Symmetry of the Relativistic Two-Body Bound State

L.P Horwitz,R.I Arshansky

doi:10.3390/sym12020313

Abstract

We show that in a relativistically covariant formulation of the two-body problem, the bound state spectrum is in agreement, up to relativistic corrections, with the non-relativistic bound-state spectrum. This solution is achieved by solving the problem with support of the wave functions in an O ( 2 , 1 ) invariant submanifold of the Minkowski spacetime. The O ( 3 , 1 ) invariance of the differential equation requires, however, that the solutions provide a representation of O ( 3 , 1 ) . Such solutions are obtained by means of the method of induced representations, providing a basic insight into the subject of the symmetries of relativistic dynamics.

Highlights

In the non-relativistic Newtonian-Galilean view, two particles may be thought of as interacting through a potential function V (x1 (t), x2 (t)); for Galilean invariance, V must be a scalar function of the difference, i.e., V (x1 (t) − x2 (t))
Two world lines with action at a distance interaction between two points μ μ x1 and x2 cannot be correlated by the variable t in every frame
The basic idea of the SHP theory is the parametrization of the world lines of particles with a universal parameter τ [5]

Summary

Introduction

In the non-relativistic Newtonian-Galilean view, two particles may be thought of as interacting through a potential function V (x1 (t), x2 (t)); for Galilean invariance, V must be a scalar function of the difference, i.e., V (x1 (t) − x2 (t)). In such a potential model, x1 and x2 are taken to be at equal time, corresponding to a correlation between the two particles consistent with the Newtonian-Galilean picture. Stueckelberg wrote an invariant Hamiltonian of the form It follows from (1) that the proper time ds2 = −dx μ dxμ satisfies pμ pμ m2.

The Two-Body Bound State

The Spectrum

Some Examples

The Induced Representation

Findings

Conclusions