Comments on Galileons

TL;DR

The paper demonstrates that Galileon equations in four dimensions can be understood as Kaluza–Klein reductions of a universal field equation (UFE) obtained by iterating the Euler operator on weight-one Lagrangians. It shows that the single-field Galileon corresponds to the Hessian determinant $\det|\Pi_{\mu\nu}|=0$ and connects these to Monge–Ampère structures through Chaundy’s implicit-solution method. The discussion situates Galileons within the broader Euler-hierarchy framework, yielding universal, dimensionally robust equations such as $\det\phi_{ij}=0$ and related bordered-determinant forms, with multi-field generalizations arising from multi-field UFEs. The work highlights deep links to developable surfaces, Dirac–Born–Infeld theories, and Lovelock gravity, suggesting a universal, symmetry-rich structure with potential integrability features and wide-ranging physical applications.

Abstract

The recent progress in the study of Galileons, i.e. equations of second order with an action invariant under a Galilean transformation is related to work on `Universal Field Equations' \cite{dbfgov} which are second order equations arising by an iterative procedure from arbitrary Lagrangians of weight one in their first derivatives. It is pointed out that the Galileon is simply a Kaluza-Klein reduction of a Universal Field Equation. An implicit solution to the equation of motion is presented, and a class of explicit solutions pointed out. The multi-field extensions of both types of equations are derived from a first order formalism, which is simply the substantive derivative of fluid dynamics.