Speaker: Roman Sverdlov, University of New Mexico

Title: Berezin integral as a limit of Riemann sum

Abstract: Berezin integral is widely used in physics, but unfortunately its properties contradict the usual properties of Riemann integral. As a result of this, it was normally assumed that it can not be described as a Riemann sum and, instead, it was viewed as a formal algebraic operation. However, Roman Sverdlov and Thomas Scanlon wrote a paper showing that Berezin integral can be modeled as a Riemann sum, after all. Accordingly, its properties were reconciled with the intuition we have from calculus. This was done in two steps. First, instead of having a single anticommuting product, they introduced an algebra consisting of two separate products: one is Clifford product and the other is anticommuting wedge product. While wedge product is used in the "finite" part of the integral, the Clifford product is used between infinitesimal and the finite part. And, secondly, instead of having a usual integral over the whole space, they replaced it with two possibilities. One possibility is an integral over the closed surface. The other is integral over space where usual measure is being replaced by weighted directed measure. In this talk some of those constructions will be outlined. The talk will be based on the first half of the paper https://aip.scitation.org/doi/full/10.1063/1.5144877 which is also available online at https://math.berkeley.edu/~scanlon/papers/BIRS-9May2020.pdf

Zoom link: https://unm.zoom.us/j/95842947813 (contact organizer for password).