Taming non-analyticities of QFT observables

TL;DR

This paper demonstrates that many quantum field theory observables, despite their non-analytic nature, are mathematically tame functions within o-minimal structures, ensuring finite geometric complexity and definability.

Contribution

It establishes that a broad class of non-analytic QFT observables are tame functions in the o-minimal structure $ ext{R}_{ ext{G}}$, linking asymptotic series and Borel resummation to definability.

Findings

Partition and correlation functions are tame in $ ext{R}_{ ext{G}}$.

Borel resummation of divergent series yields definable observables.

Examples include 0D and higher-dimensional quantum field theories.

Abstract

Many observables in quantum field theories are involved non-analytic functions of the parameters of the theory. However, it is expected that they are not arbitrarily wild, but rather have only a finite amount of geometric complexity. This expectation has been recently formalized by a tameness principle: physical observables should be definable in o-minimal structures and their sharp refinements. In this work, we show that a broad class of non-analytic partition and correlation functions are tame functions in the o-minimal structure known as $\mathbb{R}_{\mathscr{G}}$ - the structure defining Gevrey functions. Using a perturbative approach, we expand the observables in asymptotic series in powers of a small coupling constant. Although these series are often divergent, they can be Borel-resummed in the absence of Stokes phenomena to yield the full partition and correlation functions. We show that this makes them definable in $\mathbb{R}_{\mathscr{G}}$ and provide a number of motivating examples. These include certain 0-dimensional quantum field theories and a set of higher-dimensional quantum field theories that can be analyzed using constructive field theory. Finally, we discuss how the eigenvalues of certain Hamiltonians in quantum mechanics are also definable in $\mathbb{R}_{\mathscr{G}}$.