Practically linear analogs of the Born-Infeld and other nonlinear theories

TL;DR

This paper introduces practically linear analogs of nonlinear theories like Born-Infeld electrodynamics, enabling easier computation by solving linear equations while emulating nonlinear behaviors, with applications to scalar and electromagnetic theories.

Contribution

It proposes a class of practically linear theories that approximate nonlinear physics, including a novel approach for Born-Infeld electromagnetism using Legendre transforms.

Findings

Derived an exact expression for short-distance force between charges in BI theory.

Formulated a class of PL theories governed by a specific Lagrangian structure.

Demonstrated the applicability to BI electrostatics and scalar theories.

Abstract

I discuss theories that describe fully nonlinear physics, while being practically linear (PL), in that they require solving only linear differential equations. These theories may be interesting in themselves as manageable nonlinear theories. But, they can also be chosen to emulate genuinely nonlinear theories of special interest, for which they can serve as approximations. The idea can be applied to a large class of nonlinear theories, exemplified here with a PL analogs of scalar theories, and of Born-Infeld (BI) electrodynamics. The general class of such PL theories of electromagnetism are governed by a Lagrangian L=-(1/2)F_mnQ^mn+ S(Q_mn), where the electromagnetic field couples to currents in the standard way, while Qmn is an auxiliary field, derived from a vector potential that does not couple directly to currents. By picking a special form of S(Q_mn), we can make such a theory similar in some regards to a given fully nonlinear theory, governed by the Lagrangian -U(F_mn). A particularly felicitous choice is to take S as the Legendre transform of U. For the BI theory, this Legendre transform has the same form as the BI Lagrangian itself. Various matter-of-principle questions remain to be answered regarding such theories. As a specific example, I discuss BI electrostatics in more detail. As an aside, for BI, I derive an exact expression for the short-distance force between two arbitrary point charges of the same sign, in any dimension.