Murdock Grewar | Searcharxiv

arXiv2025

$\eta$ regularisation and the functional measure

In this paper, we revisit Fujikawa's path integral formulation of the chiral anomaly and develop a generalised framework for systematically defining a regularised functional measure. This construction extends the $\eta$ regularisation scheme to operator language, making the connection between spectral asymmetry and measure transformation fully explicit. Before recovering Fujikawa's expression for the chiral anomaly from the regularised measure, we explore the deeper number-theoretic structure underlying the ill-defined spectral sum associated with the anomaly, interpreting it through the lens of smoothed asymptotics. Our approach unifies two complementary perspectives: the analytic regularisation of Fujikawa and the topological characterisation given by the Atiyah-Singer index theorem. We further investigate how the measure transforms under changes to the regularisation scale and derive a function $\iota_E(\Lambda)$ that encodes this dependence, showing how its Mellin moments govern the appearance of divergences. Finally, we comment on the conceptual relationship between the regularised measure, $\eta$ regularisation, and the generalised Schwinger proper-time formalism, with a particular focus on the two-dimensional Schwinger model.

hep-th↗

arXiv2024

Practical Global Backprojection-Convolution in Transmission Cone-beam Computed Tomography

Global backprojection-convolution (GBC) is a recently developed theory for exact reconstruction in transmission cone-beam computed tomography (CBCT). It is the first exact inversion theory that applies when the X-ray source points comprise a multidimensional `source locus' $X \subset \mathbb R^3$. Theoretically, GBC is computationally highly expedient due to its structure, but producing a practical computational implementation poses a significant challenge because the method is uniquely vulnerable to four sources of discretisation error: (1) accurate discretisation of a multidimensional locus requires more points than for a 1-dimensional locus, (2) the convolution kernel has infinite range and so the backprojected volume must be of infinite size, (3) the discrete convolution kernel cannot be computed in closed form, and (4) aliasing artefacts in the backprojection are enormously magnified by the convolution step. In this article, we propose an assortment of strategies to mitigate the discretisation errors, at the level of the symbolic algorithm. As a prototype, we deploy the concept on the case where $X$ is a cylinder. The resulting algorithm is evaluated through a series of reconstructions of the 3D Shepp-Logan phantom. As additional validation, we also briefly present a reconstruction from a real experimental dataset.

physics.med-ph↗