Correlations, Long Range Order and Dimensionality

Abstract

After a retrospective look at the nature of correlations in the case of the helium A transition, we consider long range correlations and their information content for critical phenomena in general. Separation of the statistical from the geometric information leads to the resolution of a long-standing dilemma concerning the dimensionality dependence of the hyper-scaling relations. The hyper-scaling inequalities are the result of the significant correlations suffering a re-arrangement when a regular contribution dominates the singular part of the free energy. The inequalities can each be replaced by a pair of equalities. This resolution has a bearing on the special place of four dimensions as the origin for the e-expansion technique of the renormalization group approach.