zbMATH — the first resource for mathematics

Lattice gauge theories. An introduction. 4th ed. (English) Zbl 1255.81001
World Scientific Lecture Notes in Physics 82. Hackensack, NJ: World Scientific (ISBN 978-981-4365-85-7/hbk; 978-981-4365-86-4/pbk). xx, 606 p. (2012).
Three of the four fundamental forces of nature are described by gauge theories within the Standard Model of particle physics. These are the strong force, the weak force and the electromagnetic force. As the name already indicates, the strong force has the largest coupling strength. The theory of the strong force is elegantly encapsulated into a Lagrange density, known as the Lagrangian of quantum chromodynamics. At large momentum transfer the coupling strength is below one and perturbation theory offers a possibility to obtain predictions from the Lagrangian. At small momentum transfer the coupling is no longer small and perturbation theory cannot be applied. In this case lattice gauge theory offers a method to obtain predictions. Lattice gauge theory was founded by Wilson in 1974. In short, lattice gauge theory takes the QCD Lagrangian and formulates a corresponding Lagrangian on a discretized space-time. The name stems from the fact that as discretized space-time a four-dimensional lattice is used. Observables are obtained by Monte Carlo simulations on this lattice.
The book under review is the fourth edition of a text-book, addressing students starting to work in the field of lattice gauge theories. It can read with only basic knowledge of quantum field theory. The book is divided into 20 chapters. Starting with a historical introduction and a chapter on the path integral approach to quantization, the four most important examples of fields on a lattice are discussed in the following chapters. These are the scalar field, fermionic fields, abelian gauge fields and finally non-abelian gauge fields. The concept of a Wilson loop is then discussed in the context of the static quark potential. The following chapters introduce important techniques in the field of lattice gauge theories, like the extrapolation to the continuum limit, the strong coupling expansion, the hopping parameter expansion, the weak coupling expansion and Monte Carlo techniques. The last chapters of the book are devoted to lattice gauge theories at finite temperature. Compared to the third edition, the section on calorons has been extended.
The book has a clear style, puts an emphasis on the physical motivation and avoids technically difficult derivations. It is a good companion for students starting to specialise in the field of lattice gauge theories.

81-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory
81T13 Yang-Mills and other gauge theories in quantum field theory
81T25 Quantum field theory on lattices
81V05 Strong interaction, including quantum chromodynamics