Correlation Inequalities and the Lee-Yang Theorem

Abstract

Tractable nonlinear theories generally have either a small parameter and are close to a linear theory or other exactly soluble theory, or have a definite sign and are characterized by global positivity, monotonicity, or convexity properties. These general facts apply to the study of quantum physics, and just as Section 2.4 was an introduction to expansion methods, the present chapter is an introduction to convexity methods. Generally, expansion methods apply to all interactions but only in a limited parameter range (small parameters), while convexity methods apply to a limited class of interactions (the convex ones) but over all or a large part of their parameter ranges. For statistical mechanical systems, the convexity properties, known as correlation inequalities, are expressed as general inequalities between the expectation values (i.e., the moments or correlation functions) of the system. The Lee-Yang theorem is included here because its proof and usage are closely related.