On the Aharonov–Bohm Hamiltonian

Using the theory of self-adjoint extensions, we construct all the possible Hamiltonians describing the nonrelativistic Aharonov–Bohm effect. In general, the resulting Hamiltonians are not rotationally invariant so that the angular momentum is not a constant of motion. Using an explicit formula for the resolvent, we describe the spectrum and compute the generalized eigenfunctions and the scattering amplitude.     

Dynamics of the Chaplygin ball on a rotating plane

This paper addresses the problem of the Chaplygin ball rolling on a horizontal plane which rotates with constant angular velocity. In this case, the equations of motion admit area integrals, an integral of squared angular momentum and the Jacobi integral, which is a generalization of the energy integral, and possess an invariant measure. After reduction the problem reduces to investigating a three-dimensional Poincaré map that preserves phase volume (with density defined by the invariant measure). We show that in the general case the system’s dynamics is chaotic.     

NONLOCAL CURRENTS AS NOETHER CURRENTS DUE TO A NONLOCALLY TRANSFORMED GLOBAL SYMMETRY

In this paper we first present the gauge invariant conserved Noether current for the Yang-Mi11s theory, which is nonlocal is some sense. Then we introduce for the two dimensional chiral model, plain and supersymmetric, the nonlocal one-complex parameter-dependent symmetric generators, which shift the Lagrangian density by a total divergence. Both the equations of motion and the energy momentum density are tnvaunt. The associated ically Into infinite Noether current may be expanded analyt-series of nonlocal currents.     

One-dimensional non-relativistic and relativistic Brownian motions: a microscopic collision model

We study a simple microscopic model for the one-dimensional stochastic motion of a (non-)relativistic Brownian particle, embedded into a heat bath consisting of (non-)relativistic particles. The stationary momentum distributions are identified self-consistently (for both Brownian and heat bath particles) by means of two coupled integral criteria. The latter follow directly from the kinematic conservation laws for the microscopic collision processes, provided one additionally assumes probabilistic independence of the initial momenta. It is shown that, in the non-relativistic case, the integral criteria do correctly identify the Maxwellian momentum distributions as stationary (invariant) solutions. Subsequently, we apply the same criteria to the relativistic case. Surprisingly, we find here that the stationary momentum distributions differ slightly from the standard Jüttner distribution by an additional prefactor proportional to the inverse relativistic kinetic energy.     

A Symplectic Slice Theorem

We provide a model for an open invariant neighborhood of any orbit in a symplectic manifold endowed with a canonical proper symmetry. Our results generalize the constructions of Marle and Guillemin and Sternberg for canonical symmetries that have an associated momentum map. In these papers the momentum map played a crucial role in the construction of the tubular model. The present work shows that in the construction of the tubular model, the so-called Chu map, can be used instead, which exists for any canonical action, unlike the momentum map. Hamilton's equations for any invariant Hamiltonian function take on a particularly simple form in these tubular variables. As an application we will find situations, that we will call tubewise Hamiltonian, in which the existence of a standard momentum map in invariant neighborhoods is guaranteed.     

A Consistency Check for the Free Scalar Field Theory Realization of the Doubly Spacial Relativity *

We study a free scalar field theory in the framework of the Magueijo-Smolin model of the "Doubly Special Relativity" (DSR) which is a non-linear realization of the action of the Lorentz group on momentum space admitting an invariant energy cutoff. We show that unlike the standard quantum field theory, the Klein-Gordon equation obtained via Euler-Lagrange field equation and Heisenberg picture equation of motion of the field are not equivalent in this framework, at least up to the first order of the Planck length scale.     

Classification of Equivariant Star Products on Symplectic Manifolds

In this note, we classify invariant star products with quantum momentum maps on symplectic manifolds by means of an equivariant characteristic class taking values in the equivariant cohomology. We establish a bijection between the equivalence classes and the formal series in the second equivariant cohomology, thereby giving a refined classification which takes into account the quantum momentum map as well.     

