Foundations of $\eta$ regularisation in quantum field theory

Modern physics routinely extracts unique, testable numbers from expressions that are infinite and formally divergent. Our ability to accurately describe the physics of much of the microscopic world---with some of the predictions of Quantum Field Theory (QFT) experimentally tested to an accuracy of one part in a trillion---often depends on extracting such finite quantities. This raises a basic question: how does the finite emerge from out of the infinite? How can well-defined, finite predictions emerge from divergent sums and integrals? To what extent---by symmetry or dynamics or some other physical criterion---is the unique finite remainder of a divergent sum or integral physically mandated? The present project addresses this gap in a fundamental way. When $\eta$ regularisation is generalised from [analytic number theory](https://rgcsmith.com/projects/huntetaproject/) to one-loop integrals in perturbative quantum field theory (QFT), it has been shown (Padilla & Smith, 2024; Padilla & Smith, 2025) that $\eta$ regulators appear as an analytic family of regulating weight functions modifying the integration measure. The present view is that this corresponds to regularising the entire Feynman diagram, including vertices and possibly propagators as well. In previous work (Padilla & Smith, 2024; Padilla & Smith, 2025), the generalised prescription was shown to capture all conventional regularisation prescriptions (e.g., Pauli-Villars, dimensional regularisation, and denominator regularisation) commonly used in physics. We also showed that imposing gauge invariance singles out a specific, large sub-class of gauge invariant $\eta$ regulators satisfying strict consistency conditions, and that this sub-class reproduces all conventional gauge invariant regularisation schemes while yielding more general constraints, including for chiral anomalies. Example research questions being asked include, but are not limited to: - What makes a regulator physically admissible? - When are finite values uniquely fixed? - How can, if at all, number theoretic structures organise divergent quantities in QFT? Ongoing work is focused particularly on understand how, and in what way, finite parts of divergent quantities can be (or, indeed, are) uniquely fixed by fundamental physical principles: symmetry, analyticity, unitarity, anomaly structure, or more general considerations deep in high-energy sectors. This work includes extending $\eta$ regularisation to arbitrary loop order using the $\alpha,\beta,\gamma$ formalism, and rigorously derive (e.g. via the optical theorem) key consistency conditions related to the preservation of unitarity (analogous to the gauge consistency conditions in (Padilla & Smith, 2024; Padilla & Smith, 2025). This new and exciting direction may give a broader picture view of how unitarity plays a role in UV finiteness, offering also a fundamental window into understanding how unitarity can play out more generally. Additionally, generalising to $n$-loops will aid in better understanding analyticity and locality constraints, such that compatibility with the optical theorem will likely require the $\eta$ factor to satisfy specific boundary conditions. As a result, we expect to obtain new perspectives on key constraints at the foundations of a general theory of regularisation. Another important question can be framed in the context of gauge theory. Existing gauge consistency conditions (after Wu et al.) are conveniently expressed in terms of irreducible loop integrals (Padilla & Smith, 2024; Padilla & Smith, 2025). In the language of the $\eta$ formalism, this corresponds to working with a preferred ``basis” of regulators and integrands, which is calculationally convenient but not physically well motivated. This project will instead return to first principles and recompute generalised Ward identities directly in the generalised $\eta$ framework, thereby characterising the full space of $\eta$ regulators compatible with gauge invariance, allowing a more complete understanding of degeneracies in this space. An important goal is to extend this methodology to spin-two interactions to derive conditions on $\eta$ form factors that preserve linearised diffeomorphisms. In parallel, the project seeks to develop the position-space viewpoint of $\eta$ regularisation, interpreting regularisation of the functional measure as a controlled weighting of eigenmodes (Smith & Grewar, 2025). This is based on several key and pioneering observations made in my doctoral research (Smith, 2025), which opens the door for entirely new directions in the treatment of the path integral. Generalising further, it is suspected that the non-local formalism of Kleppe et al. will be an incredibly useful picture with interesting connection to the $\eta$ framework. Here, non-local formulations provide a natural viewpoint on how a generalised theory of regularisation might be expected to smooth interaction vertices and propagators in a way that hints at string theory corrections (or those of some other theory of quantum gravity), with non-locality a key feature of a potential supersymmetric regulator. We also anticipate further generalisations to curved space, with potential applications to cosmological correlators. As can be seen, a distinctive feature of the research, and its current direction, is the bridge it constructs between fundamental physics and foundational mathematics. The $\eta$ framework has established a way to reinterpret regularisation not merely as a technical prescription, but as a structured problem about the emergence of the finite from out of the infinite. ```html

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References

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