Towards a more algebraic footing for quantum field theory

TL;DR

This paper introduces algebraic and combinatorial reformulations of Fourier and Legendre transforms in quantum field theory, addressing analytic issues and providing a more robust mathematical foundation for perturbative predictions.

Contribution

It develops algebraic transform methods using formal power series to bypass convergence issues in quantum field theory, offering a new perspective on its mathematical structure.

Findings

Transforms are well-defined on formal power series coefficients.

Addresses divergence problems in generating series of Feynman graphs.

Provides algebraic tools that explain the robustness of perturbative QFT predictions.

Abstract

The predictions of the standard model of particle physics are highly successful in spite of the fact that several parts of the underlying quantum field theoretical framework are analytically problematic. Indeed, it has long been suggested, by Einstein, Schr\"odinger and others, that analytic problems in the formulation of fundamental laws could be overcome by reformulating these laws without reliance on analytic methods namely, for example, algebraically. In this spirit, we focus here on the analytic ill-definedness of the quantum field theoretic Fourier and Legendre transforms of the generating series of Feynman graphs, including the path integral. To this end, we develop here purely algebraic and combinatorial formulations of the Fourier and Legendre transforms, employing rings of formal power series. These are all-purpose transform methods and when applied in quantum field theory to the generating functionals of Feynman graphs, the new transforms are well defined and thereby help explain the robustness and success of the predictions of perturbative quantum field theory in spite of analytic difficulties. Technically, we overcome here the problem of the possible divergence of the various generating series of Feynman graphs by constructing Fourier and Legendre transforms of formal power series that operate in a well defined way on the coefficients of the power series irrespective of whether or not these series converge. Our new methods could, therefore, provide new algebraic and combinatorial perspectives on quantum field theoretic structures that are conventionally thought of as analytic in nature, such as the occurrence of anomalies from the path integral measure.