From Monte Carlo Integration to Quantum Chromo Dynamics (an introduction)

TL;DR

This paper introduces Monte Carlo integration and its application to quantum chromodynamics (QCD), explaining how regularization techniques address divergences and enable non-perturbative computations in lattice QCD.

Contribution

It presents a comprehensive introduction to Monte Carlo methods in quantum field theory, including regularization of divergences and practical lattice QCD formulations.

Findings

Regularization of divergences via distribution products

Monte Carlo methods for computing Green functions

Lattice QCD formulation with example programs

Abstract

In these lectures we provide a short introduction to the Monte Carlo integration method and its applications. We show how the origin of ultraviolet divergences if Field Theories is in the undefined formal product of distributions and how one can define the Path Integral in terms of regularized distributions in order to cancel these divergences. This technique provides the only non perturbative regularization procedure of continuum Field Theories and, at the same time, provides a practical method to compute correlation (Green) functions (using Monte Carlo integration for the regularized path integrals). We then apply these tools to formulate QCD on a lattice. Some of the examples are accompanied by complete computer programs.