EXC 2044 - T07: Field theory and randomness

Basic data for this project

Description

Quantum field theory (QFT) is the fundamental framework to describe matter at its smallest length scales. QFT has motivated groundbreaking developments in many different mathematical fields including the theory of operator algebras, probability theory, representation theory, complex algebraic geometry and others. The construction of a non-linear QFT in the physical case of 4 space-time dimensions is an outstanding open problem and part of the Yang–Mills Clay Millenium Prize Problem. In 3 space-time dimensions, mathematical models have been constructed with different techniques. All known constructions are based on a perturbative analysis around linear theories. This is not expected to work directly in 4-dimensional theories because these theories are critical: Linear and non-linear terms have the same scaling behaviour and are therefore expected to both contribute on all scales. The key goals of this topic centre around the construction of a critical (bosonic) theory. We will focus on scalar QFTs on non-commutative geometries building on the one hand on fundamental results by Wulkenhaar who found explicit formulae for the leading terms in the topological expansion of solutions and on the other hand on recent breakthrough in the probabilistic construction of QFTs by the stochastic analysis community.