Dual wavefunctions in two–dimensional quantum mechanics

It is shown that the Schrödinger equation for a large family of pairs of two–dimensional quantum potentials possess wavefunctions for which the amplitude and the phase are interchangeable, producing two different solutions which are dual to each other. This is a property of solutions with vanishing Böhm potential. These solutions can be extended to three–dimensional systems. We explicitly calculate dual solutions for physical systems, such as the repulsive harmonic oscillator and the two–dimensional hydrogen atom. These dual wavefunctions are also solutions of an analogue optical system in the eikonal limit. In this case, the potential is related to the refractive index, allowing the study of this two–dimensional dual wavefunction solutions with an optical (analogue) system.     

Elliptic waves in two-component long-wave–short-wave resonance interaction system in one and two dimensions

We consider (2+1)$(2+1)$- and (1+1)$(1+1)$-dimensional long-wave–short-wave resonance interaction systems. We construct an extensive set of exact periodic solutions of these systems in terms of Lamé polynomials of order one and two. The periodic solutions are classified into three categories as similar, mixed, superposed elliptic solutions. We also discuss the hyperbolic solutions as limiting cases.     

Measurement and “beables” in quantum mechanics

It is argued that the measurement problem reduces to the problem of modeling quasi-classical systems in a modified quantum mechanics with superselection rules. A measurement theorem is proved, demonstrating, on the basis of a principle for selecting the quantities of a system that are determinate (i.e., have values) in a given state, that after a suitable interaction between a systemS and a quasi-classical systemM, essentially only the quantity measured in the interaction and the indicator quantity ofM are determinate. The theorem justifies interpreting the noncommutative algebra of observables of a quantum mechanical system as an algebra of beables, in Bell's sense.     

Supersymmetric Chern–Simons vortex systems and extended supersymmetric quantum mechanics algebras

We study N=2$N=2$ supersymmetric Chern–Simons Higgs models in (2+1)$(2+1)$-dimensions and the existence of extended underlying supersymmetric quantum mechanics algebras. Our findings indicate that the fermionic zero modes quantum system in conjunction with the system of zero modes corresponding to bosonic fluctuations, are related to an N=4$N=4$ extended 1-dimensional supersymmetric algebra with central charge, a result closely connected to the N=2$N=2$ spacetime supersymmetry of the total system. We also add soft supersymmetric terms to the fermionic sector in order to examine how this affects the index of the corresponding Dirac operator, with the latter characterizing the degeneracy of the solitonic solutions. In addition, we analyze the impact of the underlying supersymmetric quantum algebras to the zero mode bosonic fluctuations. This is relevant to the quantum theory of self-dual vortices and particularly for the symmetries of the metric of the space of vortices solutions and also for the non-zero mode states of bosonic fluctuations.     

Travelling wave solutions and regularity results for nonlinear Newton-Schrödinger systems especially in one dimensions

Optical and Quantum Electronics - In the current paper, we try to engineer the refractive index profile in a one-dimensional photonic crystal as a powerful tool to manage the electromagnetic wave...     

Classical and quantum algebras of non-local charges in σ models

We investigate the algebras of the non-local charges and their generating functionals (the monodromy matrices) in classical and quantum non-linear models. In the case of the classical chiral models it turns out that there exists no definition of the Poisson bracket of two monodromy matrices satisfying antisymmetry and the Jacobi identity. Thus, the classical non-local charges do not generate a Lie algebra. In the case of the quantum O(N) non-linear model, we explicitly determine the conserved quantum monodromy operator from a factorization principle together withP,T, and O(N) invariance. We give closed expressions for its matrix elements between asymptotic states in terms of the known two-particleS-matrix. The quantumR-matrix of the model is found. The quantum non-local charges obey a quadratic Lie algebra governed by a Yang-Baxter equation.Laboratoire associé au CNRS No. LA 280     

Heterotic σ-models and conformal supergravity in two dimensions

The (1, 0) and (2, 0) type heterotic σ-models with Wess-Zumino term are coupled to conformal supergravity in two dimensions. There are no new restrictions on the σ-model manifolds in addition to those which arise in the globally supersymmetric cases. In the (1, 0) case possible isometries of the scalar manifold are gauged. A derivation of d = 2 conformal supergravity based on the super Lie algebra OSp(2, N)⊕OSp(2, N) (N = 1, 2) is given.     

