Relativistic Motion with Linear Dissipation

A general formalism for obtaining the Lagrangian and Hamiltonian for a one-dimensional dissipative system is developed. The formalism is illustrated by applying it to the case of a relativistic particle with linear dissipation. The relativistic wave equation is solved for a free particle with linear dissipation. PACS Numbers: 45.20.Jj, 03.65.Pm     

One-Dimensional Relativistic Dissipative System with Constant Force and its Quantization

For a relativistic particle under a constant force and a linear velocity dissipation force, a constant of motion is found. Problems are shown for getting the Hamiltonian of this system. Thus, the quantization of this system is carried out through the constant of motion and using the quantization on the velocity variable. The dissipative relativistic quantum bouncer is outlined within this quantization approach. PACS: 03.30.+p, 03.65.−w, 45.05.+x, 45.20Jj     

Constants of Motion for Several One-Dimensional Systems and Problems Associated with Getting Their Hamiltonians

The constants of motion of the following systems are deduced: a relativistic particle with linear dissipation; a no-relativistic particle with a time explicitly depending force; a no-relativistic particle with a constant force and time depending mass; and a relativistic particle under a conservative force with position depending mass. The Hamiltonian for these systems, which is determined by getting the velocity as a function of position and generalized linear momentum, can be found explicitly at first approximation for the first system. The Hamiltonians for the other systems are kept implicitly in their expressions for their constants of motion.     

On classical and quantum relativistic dynamics

A canonical formalism for the relativistic classical mechanics of many particles is proposed. The evolution equations for a charged particle in an electromagnetic field are obtained and the relativistic two-body problem with an invariant interaction is treated. Along the same line a quantum formalism for the spinless relativistic particle is obtained by means of imprimitivity systems according to Mackey theory. A quantum formalism for the spin-1/2 particle is constructed and a new definition of spin1/2 in relativity is proposed. An evolution equation for the spin-1/2 particle in an external electromagnetic field is given. The Bargmann Michel, and Telegdi equation follows from this formalism as a quasiclassical approximation. Finally, a new relativistic model for hydrogenlike atoms is proposed. The spectrum predicted is in agreement with Dirac's when radiative corrections have been added.     

Quantum Potential in Relativistic Dynamics

The experimental confirmation of nonlocality has renewed interest in Bohm's quantum potential. The construction of quantum potentials for relativistic systems has encountered difficulties which do not arise in a parametrized formulation of relativistic quantum mechanics known as Relativistic Dynamics. The purpose of this paper is to show how to construct a quantum potential in the relativistic domain by deriving a relativistically invariant quantum potential using Relativistic Dynamics. The formalism is applied to three relativistic scalar particle models: a single particle interacting with a scalar potential; N particles interacting with a scalar potential; and a single particle interacting with an electromagnetic 4-vector potential.     

Fundamental Solutions of the Axial Symmetric Goursat Problem

Abstract

Fundamental solutions (FS) with a given boundary condition on the characteristics of relativistic problems with axial symmetry are considered. This is so-called the Goursat problem (GP) or zero plane formalism in Dirac’s terminology, or modification of the proper time method in the Fock-Nambu-Schwinger formalism (FNS).

Closed FS for the Volkov problem from the point of view of GP can be found. This means that integration over proper time in a FNS integral transformation can be performed. Using the special chosen dynamic symmetry of the initial state, FS for a particle in constant magnetic or constant electric field may also be calculated.     

Hamilton–Jacobi Quantization of Systems with Time-Dependent Constraints

Hamilton–Jacobi formalism is used to investigate time-dependent constraint systems. It is proved that the generalization of Dirac's canonical quantization method in the nonstationary case can be obtained naturally in Hamilton–Jacobi formalism. The example of the relativistic particle in a plane wave is analyzed in detail.     

Coherence effects in deep inelastic scattering

A superfield action for the relativistic massless superparticle as a spinning particle is presented in the new gauge. The symmetries in the relativistic superparticle theory, that is to say, the invariances under, the reparametrization and the local supersymmetry-transformation in the parameter space, are manifest in this formalism. It is clear how these two kinds of transformations descend from a unified origin to be called “super-reparametrization”, which is a restricted form of the general coordinate transformation in the superspace. The action is manifestly invariant under these transformation by its constructions The minimal coupling with electromagnetic field is also constructed in the superfield formalism, and a manifestly invariant superfield action for the interacting superparticle is presented in our gauge. The formalism is extended to the construction of an action for theN=2 superparticle.     

Chaos in intrinsic motion of nuclei and its role in dissipative collective dynamics

We shall discuss the role of chaotic intrinsic motion in dissipative dynamics of the collective coordinates for nuclear systems. Using the formalism of linear response theory, it will be shown that the dissipation in adiabatic collective motion depends on the degree of chaos in the intrinsic dynamics of a system. This gives rise to a shape dependent dissipation rate for collective coordinates when the intrinsic motion is described by the independent particle model in a nucleus. The shape dependent chaos parameter measuring the degree of chaos in the intrinsic dynamics of the nuclear system will be obtained using the interpolating Brody distribution of nearest neighbour spacings in the single particle energy spectrum. A similar shape dependence is also found to be essential for phenomenological dissipation rates used in fission dynamics calculations.     

