Horace W. Crater

The Rest-Frame Darwin Potential from the Lienard–Wiechert Solution in the Radiation Gauge

Abstract In the semiclassical approximation in which the electric charges of scalar particles are described by Grassmann variables (Q2i=0, QiQj≠0), it is possible to re-express the Lienard–Wiechert potentials and electric fields in the radiation gauge as phase space functions, because the difference among retarded, advanced, and symmetric Green functions is of order Q2i. By working in the rest-frame instant form of dynamics, the elimination of the electromagnetic degrees of freedom by means of suitable second classs contraints leads to the identification of the Lienard–Wiechert reduced phase space containing only N charged particles with mutual action-at-a-distance vector and scalar potentials. A Darboux canonical basis of the reduced phase space is found. This allows one to re-express the potentials for arbitrary N as a unique effective scalar potential containing the Coulomb potential and the complete Darwin one, whose 1/c2 component agrees with the known expression. The effective potential gives the classical analogue of all static and non-static effects of the one-photon exchange Feynman diagram of scalar electrodynamics.

Hamiltonian relativistic two-body problem: center of mass and orbit reconstruction

After a short review of the history and problems of relativistic Hamiltonian mechanics with action-at-a-distance inter-particle potentials, we study isolated two-body systems in the rest-frame instant form of dynamics. We give explicit expressions of the relevant relativistic notions of center of mass, we determine the generators of the Poincare’ group in presence of interactions and we show how to do the reconstruction of particles’ orbits from the relative motion and the canonical non-covariant center of mass. In the case of a simple Coulomb-like potential model, it is possible to integrate explicitly the relative motion and show the two dynamical trajectories.

Two-body Dirac equations

Abstract P. A. M. Dirac took the matrix square root of the Klein-Gordon equation to obtain his relativistic wave equation for a single spin-one-half particle. In this paper, we use Diracs constraint mechanics and supersymmetry to perform the same operation on the relativistic description of two spinless particles to obtain consistent descriptions of two interacting particles, either or both of which may have spin one-half. The resulting coupled quantum wave equations correctly incorporate relativistic kinematics as well as heavy-particle limits to one-body Dirac or Klein-Gordon equations. The 16-component wave equations for the system of two spin one-half particles separate exactly into four decoupled four-component equations for the analogs of “upper” and “lower” components of the Dirac equation. Perturbative treatment of our equations through 0( α 4 ) automatically reproduces the appropriate fine structure. Furthermore, like the decoupled forms of Diracs equation, the two-body versions have spin-dependent pieces that make non-perturbative quantum-mechanical sense. This feature eliminates the need for extra smoothing parameters in the potential or finite particle size in phenomenological applications.

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

A new formulation of relativistic quantum mechanics is proposed in the framework of the rest-frame instant form of dynamics, where the world-lines of the particles are parametrized in terms of the Fokker-Pryce center of inertia and of Wigner-covariant relative 3-coordinates inside the instantaneous Wigner 3-spaces, and where there is a decoupled (non-covariant and non-local) canonical relativistic center of mass. This approach: (a) allows us to make a consistent quantization in every inertial frame; (b) leads to a description of both bound and scattering states; (c) offers new insights on the relativistic localization problem; (d) leads to a non-relativistic limit with a Hamilton-Jacobi treatment of the Newton center of mass; (e) clarifies non-local aspects (spatial non-separability) of relativistic entanglement connected with Lorentz signature and not present in its non-relativistic treatment.

Relativistic quark potential for the vector mesons

Abstract We extend the heavy quark potential recently proposed by Richardson to the light quarks by using Diracs theory of constraints. Our relativistic version reproduces the ground state spectrum of the light vector mesons while slightly improving Richardsons own results for the ψ and ϒ systems.

Scalar interactions of supersymmetric relativistic spinning particles

We introduce scalar interactions for the relativistic spinning particle in such a way as to preserve a supersymmetry that leaves a special position‐variable invariant. This generates systems of particles in scalar interaction with a supersymmetry for each spinning particle. For two‐particle systems the supersymmetry eliminates all spin complications and reduces consistency problems to those of a purely bosonic system. Once the latter are disposed of, our approach leads to consistent systems of quantum mechanical wave equations.

Towards relativistic atomic physics. Part II. Collective and relative relativistic variables for a system of charged articles plus the electromagnetic field

In this second paper, we complete the classical description of an isolated system of “charged positive-energy particles, with Grassmann-valued electric charges and mutual Coulomb interaction, plus a transverse electromagnetic field” in the rest-frame instant form of dynamics. In particular, we show how to determine a collective variable associated with the internal 3-center of mass on the instantaneous 3-spaces, to be eliminated with the constraints K(int)≈0. Here, K(int) is the Lorentz boost generator in the unfaithful internal realization of the Poincare group and its vanishing is the gauge-fixing to the rest-frame conditions P(int)≈0. We show how to find this collective variable for the following isolated systems: (a) charged particles with a Coulomb plus Darwin mutual interaction; (b) transverse radiation field; (c) charged particles with a mutual Coulomb interaction plus a transverse electro-magnetic field. Then we define the Dixon multipolar expansion for the open particle subsystem. We also define ...

Extension of two‐body Dirac equations to general covariant interactions through a hyperbolic transformation

In previous work, Dirac’s constraint mechanics and supersymmetry were used to construct two‐body Dirac equations for particles interacting through world scalar and vector potentials. The resulting compatible 16‐component wave equations, S1ψ=γ51(γ1⋅(p1−A1) +m1+S1)ψ=0, S2ψ=γ52(γ2⋅(p2−A2) +m2+S2)ψ=0, yield Schrodinger‐like equations in the center‐of‐momentum system with simpler structure than that possessed by the Bethe–Salpeter equation or any of its standard three‐dimensional truncations. For strong interactions, these equations have yielded a relativistic quark model for meson spectroscopy, while for electromagnetic interactions they have yielded a family of exact singlet positronium solutions. This paper uncovers a hyperbolic interaction structure in these equations. This structure is used to generalize these equations to include pseudoscalar, pseudovector, and tensor interactions.

Massless particles plus matter in the rest-frame instant form of dynamics

After introducing the parametrized Minkowski theory describing a positive-energy scalar massless particle, we study the rest-frame instant form of dynamics of such a particle in the presence of another massive one (to avoid the front form of dynamics). Then we describe massless particles with Grassmann-valued helicity and their quantization.

A relativistic version of a two-level atom in the rest-frame instant form of dynamics

We define a relativistic version of a two-level atom, in which an extended atom is replaced by a point particle carrying suitable Grassmann variables for the description of the two-level structure and that of the electric dipole. After studying the isolated system ‘atom plus the electro-magnetic field’ in the electric-dipole representation as a parameterized Minkowski theory, we give its restriction to the inertial rest frame and the explicit form of the Poincare generators. After quantization, we get a two-level atom with a spin-1/2 electric dipole and the relativistic generalization of the Hamiltonians of the Rabi and Jaynes–Cummings models.