The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

In this chapter, we introduce a few examples of causal variational principles and analyze them in detail.

Causal variational principles are introduced as a generalization of the causal action principle. An overview of different cases and settings is given.

The discussion of any new topic necessarily makes use of knowledge that is to some extent assumed to have been already acquired. One cannot start from the very beginning and teach the whole of physics every time something new is to be introduced – even though the Landau and Lifshitz series of books comes close to pulling it off. More pragmatically, I would like to make sure that we are all on the proverbial same page with some of the basic notions. And where, you may ask, are these basic notions learned? I have in mind what can be called the canon of physics, that is, the books where we, as students, first studied the basic concepts and equations, the books that everyone has read to study, say, classical mechanics, electromagnetism, quantum mechanics, thermodynamics and statistical mechanics – as well as the mathematical tools necessary to comprehend the equations and the statistics to make sense of the data analysis. I have them (most of them, at least) in my office, and so do most of the physicists I know. The covers of some of them are shown in Figure 2.1.

This chapter is devoted to the topological and geometric structures of a causal fermion system.