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

Abstract

Starting from the tensor product of N irreducible positive energy representations of the PoincarC 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 ((I k I2 + 1)1/2, k) and a free "reduced Hamiltonian"Hro, which is a positive operator acting only on internal variables, and then replace HrO by an interacting reduced Hamiltonian H, = H> + V, where V commutes with the Lorentz group and is such that H, is a positive operator. The resulting product form is shown to imply that the wave operators intertwine 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 Schrodinger 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 spin-& 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 spin-$ 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.

Sine-Gordon Solitons vs. Relativistic Calogero-Moser Particles

“…The sine-Gordon equation (8~ -8~)4' =sin</> (1) defines a relativistically invariant field theory that has been studied very extensively. Indeed, there are hundreds of papers that have a bearing on it, yielding information from a great many angles.…”

“…In most of these papers, one of two possible interpretations of {l} is chosen, namely as a classical or as a quantum equation. Accordingly, one either views (1) as a nonlinea.r evolution equation for a real-valued function </J(t, y), (t,y) E R. 2, or as an interacting relativistic quantum field theory in two space-time dimensions.…”

“…Beginning with the classical level, we illustrate the N-soliton solutions to (1) with Fig. 1, which depicts a 3-soliton collision.…”

“…below.) Second, in (1) no scale parameters occur, and accordingly we need a special choice for our scale parameters M, c andµ to make contact with (1), viz., M = c = µ, = 1.…”

“…It is by no means obvious, but true that </; (t, y) is an N-soliton solution to (1). Furthermore, one obtains all N-soliton solutions by letting u vru:y over n (12).…”