Nonlinear electrodynamics of Born-Infeld

Nonlinear theory developed by Born and Infeld in the years 1930 in order to remove the problem of the infinite clean energy of a existing electron in the linear theory of Maxwell.

These authors hoped to thus solve the problem of infinite the who systematically appeared in the quantum news electrodynamic obtained by the canonical quantification of the theory of Maxwell.

Problem of the infinite clean energy of an electron

Let us suppose a spherical electron of fixed ray $a$ in a point of the space, taken as origin of the coordinates. This static electron of load $q = - \, e$ creates, in the linear theory of Maxwell, a Coulomb field in the surrounding space, which is worth at the point $M$ located by the vector position $\ vec {OM} = \ vec {R} = R \ \ vec {U} _r$ where $\ vec {U} _r$ is the radial unit vector:

This electric field is associated, in the linear theory of Maxwell, the voluminal density of energy :

Let us calculate the total energy then in space surrounding the spherical electron of ray $a$ ; it comes in spherical coordinates:

Knowing that the angular integral is worth:

and that the radial integral is worth:

total energy in space surrounding the spherical electron of ray $a$ is worth:

This expression diverges in extreme cases from the specific electron, where: $a \to 0$ ! It is to eliminate this problem that Born and Infeld developed a nonlinear theory of electromagnetism.

Theory of Born-Infeld

Taking as a starting point ideas introduced by Mie in 1912-1913, max Born will develop as from 1933 a nonlinear theory of the electrodynamics. This construction will be done quickly in collaboration with Infeld.

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