Combinatorial aspects of the Legendre and Fourier transforms in perturbative quantum field theory

David M. Jackson, Achim Kempf, Alejandro H. Morales

Published 2018-05-24Version 1

The predictions of the standard model of particle physics are highly successful in spite of the fact that parts of the underlying quantum field theoretical framework, such as the path integral, are analytically problematic or even ill defined. Here, we pursue the idea that the reason for the robustness of the quantum field theoretical framework to these analytic issues is that there exist underlying algebraic and combinatorial structures that are robust with respect to analytic issues. To this end, we consider generating series such as the partition function, $Z$, and the effective action, $\Gamma$, not as functionals (because they generally do not converge), but as mathematically well-defined formal power series. The central idea is that all physical information is encoded in the coefficients of these formal power series independently of the convergence properties of these series. We then show that the fundamental Legendre and Fourier transforms between these generating series can be defined unambiguously as well-defined maps between the coefficients of these formal power series.