Positive energy theorem for $P(X,\hspace{0.17em}\varphi )$ theories

We describe a positive energy theorem for Einstein gravity coupled to scalar fields with first-derivative interactions, so-called $P(X,\varphi )$ theories. We offer two independent derivations of this result. The first method introduces an auxiliary field to map the theory to a Lagrangian describing two canonical scalar fields, where one can apply a positive energy result of Boucher and Townsend. The second method works directly at the $P(X,\varphi )$ level and uses spinorial arguments introduced by Witten. The latter approach follows that of recent work by Nozawa and Shiromizu [Phys. Rev. D 89, 023011 (2014)], but the end result is considerably less restrictive. We point to the technical step where our derivation deviates from theirs, which substantially expands the class of Lagrangians encompassed by the theorem. One of the more interesting implications of our analysis is to show it is possible to have positive energy in cases where dispersion relations following from locality and S-matrix analyticity are violated. This indicates that these two properties are logically distinct; i.e., it is possible to have positive energy even when the S-matrix is nonanalytic and vice versa.