Operator equations and Moyal products–metrics in quasi-Hermitian quantum mechanics

The Moyal product is used to cast the equation for the metric of a non-Hermitian Hamiltonian in the form of a differential equation. For Hamiltonians of the form p²+V(ix)$p^2+V(ix)$ with V polynomial this is an exact equation. Solving this equation in perturbation theory recovers known results. Explicit criteria for the hermiticity and positive definiteness of the metric are formulated on the functional level.     

Remote interactions between two d-dimensional distributed quantum systems： nonlocal generalized quantum control-NOT gate and entanglement swapping

We present a systematic simple method to implement a generalized quantum control-NOT （CNOT） gate on two d-dimensional distributed systems. First, we show how the nonlocal generalized quantum CNOT gate can be implemented with unity fidelity and unity probability by using a maximally entangled pair of qudits as a quantum channel. We also put forward a scheme for probabilistically implementing the nonlocal operation with unity fidelity by employing a partially entangled qudit pair as a quantum channel. Analysis of the scheme indicates that the use of partially entangled quantum channel for implementing the nonlocal generalized quantum CNOT gate leads to the problem of ＇the general optimal information extraction＇. We also point out that the nonlocal generalized quantum CNOT gate can be used in the entanglement swapping between particles belonging to distant users in a communication network and distributed quantum computer.[第一段]     

Locally supersymmetric σ-model with Wess-Zumino term in two dimensions and critical dimensions for strings

We construct an N = 1 locally supersymmetric σ-model with a Wess-Zumino term coupled to supergravity in two dimensions. If one takes the σ-model manifold to be the product of d-dimensional Minkowski space M^d and a group manifold G, and if the radius of G is quantized in appropriate units of the string tension, then the model describes a Neveu-Schwarz-Ramond (NSR)-type string moving on M_d × G. (Our model generalizes earlier work of refs. [1,2] which do not contain a Wess-Zumino term and that of refs. [5,6] which is not locally supersymmetric.) The zweibein and the gravitino field equations yield constraints which generalize those of the NSR model to the case of a non-abelian group manifold. In particular, the fermionic constraint contains a new term trilinear in the fermionic fields. We quantize the theory in the light-cone gauge and derive the critical dimensions. We compute the mass spectrum of a closed string moving on M_d × G and show that massless fermions do not arise for non-abelian G for the spinning string, in agreement with the result of Friedan and Shenker [22].     

Combining relativity and quantum mechanics: Schrödinger's interpretation of ψ

The incongruence between quantum theory and relativity theory is traced to the probability interpretation of the former. The classical continium interpretation of removes the difficulty. How quantum properties of matter and light, and in particular the radiative problems, like spontaneous emission and Lamb shift, may be accounted in a first quantized Maxwell-Dirac system is discussed.     

Classical and quantum Hořava–Lifshitz cosmology in a minisuperspace perspective

We study the classical and quantum models of a Friedmann-Robertson-Walker (FRW) cosmology in the framework of the gravity theory proposed by Ho?ava, the so-called Ho?ava–Lifshitz theory of gravity. Beginning with the ADM representation of the action corresponding to this model, we construct the Lagrangian in terms of the minisuperspace variables and show that in comparison with the usual Einstein-Hilbert gravity, there are some correction terms coming from the Ho?ava theory. Either in the matter free or in the case when the considered universe is filled with a perfect fluid, the exact solutions to the classical field equations are obtained for the flat, closed and open FRW model and some discussions about their possible singularities are presented. We then deal with the quantization of the model in the context of the Wheeler–DeWitt approach of quantum cosmology to find the cosmological wave function. We use the resulting wave functions to investigate the possibility of the avoidance of classical singularities due to quantum effects.     

Properties of ground state and anomalous quantum fluctuations in one-dimensional polaron-soliton systems—the effects of electron-two-phonon interaction and non-adiabatic quantum correlations

Based on the Holstein model Hamiltonian of one-dimensional molecular crystals, by making use of the expansion approach of the correlated squeezed-coherent states of phonon instead of the two-phonon coherent state expansion scheme, the properties of the ground state and the anomalous quantum fluctuations are investigated in a strongly coupled electron-phonon system with special consideration of the electron-two-phonon interaction. The effective renormalization (αi) of the displacement of the squeezed phonons with the effect of the squeezed-coherent states of phonon and both the electron-displaced phonon and the polaron-squeezed phonon correlations have been combined to obtain the anomalous quantum fluctuations for the corrections of the coherent state. Due to these non-adiabatic correlations, the effective displacement parameter αi is larger than the ordinary parameter α (0) i . In comparison with the electron-one-phonon interaction (g) corrected as αig, we have found the electron-two-phonon interaction (g1) corrected as αi2 g1 is enhanced significantly. For this reason, the ground state energy (E(2) 0 ) contributed by the electron-two-phonon interaction is more negative than the single-phonon case (E(1) 0 ) and the soliton solution is more stable. At the same time, the effects of the electron-two-phonon interaction greatly increase the polaron energy and the quantum fluctuations. Furthermore, in a deeper level, we have considered the effect of the polaron-squeezed phonon correlation (f-correlation). Since this correlation parameter f > 1, this effect will strengthen the electron-one and two-phonon interactions by fαig and f2αi2 g1, respectively. The final results show that the ground state energy and the polaron energy will appear more negative further and the quantum fluctuations will gain further improvement.     

