Three-Parameter (Two-Sided) Deformation of Heisenberg Algebra

Abstract

A 3-parametric two-sided deformation of Heisenberg algebra (HA), with p, q-deformed commutator in the l.h.s. of basic defining relation and certain deformation of its r.h.s., is introduced and studied. The third deformation parameter µ appears in an extra term in the r.h.s. as pre-factor of Hamiltonian. For this deformation of HA we find novel properties. Namely, we prove it is possible to realize this (p, q, µ)-deformed HA by means of some deformed oscillator algebra. Also, we find the unusual property that the deforming factor µ in the considered deformed HA inevitably depends explicitly on particle number operator N . Such a novel N -dependence is special for the two-sided deformation of HA treated jointly with its deformed oscillator realizations.

Pseudo-Hermitian position and momentum operators, Hermitian Hamiltonian, and deformed oscillators

“…Consider first the q-deformed extension of HA. The desired DSF can be derived [8] if the operator functions H(N), G(N) are known. To find these, we set the (nonlinear) relation expressing the position/momentum operators through a − , a + , namely…”

“…In the last two decades, great deal of attention was devoted to miscellaneous generalizations of the Heisenberg algebra (HA) which use appropriate extensions [1,2,3,4,5,8,6,7] of the relation [X, P ] = i . Diversity of respective modifications of the famous position-momentum uncertainty relation, including those which imply minimal length, were also under study, see e.g.…”

“…Recently, in ref. [8] we introduced and studied the two-parameter (p, q)and three-parameter (p, q, µ)-deformations of HA. In that paper (see also the two subsequent ones [13,14]), the explicit mapping onto certain q-or (p, q)deformed oscillator (DO) algebras was obtained.…”

Nonstandard Deformed Oscillators from q- and p,q-Deformations of Heisenberg Algebra

“…The obtaining and analysis of modified/generalized versions of the famous Heisenberg uncertainty relation or principle corresponds to diverse generalizations of the standard Heisenberg algebra (HA) with basic relation [X, P ] = i for the position and momentum operators. This direction of research is under development for more than three decades (with some early works quoted as [7,12,28,29,34,37,38]), and still remains to be a hot topic, see, e.g., [5,8,13,16,17,32] for more recent papers. Physically, deformations of the Heisenberg algebra and the related generalized uncertainty relation find their motivation in quantum gravity, in string theory, noncommutative geometry etc.…”

“…In that DHA, first, the commutator in the l.h.s. of its defining relation is replaced with q-commutator like in [12,38] or with q, p-commutator as in [16]. In addition, the r.h.s.…”