Treatment of Dirac Fields in Light—Front Coordinates by Dirac‘s Method for Constrained Hamiltonian Systems

Dirac‘s method which itself is for constrained Boson fields and particle systems is followed and developed to treat Dirac fields in light-front coordinates.     

Hamiltonian and Thermodynamic Modeling of Quantum Turbulence

The state variables in the novel model introduced in this paper are the fields playing this role in the classical Landau-Tisza model and additional fields of mass, entropy (or temperature), superfluid velocity, and gradient of the superfluid velocity, all depending on the position vector and another tree dimensional vector labeling the scale, describing the small-scale structure developed in ⁴He superfluid experiencing turbulent motion. The fluxes of mass, momentum, energy, and entropy in the position space as well as the fluxes of energy and entropy in scales, appear in the time evolution equations as explicit functions of the state variables and of their conjugates. The fundamental thermodynamic relation relating the fields to their conjugates is left in this paper undetermined. The GENERIC structure of the equations serves two purposes: (i) it guarantees that solutions to the governing equations, independently of the choice of the fundamental thermodynamic relation, agree with the observed compatibility with thermodynamics, and (ii) it is used as a guide in the construction of the novel model.     

How to Find Separation Coordinates for the Hamilton–Jacobi Equation: A Criterion of Separability for Natural Hamiltonian Systems

The method of separation of variables applied to the natural Hamilton–Jacobi equation (u/q _i )²+V(q)=E consists of finding new curvilinear coordinates x _i (q) in which the transformed equation admits a complete separated solution u(x)=u _(i)(x _i ;). For a potential V(q) given in Cartesian coordinates, the main difficulty is to decide if such a transformation x(q) exists and to determine it explicitly. Surprisingly, this nonlinear problem has a complete algorithmic solution, which we present here. It is based on recursive use of the Bertrand–Darboux equations, which are linear second order partial differential equations with undetermined coefficients. The result applies to the Helmholtz (stationary Schrödinger) equation as well.     

Hamiltonian Feynman-Kac and Feynman Formulae for Dynamics of Particles with Position-Dependent Mass

A Feynman formula is a representation of the semigroup, generated by an initial-boundary value problem for some evolutionary equation, by a limit of integrals over Cartesian powers of some space E, the integrands being some elementary functions. The multiple integrals in Feynman formulae approximate integrals with respect to some measures or pseudomeasures on sets of functions which take values in E and are defined on a real interval. Hence Feynman formulae can be used both to calculate explicitly solutions for such problems, to get some representations for these solutions by integrals over functions taking values in E (such representations are called Feynman-Kac formulae), to get approximations for transition probability of some diffusion processes and transition amplitudes for quantum dynamics and to get computer simulations for some stochastic and quantum dynamics. The Feynman formula is called a Hamiltonian Feynman formula if the space, Cartesian products of which are used, is the phase space of a classical Hamiltonian system; the corresponding Feynman-Kac formula is called a Hamiltonian Feynman-Kac formula. In the latter formula one integrates over functions taking values in the same phase space. In a similar way one can define Lagrangian Feynman formulae and Lagrangian Feynman-Kac formulae substituting the phase space by the configuration space.     

Extended Nonlocal Chiral-Quark Model for the Heavy–Light Quark Systems

In this talk, we report the recent progress on constructing a phenomenological effective model for the heavy–light quark systems, which consist of (u, d, s, c, b) quarks, i.e. extended nonlocal chiral-quark model. We compute the heavy-meson weak-decay constants to verify the validity of the model. From the numerical results, it turns out that ${( f_D, f_B, f_{D_s},f_{B_s}) = (207.54, 208.13, 262.56, 262.39)}$ MeV. These values are in relatively good agreement with experimental data and various theoretical estimations.     

Parametrization of Projector-Based Witnesses for Bipartite Systems

Entanglement witnesses are nonpositive Hermitian operators which can detect the presence of entanglement. In this paper, we provide a general parametrization for orthonormal basis of ℂ ⁿ and use it to construct projector-based witness operators for entanglement detection in the vicinity of pure bipartite states. Our method to parameterize entanglement witnesses is operationally simple and could be used for doing symbolic and numerical calculations. As an example we use the method for detecting entanglement between an atom and the single mode of quantized field, described by the Jaynes-Cummings model. We also compare the detection of witnesses with the negativity of the state, and show that in the vicinity of pure stats such constructed witnesses able to detect entanglement of the state.     

Polarization response explored by joint Hamiltonian and stochastic approach

The effect of thermal fluctuations on systems modeled by anharmonic nonconservative Hamiltonians is investigated in the framework of interacting lattice cells each cell obeying to Langevin dynamics. Representative examples addressed to critical phenomena in ferroelectrics include polarization response of a single lattice cell, ergodicity breaking, and birth of a domain, and the effect of nonlocal electroelastic interaction all derived combining the Fokker–Planck, imaginary time Schrödinger, and symplectic integration techniques.     

Renormalization and Periodic Orbits¶for Hamiltonian Flows

We consider a renormalization group transformation for analytic Hamiltonians in two or more dimensions, and use this transformation to construct invariant tori, as well as sequences of periodic orbits with rotation vectors approaching that of the invariant torus. The construction of periodic and quasiperiodic orbits is limited to near-integrable Hamiltonians. But as a first step toward a non-perturbative analysis, we extend the domain of to include any Hamiltonian for which a certain non-resonance condition holds. Received: 5 October 1999 / Accepted: 2 February 2000     

Probability Description and Entropy of Classical and Quantum Systems

Tomographic approach to describing both the states in classical statistical mechanics and the states in quantum mechanics using the fair probability distributions is reviewed. The entropy associated with the probability distribution (tomographic entropy) for classical and quantum systems is studied. The experimental possibility to check the inequalities like the position–momentum uncertainty relations and entropic uncertainty relations are considered.     