Particle dynamics in a relativistic invariant stochastic medium

The dynamics of particles moving in a medium defined by its relativistically invariant stochastic properties is investigated. For this aim, the force exerted on the particles by the medium is defined by a stationary random variable as a function of the proper time of the particles. The equations of motion for a single one-dimensional particle are obtained and numerically solved. A conservation law for the drift momentum of the particle during its random motion is shown. Moreover, the conservation of the mean value of the total linear momentum for two particles repelling each other according to the Coulomb interaction also follows. Therefore, the results indicate the realization of a kind of stochastic Noether theorem in the system under study.     

Quantum entanglement of moving bodies

We study the properties of quantum entanglement in moving frames, and show that, because spin and momentum become mixed when viewed by a moving observer, the entanglement between the spins of a pair of particles is not invariant. We give an example of a pair, fully spin entangled in the rest frame, which has its spin entanglement reduced in all other frames. Similarly, we show that there are pairs whose spin entanglement increases from zero to maximal entanglement when boosted. While spin and momentum entanglement separately are not Lorentz invariant, the joint entanglement of the wave function is.     

Breakdown of Casimir invariance in curved space‐time

It is shown that the commonly accepted definition for the Casimir scalar operators of the Poincaré group does not satisfy the properties of Casimir invariance when applied to the non‐inertial motion of particles while in the presence of external gravitational and electromagnetic fields, where general curvilinear co‐ordinates are used to describe the momentum generators within a Fermi normal co‐ordinate framework. Specific expressions of the Casimir scalar properties are presented. While the Casimir scalar for linear momentum remains Lorentz invariant in the absence of external fields, this is no longer true for the spin Casimir scalar. Potential implications are considered for the propagation of photons, gravitons, and gravitinos as described by the spin‐3/2 Rarita‐Schwinger vector‐spinor field. In particular, it is shown that non‐inertial motion introduces a frame‐based effective mass to the spin interaction, with interesting physical consequences that are explored in detail.     

Equations of Relativistic and Quantum Mechanics and Exact Solutions of Some Problems

Relativistic invariant equations are proposed for the action function and the wave function based on the invariance of the representation of the generalized momentum. The equations have solutions for any values of the interaction constant of a particle with a field, for example, in the problem of a hydrogen-like atom, when the atomic number of the nucleus Z > 137. Based on the parametric representation of the action, the expression for the canonical Lagrangian, the equations of motion and the expression for the force acting on the charge during motion in an external electromagnetic field are derived. The Dirac equation with the correct inclusion of the interaction for a particle in an external field is presented. In this form, the solutions of the equations are not limited by the value of the interaction constant. The solutions of the problem of charge motion in a constant electric field, problems for a particle in a potential well, and penetration of a particle through a potential barrier, as well as problem of a hydrogen atom are presented.     

Energy–momentum tensor for the electromagnetic field in a dielectric

The total momentum of a thermodynamically closed system is unique, as is the total energy. Nevertheless, there is continuing confusion concerning the correct form of the momentum and the energy–momentum tensor for an electromagnetic field interacting with a linear dielectric medium. Rather than construct a total momentum from the Abraham momentum or the Minkowski momentum, we define a thermodynamically closed system consisting of a propagating electromagnetic field and a negligibly reflecting dielectric and we identify the Gordon momentum as the conserved total momentum by the fact that it is invariant in time. In the formalism of classical continuum electrodynamics, the Gordon momentum is therefore the unique representation of the total momentum in terms of the macroscopic electromagnetic fields and the macroscopic refractive index that characterizes the material. We also construct continuity equations for the energy and the Gordon momentum, noting that a time variable transformation is necessary to write the continuity equations in terms of the densities of conserved quantities. Finally, we use the continuity equations and the time–coordinate transformation to construct an array that has the properties of a traceless, symmetric energy–momentum tensor.     

Particle Dynamics of a Phase Space Distribution Function

No Heading A hydrodynamic analogy for quantum mechanics is used to develop a phase-space representation in terms of a quasi-probability distribution function. Averages over phase space using this approach agree with the usual expectation values of quantum mechanics for a certain class of observables. We also derive the equations of motion that particles in an ensemble would have in phase space in order to mimic the time development of this probability distribution, thus giving the position and momentum of particles in the ensemble as a function of time. The equations of motion separate into position and momentum components. The position component reproduces the de Broglie-Bohm equation of motion. As a simple example, we calculate the phase space trajectories and entropy of a free particle wave packet.     

