Born–Infeld model
Encyclopedia
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 30s 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. The Born-Infeld electrodynamics possesses a whole series of physically interesting properties:
First of all the total energy of the electromagnetic field is finite and the electric field is regular everywhere.
Second it 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 completely exceptional regular nonlinear electrodynamics.
Finally (and more technically) Born-Infeld 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.
notation here as this theory is fully relativistic.
The Lagrangian density is
where η is the Minkowski metric, F is the Faraday tensor and both are treated as square matrices so that we can take the determinant
of their sum; 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
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:
where T is the tension of the D-brane.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, the Born–Infeld model is a particular example of what is usually known as a nonlinear electrodynamics. It was historically introduced in the 30s 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. The Born-Infeld electrodynamics possesses a whole series of physically interesting properties:
First of all the total energy of the electromagnetic field is finite and the electric field is regular everywhere.
Second it displays good physical properties concerning wave propagation, such as the absence of shock waves
Shock Waves
Shock Waves, , is a 1977 horror movie written and directed by Ken Wiederhorn...
and birefringence
Birefringence
Birefringence, or double refraction, is the decomposition of a ray of light into two rays when it passes through certain anisotropic materials, such as crystals of calcite or boron nitride. The effect was first described by the Danish scientist Rasmus Bartholin in 1669, who saw it in calcite...
. A field theory showing this property is usually called completely exceptional and Born-Infeld theory is the only completely exceptional regular nonlinear electrodynamics.
Finally (and more technically) Born-Infeld theory can be seen as a covariant generalization of Mie
Mie
Mie may refer to:* KMIE, the ICAO code for the Delaware County Airport near Muncie, Indiana* Mie Prefecture, Japan* Mie District, Mie, a district of Japan* Mie, Ōita, a town of Japan* Mie University, a university in Tsu, Mie, Japan...
'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
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
.
The model is named after physicists Max Born
Max Born
Max Born was a German-born 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 number of notable physicists in the 1920s and 30s...
and Leopold Infeld
Leopold Infeld
Leopold Infeld was a Polish physicist who worked mainly in Poland and Canada . He was a Rockefeller fellow at Cambridge University and a member of the Polish Academy of Sciences....
who first proposed it.
Equations
We will use the relativisticTheory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
notation here as this theory is fully relativistic.
The Lagrangian density is
where η is the Minkowski metric, F is the Faraday tensor and both are treated as square matrices so that we can take the determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of their sum; b is a scale parameter. The maximal possible value of the electric field in this theory is b, and the self-energy
Self-energy
In theoretical physics and quantum field theory a particle's self-energy \Sigma represents the contribution to the particle's energy, or effective mass, due to interactions between the particle and the system it is part of...
of point charges is finite. For electric and magnetic fields much smaller than b, the theory reduces to Maxwell electrodynamics
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
.
In 4-dimensional spacetime the Lagrangian can be written as
where E is the electric field, and B is the magnetic field.
In string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, gauge fields on a D-brane
D-brane
In string theory, D-branes 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 Dai, Leigh and Polchinski, and independently by Hořava in 1989...
(that arise from attached open strings) are described by the same type of Lagrangian:
where T is the tension of the D-brane.