Field Redefinitions Can Be Nonlocal

TL;DR

This work demonstrates that the conventional locality restrictions on field redefinitions are not fundamental: the $S$-matrix and correlation functions are invariant under highly general transformations, including nonlocal and time-dependent ones, provided the full set of effects (Jacobian determinants, ghosts, and tadpoles) is correctly accounted. The authors give a diagrammatic, all-orders proof for a real scalar theory and extend it to fermions and gauge bosons, then show how certain redefinitions can be resummed into the propagator, with explicit formulations for linear and shift transformations. The implications span correlation functions, LSZ reductions, and Schwinger-Dyson equations, yielding soft-theorem structures and enabling perturbative calculations away from field-space minima. Concrete relativistic and nonrelativistic examples illustrate nonlocal and time-dependent redefinitions in action, underlining the practical flexibility and broad applicability of these ideas within EFTs and beyond.

Abstract

We revisit the lore establishing the allowed space of field redefinitions and show that there are essentially no restrictions. Our conclusions hold to all orders in perturbation theory and for any dispersion relation. Field redefinitions can be nonlocal, symmetry breaking, or in certain cases have explicit dependence on spacetime. We address field redefinitions that can be resummed into the propagator, which demonstrates how to perform perturbative calculations away from the minimum in field space. Field redefinitions are used to derive higher-order Schwinger-Dyson equations, which imply multiparticle soft theorems. Non-standard field redefinitions are showcased using both relativistic and nonrelativistic examples.