Foundations of regularisation and resurgence

This project explores connections between the smoothed asymptotics arising in $\eta$ regularisation and Borel summation methods essential to resurgence (Smith, 2025), suggesting a deeper link between regularisation and non-perturbative physics.

The key interest is in establising a proof of such a connection, which would offer entirely new perspectives on physical mechanisms for the extraction of finite, testable values from expressions that are formally divergent (Smith, 2025). Specifically, the goal is to investigate how to promote $\eta$ regularisation to a fully non-perturbative setting, where perturbative series and instanton sectors are combined into a trans-series. The objective is to then identify extremely general classes of $\eta$ regulators that naturally encode resurgent structures, and to clarify how non-perturbative contributions can ensure finiteness of generic divergent sums.

This work involves exploring connections with quasi-asymptotics, Stieltjes integrals and Tauberian theorems, and testing the resulting framework in simple quantum mechanical and field theoretic models (Smith, 2025).

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