BornInfeld

Born-Infeld theory is a nonlinear generalization of classical electromagnetism proposed by Max Born and Leopold Infeld in 1934 to address the divergent self-energy of point charges. The theory introduces a maximal field strength, b, and modifies the Lagrangian so that, at weak fields, it reproduces Maxwell electrodynamics, while at strong fields it tames singularities.

In four-dimensional spacetime, the Born-Infeld Lagrangian density can be written as

L_BI = b^2 [1 - sqrt(1 + (F_{μν} F^{μν})/(2 b^2) - (F_{μν} ˜F^{μν})^2/(16 b^4))].

Here F_{μν} is the electromagnetic field strength, and ˜F^{μν} is its Hodge dual. In the limit F_{μν} F^{μν} ≪ b^2 and F_{μν} ˜F^{μν} ≈ 0, L_BI reduces to the Maxwell Lagrangian (-1/4) F_{μν} F^{μν}. For a static point charge, the electric field remains finite at the origin, and the total self-energy is finite.

Born-Infeld theory is notable for preserving causality and, in flat spacetime, for avoiding vacuum birefringence at leading order, with nonlinear effects arising only at high field strengths. It also predicts deviations from Coulomb’s law at short distances.

A central modern connection is the Dirac-Born-Infeld (DBI) action in string theory, where the same functional form describes the effective action of a D-brane’s worldvolume gauge field. The BI action thus appears in D-brane dynamics and related cosmological models, including DBI inflation. The parameter b is related to the underlying string tension and background fields.