# Born–Infeld model

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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 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]

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

currently excluded by experiment [2].

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 [3] 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.

The model is named after physicists Max Born and Leopold Infeld, who first proposed it.

## 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

Notes
1. ^ M. Born, L. Infeld, "Foundations of the New Field Theory", Proc. R. Soc. Lond.,144 (1934) pp. 425-451. doi:10.1098/rspa.1934.0059
2. ^ Rafelski, Johann (2017). Relativity Matters. Springer. pp. 432–435. doi:10.1007/978-3-319-51231-0 .
3. ^ Iwo Bialynicki-Birula, "Nonlinear Electrodynamics: Variations of a theme by Born and Infeld", World Scientific, (1983)