Table of Contents

Definitions and group structure.- The duality transformation.- Duality relation for the correlation functions.- Phase transitions with spontaneous symmetry breakdown. Ergodic decomposition.- Series and cluster expansion an application of universality hypothesis a “generalized” droplet-model.- The partial trace transformation. Equation for the correlation functions and representation of the symmetry group.- Invariant equilibrium states and duality transformation for infinite systems.- Asano contraction and group structure. Analyticity properties of the free energy.- Analyticity and uniqueness of the invariant equilibrium state.- Definitions and group structure for systems with constraints.- Expansions for the partition function.- Partial trace method and equilibrium equations.- Duality transformation restricted to finite bonds.- Asano contractions and unicity of state.- General framework of higher spin systems.- Physical implications of the group structure.- Spin 1 lattice systems.- The duality transformation.- Zeroes of the partition function.