Approximately relativistic Hamiltonians for interacting particles

Abstract

In the relativistic canonical formalism of Bakamjian and Thomas describing direct particle interactions the generators are defined in terms of the total momentum, the center-of-mass position, and a complete set of additional intrinsic canonical variables. In the interaction region of phase space the transformation linking these variables to individual particle coordinates and momenta is not determined by basic principles. In this paper canonical transformations to single-particle variables valid to order ${c}^{\ensuremath{-}2}$ and the corresponding approximate Hamiltonians are constructed for a two-particle system; approximate many-body Hamiltonians are then constructed from the two-body ones, maintaining the Lie algebra of the Poincar\'e group to the same order. If, and only if, the nonrelativistic limit of the potential is velocity independent (except for a possible spin-orbit interaction) it is possible to require, to order ${c}^{\ensuremath{-}2}$, transformation properties of the position operators corresponding to the classical world-line conditions. This requirement implies restrictions on admissible canonical transformations to single-particle variables. The cluster separability condition is then automatically satisfied. In the classical limit the class of approximately relativistic Hamiltonians for spinless particles is identical with that obtained by Woodcock and Havas from expansion of an exact Poincar\'e-invariant Fokkertype variational principle automatically satisfying the world-line conditions. Conversely, direct quantization of their classical Hamiltonians is shown to lead to the approximate quantum-mechanical ones resulting from the Bakamjian-Thomas theory. The relation of these results to various approximately relativistic Hamiltonians built up by several authors starting from the nonrelativistic theory is discussed, as well as their implications for phenomenological nucleon-nucleon potentials.