Euclidean Quantum Field Theory on Commutative and Noncommutative Spaces

Abstract

I give an introduction to Euclidean quantum field theory from the point of view of statistical physics, with emphasis both on Feynman graphs and on the Wilson-Polchinski approach to renormalisation. In the second part I discuss attempts to renormalise quantum field theories on noncommutative spaces. 1 From Classical Actions to Lattice Quantum Field Theory 1.1 Introduction Ignoring gravity, space-time is described by Minkowski space given by the metric g µν = diag(1, −1, −1, −1). In particular, time plays a very different rôle than space. Looking at a classical field theory modelled on Minkowski space, the resulting field equations are hyperbolic ones. The formulation of the associated quantum field theory requires a sophisticated mathematical machinery. The classical reference is [1]. A comprehensive treatment can be found in [2]. From our point of view, it is much easier for a beginner to first study quantum field theory in Euclidean space E 4 given by the metric g µν = δ µν = diag(1, 1, 1, 1). Euclidean quantum field theory is more than just a bad trick. It has a physical interpretation as a spin system treated in the language of statistical mechanics [3, 4]. Applications to physical models are treated in [5]. There are rigorous theorems which under certain conditions allow to translate quantities computed within Euclidean quantum field theory to the Minkowskian version [6]. Eventually, from a practical point of view, computa-tions of phenomenological relevance are almost exclusively performed in the Euclidean situation, making use of the possibility to translate them into the Minkowskian world. Our presentation of the subject is inspired by [7].