Extended WKB Method,Resonances and Supersymmetric Radial Barriers

Semiclassical approximations are implemented in the calculation of position and width of low-energy resonances for radial barriers. The numerical integrations are delimited by τ/τ _life≪8, with τ the period of a classical particle in the barrier trap and τ _life the resonance lifetime. These energies are used in the construction of ‘haired’ short-range potentials as the supersymmetric partners of a given radial barrier. The new potentials could be useful in the study of the transient phenomena which give rise to the Moshinsky’s diffraction in time.     

Hamiltonian Dynamics and Spectral Theory for Spin–Oscillators

We study the Hamiltonian dynamics and spectral theory of spin-oscillators. Because of their rich structure, spin-oscillators display fairly general properties of integrable systems with two degrees of freedom. Spin-oscillators have infinitely many transversally elliptic singularities, exactly one elliptic-elliptic singularity and one focus-focus singularity. The most interesting dynamical features of integrable systems, and in particular of spin-oscillators, are encoded in their singularities. In the first part of the paper we study the symplectic dynamics around the focus-focus singularity. In the second part of the paper we quantize the coupled spin-oscillators systems and study their spectral theory. The paper combines techniques from semiclassical analysis with differential geometric methods.     

Extreme Value Laws in Dynamical Systems for Non-smooth Observations

We prove the equivalence between the existence of a non-trivial hitting time statistics law and Extreme Value Laws in the case of dynamical systems with measures which are not absolutely continuous with respect to Lebesgue. This is a counterpart to the result of the authors in the absolutely continuous case. Moreover, we prove an equivalent result for returns to dynamically defined cylinders. This allows us to show that we have Extreme Value Laws for various dynamical systems with equilibrium states with good mixing properties. In order to achieve these goals we tailor our observables to the form of the measure at hand.     

Compact and efficient modulators for 100 Gb/s ETDM for telecom and interconnect applications

State of the art and prospects regarding semiconductor compact modulators and transmitters for on–off keying and more advanced modulations formats for output bitrates of 100 Gb/s and above are discussed. The implementation of a monolithically integrated transmitter comprising laser and light-intensity modulator is described and the prospects for a fully integrated transmitter for more advanced modulation formats elucidated, all for 100 Gb/s output bitrate     

A Lagrangian Formalism Based on the Velocity-Determined Virtual Displacements for Systems with Nonholonomic Constraints

We develop a Lagrangian formulation for classical systems with a general nonholonomic constraints by utilizing the so-called velocity-determined virtual-displacement conditions, i.e. by assuming the virtual displacements to be along the direction of the velocities in a special reference frame. It is shown that our general scheme encompasses as special cases the Chetaev and Voronets approaches when the constraints are homogeneous or linear in relative velocities.     

Hamiltonian Systems and Darboux Transformation Associated with a 3 × 3 Matrix Spectral Problem

Starting from a 3 × 3 matrix spectral problem, we derive a hierarchy of nonlinear equations. It is shown that the hierarchy possesses bi-Hamiltonian structure. Under the symmetry constraints between the potentials and the eigenfunctions, Lax pair and adjoint Lax pairs including partial part and temporal part are nonlinearied into two finitedimensional Hamiltonian systems (FDHS) in Liouville sense. Moreover, an explicit N-fold Darboux transformation for CDNS equation is constructed with the help of a gauge transformation of the spectral problem.     

Measure Synchronization of High-Cycle Islets in Coupled Hamiltonian Systems

Measure synchronization is a new phenomenon found in coupled Hamiltonian systems recently and it is interesting to understand its properties comprehensively. We discuss the measure synchronization of a coupled pair of standard maps in high period quasi-period orbits, and the measure synchronization transition is associated with the transition of coupled systems from quasi-periodicity to chaos. This behaviour is very different from that found by Hampton and Zanette [Phys. Rev. Lett. 83 (1999) 2179].     

Hamiltonian Systems and Darboux Transformation Associated with a 3 × 3 Matrix Spectral Problem

Starting from a 3 × 3 matrix spectral problem, we derive a hierarchy of nonlinear equations. It is shown that the hierarchy possesses bi-Hamiltonian structure. Under the symmetry constraints between the potentials and the eigenfunctions, Lax pair and adjoint Lax pairs including partial part and temporal part are nonlinearied into two finitedimensional Hamiltonian systems (FDHS) in Liouville sense. Moreover, an explicit N-fold Darboux transformation for CDNS equation is constructed with the help of a gauge transformation of the spectral problem.     

A Resonance Theory for Open Quantum Systems with Time-Dependent Dynamics

We develop a resonance theory to describe the evolution of open systems with time-dependent dynamics. Our approach is based on piecewise constant Hamiltonians: we represent the evolution on each constant bit using a recently developed dynamical resonance theory, and we piece them together to obtain the total evolution. The initial state corresponding to one time-interval with constant Hamiltonian is the final state of the system corresponding to the interval before. This results in a non-Markovian dynamics. We find a representation of the dynamics in terms of resonance energies and resonance states associated to the Hamiltonians, valid for all times t≥0 and for small (but fixed) interaction strengths. The representation has the form of a path integral over resonances. We present applications to a spin-fermion system, where the energy levels of the spin may undergo rather arbitrary crossings in the course of time. In particular, we find the probability for transition between ground- and excited state at all times.