Form factors in the “point form” of relativistic quantum mechanics: Single- and two-particle currents

Electromagnetic and Lorentz-scalar form factors are calculated for a bound system of two spin-less particles exchanging a zero-mass scalar particle. Different approaches are considered including solutions of a Bethe-Salpeter equation, a point form approach to relativistic quantum mechanics and a non-relativistic one. The comparison of the Bethe-Salpeter results, which play the role of an experiment here, with the ones obtained in point form in single-particle approximation, evidences sizable discrepancies, pointing to large contributions from two-body currents in the latter approach. These ones are constructed using two constraints: ensuring current conservation and reproducing the Born amplitude. The two-body currents so obtained are qualitatively very different from standard ones. Quantitatively, they turn out not to be sufficient to remedy all the shortcomings of the point form form factors evidenced in impulse approximation.Received: 3 July 2003, Revised: 28 October 2003, Published online: 23 July 2004PACS: 11.10.St Bound and unstable states; Bethe-Salpeter equations - 13.40.Gp Electromagnetic form factors - 12.39.Ki Relativistic quark modelL. Theußl: Present address: TRIUMF, 4004 Wesbrook Mall, Vancouver, B.C., Canada, V6T 2A3     

Arnol’d diffusion in a system with 2.5 degrees of freedom: Classical and quantum mechanical approaches

Arnol’d diffusion, a universal phenomenon in nonlinear dynamics, is analyzed for a model system with 2.5 degrees of freedom. Only the three primary order resonances are taken into account, and the results obtained by using classical and quantum mechanical approaches are compared. It is shown that the parameter dependence of the rate of quantum Arnol’d diffusion is similar to the classical one, but the quantum diffusion coefficient is smaller by approximately an order of magnitude. It is found that the existence of a threshold with respect to perturbation parameters, pointed out earlier, is not an indispensable feature of quantum Arnol’d diffusion. It is shown that a quantum system with weakly overlapping resonances can exhibit mixed dynamics that has no classical counterpart (diffusion along a resonance superimposed by oscillations across the overlapped resonances).     

Classical mechanics via general relativity and Maxwell’s theory: a bit of magic

In the 1950’s Herman Bondi observed that a very effective way to study gravitational radiation was to use null surfaces as part of the coordinate system for analyzing the Einstein (Einstein–Maxwell) equations. A particular class of such surfaces, (referred to as Bondi null surfaces) with their associated null tetrad, has now been the main tool for this analysis for many years; their use—until recently—has been almost ubiquitous. Several years ago we realized that there was an attractive alternative to the use of Bondi coordinates, namely to use coordinates (in the asymptotic null future space–time region) that were as close to ordinary flat-space light-cones as possible—very different from Bondi surfaces. There were initially serious impediments to this program: these new null surfaces (referred to as asymptotically shear-free surfaces, ASF) were determined by solving a non-linear differential equation (the “good-cut” equation) whose solutions were most often complex. Eventually these problems were overcome and the program was implemented. In a series of papers, using the ASF null surfaces to study the asymptotically flat Einstein (or Einstein–Maxwell) equations, a variety of surprising (strange) results were obtained. Using only the Einstein and Maxwell equations, we found a large number of the basic relations of classical mechanics. They included very detailed conservation laws, well know kinematic relations and dynamic equations and even the Abraham–Lorentz–Dirac radiation reaction force and the rocket force. As interesting as these were, they came with a serious enigma. These relations from classical mechanics had no relationship with the physical space–time. The space for the action of these relations was the parameter space of solutions of the good-cut equation—a complex space, known as H-space. The enigma—what possible relationship did these standard appearing classical relations have with physical space–time? It is the purpose of this work to establish such a relationship—objects in H-space are related to structures in physical space–time. For example, a complex world-line in H-space becomes in physical space–time an asymptotically shear-free null geodesic congruence where its twist describes its intrinsic spin and if charged, its magnetic dipole.