# 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 Albert 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.

The compatibility of Born–Infeld theory with high-precision atomic experimental data requires a value of a limiting field some 200 times higher than that introduced in the original formulation of the theory.[3]

Since 1985 there was a revival of interest on Born–Infeld theory and its nonabelian extensions, as they were found in some limits of string theory. It was discovered by E.S. Fradkin and A.A. Tseytlin[4] that the Born–Infeld action is the leading term in the low-energy effective action of the open string theory expanded in powers of derivatives of gauge field strength.

## 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 and ${\displaystyle 2\pi \alpha '}$ is the invert of the string tension.[5][6]

## References

1. ^ Born, M.; Infeld, L. (1934). "Foundations of the New Field Theory". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 144 (852): 425–451. Bibcode:1934RSPSA.144..425B. 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).
3. ^ Soff, Gerhard; Rafelski, Johann; Greiner, Walter (1973). "Lower Bound to Limiting Fields in Nonlinear Electrodynamics". Physical Review A. 7 (3): 903–907. doi:10.1103/PhysRevA.7.903. ISSN 0556-2791.
4. ^ Fradkin, E.S.; Tseytlin, A.A. (1985). "Non-linear electrodynamics from quantized strings". Physics Letters B. 163 (1–4): 123–130. Bibcode:1985PhLB..163..123F. doi:10.1016/0370-2693(85)90205-9.
5. ^ Leigh, R.G. (1989). "DIRAC-BORN-INFELD ACTION FROM DIRICHLET σ-MODEL". Modern Physics Letters A. 04 (28): 2767–2772. doi:10.1142/S0217732389003099.
6. ^ Tseytlin, A. A. (2000). "Born-Infeld Action, Supersymmetry and String Theory". The Many Faces of the Superworld. pp. 417–452. arXiv:hep-th/9908105. doi:10.1142/9789812793850_0025. ISBN 978-981-02-4206-0. S2CID 9569497.