Wilson Lines in Quantum Field Theory

Igor Olegovich Cherednikov, Tom Mertens, Frederik Van der Veken

• Publisher: De Gruyter
• Year: 2019
• ISBN:
  9783110651034
• Edition: 2
• Topic: Physical Sciences
• Subtopic: Mathematical & Computational Physics

The objective of this book is to get the reader acquainted with theoretical and

mathematical foundations of the concept of Wilson loops in the context of modern

quantum fi eld theory. It offers an introduction to calculations with Wilson lines, and

shows the recent development of the subject in different important areas of research

within the historical context.

1 Introduction: What are Wilson lines?

The idea of gauge symmetry suggests that any field theory must be invariant under the local (i. e., depending on space-time points) transformations of field functions

(1.1)$\begin{array}{c}\mathit{\psi }\mathrm{(}\mathit{x}\mathrm{)}\to \mathit{U}\mathrm{(}\mathit{x}\mathrm{)}\mathit{\psi }\mathrm{(}\mathit{x}\mathrm{)}\mathrm{,}\bar{\mathit{\psi }}\mathrm{(}\mathit{x}\mathrm{)}\to \bar{\mathit{\psi }}\mathrm{(}\mathit{x}\mathrm{)}{\mathit{U}}^{\mathrm{†}}\mathrm{(}\mathit{x}\mathrm{)}\mathrm{,}\end{array}$

where the matrices $\mathit{U}\mathrm{(}\mathit{x}\mathrm{)}$ belong to the fundamental representation of a given Lie group. In other words, the Lagrangian has to exhibit local symmetry. We shall mostly deal with special unitary groups, $\mathrm{SU}\mathrm{(}{\mathit{N}}_{\mathit{c}}\mathrm{)}$, which are used in Yang–Mills theories. Although a number of important results can be obtained by using only the general form of this gauge transformation, it will be sometimes helpful to use the parameterization¹

(1.2)$\begin{array}{c}\mathit{U}\mathrm{(}\mathit{x}\mathrm{)}={\mathit{e}}^{\pm \mathit{i}\mathit{g}\mathit{\alpha }\mathrm{(}\mathit{x}\mathrm{)}}\mathrm{,}\end{array}$

where

$\begin{array}{c}\mathit{\alpha }\mathrm{(}\mathit{x}\mathrm{)}={\mathit{t}}^{\mathit{a}}{\mathit{\alpha }}^{\mathit{a}}\mathrm{(}\mathit{x}\mathrm{)}\mathrm{,}{\mathit{t}}^{\mathit{a}}={\displaystyle \frac{{\mathit{\lambda }}^{\mathit{a}}}{2}}\mathrm{,}\end{array}$

and ${\mathit{\lambda }}^{\mathit{a}}$ are the generators of the Lie algebra of the group U.

Consider for simplicity the free Lagrangian for a single massless fermion field $\mathit{\psi }\mathrm{(}\mathit{x}\mathrm{)...}$

Table of contents

• Title Page
• Copyright
• Contents
• Preface
• 1 Introduction: What are Wilson lines?
• 2 Prolegomena to the mathematical theory of Wilson lines
• 3 The group of generalized loops and its Lie algebra
• 4 Shape variations in the loop space
• 5 Wilson lines in high-energy QCD
• A Mathematical vocabulary
• B Notations and conventions in quantum field theory
• C Color algebra
• D Brief literature guide
• Bibliography
• Subject Index