Essentials
of Lattice
Field Theory

Conveners

Margarita García Pérez, Instituto de Física Teórica UAM-CSIC
Christof Gattringer, Austrian Science Fund FWF & University of Graz
Simon Hands, University of Liverpool

General motivation

Quantum Chromodynamics (QCD) is the quantum field theory that describes the interactions between quarks and gluons. It is a strongly interacting theory at low energies, and therefore an ab-initio description of low-energy properties, such as the spectrum of mesons and baryons, the confinement of quarks, or chiral symmetry breaking, requires the use of a non-perturbative regulator.

In this unit on the fundamentals of lattice field theory, you will learn how to define a field theory on a discretised space-time grid and how to formally recover the continuum limit; the emphasis is on the Lagrangian formalism, but the Hamiltonian formalism is also introduced.

Prerequisites

Basic knowledge of relativistic field theory and in particular Quantum Chromodynamics.

Lectures

1 Introduction

Outline and delimitation of this section

2 Euclidean n-point functions

Euclidean n-point functions are a powerful tool in quantum field theory.
They can be used to compute energy levels and matrix elements.
Here we define Euclidean n-point functions and discuss their properties.

Slides

3 Euclidean n-point functions in QM

Euclidean n-point functions have a representation in terms of euclidean path integrals.
We work out the path integral representation for the example of 3d Quantum Mechanics.

Slides

4 Path integral for the scalar field

We work out the path integral representation of euclidean n-pointfunctions of the scalar field.
We adopt euclidean path integrals as our new method for quantizing field theories

Slides

5 Solving the free boson theory

Free lattice field theories can be solved in closed form.
We discuss the necessary Gaussian integrals and introduce source terms for generating n-point functions.
The Wick theorem for n-point functions is derived.
We evaluate the boson propagator using Fourier transformation on the lattice.