The electromagnetic mass of the electron in Born–Infeld nonlinear electrodynamics: the absolute-field constant, the point-charge dynamics, and the quantum many-particle theory

A 1933 Nature letter by Max Born and Leopold Infeld, 'Electromagnetic Mass', proposed nonlinear modifications of Maxwell's equations in which the electron's inertia is the finite energy of its own field, introducing a new fundamental constant, the 'absolute field' $a$ (with an effective electron radius $r_{0}=\sqrt{e/a}$). The letter itself marked the theory's provisional character: the Lorentz force law was derived only for external fields of wavelengths large compared with $r_{0}$, and the mass was declared calculable from the charge and the field constants but not confronted numerically with the measured electron mass. The letter is purely classical, but a quantum many-particle extension was an acknowledged open goal of Born's wider program: his August 1933 letter motivated the new field law by the failure of combining Maxwell's equations with the quantum theory, and the authors' 1934–1935 papers on the quantization of the new field equations pursued that extension without completing it. A precise open core remains: the value of the absolute-field constant remains undetermined (Born's own identification is inconclusive; experiments give only lower bounds); whether the electron's measured mass is numerically reproduced by electromagnetic field energy in a dynamically consistent treatment is unsettled; the point-charge self-force in the nonlinear equations is not rigorously established; and no complete quantum many-particle theory of the Born–Infeld fields exists.

Background

The idea that the electron's inertia might itself be electromagnetic is as old as classical electron theory. In Maxwell's electrodynamics the Coulomb $1/r^{2}$ field of a point charge carries a self-energy $\int E^{2}\,d^{3}x/2$ that diverges at $r=0$, so a point electron would have infinite 'electromagnetic mass'; classical electron theorists around 1900 therefore treated the electron as a charged sphere whose mass is its field energy $E/c^{2}$, an approach plagued by the '4/3 problem' (the field momentum computed from Poynting's vector gives $(4/3)E/c^{2}$ unless the internal stresses holding the charge together are handled consistently). In August 1933 Max Born, then in Cambridge, proposed modifying Maxwell's equations by a nonlinear field law containing a new fundamental constant with the dimensions of field strength — the 'absolute field' $a$ — which bounds the electric field of a point charge and makes its electrostatic energy finite ('Modified Field Equations with a Finite Radius of the Electron', Nature 132, 282). The joint letter with Leopold Infeld, 'Electromagnetic Mass' (Nature 132, 970; 23 December 1933 — the source letter), announced that by these modified equations 'it is possible to revive the old idea of the electromagnetic origin of inertia': the electron's mass 'can then be calculated from its charge and the constants of the field equations (the velocity of light $c$, and the absolute field $a$)'. The letter introduces the stress-energy-momentum tensor $S$ (Maxwell stresses, momentum and energy densities) in terms of a Lagrangian and a Hamiltonian density, relates the mass of an electron at rest to the total field energy with effective radius $r_{0}=\sqrt{e/a}$, and derives the Lorentz equations of motion — but only 'for external fields which contain only those wave-lengths which are large compared with $r_{0}$'. The full classical theory, expounding the 'unitarian standpoint' (a single physical entity, the electromagnetic field; particles are its singularities and mass is field energy), followed as 'Foundations of the new field theory' (Proc. R. Soc. A 144, 425; 1934). In modern notation the Born–Infeld Lagrangian density (in units where Maxwell's is $(E^{2}-B^{2})/2$) is

$$\mathcal{L}_{BI}=a^{2}\left(1-\sqrt{1-\frac{E^{2}-B^{2}}{a^{2}}-\frac{(E\cdot B)^{2}}{a^{4}}}\right)$$

which reduces to Maxwell's theory for fields weak compared with $a$ and caps the electric field at $a$. Nonlinear electrodynamics (NLED) is the general class of such classical field theories with a nonlinear field law. The program's later history defines the present status of the question. The quantum many-particle problem was never completed within the theory; instead, renormalized quantum electrodynamics (QED) absorbed the divergent self-energy of the electron into mass renormalization — redefining the observed mass so that the divergence is absorbed into it, the electron mass thereby entering as an input parameter rather than a computed quantity — so that the unitarian claim lost its mainstream role; Dirac's contemporaneous classical point-electron theory took the same route. The Born–Infeld action was, however, revived from 1985 as the exact low-energy effective action of open strings — later understood as the gauge theory on D-branes, membrane-like extended objects of string theory, where $a$ is set by the string tension — and a mathematical-physics continuation of Born's program (notably M. K.-H. Kiessling and collaborators) has since constructed, for restricted configurations, a classically consistent electrodynamics of point charges obeying the nonlinear Born–Infeld field equations, together with a partial quantization of de Broglie–Bohm type (a formulation of quantum mechanics in which particles follow deterministic trajectories guided by the wave function), covering spinless charges and no photon sector; the general point-charge law of motion, including the small-scale self-force and the well-definedness of the electromagnetic force, remains open. Both the exact value of $a$ and the numerical identification of the electron's mass with its field energy remain unsettled — the point at which the question is posed. Two empirical facts frame any answer: the Euler–Heisenberg effective Lagrangian (the low-energy effective action of QED in slowly varying fields, itself a nonlinear electrodynamics induced by virtual electron–positron pairs) shows that nonlinear corrections of quantum origin exist independently of $a$; and the observed absence of deviations from the QED prediction in light-by-light scattering (direct photon–photon scattering) currently constrains the scale associated with $a$ only from below.

Problem Statement

The source letter claims that in the modified field equations 'the mass of the electron can then be calculated from its charge and the constants of the field equations' — the velocity of light $c$ and the absolute field $a$ — while deriving the equations of motion only under the restriction to external wavelengths large compared with $r_{0}=\sqrt{e/a}$. The problem is to complete, or definitively refute, this electromagnetic-mass program:

(1) Dynamics beyond the approximation: establish the exact law of motion of a point charge in the nonlinear Born–Infeld field equations — including the self-force (the force exerted on the charge by its own field) for field structures varying on scales comparable to $r_{0}$, removing the source derivation's long-wavelength restriction — and establish the well-definedness of the electromagnetic force on the point charge.

(2) The constant: determine the value of $a$.

(3) The mass: in a dynamically consistent treatment of the point electron, decide whether its electromagnetic energy reproduces the measured electron rest mass $m_{e}$ numerically. (The source identification is the static one — the electron rest energy equals the electrostatic field energy of the point charge at rest — which fixes only the ratio of $a$ to the charge coupling and is known not to be dynamically justified as it stands.)

(4) The quantum theory: construct the consistent quantum many-particle theory of the nonlinear fields interacting with point charges — an open goal of Born's program rather than of the purely classical December letter, evidenced by his August 1933 letter and by the authors' 1934–1935 quantization papers, and still only partially achieved.

These four items serve one objective — the unitarian claim that the electron's inertia is of electromagnetic origin and calculable from the charge and the field constants alone. The four items are jointly posed: the objective concerns their joint resolution, and resolving some of them without the others does not achieve it. Whether the program can be completed in this way is itself open — it may be completed, or it may be rigorously shown that it cannot be. Any value of $a$ invoked is bound by the empirical lower bounds and by the internal consistency of the theory.

References

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