Towards relativistic atomic physics. Part 1. The rest-frame instant form of dynamics and a canonical transformation for a system of charged particles plus the electro-magnetic field

Abstract

A complete exposition of the rest-frame instant form of dynamics for arbitrary isolated systems (particles, fields, strings, fluids) admitting a Lagrangian description is given. The starting point is the parametrized Minkowski theory describing the system in arbitrary admissible non-inertial frames in Minkowski space-time, which allows one to define the energy-momentum tensor of the system and to show the independence of the description from the clock synchronization convention and from the choice of the 3-coordinates. The restriction to the inertial rest frame, centered on the inertial observer having the Fokker-Pryce center-of-inertia world-line, and the study of relativistic collective variables replacing the non-relativistic center of mass lead to the description of the isolated system as a decoupled globally-defined non-covariant canonical external center of mass carrying a pole-dipole structure (the invariant mass M and the rest spin S of the system) and an external realization of the Poincare' group. M c and S are the energy and angular momentum of a unfaithful internal realization of the Poincare' group built with the energy-momentum tensor of the system and acting inside the instantaneous Wigner 3-spaces where all the 3-vectors are Wigner covariant. The vanishing of the internal 3-momentum and of the internal Lorentz boosts eliminate the internal 3-center of mass inside the Wigner 3-spaces, so that at the end the isolated system is described only by Wigner-covariant canonical internal relative variables.Then an isolated system of positive-energy charged scalar articles with mutual Coulomb interaction plus a transverse electro-magnetic field in the radiation gauge is investigated as a classical background for defining relativistic atomic physics. The electric charges of the particles are Grassmann-valued to regularize the self-energies. The external and internal realizations of the Poincare' algebra in the rest-frame instant form of dynamics are found. This allows one to define explicitly the rest-frame conditions and their gauge-fixings (needed for the elimination of the internal 3-center of mass) for this isolated system. It is shown that there is a canonical transformation which allows one to describe the isolated system as a set of Coulomb-dressed charged particles interacting through a Coulomb plus Darwin potential plus a free transverse radiation field: these two subsystems are not mutually interacting (the internal Poincare' generators are a direct sum of the two components) and are interconnected only by the rest-frame conditions and the elimination of the internal 3-center of mass. Therefore in this framework with a fixed number of particles there is a way out from the Haag theorem, at least at the classical level.

Relativistic quantum mechanics and relativistic entanglement in the rest-frame instant form of dynamics

“…This allows us to define the rest-frame instant form of dynamics for arbitrary isolated systems: a complete exposition of all its properties has been done in Ref. [8] and extended to non-inertial rest frames in Ref. [9].…”

From Clock Synchronization to Dark Matter as a Relativistic Inertial Effect

“…[28] it is shown that by using the previous results one can find a canonical transformation from the canonical basis η i (τ ), κ i (τ ), A ⊥ (τ, σ r ), π ⊥ (τ, σ r ), in which the internal Poincaré generators have the expression in the case N=2 (B = ∂ × A ⊥ , c(σ) = −1/4π |σ|)…”

“…[26,28,29,33], the three collective variables can be expressed as known functions of the Lorentz-scalar rest time τ = c T s = h ·x = h · Y = h · R, of canonically conjugate Jacobi data (frozen Cauchy data) z = M c x N W (0) and h = P /M c 14 , of the invariant mass M c = √ ǫ P 2 of the system and of its rest spinS.…”

“…B) The evolution equation for the four basic gauge variables θ i (τ, σ u ) and 3 K(τ, σ u ) (the equation for the York time is the Raychaudhuri equation 28 ): these equations determine the lapse and the shift functions once four gaugefixings for the basic gauge variables are given.…”

“…In the electro-magnetic case in SR [28] the regularized coupled secondorder equations of motion of the particles obtained by using the LienardWiechert solutions for the electro-magnetic field are independent by the type of Green function (retarded or advanced or symmetric) used. The electromagnetic retardation effects, killed by the Grassmann regularization, are connected with QED radiative corrections to the one-photon exchange diagram.…”