Geometric Amplitudes: A Covariant Functional Approach for Massless Scalar Theories

Functional geometry is a framework using concepts from geometry to understand the invariance of amplitudes in quantum field theory under a large class of field redefinitions, including those involving derivatives. It is inspired by recursion relations among correlation functions, where higher-point functions depend iteratively upon smaller correlators. Previous work has shown that, with suitable modifications, these correlation functions become covariant under field redefinitions, provided they are evaluated at the physical ``on-shell" point. In this paper, we show how to further modify correlation functions in massless scalar field theories to achieve ``off-shell" covariance. We investigate the conditions required for the framework to work and discuss the geometric interpretation of this construction -- which prioritizes the covariant transformation of observables under field redefinitions over the role of a metric tensor and its derivatives. While analogous modifications may exist for massive theories, we show that framework developed here does not extend straightforwardly to that case.