q-deformed phase space and its lattice structure

P. Narayana Swamy

S. Naka, H. Toyoda, Takahiro Takanashi

Dynamical systems associated with a q-deformed two dimensional phase space are studied as effective dynamical systems described by ordinary variables.In quantum theory, the momentum operator in such a deformed phase space becomes a difference operator instead of the differential operator.Then, using the path integral representation for such a dynamical system, we derive an effective short-time action, which contains interaction terms even for a free particle with q-deformed phase space.We also analyze the eigenvalue problem for a particle with q-deformed phase space confined in a compact space.Under some boundary conditions of the compact space, there arises fairly different structures from that of the q = 1 case in the energy spectrum of the particle and in the corresponding eigenspace.

D. Galetti, Jose Tadeu Lunardi, B. M. Pimentel +1 more

Sergei V. Shabanov

It is shown that q-deformed quantum mechanics (systems with q-deformed Heisenberg commutation relations) can be interpreted as an ordinary quantum mechanics on Kähler manifolds, or as a quantum theory with second (or first)-class constraints. 1. The q-deformed Heisenberg-Weyl algebras [1], [2] exhibiting the quantum group symmetries [3],[4] have attracted much attention of physicists and mathematicians. In particular, some works have been devoted to establishing potentials for ordinary quantum systems which exhibit a q-deformed energy spectrum (see, for example, [5] and references therein) in order to obtain classical dynamical systems associated with the q-deformed Heisenberg-Weyl algebras. In the present paper, we analyze the correspondence limit, when ¯h → 0 (rather than q → 1), of multimode q-deformed Heisenberg-Weyl algebras [1], [2] regardless of the specific form of a Hamiltonian. We show that classical systems associated with the q-deformed Heisenberg-Weyl algebras in this way possess a non-trivial symplectic structure which, in turn, is related to the Kähler symplectic structure [6]. From the other hand, a natural (physical) reason for a dynamical system to have a nontrivial symplectic structure is the existence of ”frozen ” , non-dynamical degrees of freedom or, in other words, constraints [7]. Dirac pointed out that a nontrivial symplectic structure might naturally occur in dynamical systems through second-class constraints [7]. We shall demonstrate below how constrained dynamics can be associated with the q-deformed Heisenberg-Weyl algebra. As was shown in [8], a non-commutative phase space does not necessarily emerges in the formal classical limit of q-deformed quantum mechanics, provided the deformation parameter q is a function of the Planck constant. In the latter case, q-deformed Heisenberg-Weyl algebras yield quadratic symplectic structures on a commutative phase space. For example, the one-mode q-deformed Heisenberg-Weyl algebra defined by the commutation relation ˆ b ˆ b − q

A. Lavagno, Antonio Maria Scarfone, P. Narayana Swamy

Hongjuan Meng, Yushan Zhou, Xue-Ping Ren +6 more

We investigate the existence and dynamical stability of multipole gap solitons in Bose–Einstein condensate loaded in a deformed honeycomb optical lattice. Honeycomb lattices possess a unique band structure, the first and second bands intersect at a set of so-called Dirac points. Deformation can result in the merging and disappearance of the Dirac points, and support the gap solitons. We find that the two-dimensional honeycomb optical lattices admit multipole gap solitons. These multipoles can have their bright solitary structures being in-phase or out-of-phase. We also investigate the linear stabilities and nonlinear stabilities of these gap solitons. These results have applications of the localized structures in nonlinear optics, and may helpful for exploiting topological properties of a deformed lattice.