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