Nuovo Cimento B (11) 101, 721-749 (1988)

Classical Poincaré and Galilei invariant Hamiltonian two-particle interactions with commuting position variables

H.P. Künzle

Department of Mathematics, University of Alberta
Edmonton, Canada T6G~2G1
e-mail:HP.Kunzle@UAlberta.ca

Abstract

A five-dimensional space-time formalism is used to describe relativistic and nonrelativistic two-particle interactions in a unified way. The system is assumed to have six degrees of freedom and to admit a Poincaré or Galilei invariant symplectic structure such that position coordinates commute under Poisson brackets if they are "measured" on the standard simultaneous intitial-value surface in the Galileian and on an advanced-retarded surface in the Poincaré invariant case. The most general of such interactions is found. It is characterized uniquely in the relativistic case by one function of three variables and a constant that vanishes when invariance under space reflection is required. The Galilei invariant systems that are limits of such relativistic ones comprise all the standard classical interactions.