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 $\mathcal{L}=-(1/2)F_{\mu \nu }{Q}^{\mu \nu }+\tilde{\mathcal{S}}(Q_{\mu \nu })$, where $F_{\mu \nu }=A_{\nu ,\mu }-A_{\mu ,\nu }$, and $A_{\mu }$ couples to currents in the standard way, while $Q_{\mu \nu }=B_{\nu ,\mu }-B_{\mu ,\nu }$ is an auxiliary field that does not couple directly to currents. By picking a special form of $\tilde{\mathcal{S}}(Q_{\mu \nu })$, we can make such a theory similar in some regards to a given fully nonlinear theory, governed by the Lagrangian $-\tilde{\mathcal{U}}(F_{\mu \nu })$. For example, by similar we may imply that the theories are equivalent to second order in the expansion for weak fields, and that they are also equivalent for static configurations with one-dimensional symmetry (e.g., near point charges). A particularly felicitous choice, which implies the above similarities, is to take $\tilde{\mathcal{S}}$ as the Legendre transform of $\tilde{\mathcal{U}}$ in the variables $F_{\mu \nu }$. For the BI theory, this Legendre transform has the same form as the BI Lagrangian itself: $\tilde{\mathcal{S}}(Q_{\mu \nu },E_0^2)=\tilde{\mathcal{U}}(Q_{\mu \nu },-E_0^2)$ ($E_0$ is the limiting field of the BI theory). 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.

• Received 9 February 2012

DOI:https://doi.org/10.1103/PhysRevD.85.105018