Computational Methods in Quantum Field Theory

TL;DR

This paper reviews computational methods in quantum field theory, focusing on lattice models, Monte Carlo algorithms, and the emergence of Yang-Mills theories from gauge-invariant lattice formulations.

Contribution

It provides an overview of modern algorithms and concepts for simulating quantum field theories on lattices, including dualities and gauge symmetry incorporation.

Findings

Illustrates the use of the Ising model for key concepts

Discusses the implementation of gauge symmetries in lattice models

Explores the continuum limit leading to quantum Yang-Mills theories

Abstract

After a brief introduction to the statistical description of data, these lecture notes focus on quantum field theories as they emerge from lattice models in the critical limit. For the simulation of these lattice models, Markov chain Monte-Carlo methods are widely used. We discuss the heat bath and, more modern, cluster algorithms. The Ising model is used as a concrete illustration of important concepts such as correspondence between a theory of branes and quantum field theory or the duality map between strong and weak couplings. The notes then discuss the inclusion of gauge symmetries in lattice models and, in particular, the continuum limit in which quantum Yang-Mills theories arise.