Magneto-dimensional resonance. Pseudospin phase and hidden quantum number

The Schrödinger operator for a spinless charge inside a layer with parabolic confinement profile and homogeneous magnetic field is considered. The Lorentz (cyclotron) and the confinement frequencies are assumed to be equal to each other. After inclination of the layer normal from the magnetic field direction there appears a pseudospin su(2)-field removing the resonance degeneracy of Landau levels. Under deviations of the layer surface from the plane shape, a longitudinal geometric current is created. In circulations around surface warping, there is a nontrivial quantum phase transition generated by an element of the π₁-homotopy group and a hidden degree of freedom (spectral degeneracy) associated with a “charge” of geometric poles on the layer. The quantization rule contains an additional parity index related to the algebraic number of geometric poles and the Landau level number. The resonance pseudospin phase-shift represents an example of general Aharonov–Bohm type topologic phenomena in quantum (semiclassical or adiabatic) systems with delta-function singularities in symplectic structure.     

Quantum geometry and quantum mechanics of integrable systems. II

For a generic quantum integrable system, we describe the asymptotics of the eigenstate density and of the trace of the evolution operator in all orders of the quantization parameter. This is done by using quantum symplectic geometry, which makes the given quantum system to be equivalent to a deformed classical system with arbitrary accuracy with respect to the quantization parameter. The asymptotics is explicitly given via the deformed symplectic form, deformed Liouville-Arnold tori, and deformed Maslov class.     

The geometric phase of the quantum systems with slow but finite rate of the external time-dependent field

With the help of the time-dependent gauge transformation technique, we have studied the geometric phase of a spin-half particle in a rotating magnetic field. We have found that the slow but finite frequency of the rotating magnetic field will make the difference between the adiabatic geometric phase and the exact geometric phase. When the frequency is much smaller than the energy space and the adiabatic condition is perfectly guaranteed, the adiabatic approximation geometric phase is exactly consistent with the adiabatic geometric phase. A simple relation for the accuracy of the adiabatic approximation is given in terms of the changing rate of the frequency of the rotating magnetic field and the energy level space.     

Nonadiabatic conditional geometric phase shift with NMR

A conditional geometric phase shift gate, which is fault tolerant to certain types of errors due to its geometric nature, was realized recently via nuclear magnetic resonance (NMR) under adiabatic conditions. However, in quantum computation, everything must be completed within the decoherence time. The adiabatic condition makes any fast conditional Berry phase (cyclic adiabatic geometric phase) shift gate impossible. Here we show that by using a newly designed sequence of simple operations with an additional vertical magnetic field, the conditional geometric phase shift gate can be run nonadiabatically. Therefore geometric quantum computation can be done at the same rate as usual quantum computation.     

Symplectic Geometry,Hopf Algebraic Structures of Classical and Quantum q-Deformations of SU(2) Algebra

By deforming the symplectic structure on S², we get the q-deformation of SU(2) algebra at classical level, SU_q,h→0(2), in a Hamiltonian approach. Furthermore, we construct a set of operators on the line bundle over the deformed symplectic manifo1d.S_q² such that they form SU_q,h→0(2) in Lie brackets and set up a nontrivial Hopf algebra with a parameter q only in such a classical Hamiltonian system. We also show that the deformations from S_q² to S_q² are a set of quasiconformal transformations. The quantization via geometric approach of the system gives rise to the quantum q-deformed algebra SU_q,h(2), wnich has a Hopf algebraic structure with two independent parameters q and h.     

Index Theory and Adiabatic Limit in QFT

The paper has the form of a proposal concerned with the relationship between the three mathematically rigorous approaches to quantum field theory: (1) local algebraic formulation of Haag, (2) Wightman formulation and (3) the perturbative formulation based on the microlocal renormalization method. In this project we investigate the relationship between (1) and (3) and utilize the known relationships between (1) and (2). The main goal of the proposal lies in obtaining obstructions for the existence of the adiabatic limit (confinement problem in the phenomenological standard model approach). We extend the method of deformation of Dütsch and Fredenhagen (in the Bordeman-Waldmann sense) and apply Fedosov construction of the formal index—an analog of the index for deformed symplectic manifolds, generalizing the Atiyah-Singer index. We present some first steps in realization of the proposal.     

