The spinning-electron solution in Born's nonlinear electrodynamics and the concurrent determination of the absolute field constant b

The absolute field constant $b$ of Born's nonlinear electrodynamics was fixed in 1934 from the electrostatic self-energy of the electron, a calculation that neglected the electron's spin entirely; the 1935 Born–Schrödinger correction kept only the spin's magnetostatic energy and found an electron radius an order of magnitude larger and an absolute field constant roughly twenty times smaller. What has never been obtained is the concurrent, self-consistent solution: a stationary finite-energy configuration of the Maxwell–Born–Infeld equations carrying the electron's charge $e$, rest energy $mc^2$, and Bohr-magneton magnetic moment together. The 1930s ring-singularity attempts stalled, and the modern mathematical literature establishes existence only perturbatively for small regular sources, leaves uniqueness open when both charge and current are present, and calls the value of Born's constant a burning open question. The problem is to construct this spinning-electron solution and derive from it an accurate $b$ and electron radius, or prove rigorously that none exists.

Background

Born's nonlinear electrodynamics is the classical field theory proposed by Max Born in 1933 and elaborated with Leopold Infeld in 1934, intended to remove the infinite self-energy of the point electron. Its Lagrangian replaces Maxwell's quadratic energy density by a square-root form containing a single new universal constant $b$, the absolute field: in a purely electrostatic situation $b$ is the upper limit of the electric field strength, so the Coulomb field of a point charge is capped and smoothed and its total energy is finite. In the general case $b$ limits a certain invariant of the field, computed in the Lorentz frame in which the Poynting vector (the electromagnetic energy-flux density) vanishes at the point considered.

Born and Infeld fixed the numerical value of $b$ by identifying the rest energy of the electron with the total electrostatic energy of the spherically symmetric solution carrying the elementary charge $e$. This gives $b = 3.96\times 10^{15}$ E.S.U. (electrostatic units, the Gaussian unit of field strength) and an effective electron radius $r_{el} = 1.236\,r_0$, defined as the length $\sqrt{e/b}$ at which the Coulomb field of the point charge $e$ would reach $b$; here $r_0 = e^2/(mc^2) \approx 2.8$ fm is the classical electron radius, and the factor 1.2361 comes from the Born–Infeld energy integral, which evaluates to $\tfrac{1}{6}B(1/4,1/4)\sqrt{e^3 b}$ with $B$ the Euler Beta function. In the modern dimensionless convention this corresponds to an aether constant $\beta_B \approx 1.236\,\alpha$, with $\alpha = e^2/\hbar c \approx 1/137$ the Sommerfeld fine-structure constant.

The electrostatic calculation, however, neglects the electron's spin entirely. The electron has spin $\hbar/2$ and, with it, a magnetic moment equal to the Bohr magneton $\mu_B = e\hbar/(2mc)$, whose magnetic field carries energy. In a Nature letter of March 1935 Born and Schrödinger therefore re-estimated $b$ from the opposite side: assuming the rest energy to be dominated by the magnetic dipole energy, $\mu_B^2/(2r'^3) = mc^2$, they obtained a revised radius $r' = r_0/(2\alpha^{2/3}) \approx 13\,r_0 \approx 11\,r_{el}$ (the magnetostatic-energy radius, not the length $\sqrt{e/b_{BS}}$), an order of magnitude larger, corresponding to $\beta_{BS} \approx 4.74\,\beta_B$ in Kiessling's notation — in this convention $b \propto 1/\beta^2$, so $b_{BS} \approx b/4.74^2$, i.e. an absolute field constant roughly twenty times smaller. Each estimate keeps exactly the energy the other discards, so neither is trustworthy; the letter itself presents its magnetostatic figure as a very rough estimate.

In the modern formulation the theory is the Maxwell–Born–Infeld (MBI) system: Maxwell's evolution equations with a nonlinear aether law linking the fields, sourced by a charge density $\rho$ and current density $j$. An electron at rest corresponds to a stationary (time-independent) finite-energy solution. What is known about these is sharply asymmetric: if $j = 0$ the unique stationary solution is purely electrostatic, with existence and uniqueness proved even for point-charge sources; if $\rho = 0$ it is purely magnetostatic and unique; but if both $\rho$ and $j$ are present — precisely the situation of a spinning charged object — only a perturbative existence theorem for sufficiently small, regular sources is known, uniqueness is open, and no closed-form solution is known. The self-consistent spinning-electron solution, and with it an accurate value of $b$ and the electron radius, has accordingly never been obtained.

Problem Statement

Within Born's nonlinear electrodynamics (the Born–Infeld theory), construct the full self-consistent solution — the stationary fields the Maxwell–Born–Infeld equations determine from the electron's own source, with the energy balance imposed, the source being fixed by the statement rather than chosen by the solver — for the electron at rest including its spin: a stationary, finite-energy solution of the Maxwell–Born–Infeld field equations whose source is the electron itself in the point-source limit — a point charge $e$ carrying an intrinsic point magnetic dipole moment $\mu_B$, taken as the limit of regular sources, so that the source has no free shape or scale — and whose total field energy — with no bare mechanical mass contribution — equals the rest energy $mc^2$, with the electrostatic (charge) energy and the magnetostatic (spin) energy treated concurrently rather than one of them being neglected. From this solution, derive the value of the absolute field constant $b$ and the electron radius it implies, where the electron radius is defined as the length $\sqrt{e/b}$ at which the self-field of the charge reaches $b$, replacing the mutually inconsistent one-sided estimates — the spinless electrostatic value $b = 3.96\times 10^{15}$ E.S.U. and the Born–Schrödinger magnetostatic estimate. Alternatively, a rigorous proof that no value of $b$ admits such a stationary finite-energy solution for this point source with total field energy $mc^2$ would equally settle the question, by refuting the classical determination. The sought solution is the stationary field configuration generated by this source (its charge density $\rho$ and current density $j$), whose total charge, energy and magnetic moment are those of the electron, and which determines the absolute field constant $b$ and the electron radius defined above.

References

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