q-oscillators, (non-)Kähler manifolds and constrained dynamics

It is shown that q-deformed quantummechanics (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 h̄ → 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 b̂b̂ − qb̂b̂ = h̄ , (1) where b̂ and b̂ are creation and destruction operators, turns into the following symplectic structure [8] {b, b∗} = −i(1− b∗b/β) , (2) where β is a constant and b and b∗ are commutative complex coordinates on a phasespace plane. Indeed, taking the classical limit [̂b, b̂]/ih̄ = −i(1− (1−q)b̂b̂/h̄) → {b, b∗} as h̄ → 0, and b̂, b̂ are simultaneously replaced by classical holomorphic variables b, b∗, respectively, we see that Eq.(2) results from (1) if one assumes 1 − q = h̄/β + O(h̄). Alexander von Humboldt fellow