Complex phase space and Weyl’s commutation relations

Alfredo Luis, Gonzalo Donoso

Yan Jiang-feng

In this paper,Moyal-Weyl multiplcation sign and Bopp shift are introduced.By considering the relation of non-commutative in coordinate-coordinate and momentum-momentum,the Wigner function of charged harmonic oscillator in non-commutative phase spaces are discussed.By using algebraic relation of quantum mechanics in non-commutative phase space,it has been found that the Wigner function depends on the addition item of non-commutative parameters,compared with the commutative space.

A. D. Jannussis, P. Filippakis, Loucas C. Papaloucas

Vladimir Budanov

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

Evgenii Igorevich Zelenov

The standard C*-formalism of representations of commutations relations for systems of infinitely many degrees of freedom over a real numbers field is extended to the case of a p-adic numbers field. It is proved that any positive function on the Weyl algebra over a p-adic symplectic space is regular and restriction of the generating function of a state of the Weyl algebra on any nongenerate finite-dimensional subspace of p-adic symplectic space is continuous. Some nontrivial class of nonequivalent representations of commutations relations is described and analogies of Fock representations and representations with number operators are constructed. The criteria for the existence of free-field unitary quantization of symplectic transformations is given.