theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experi ...

, 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 quantum field theory, the energy that a particle has as a result of changes that it causes in its environment defines self-energy \Sigma, and represents the contribution to the particle's energy, or effective mass, due to interactions between ...

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 Max Born (; 11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a ...

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 :

E_ = 1.187 \times 10^ \, \mathrm / \mathrm.

Born–Infeld electrodynamics displays good physical properties concerning wave propagation, such as the absence of

shock waves In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a med ...

and

birefringence Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefrin ...

. A field theory showing this property is usually called completely exceptional, and Born–Infeld theory is the only completely exceptional ''regular'' nonlinear electrodynamics. This theory can be seen as a covariant generalization of Mie's theory and very close to

Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...

's idea of introducing a nonsymmetric

metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allo ...

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. 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 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 Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...

is :

\mathcal = -b^2 \sqrt + b^2,

where ''η'' is the

Minkowski metric In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...

, ''F'' is the Faraday tensor (both are treated as square matrices, so that we can take the

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...

of their sum), and ''b'' is a scale parameter. The maximal possible value of the electric field in this theory is ''b'', and the

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 :

\mathcal = -b^2 \sqrt + b^2,

where E is the electric field, and B is the magnetic field. In string theory, gauge fields on a

D-brane In string theory, D-branes, short for ''Dirichlet membrane'', are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Jin Dai, Leigh, and Polchi ...

(that arise from attached open strings) are described by the same type of Lagrangian: :

\mathcal = -T \sqrt,

where ''T'' is the tension of the D-brane and

2\pi \alpha'

is the invert of the string tension.

References

Quantum field theory Quantum electrodynamics String theory Max Born {{string-theory-stub