辛几何算法在计算非磁化等离子体中波传播轨迹时的应用

为了在数值计算中保持哈密顿系统的辛几何结构不变,利用辛几何算法得到了在线性哈密顿系统中射线追踪方程的一般辛差分格式。通过具体算例,利用辛几何算法计算了波在非磁化等离子体中的传播轨迹,并且与传统Runge-Kutta-Fehlberg算法所得结果进行了比较。利用辛几何算法所得传播轨迹与解析解一致,其色散函数值的误差随时间线性增长,能在长时间内保持色散函数值在一个很小的误差范围内。利用传统的Runge-Kutta-Fehlberg算法所得传播轨迹与解析解不一致,其误差随时间做大幅振荡增加。计算结果表明辛几何算法在保持传播轨迹和色散函数值方面具有独特的优势。     

Quantization of Multiply Connected Manifolds

The standard (Berezin-Toeplitz) geometric quantization of a compact Kähler manifold is restricted by integrality conditions. These restrictions can be circumvented by passing to the universal covering space, provided that the lift of the symplectic form is exact. I relate this construction to the Baum-Connes assembly map and prove that it gives a strict quantization of the original manifold. I also propose a further generalization, classify the required structure, and provide a means of computing the resulting algebras. These constructions involve twisted group C*-algebras of the fundamental group which are determined by a group cocycle constructed from the cohomology class of the symplectic form. This provides an algebraic counterpart to the Morita equivalence of a symplectic manifold with its fundamental group.     

Evidence for symplectic symmetry in Ab initio no-core shell model results for light nuclei

Clear evidence for symplectic symmetry in low-lying states of 12C and 16O is reported. Eigenstates of 12C and 16O, determined within the framework of the no-core shell model using the J-matrix inverse scattering potential with A相似文献   

Measures in the geometric quantization of field theories

In this article, it is shown that for the standard symplectic form on the space of compactly supported sections of a symplectic fibre bundle, there is no locally-finite Borel measure which is preserved by the Hamiltonian flows of even a quite restricted set of functions on this space. As this means that some of the operators arising in geometric quantization associated to classical observables would not be Hermitean, the result suggests that one should consider quotients by gauge groups as classical phase spaces to avoid this problem.     

Diffusion of magnetic field lines in a confined RFP plasma

Summary  A volume-preserving symplectic map is proposed to describe the magnetic field lines when the Taylor equilibriumis perturbed in a generic way. The standard scenario is observed by varying the perturbation strength, but the statistical properties in the chaotic regions are not simple due to the presence of boundaries and remnants of invariant structures. Simpler models of volume-preserving maps are proposed. The slowly modulated standard map captures the basic topological and statistical features. The diffusion is analytically described for large perturbations (above the break-up of the last KAM torus) in terms of correlation functions and for small perturbations using the adiabatic theory, provided that the modulation is sufficiently slow.     

An improved symplectic precise integration method for analysis of the rotating rigid–flexible coupled system

This paper presents an improved symplectic precise integration method (PIM) to increase the accuracy and keep the stability of the computation of the rotating rigid–flexible coupled system. Firstly, the generalized Hamilton's principle is used to establish a coupled model for the rotating system, which is discretized and transferred into Hamiltonian systems subsequently. Secondly, a suitable symplectic geometric algorithm is proposed to keep the computational stability of the rotating rigid–flexible coupled system. Thirdly, the idea of PIM is introduced into the symplectic geometric algorithm to establish a symplectic PIM, which combines the advantages of the accuracy of the PIM and the stability of the symplectic geometric algorithm. In some sense, the results obtained by this method are analytical solutions in computer for a long span of time, so the time-step can be enlarged to speed up the computation. Finally, three numerical examples show the stability of computation, the accuracy of solving stiff equations and the capability of solving nonlinear equations, respectively. All these examples prove the symplectic PIM is a promising method for the rotating rigid–flexible coupled systems.     

