Bertrand’s Theorem and the Double Copy of Relativistic Field Theories

Which relativistic field theories give rise to Kepler dynamics in thetwo-body problem? We consider a class of Hamiltonians that is the uniquerelativistic extension of the Kepler problem preserving its so(4) algebra, andhave orbits related through time reparametrisation to orbits of the originalKepler problem. For three explicit examples, we give a natural interpretationin terms of spin-0,-1 and -2 interacting field theories in 5D. These areorganically connected via the classical double copy, which therefore preservesmaximal superintegrability.