Electromagnetic Field Theory without Divergence Problems 2. A Least Invasively Quantized Theory

The classical Maxwell--Born--Infeld field equations coupled with a Hamilton--Jacobi law of point charge motion are partially quantized by coupling the Hamilton-Jacobi phase function with an amplitude function, which combines with the phase function into a single complex wave function satisfying a relativistic Klein--Gordon equation self-consistently coupled to the evolution equations for the electromagnetic fields with generic point source (explicitly worked out for one particle; options for many particles briefly discussed). Radiation-free stationary states exist. The hydrogen spectrum with infinitely massive nucleus is discussed in some detail and upper estimates for Born's `aether constant' obtained. In the nonrelativistic limit the model reduces to the de-Broglie--Bohm formulation of quantum mechanics.Comment: Corrections at galley stage incorporated (mostly minor corrections, except for a blunder in the estimate of the error term U to the Coulomb interaction) 38p; to appear in JSP vol. 116, issue dedicated to Elliott H. Lieb on his 70th birthday. Part I is math-ph/030607

Electromagnetic field theory without divergence problems: 1. The Born Legacy

A fully consistent classical relativistic electrodynamics with spinless point charges is constructed. The classical evolution of the electromagnetic fields is governed by the nonlinear Maxwell--Born--Infeld field equations, the classical evolution of the point charges by a many-body Hamilton--Jacobi law of motion. The Pauli principle for bosons can be incorporated in the classical Hamilton--Jacobi formalism. The Cauchy problem is explained and illustrated with examples. The question of charge-free field solitons is addressed also and it is shown that if they exist, their peak field strengths must be enormous. The value The value of Born's constant is shown to be a subtle open issue.Comment: Minor corrections at galley stage incorporated. 66p; to appear in JSP vol. 116, issue dedicated to Elliott H. Lieb on his 70th birthday. Part II is math-ph/031103

Symmetry Results for Finite-Temperature, Relativistic Thomas-Fermi Equations

In the semi-classical limit, the quantum mechanics of a stationary beam of counter-streaming relativistic electrons and ions is described by a nonlinear system of finite-temperature Thomas-Fermi equations. In the high temperature / low density limit these Thomas-Fermi equations reduce to the (semi-)conformal system of Bennett equations discussed earlier by Lebowitz and the author. With the help of a sharp isoperimetric inequality it is shown that any hypothetical particle density function which is not radially symmetric about and decreasing away from the beam's axis would violate the virial theorem. Hence, all beams have the symmetry of the circular cylinder.Comment: Final version. To appear in Commun. Math. Phys. (LaTeX, 26 pages