Form Of Relativistic Dynamics Research Articles

Descriptions of Relativistic Dynamics with World Line Condition

In this paper, a generalized form of relativistic dynamics is presented. A realization of the Poincaré algebra is provided in terms of vector fields on the tangent bundle of a simultaneity surface in R 4 . The construction of this realization is explicitly shown to clarify the role of the commutation relations of the Poincaré algebra versus their description in terms of Poisson brackets in the no-interaction theorem. Moreover, a geometrical analysis of the “eleventh generator” formalism introduced by Sudarshan and Mukunda is outlined, this formalism being at the basis of many proposals which evaded the no-interaction theorem.

Triton Binding Energy of Kharkov Potential

The Kharkov potential is a recent field theoretical model of nucleon–nucleon (NN) interaction that has been built up in the framework of the instant form of relativistic dynamics starting with the total Hamiltonian of interacting meson and nucleon fields and using the method of unitary clothing transformations. The latter connect the representation of “bare” particles and the representation of “clothed” particles, i.e., the particles with physical properties. Unlike many available NN potentials each of which is the kernel of the corresponding nonrelativistic Lippmann–Schwinger (LS) equation this potential being dependent in momentum space on the Feynman-like propagators and covariant cutoff factors at the meson–nucleon vertices is the kernel of relativistic integral equations for the NN bound and scattering states. Therefore we do not need to invent any transform of a given nonrelativistic potential to its relativistic counterpart. As a feasible study, we have started with the so-called 5ch Faddeev calculation for three-nucleon bound state (triton) and obtained a reasonable value of its binding energy ( $$-7.42$$ MeV).

Interpolation Between the Instant Form and the Front Form of Relativistic Dynamics

The instant form and the front form of relativistic dynamics proposed by Dirac (Rev Mod Phys 21:392–399, 1949) can be linked by introducing an interpolating angle. The instant form dynamics (IFD) provides a traditional approach evolved from non-relativistic dynamics and makes a close contact with Euclidean space developing temperature-dependent quantum field theory, lattice QCD, etc. The front form dynamics, now known as the light-front dynamics (LFD), however, works strictly in Minkowski space and provides useful frameworks to study deep inelastic scattering, parton distribution functions, deeply virtual Compton scattering, generalized parton distributions, etc. We present the recent development of the interpolation between IFD and LFD and discuss ramifications from this development.

Scattering Amplitudes Interpolating Between Instant Form and Front Form of Relativistic Dynamics

Among the three forms of relativistic Hamiltonian dynamics proposed by Dirac in 1949, the front form has the largest number of kinematic generators. This distinction provides useful consequences in the analysis of physical observables in hadron physics. Using the method of interpolation between the instant form and the front form, we introduce the interpolating scattering amplitude that links the corresponding time-ordered amplitudes between the two forms of dynamics and provide the physical meaning of the kinematic transformations as they allow the invariance of each individual time-ordered amplitude for an arbitrary interpolation angle. We discuss the rationale for using front form dynamics, nowadays known as light-front dynamics (LFD), and present a few explicit examples of hadron phenomenology that LFD uniquely can offer from first-principles quantum chromodynamics. In particular, model-independent constraints are provided for the analyses of deuteron form factors and the N Δ transition form factors at large momentum transfer squared Q2. The swap of helicity amplitudes between the collinear and non-collinear kinematics is also discussed in deeply virtual Compton scattering.

Clothing of particles in the meson-nucleon system: Mass and charge renormalization

Generators of the Poincare group have been constructed in the instant form of relativistic dynamics in the second and third orders in the coupling constant, using the method of unitary clothing transformations. It is shown that the algebra of group generators is satisfied in the corresponding orders. The relativistic invariance of the calculated corrections to the particle masses and coupling constant is substantiated.

CLASSICAL RELATIVISTIC SYSTEMS OF CHARGED PARTICLES IN THE FRONT FORM OF DYNAMICS AND THE LIOUVILLE EQUATION

Classical relativistic system of point particles coupled with an electromagnetic field is considered in the three-dimensional representation. The gauge freedom connected with the chronometrical invariance of the four-dimensional description is reduced by use of the geometrical concept of the forms of relativistic dynamics. The remainder gauge degrees of freedom of the electromagnetic potential are analysed within the framework of Dirac’s constrained Hamiltonian mechanics in the front form of dynamics. The results are applied to the problems of relativistic statistical mechanics. Based on the corresponding Liouville equation the classical partition function of the system is written down in a gauge-invariant manner and an integration over field variables is performed.

Group quantization of parametrized systems. I. Time levels

A method of quantizing parametrized systems is developed that is based on a kind of ‘‘gauge invariant’’ quantities—the so-called perennials (a perennial must also be an ‘‘integral of motion’’). The problem of time in its particular form (frozen time formalism, global problem of time, multiple choice problem) is met, as well as a related difficulty characteristic for this type of theory: the paucity of perennials. The present paper is an attempt to find some remedy in the ideas on ‘‘forms of relativistic dynamics’’ by Dirac. Some aspects of Dirac’s theory are generalized to all finite-dimensional first-class parametrized systems. The generalization is based on replacing the Poincaré group and the algebra of its generators as used by Dirac by a canonical group of symmetries and by an algebra of elementary perennials. A number of insights are gained; the following are the main results. First, conditions are revealed under which the time evolution of the ordinary quantum mechanics, or a generalization of it, can be constructed. The construction uses a kind of gauge and time choice and it is described in detail. Second, the theory is structured so that the quantum mechanics resulting from different choices of gauge and time are compatible. Third, a practical way is presented of how a broad class of problems can be solved without the knowledge of explicit form of perennials.

