A positive energy dynamics and scattering theory for directly interacting relativistic particles

Starting from the tensor product of N irreducible positive energy representations of the Poincaré group describing N free relativistic particles with arbitrary spins and positive masses, we construct an interacting positive energy representation by modifying the total 4-momentum operator. We first make a transformation to a Hilbert space on which the free total 4-momentum operator equals the product of a dimensionless center-of-mass 4-vector ($\text{(|k|}{}^2\text{+ 1)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, k) and a free “reduced Hamiltonian” H_r⁰, which is a positive operator acting only on internal variables, and then replace H_r⁰ by an interacting reduced Hamiltonian H_r = H_r⁰ + V, where V commutes with the Lorentz group and is such that H_r is a positive operator. The resulting product form is shown to imply that the wave operators interwine the free and interacting representations so that the S-operator is Lorentz invariant. From a physical point of view the scheme is related to the framework first introduced by Bakamjian and Thomas, in which the Hamiltonian and boost generators are modified, but the above procedure makes a mathematically rigorous discussion much simpler. In the spin-zero case we introduce a natural generalization of the pair potentials of nonrelativistic N-particle Schrödinger theory to the present relativistic setting, study its scattering theory, and point out some problems that do not have analogs at the nonrelativistic level. In the $\text{spin}\text{-}\text{1}\text{2}$ case we propose, inspired by the Dirac equation, explicit reduced Hamiltonians to describe atomic energy levels and present arguments making plausible that their eigenvalues are in closer agreement with the experimental data than their nonrelativistic counterparts. We also consider extensions to arbitrary spin and, in the $\text{spin}\text{-}\text{1}\text{2}$ case, coupling of a quantized radiation field. In view of eventual applications to “completely integrable” one-dimensional field theories the case of one space dimension is studied as well, both in quantum mechanics and in classical mechanics.