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.
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.
Quantum and classical mechanics of a system of q-deformed bosonic oscillators are considered. The q-deformed Heisenberg-Weyl algebra of creation and destruction operators is realized by differential operators in a space of functions of real commutative variables. The corresponding Hilbert space is constructed. In this approach, the deformation parameter turns out to be a function of the Planck constant, an oscillator frequency and a parameter with dimension of length. The Hamiltonian path integral is derived and its semiclassical approximation is investigated to obtain the corresponding q-deformed classical theory. The phase space spanned by the usual commutative coordinates is shown to be a cylinder in classical theory. The dimensional parameter introduced determines its radius. It is argued that the q-deformation can be associated with a special non-canonical transformation. The principle of least action for the classical q-deformed system is formulated. A representation of U q (m) in a space of functions on a phase space spanned by commutative coordinates is constructed.
We discuss the possibility of interpreting a q-deformed non-interacting system as incorporating the effects of interactions among its particles. This can be accomplished, for instance, in an ensemble of q-Bosons by means of the virial expansion of a real gas in powers of the deformed parameter. The lowest order virial coefficient reduces to the case of the standard, non-interacting Bose gas, while the higher order virial coefficients contain effects arising from the interaction. The same picture can be drawn in a quantum mechanical system where it is shown that the q-deformed momentum can be expanded in a series contains high-order powers of the standard quantum phase-space variables. Motivated by this result, we introduce, in the classical framework, a transformation relating the momentum of a free system with the momentum of an interacting system. It is shown that the canonical quantization applied to the interacting system imply a q-deformed quantization for the free system.
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
Abstract In Chapter 2, a path integral representation of the quantum operator e-β H in the case of Hamiltonians H of the separable form p2/2m + V(q) has been constructed. Here, the construction is extended to Hamiltonians that are more general functions of phase space variables. This results in integrals over paths in phase space involving the action expressed in terms of the classical Hamiltonian H(p,q). However, it is shown that, in the general case, the path integral is not completely defined, and this reflects the problem that the classical Hamiltonian does not specify completely the quantum Hamiltonian, due to the problem of ordering quantum operators in products. When the Hamiltonian is a quadratic function of the momentum variables, the integral over momenta is Gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is ambiguous, since a problem of operator ordering arises, and the ambiguity must be fixed. Hamiltonians that are general quadratic functions provide other important examples, which are analysed thoroughly. Such Hamiltonians appear in the quantization of the motion on Riemannian manifolds. There, the problem of ambiguities is even more severe. The problem is illustrated by the analysis of the quantization of the free motion on the sphere SN−1.
We develop a path integral representation for the dynamics of quantum systems with a finite-dimensional Hilbert space, formulated entirely within a discrete phase space. Starting from the discrete Wigner function on Zd×Zd (d an odd prime) and the associated Weyl transform built from generalized displacement operators, we derive an exact kernel that propagates the discrete Wigner function in time and, by iterating its composition law through a short-time approximation, obtain a sum-over-paths expression weighted by a discrete phase-space action -- the natural finite-dimensional counterpart of Marinov's functional. For Hamiltonians linear in the phase-space coordinates and at times strictly commensurate with the lattice, the fluctuation sum collapses to a deterministic shift, realizing the discrete analog of classical Hamiltonian flow. Applying the formalism to one and to two interacting qutrits (d=3), we show that the full entanglement dynamics -- captured by a closed-form linear entropy valid for all times -- requires the coherent contribution of all fluctuation sectors; the boundary-term (mean-field) sector alone fails to reproduce it. For a non-stabilizer Hamiltonian, where the short-time kernel is only approximate, the time-sliced path integral converges to the exact dynamics, including the dynamical generation of Wigner negativity. We discuss implications for the semiclassical simulation of many-body spin systems and for the characterization of non-classicality through Wigner negativity.
This is the second part of a paper about a q-deformed analog of non-relativistic Schroedinger theory. It applies the general ideas of part I and tries to give a description of one-particle states on q-deformed quantum spaces like the braided line or the q-deformed Euclidean space in three dimensions. Hamiltonian operators for the free q-deformed particle in one as well as three dimensions are introduced. Plane waves as solutions to the corresponding Schroedinger equations are considered. Their completeness and orthonormality relations are written down. Expectation values of position and momentum observables are taken with respect to one-particle states and their time-dependence is discussed. A potential is added to the free-particle Hamiltonians and q-analogs of the Ehrenfest theorem are derived from the Heisenberg equations of motion. The conservation of probability is proved.