[Paper Review] Algebraic combinatorial Fourier and Legendre transforms with applications in perturbative quantum field theory

D. M. Jackson, Achim Kempf+1 more|arXiv (Cornell University)|May 24, 2018

Advanced Mathematical Theories and Applications17 references4 citations

TL;DR

This paper introduces algebraic and combinatorial formulations of Fourier and Legendre transforms using formal power series rings to resolve analytic ill-definedness in perturbative quantum field theory (QFT). By operating solely on power series coefficients, the transforms remain well-defined even when generating series of Feynman graphs diverge, offering a robust algebraic framework that explains the success of QFT predictions despite analytic shortcomings.

ABSTRACT

Curiously, 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, Schrodinger 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, i.e., their applicability is not restricted to QFT. When applied in QFT 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 QFT 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. In contrast, the use of formal power series in QFT by Bogolubov, Hepp, Parasiuk and Zimmermann concerned a different kind of divergencies, namely the UV divergencies of loop integrals and their renormalization. Our new methods could provide new algebraic and combinatorial perspectives on QFT structures that are conventionally thought of as analytic in nature, such as the occurrence of anomalies from the path integral measure.

Motivation & Objective

• To address the analytic ill-definedness of quantum field theoretic Fourier and Legendre transforms of Feynman graph generating series.
• To develop algebraic and combinatorial alternatives to analytic transforms, avoiding reliance on convergence.
• To provide a well-defined framework for transforms in QFT that operates purely on power series coefficients.
• To offer new algebraic perspectives on structures traditionally viewed as inherently analytic, such as anomalies from path integral measures.

Proposed method

• Formal power series rings are used as the foundational algebraic structure for defining transforms.
• The Fourier and Legendre transforms are redefined to act on coefficients of formal power series, independent of series convergence.
• Transforms are constructed using combinatorial operations on series coefficients, ensuring well-definedness regardless of divergence.
• The approach generalizes beyond QFT, applying to any context involving generating series.
• The method avoids traditional renormalization techniques for UV divergences, focusing instead on analyticity issues in transform definitions.
• The framework enables consistent manipulation of generating functionals of Feynman graphs without requiring convergence of the underlying series.

Experimental results

Research questions

• RQ1 How can Fourier and Legendre transforms be redefined algebraically to remain well-defined when the generating series of Feynman graphs diverge?
• RQ2 What algebraic structures can replace analytic transforms in QFT while preserving physical consistency?
• RQ3 Can formal power series transforms explain the robustness of perturbative QFT predictions despite analytic pathologies?
• RQ4 In what ways do algebraic transforms reveal new insights into analytically problematic structures like anomalies from path integral measures?
• RQ5 How do these algebraic transforms compare to conventional analytic formulations in terms of mathematical rigor and applicability?

Key findings

• The proposed algebraic Fourier and Legendre transforms are well-defined on formal power series, even when the series diverge.
• The transforms operate solely on coefficients, eliminating dependence on convergence of the generating series.
• The method provides a robust framework that explains the success of perturbative QFT despite analytic issues in standard formulations.
• The approach offers new algebraic interpretations of structures previously considered inherently analytic, such as anomalies in path integral measures.
• The transforms are general-purpose tools applicable beyond QFT, extending to any context involving generating functions.
• The framework complements but does not replace traditional renormalization, addressing a different class of analytic problems—those in transform definitions rather than loop integrals.