Title: Berezin integral as a limit of Riemann sum

Abstract:  Berezin integration is an integral over anticommutting variables (which are also called Grassmann variables). The rules of Berezin integration are counterintuitive. For example, integral of even function is 0 while integral of odd function is not. Integral of e^{k \theta} is k rather than k^{-1}, and so forth. Conventionally, people answer those types of questions by saying that Berezin integral has nothing to do with Riemann sum, and it is just a symbolic operation. In fact, they claim that Grassmann numbers themselves are not literal mathematical objects either. I find this troubling, particularly since they introduce a space (which they call superspace) some of which coordinates are Grassmann variables. So if Grassmann variables are just symbols, how can superspace be real? They claim that superspace isn't real either: its just a mathematical tool. I do not like this way of thinking. 

In light of this, I wrote a paper, that I co-authored with Professor Thomas Scanlon from Math department at UC Berkeley, where I have shown how it is possible to model Grassmann numbers as literal mathematical objects, Berezin integral as a literal limit of the Riemann sum, and superspace as a literal space (the official publication of that paper is at https://aip.scitation.org/doi/abs/10.1063/1.5144877?journalCode=jmp and free access version is posted in Professor Scanlon's website, which is https://math.berkeley.edu/~scanlon/papers/BIRS-9May2020.pdf).

Our approach was that, in addition to anticommutting product (that traditionally in physics isn't denoted by any symbol at all, but we chose to denote by the wedge) we added Clifford product (that we denote by a star). Instead of viewing Grassmann numbers as numbers, we view them as vectors in a space whose dimension, D, is very large. One of our approaches is to identify Berezin integral as a surface integral. In this case, our answer approximates conventional one if the volume bounded by that surface is approximately 1/D. Our other approach is to take the volume integral, but instead of regular measure use ``directed measure" so that the volume element has a direction just like a surface area element does. In this case, by choosing appropriate measure that approaches 0 at infinity, we again recover the properties of the Berezin integral. 

In this talk I will discuss those particular models of Berezin integral. Then, if time allows, I will show how to extend it to spinors and also how to model superspace in a way that would make geometric sense.

About the Author: Roman Sverdlov  received his Ph.D. in Physics from University of  Michigan in 2009 under the supervision of Luca Bombelli and Marc Ross. He spent five years in India first as a postdoc at the  Raman Research Institute (Bangalore), second with a visitting position at Institute of Mathematical Sciences (Chennai), and finally as postdoc at IISER Mohali. In 2014 he returned to the US with a visitting position at the Department of Physics at the University of Mississippi for a year and started  a  Ph.D. in Mathematics  before   transferring to New Mexico in 2016. He is now on his fifth year as a PhD student in Mathematics at UNM working under the direction of Terry Loring.