Geometric Amplitudes: A Covariant Functional Approach for Massless Scalar Theories

TL;DR

This paper develops a covariant geometric framework for massless scalar quantum field theories, enabling off-shell invariance of correlation functions under field redefinitions, with a focus on geometric interpretation and limitations for massive theories.

Contribution

It introduces modifications to correlation functions that achieve off-shell covariance in massless scalar theories, expanding the geometric understanding of amplitude invariance.

Findings

Correlation functions can be modified for off-shell covariance.

The framework is specific to massless scalar theories and does not directly extend to massive cases.

The geometric interpretation emphasizes observable covariance over metric dependence.

Abstract

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.