On Covariance for Entire-Function Deformations of Relativistic Field Theories

We prove a what we call a general covariance theorem for entire-function deformations of relativistic field theories, the result is that if an undeformed theory is covariant or invariant under some symmetry group, and the deformation is defined by an entire function of a differential operator that intertwines the symmetry group-action, then the deformed theory remains covariant or invariant with respect to that same symmetry group. We will establish this result in both abstract and geometric forms and derive corollaries for Poincar\'e symmetry, gauge covariance, and continuum diffeomorphism covariance. In the gauge-fixed sector we as well prove a finite-dimensional BRST theorem with Ward and Slavnov--Taylor identities for admissible truncations in the theory.