Relative motion in spacetime

In Minkowski spacetime, we consider an isolated system made of two pointlike bodies interacting at a distance, in the nonradiative approximation. Our framework is the covariant and a priori Hamiltonian formalism of "predictive relativistic mechanics", founded on the equal-time condition. The issue of an equivalent one-body description is discussed. We distinguish two different concepts: on the one hand an almost purely kinematic relative particle, on the other hand an effective particle which involves an explicit dynamical formulation; several versions of the latter are possible. Relative and effective particles have the same orbit, but may differ by their schedules.Comment: 26 pages, no figure. Several inadequacies and various misprints corrected. An example, an Appendix, one reference added; some obscure points clarifie

Binding Energy in Two and Three-Body Relativistic Dynamics

Two-body and three-body systems of scalar bosons are considered in the framework of covariant constraint dynamics. The reduced equation obtained after eliminating redundant degrees of freedom can be viewed as an eigenvalue equation for an observable which is intimately related with the relative motion. We display the connection of this observable with binding energy.Comment: 12 pages, LaTeX. Talk presented at the Workshop "Critical Stability of Few-Body Quantum Systems", Les Houches Oct. 8-12, 200

Covariant Model for Relativistic Three-Body Systems

The system is described by three mass-shell constraints. After a nonlinear transformation of the momenta, the analytic form taken by admissible interactions (allowing compatibility) is characterized in terms of the new variables. These variables mix two-body clusters, which results in automatically incorporating three-body forces. Two superfluous degrees of freedom are eliminated, which yields a reduced equation for a wave function depending on three-dimensional arguments. When at least two masses are equal, this picture has a reasonable nonrelativistic limit. At first post-Galilean order and provided the interaction is not too much energy-dependent, the relativistic correction is tractable like a conventional perturbation problem. A covariant version of harmonic oscillator is given as a toy model.Comment: Latex without style files: 5 pages. Talk given at "Quark Confinement and the Hadron Spectrum VI", Villasimius, Cagliari, Sardinia, Italy. 21-25 September 200

Scalar products of elementary distributions

The field of real numbers being extended as a larger commutative field, we investigate the possibility of defining a scalar product for the distributions of finite discrete support. Then we focus on the most simple possible extension (which is an ordered field), we provide explicit formulas for this scalar product, and we exhibit a structure of positive definite inner-product space. In a one-dimensional application to the Schroedinger equation, the distributions supported by the origin are embedded into a bra-ket vector space, where the "singular" potential describing point interaction is defined in a natural way. A contact with the hyperreal numbers that arise in nonstandard analysis is possible but not essential, our extensions of $\bf R$ and $\bf C$ being obtained by a quite elementary method.Comment: 27 page

Direct Interactions in Relativistic Statistical Mechanics

Directly interacting particles are considered in the multitime formalism of predictive relativistic mechanics. When the equations of motion leave a phase-space volume invariant, it turns out that the phase average of any first integral, covariantly defined as a flux across a $7n$-dimensional surface, is conserved. The Hamiltonian case is discussed, a class of simple models is exhibited, and a tentative definition of equilibrium is proposed.Comment: Plain Tex file, 26 page