dimensional q-deformed oscillator [13,14,15] and the q-deformed parafermions [16,17] are given by the characteristic functions having the property (2.5). The form of the degenerated spectra indicates that these systems may possess the supersymmetry of two types.…”

“…, one can also represent the supercharges as 17) with ϕ(N) = F (N). Now let us turn to the case of odd p, where the system described by the Hamiltonian…”

Maths-type q-deformed coherent states for q>1

“…where b † is the Hermitian conjugate of b and q is some real constant [4,5,6], are often termed maths-type q-bosons [7] because the 'basic' numbers and special functions associated with them have been investigated in the mathematical literature for over 150 years (see e.g. [8]).…”

Unified (p, q; α, β, ν; γ) −deformed oscillator algebra: Irreducible representations and induced deformed harmonic oscillator

“…The common used deformation of this Heisenberg-Weyl algebra is defined as the algebra generated by the set of operators {I, a, a † , N } and the relations [13] […”