Covariant Poisson Brackets in Geometric Field Theory

We establish a link between the multisymplectic and the covariant phase space approach to geometric field theory by showing how to derive the symplectic form on the latter, as introduced by Crnkovi-Witten and Zuckerman, from the multisymplectic form. The main result is that the Poisson bracket associated with this symplectic structure, according to the standard rules, is precisely the covariant bracket due to Peierls and DeWitt.     

Geometric algebra and star products on the phase space

Superanalysis can be deformed with a fermionic star product into a Clifford calculus that is equivalent to geometric algebra. With this multivector formalism it is then possible to formulate Riemannian geometry and an inhomogeneous generalization of exterior calculus. Moreover, it is shown here how symplectic and Poisson geometry fit in this context. The application of this formalism together with the bosonic star product formalism of deformation quantization leads then on space and space-time to a natural appearance of spin structures and on phase space to BRST structures that were found in the path integral formulation of classical mechanics. Furthermore it will be shown that Poincaré and Lie-Poisson reduction can be formulated in this formalism.     

电磁波导的辛分析与对偶棱边元

将电磁波导的控制方程导向了Hamilton体系、辛几何的形式.以电磁场的横向分量组成对偶向量并采用分离变量法，可以得到Hamilton算子矩阵的辛本征值问题.共轭辛正交归一关系、辛本征解展开定理等均可在此应用.对于复杂横截面和填充非均匀材料的电磁波导，提出对偶棱边元，对截面半解析离散后即可进行数值求解.对偶棱边元克服了结点基有限元求解电磁场问题的困难，与常规棱边元相比在某些方面具有一定的优势. 关键词： 电磁波导 Hamilton体系 对偶变量 棱边元     

Variational approaches to the problems of plasma stability and of nonlinear plasma dynamics

A variational method is developed in order to investigate the nonlinear dynamics and stability of plasma using hydrodynamic plasma models, namely, the one-fluid, Hall, and electron MHD models. The key idea of the method is to adequately take into account variational symmetries and the associated conservation laws inherent in these hydrodynamic models. This approach is applied to derive variational criteria for the stability of a steadily moving plasma and to propose a variational method of the adiabatic separation of fast and slow motions, which makes it possible to simplify (reduce) the basic hydrodynamic models.     

Extended Born-Oppenheimer molecular dynamics

A Lagrangian generalization of time-reversible Born-Oppenheimer molecular dynamics Niklasson et al. [Phys. Rev. Lett. 97, 123001 (2006)10.1103/PhysRevLett.97.123001] is proposed. The formulation enables the application of higher-order symplectic or geometric integration schemes that are stable and energy conserving even under incomplete self-consistency convergence. It is demonstrated how the accuracy is improved by over an order of magnitude compared to previous formulations at the same level of computational cost. The proposed Lagrangian includes extended electronic degrees of freedom as auxiliary dynamical variables in addition to the nuclear coordinates and momenta. While the nuclear degrees of freedom propagate on the Born-Oppenheimer potential energy surface, the extended auxiliary electronic degrees of freedom evolve as a harmonic oscillator centered around the adiabatic propagation of the self-consistent ground state.     

An adiabatic energy-level expression for anharmonic hydrogen-bonded systems

An adiabatic energy-level expression is obtained for hydrogen-bonded systems X-H----Y without neglecting the anharmonicity of the proton motion, treating the anharmonicity of the X----Y motion perturbatively or omitting many of the terms that couple the fast and slow motions. The energy-level expression is applied to FH----O (CH₃)₂. The reliability of the adiabatic energy-level expression is tested by comparing the adiabatic energy levels with variational energy levels.