# Born–Infeld model

In theoretical physics, the Born–Infeld model is a particular example of what is usually known as a nonlinear electrodynamics. It was historically introduced in the 1930s to remove the divergence of the electron's self-energy in classical electrodynamics by introducing an upper bound of the electric field at the origin.

## Overview

Born–Infeld electrodynamics is named after physicists Max Born and Leopold Infeld, who first proposed it. The model possesses a whole series of physically interesting properties.

In analogy to a relativistic limit on velocity, Born-Infeld theory proposes a limiting force via limited electric field strength. A maximum electric field strength produces a finite electric field self-energy, which when attributed entirely to electron mass-produces maximum field [1]

${\displaystyle E_{\rm {BI}}=1.187\times 10^{20}\,\mathrm {V} /\mathrm {m} .}$

Born–Infeld electrodynamics displays good physical properties concerning wave propagation, such as the absence of shock waves and birefringence. A field theory showing this property is usually called completely exceptional, and Born–Infeld theory is the only [2] completely exceptional regular nonlinear electrodynamics.

This theory can be seen as a covariant generalization of Mie's theory and very close to Einstein's idea of introducing a nonsymmetric metric tensor with the symmetric part corresponding to the usual metric tensor and the antisymmetric to the electromagnetic field tensor.

During the 1990s there was a revival of interest on Born–Infeld theory and its nonabelian extensions, as they were found in some limits of string theory.

## Equations

We will use the relativistic notation here, as this theory is fully relativistic.

The Lagrangian density is

${\displaystyle {\mathcal {L}}=-b^{2}{\sqrt {-\det \left(\eta +{\frac {F}{b}}\right)}}+b^{2},}$

where η is the Minkowski metric, F is the Faraday tensor (both are treated as square matrices, so that we can take the determinant of their sum), and b is a scale parameter. The maximal possible value of the electric field in this theory is b, and the self-energy of point charges is finite. For electric and magnetic fields much smaller than b, the theory reduces to Maxwell electrodynamics.

In 4-dimensional spacetime the Lagrangian can be written as

${\displaystyle {\mathcal {L}}=-b^{2}{\sqrt {1-{\frac {E^{2}-B^{2}}{b^{2}}}-{\frac {(\mathbf {E} \cdot \mathbf {B} )^{2}}{b^{4}}}}}+b^{2},}$

where E is the electric field, and B is the magnetic field.

In string theory, gauge fields on a D-brane (that arise from attached open strings) are described by the same type of Lagrangian:

${\displaystyle {\mathcal {L}}=-T{\sqrt {-\det(\eta +2\pi \alpha 'F)}},}$

where T is the tension of the D-brane.

## References

1. ^ M. Born, L. Infeld, Foundations of the New Field Theor, Proc. R. Soc. Lond.,144 (1934) pp. 425-451. doi:10.1098/rspa.1934.0059
2. ^ Bialynicki-Birula, I, in Festschrift of J. Lopuszanski, Quantum Theory of Particles and Fields, Eds. B. Jancewicz and J. Lukierski, p. 31 - 42, World Scientific, Singapore (1983).