Introduction to mathematically rigorous quantum field theory

When

February 28th, 2025 | 2:30 p.m. – 3:30 p.m.
Refreshments provided

Where

University Hall 4002

Speaker

Abdelmalek Abdesselam, University of Virginia

Abstract

Quantum field theory (QFT) is a vast subject that can be approached from many angles. While much remains to be done, as far as the rigorous construction and study of models of QFT, much has already been done. In recent years for instance, thanks to the work of Hairer, Gubinelli, and many others, there has been tremendous progress leading to the construction of QFTs as invariant measures for Markov processes given by singular SPDEs. In the so-called Euclidean or probability theory approach to mathematical QFT, the main goal is to construct certain probability measures on the space of Schwartz distributions. After a brief general introduction, I will focus on one such family of measures, the fractional scalar QFT model with quartic interaction in three dimensions. I will discuss the main open problems related to proving the conjectured conformal invariance of the model, i.e., a far-reaching generalization of the familiar time inversion invariance of Brownian motion. I will also introduce the hierarchical simplified version of the model where the random scalar field lives on the ends or leaves of an infinite "Tits-Bruhat" tree instead of 3d Euclidean space. All conjectured properties of the Euclidean model, have a natural translation in the hierarchical context which thus provides an ideal testing ground for tools and methods, in particular, the renormalization group which I will also introduce.

*All UAB students, faculty and guests are invited.