Canonical realizations of the Poincaré group. II. Space–time description of two particles interacting at a distance, Newtonian-like equations of motion and approximately relativistic Lagrangian formulation

Abstract

The physical meaning of the relativistic action‐at‐a‐distance dynamics for two particles in a canonical framework is investigated on the basis of a general formalism introduced in previous works. Starting from the well‐known prescription given by Bakamjian and Thomas in terms of ’’center‐of‐mass’’ (Q,P) and ’’internal’’ (ρ,π) canonical coordinates, we show how to construct physical, i.e., covariant, position vectors x_τ (Q, P, ρ, π) (τ=1,2) which approach the free particle coordinates in the limit ρ→∞ for short range forces; this procedure is actually performed by means of a 1/c² power expansion for any interaction potential U (ρ,π). In force of the zero‐interaction theorem the physical coordinates, which do satisfy the world‐line condition to any order in 1/c², cannot play the role of canonical variables, i.e., the ’’localizability,’’ {x_τi, x_τj}=0 (τ=1,2), and the ’’causality’’ conditions {x_τi, x_τ′j}=0 (τ,τ′=1,2; τ≠τ′) cannot be simultaneously satisfied. It is possible, however, to satisfy the former set of equations to any order in 1/c² by exploiting the arbitrariness lying in the definition of x₁ and x₂. By means of a suitable choice of a ’’gauge’’ for the internal variables, the remaining freedom is then shown to consist of the appearance of a single scalar function Λ (ρ,π). This function, entering the defining relations of x₁, x₂ in terms of the canonical variables Q, P, ρ, π, plays the role of an additional interaction potential which is effective for the space–time description of the particles in the interaction region, but does not affect the scattering properties of the system. On the other hand, assuming a static nonrelativistic limit of the canonical potential, U⁽⁰⁾=U⁽⁰⁾(ρ), the ’’causality’’ conditions are necessarily violated at the order of the radiation effects (1/c⁴). In terms of x₁, x₂, the equations of motion assume a Newtonian‐like structure m_τẍ_τ=F_τ[x₁−x₂,v₁,v₂] (τ=1,2), of the Currie type or a variety of manifestly covariant forms m_τd²x^μ_τ/ds²_τ=S^μ _νf^ν [x₁(s₁),x₂(s₂),u₁(s₁),u₂(s₂)], where S^μ _ν is the Lorentz transformation which connects the laboratory frame with the Lorentz frame in which x₁(s₁) and x₂(s₂) are simultaneous. A final point is the derivation of the Newtonian‐like equations of motion from a true Lagrangian variational principle δFL [x₁,x₂,v₁,v₂]dt=0. It is shown in general that if U⁽⁰⁾(ρ) ≡0, this can be done only up to the post‐Newtonian approximation, essentially because of the violation of the ’’causality’’ conditions at the order 1/c⁴. Then a general form of approximately relativistic Lagrangian for two particles is derived which actually contains all the examples quoted in the literature, among which the well‐known Darwin–Breit and the Einstein–Infeld–Hoffmann Lagrangians. This investigation appears to disprove the widespread opinion according to which the zero‐interaction theorem prevents the existence of invariant world lines and/or renders the relativity principle vacuous within a Hamiltonian framework.