Nonlocal conserved quantities,balance laws,and equations of motion

A formalism is developed whereby balance laws are directly obtained from nonlocal (integrodifferential) linear second-order equations of motion for systems described by several dependent variables. These laws augment the equations of motion as further useful information about the physical system and, under certain conditions, are shown to reduce to conservation laws. The formalism can be applied to physical systems whose equations of motion may be relativistic and either classical or quantum. It is shown to facilitate obtaining global conservation laws for quantities which include energy and momentum. Applications of the formalism are given for a nonlocal Schrödinger equation and for a system of local relativistic equations of motion describing particles of arbitrary integral spin.     

Relativistic version of the imaginary-time formalism

A relativistic version of the quasiclassical imaginary-time formalism is developed. It permits calculation of the tunneling probability of relativistic particles through potential barriers, including barriers lacking spherical symmetry. Application of the imaginary-time formalism to concrete problems calls for finding subbarrier trajectories which are solutions of the classical equations of motion, but with an imaginary time (and thus cannot be realized in classical mechanics). The ionization probability of an s level, whose binding energy can be of the order of the rest energy, under the action of electric and magnetic fields of different configuration is calculated using the imaginary-time formalism. Besides the exponential factor, the Coulomb and pre-exponential factors in the ionization probability are calculated. The Hamiltonian approach to the tunneling of relativistic particles is described briefly. Scrutiny of the ionization of heavy atoms by an electric field provides an additional argument against the existence of the “Unruh effect.” Zh. éksp. Teor. Fiz. 114, 798–820 (September 1998)     

Wigner's pseudo-particle relativistic dynamics in external potential field

The Wigner's pseudo-particle formalism has been generalized to describe quantum dynamics of relativistic particle in external potential field. As a simplest application of the developed formalism the time evolution of the 1D relativistic quantum harmonic oscillator been considered. Due to the complex structure of the evolution equation for Wigner function, the only numerical treatment is possible by combining Monte Carlo and molecular dynamics methods. Relativistic dynamics results in appearance of the new physical effects as opposed to non-relativistic case. Interesting is the complete changing of the shape of the momentum and coordinate distribution functions as well as formation of ‘unexpected’ protuberances. To analyze the influence of relativistic effects on average values of quantum operators, the dependencies on time of average momentum, position, their dispersions and energy have been compared for the non-relativistic and relativistic dynamics.     

Mach's principle and Newtonian mechanics

The union of Mach's principle and Newtonian mechanics gives rise to Relational Mechanics. We find that the characteristics of the revised mechanics are: (1) freedom from any reference to absolute space; (2) the identity of inertial and gravitational mass; (3) the relative acceleration of a body in a gravitational field dependent on the mass of the body. All these results are valid in the context of a Newtonian mechanics which is being developed in the center-of-mass system of all the particles. The conservation of linear momentum, energy, angular momentum are expressed in relational terms, i.e., no reference is made to absolute space. Relational Mechanics is a classical relativistic theory which can be formulated to satisfy Einsteinian relativistic requirements. The Hamiltonian formalism for Relational Mechanics is discussed. Preliminary report Bull. Am. Phys. Soc.14, 15 (1969)     

Quantum Theory and the Category of Complex Numbers

The problem of developing a formalism of quantum theory, which is both consistent with the reality of the measurements and with the invariance properties of relativistic theories, is considered. A solution is found by using a real formulation of quantum mechanics, such that there exists an interpretation of the real properties of a physical system at all times. It is demonstrated also that several concepts in quantum field theory can be recast in a real formalism. PACS: 03.65.Ca; 11.55.Ds.     

A Classical and Quantum Relativistic Interacting Variable-Mass Model

A classical and quantum relativistic interacting particle formalism is revisited. A Hilbert space is achieved through the use of variable individual particle rest masses, but no c-number mass parameter is required for the relativistic free particle. Boosted center of momentum states feature in both the free and interacting model. The implications of a failure to impose simultaneity conditions at the classical level are explored. The implementation of these conditions at the quantum level leads to a finite uncertainty in interaction times, perhaps more closely modeling the exchange of virtual particles in quantum field theory. This work is compared and contrasted with other variable mass models in the literature.     

On the nature of the so-called generic instabilities in dissipative relativistic hydrodynamics

It is shown that the so-called generic instabilities that appear in the framework of relativistic linear irreversible thermodynamics (LIT), describing the fluctuations of a simple fluid close to equilibrium, arise due to the coupling of heat with hydrodynamic acceleration which appears in Eckart’s formalism of relativistic irreversible thermodynamics. Further, we emphasize that such behavior should be interpreted as a contradiction to the postulates of LIT, namely a violation of Onsager’s hypothesis on the regression of fluctuations, and not as fluid instabilities. Such contradictions can be avoided within a relativistic linear framework if a Meixner-like approach to the phenomenological equations is employed.     

The relativistic symmetry group of spin and particle number

We suggest that spin and particle number are more naturally associated with the generators of a group than are spin and mass value (as is usually assumed). This is examined using the hypercomplex number formalism of relativistic physics.     

Quantum mechanics of relativistic spinless particles

A relativistic one-particle, quantum theory for spin-zero particles is constructed uponL ²(x, ct), resulting in a positive definite spacetime probability density. A generalized Schrödinger equation having a Hermitian HamiltonianH onL ²(x, ct) for an arbitrary four-vector potential is derived. In this formalism the rest mass is an observable and a scalar particle is described by a wave packet that is a superposition of mass states. The requirements of macroscopic causality are shown to be satisfied by the most probable trajectory of a free tardyon and a nontrivial framework for charged and neutral particles is provided. The Klein paradox is resolved and a link to the free particle field operators of quantum field theory is established. A charged particle interacting with a static magnetic field is discussed as an example of the formalism.