Statistical approach to quantum field theory. An introduction
type: Item
instance of: scholarly article
title: Statistical approach to quantum field theory. An introduction
zbMATH Open document ID: 1268.81005
DOI: 10.1007/978-3-642-33105-3
author: Andreas Wipf
published in: Lecture Notes in Physics
publication date: 2012-09-12T00:00:00Z
review text: It is known that statistical field theory studies the behavior of classical or quantum systems consisting of a large number of degrees of freedom in and out of equilibrium. On the other hand, quantum field theory deals with a theoretical framework for constructing quantum mechanical models of systems with an infinite number of degrees of freedom. In the past decades, the powerful methods in statistical physics and Euclidean quantum field theory have common tools based on the use of path integrals. The book consists of 15 chapters. At the end of many chapters, one can find listings of computer programs either written in C or for \texttt{Matlab}. In Chapters 2 and 3, a self contained introduction to path integrals in Euclidean quantum mechanics and statistical mechanics is provided. The high-dimensional integrals can be estimated with the help of Monte Carlo simulations based on Markov processes. The method is first introduced and then applied to ordinary integrals and to quantum mechanical systems (see Chapter 4). Chapter 5 is devoted to scalar fields at low and finite temperatures. In Chapter 6, several classical spin models such as the Ising model, the Potts model, the \(\mathbb{Z}_q\)-model, the \(\mathrm{U}(1)\)-model, the \(\mathrm{O}(N)\) model, etc., are provided. Despite of their simplicity, the models are often used to illustrate the key features of statistical systems and the methods available to understand these features. Chapter 6 is devoted to mean field approximation. Basically, the above mentioned models are studied in detail. In Chapter 8, the transfer matrix method is studied. As an illustration, the Ising and Potts models are considered. Moreover, models of Euclidean quantum mechanics are studied as well. Chapter 9 is devoted to high-temperature and low-temperature expansions of the partition functions of Ising and \(\mathrm{O}(N)\) models. In Chapter 10, Peierls argument is considered. As an alternative to the lattice formulation of quantum field t…
Mathematics Subject Classification ID: 81-01
Mathematics Subject Classification ID: 82B10
Mathematics Subject Classification ID: 82C10
Mathematics Subject Classification ID: 82Dxx
Mathematics Subject Classification ID: 81S40
Mathematics Subject Classification ID: 82B80
Mathematics Subject Classification ID: 82B28
zbMATH DE Number: 6081519
zbMATH Keywords: models
zbMATH Keywords: path integral
zbMATH Keywords: renormalization
zbMATH Keywords: fermion
zbMATH Keywords: gauge theory
reviewed by: Farruh Mukhamedov
describes a project that uses: Matlab
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-642-33105-3
OpenAlex ID: W3211055271
description: scientific article; zbMATH DE number 6081519