Relativistic effects in systems of three constituent quarks

We investigate the most general form of the three-body mass operator in the instant form of relativistic dynamics. It is shown that this operator is defined not only by the two-body mass operators and the three-body interaction operator but also by some three extra unitary operators. The latter are nontrivial already on the one-gluon-exchange level and, therefore, must be present in any relativistic constituent quark model. The restrictions imposed by the relativistic invariance on the form of the two-body energy operator in first order in 1/[ital m][sup 2] are discussed in detail. We write down the explicit expression for the three-quark mass operator in this order and discuss the dependence of this operator on the total momenta of each pair of quarks in the c.m. frame of three quarks.

Relativistic kinematics of a solid medium in a three-dimensional description

We propose a three-dimensional formulation of the relativistic kinematics of a solid medium based on the geometric concept of the forms of relativistic dynamics. We introduce the concepts of instantaneous configuration of a body, the velocity field, and different measures of deformations and the deformation velocity tensor. We consider the definition of rigid motion in a given form of dynamics. We study the transformation properties of the kinematic characteristics introduced and also those of the quantities that arise under three-dimensional decomposition of the energy-momentum tensor of the medium with respect to the transformations of the Poincare group with recomputation to a new simultaneity.

Transformation properties of certain characteristics of the motion of a deformable solid body in different forms of relativistic dynamics

We establish the transformation properties of certain characteristics of the motion of a deformable solid body (velocities, strains, stresses) with respect to a realization of the Poincare group by nonpoint infinitesimal transformations of the configuration space—a three-dimensional Riemannian manifold with the metric induced by the imbedding in the Minkowski space for an arbitrary form of relativistic dynamics.

Group-theoretic approach to the construction of relativistic lagrangian mechanics of a system of particles

Mathematical aspects of the Lagrangian formalism of relativistic mechanics of a system of interacting particles are considered. A geometric definition of a form of relativistic dynamics possessing the spacelike or isotropic foliation of Minkowski space is introduced. A realization of the Lie algebra of the Poincare group by means of Lie-Backlund vector fields on a general jet continuation of the configuration space is constructed. Invariance conditions of Lagrangian relativistic mechanics are formulated and investigated; the characteristic features of this formalism, which arises as a consequence of the demands of Poincare invariance, are described.

Interpolating between instant and light front dynamics

The “instant” and “light front” forms of relativistic dynamics postulated by Dirac refer to physical conditions on flat spacelike surfaces in the Minkowski space. We consider another flat surface in the Minkowski space that upon parametric rotation interpolates between “instant” and “light front” surfaces. A projection of the covariant perturbation theory on the new surface is then considered. One limit of the theory reproduces the usual instant dynamics, whereas a different limit gives a light front dynamics. Neither infinite momemtum frame limit nor infinite momenta are involved. Although the interpolating surface is the “instant” surface in the moving system of reference, we argue why our procedure should not be associated with a Lorentz transformation.

Modifications of forms of relativistic dynamics associated with Poincar� transformations

Modifications of forms of relativistic dynamics associated with Poincar� transformations

Form of relativistic dynamics with world lines

In any Hamiltonian relativistic theory there are ten generators of the Poincar\'e group which are realized canonically. The dynamical evolution is described by a Hamiltonian which is one of the ten generators in Dirac's generator formalism. The requirement that the canonical transformations reproduce the geometrical transformation of world points generates the world-line conditions. The Dirac identification of the Hamiltonian and the world-line conditions together lead to the no-interaction theorem. Interacting relativistic theories with world-line conditions should go beyond the Dirac theory and have eleven generators. In this paper we present a constraint dynamics formalism which describes an eleven-generator theory of $N$ interacting particles using $8(N+1)$ variables with suitable constraints. The $(N+1)\mathrm{th}$ pair of four-vectors is associated with the uniform motion of a center which coincides with the center of energy for free particles. In such theories dynamics and kinematics cannot be separated out in a simple fashion.

Physical equivalence of the three forms of relativistic dynamics and addition of interactions in the front and instant forms

Physical equivalence of the three forms of relativistic dynamics and addition of interactions in the front and instant forms

Physical equivalence of the point and instantaneous forms of relativistic dynamics

Physical equivalence of the point and instantaneous forms of relativistic dynamics

New form of dynamics and current algebra

We study a new form of dynamical system, in which the commutation relations for the dynamical variables of a quantized field are defined on a “lightlike surface τ≡ (t+z) 2 =0 rather than at one instant of time t = 0. We clarify the physical implications of the use of the new variables x 1 = x, x 2 = y, x += (t+z) 2 , x −= (t−z) 2 and explore its significance as a new form of relativistic dynamics, which holds in any Lorentz frame but not in the so-called “infinite momentum frame.” Using the quark model, we build up a new algebra of currents, in which the current commutators are defined at equal τ. The sum rules and other results of the usual current algebra can be obtained without taking the unjustifiable limit of infinite momentum. In particular, we obtain the Gell-Mann-Okubo mass formulas in quadratic form for both mesons and baryons without the trouble due to momentum dependence. We derive the reduction formula and find the physical high energy limit (not the Bjorken limit) of an amplitude is determined by the equal τ commutator.

Canonical formalism in special relativity

A covariant hamiltonian description was introduced in the dynamics of charges and electromagnetic interaction. By a canonical transformation this hamiltonian formalism was transformed to obtain the Dirac generators for any form of relativistic dynamics, as coefficients of a first degree polynomial in the ten translation and rotation velocities of the Poincaré transformation. The Currie's world line conditions were generalized to any form of the dynamics. The explicit relation between the covariant field variables and the more usual 3-dimensional Fourier variables was derived.