On the treatment of the center of mass motion in nuclear mean field theories

It is shown how the spurious components due to the center of mass motion can be eliminated from general Hartree-Fock-Bogoliubov quasi-particle configurations with the help of projection techniques. The problem how to restore the additional symmetries being broken by such configurations is discussed. An explicit formulation is given for the spherical Hartree-Fock problem with center of mass momentum projection before the variation. As an example for the application of this method the ground state of⁴He is studied using two different interactions, a microscopic two-body one as well as a phenomenological one including a Skyrme-type three-body force. The results are compared to those of the usual approximate treatment of the center of mass motion in Hartree-Fock calculations. It turns out that, at least for the chosen example, the latter yields a rather reasonable approximation to the correct total energy, single particle energy and even the mass density provided that it is calculated from a translationally invariant density operator.     

Orbital angular momentum exchange in the interaction of twisted light with molecules

In the interaction of molecules with light endowed with orbital angular momentum, an exchange of orbital angular momentum in an electric dipole transition occurs only between the light and the center of mass motion; i.e., internal "electronic-type" motion does not participate in any exchange of orbital angular momentum in a dipole transition. A quadrupole transition is the lowest electric multipolar process in which an exchange of orbital angular momentum can occur between the light, the internal motion, and the center of mass motion. This rules out experiments seeking to observe exchange of orbital angular momentum between light beams and the internal motion in electric dipole transitions.     

Integrable vs. nonintegrable geodesic soliton behavior

We study confined solutions of certain evolutionary partial differential equations (PDE) in 1+1 space–time. The PDE we study are Lie–Poisson Hamiltonian systems for quadratic Hamiltonians defined on the dual of the Lie algebra of vector fields on the real line. These systems are also Euler–Poincaré equations for geodesic motion on the diffeomorphism group in the sense of the Arnold program for ideal fluids, but where the kinetic energy metric is different from theL² norm of the velocity. These PDE possess a finite-dimensional invariant manifold of particle-like (measure-valued) solutions we call “pulsons”. We solve the particle dynamics of the two-pulson interaction analytically as a canonical Hamiltonian system for geodesic motion with two degrees of freedom and a conserved momentum. The result of this two-pulson interaction for rear-end collisions is elastic scattering with a phase shift, as occurs with solitons. The results for head-on antisymmetric collisions of pulsons tend to be singularity formation. Numerical simulations of these PDE show that their evolution by geodesic dynamics for confined (or compact) initial conditions in various nonintegrable cases possesses the same type of multi-soliton behavior (elastic collisions, asymptotic sorting by pulse height) as the corresponding integrable cases do. We conjecture this behavior occurs because the integrable two-pulson interactions dominate the dynamics on the invariant pulson manifold, and this dynamics dominates the PDE initial value problem for most choices of confined pulses and initial conditions of finite extent.     

Central region in relativistic heavy ion collisions; results from hydrodynamic calculations and cascade simulation

Results of three dimensional hydrodynamic calculations with boost invariant longitudinal expansion are presented with special emphasis on the transverse momentum spectia of hadrons and production rates of dileptons. The effect and signatures of transverse collective motion are discussed in detail. The hydrodynamic results which are based on the assumption of the existence of a first order phase transition and formation of an equilibrium mixed phase are compared with cascade simulation where the mixed phase is modelled in terms of plasma droplets embedded in a pion gas. The comparison shows a great deal of similarity between the two approaches lending further support for the hydrodynamic approach.     

On the quantum motion of a generalized time-dependent forced harmonic oscillator

In this work, we use linear invariants and the dynamical invariant method to obtain exact solutions of the Schrödinger equation for the generalized time-dependent forced harmonic oscillator in terms of solutions of a second order ordinary differential equation that describes the amplitude of the classical unforced damped oscillator. In addition, we construct Gaussian wave packet solutions and calculate the fluctuations in coordinate and momentum as well as the quantum correlations between coordinate and momentum. It is shown that the width of the Gaussian packet, fluctuations and correlations do not depend on the external force. As a particular case, we consider the forced Caldirola-Kanai oscillator.