Separable interactions in classical relativistic Hamiltonian mechanics (original) (raw)
Relativistic and separable classical Hamiltonian particle dynamics
Annals of Physics, 1981
We show within the Hamiltonian formalism the existence of classical relativistic mechanics of N scalar particles interacting at a distance which satisfies the requirements of Poincart invariance, separability, world-line invariance and Einstein causality. The line of approach which is adopted here uses the methods of the theory of systems with constraints applied to manifestly covariant systems of particles. The study is limited to the case of scalar interactions remaining weak in the whole phase space and vanishing at large space-like separation distances of the particles. Poincare invariance requires the inclusion of many-body, up to N-body, potentials. Separability requires the use of individual or two-body variables and the construction of the total interaction from basic two-body interactions. Position variables of the particles are constructed in terms of the canonical variables of the theory according to the world-line invariance condition and the subsidiary conditions of the non-relativistic limit and separability. Positivity constraints on the interaction masses squared of the particles ensure that the velocities of the latter remain always smaller than the velocity of light.
The European Physical Journal Plus, 2012
An open issue in classical relativistic mechanics is the consistent treatment of the dynamics of classical N -body systems of mutually-interacting particles. This refers, in particular, to charged particles subject to EM interactions, including both binary and self interactions (EM-interacting N -body systems). The correct solution to the question represents an overriding prerequisite for the consistency between classical and quantum mechanics. In this paper it is shown that such a description can be consistently obtained in the context of classical electrodynamics, for the case of a N -body system of classical finite-size charged particles. A variational formulation of the problem is presented, based on the N -body hybrid synchronous Hamilton variational principle. Covariant Lagrangian and Hamiltonian equations of motion for the dynamics of the interacting N -body system are derived, which are proved to be delay-type ODEs. Then, a representation in both standard Lagrangian and Hamiltonian forms is proved to hold, the latter expressed by means of classical Poisson Brackets. The theory developed retains both the covariance with respect to the Lorentz group and the exact Hamiltonian structure of the problem, which is shown to be intrinsically nonlocal. Different applications of the theory are investigated. The first one concerns the development of a suitable Hamiltonian approximation of the exact equations that retains finite delay-time effects characteristic of the binary and self EM interactions. Second, basic consequences concerning the validity of Dirac generator formalism are pointed out, with particular reference to the instant-form representation of Poincarè generators. Finally, a discussion is presented both on the validity and possible extension of the Dirac generator formalism as well as the failure of the so-called Currie "no-interaction" theorem for the non-local Hamiltonian system considered here.
Relativistic N-body problem in a separable two-body basis
Physical Review C, 2001
We use Dirac's constraint dynamics to obtain a Hamiltonian formulation of the relativistic N -body problem in a separable two-body basis in which the particles interact pair-wise through scalar and vector interactions. The resultant N -body Hamiltonian is relativistically covariant. It can be easily separated in terms of the center-of-mass and the relative motion of any twobody subsystem. It can also be separated into an unperturbed Hamiltonian with a residual interaction. In a system of two-body composite particles, the solutions of the unperturbed Hamiltonian are relativistic two-body internal states, each of which can be obtained by solving a relativistic Schrödingerlike equation. The resultant two-body wave functions can be used as basis states to evaluate reaction matrix elements in the general N -body problem.
A model forN classical relativistic particles
Il Nuovo Cimento A, 1981
A model for ~V classical relativistic particles with actionat-a.distance interaction is proposed. It is the generalization of previous models for two particles. As in these models the interaction acts instantaneously among the particles in the centre-of-mass frame. There is a universal arbitrary potential, and therefore the system turns out to be only quasi-separable for long-range interactions. However, we find that there is separability for finite-range interactions. l. -Introduction.
Four-body problem in d-dimensional space: Ground state, (quasi)-exact-solvability. IV
Journal of Mathematical Physics
Due to its great importance for applications, we generalize and extend the approach of our previous papers to study aspects of the quantum and classical dynamics of a 4-body system with equal masses in d-dimensional space with interaction depending only on mutual (relative) distances. The study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only. The ground state (and some other states) in the quantum case and some trajectories in the classical case are of this type. We construct the quantum Hamiltonian for which these states are eigenstates. For d ≥ 3, this describes a six-dimensional quantum particle moving in a curved space with special d-independent metric in a certain d-dependent singular potential, while for d = 1 it corresponds to a three-dimensional particle and coincides with the A 3 (4-body) rational Calogero model; the case d = 2 is exceptional and is discussed separately. The kinetic energy of the system has a hidden sl(7, R) Lie (Poisson) algebra structure, but for the special case d = 1 it becomes degenerate with hidden algebra sl(4, R). We find an exactlysolvable four-body S 4-permutationally invariant, generalized harmonic oscillator-type potential as well as a quasi-exactly-solvable four-body sextic polynomial type potential with singular terms. Naturally, the tetrahedron whose vertices correspond to the positions of the particles provides pure geometrical variables, volume variables, that lead to exactly solvable models. Their generalization to the n-body system as well as the case of non-equal masses is briefly discussed.
Comparison of several approaches to the relativistic dynamics of directly interacting particles
Annals of Physics, 1983
Several approaches to the relativistic dynamics of directly interacting particles are compared. The equivalence between constrained Hamiltonian relativistic systems and a priori Hamiltonian predictive ones is completely proved. Coordinate transformations are obtained to express these systems in the framework of noncovariant predictive mechanics. The world line condition for constrained Hamiltonian relativistic systems is analyzed and is proved to be also necessary in the predictive Hamiltonian framework.
Condensed Matter Physics, 2001
The procedure of reducing canonical field degrees of freedom for a system of charged particles plus field in the constrained Hamiltonian formalism is elaborated up to the first order in the coupling constant expansion. The canonical realization of the Poincaré algebra in the terms of particle variables is found. The relation between covariant and physical particle variables in the Hamiltonian description is written. The system of particles interacting by means of scalar and vector massive fields is also considered. The first order approximation in c −2 is examined. An application to calculating the relativistic partition function of an interacting particle system is discussed.
Journal of Mathematical Physics, 2000
This selective review is written as an introduction to the mathematical theory of the Schro ¤ d inger ¥ equation for N ¦ particles. § Characteristic for these systems are the cluster properties of the potential in configuration space, which are expressed in a simple © geometric language. The methods developed over the last 40 years to deal with this primary aspect are described by giving full proofs of a number of basic and by now classical results. The central theme is the interplay between the spectral theorÿ of N ¦-body Hamiltonians and the space-time and phase-space analysis of bound states and scattering states.