Conclusion

Abstract

Throughout this volume we have tried to demonstrate the utility of operator algebras and their states in the analysis of quantum statistical mechanics. The most detailed and interesting justification of these techniques is provided by the theory of quantum spin systems discussed in the first half of this chapter. A large class of these models, including all the basic examples such as the Heisenberg and Ising models, can be described by C* -dynamical systems (𝕬, τ) and states of these systems correspond to the physical states described by the model. A global viewpoint of this type is essential if one desires understanding of such basic questions as the nature of thermodynamic phases, mixture properties of the phases, etc., and this is perhaps the greatest single advantage of the algebraic methods. Traditionally, equilibrium states had been described by a variety of methods, e.g., implicit or explicit thermodynamic limits of the Gibbs ensembles, the principle of maximum entropy, etc., but in all these methods the affine properties of the states were unclear. Phase transitions were partially understood in terms of nondifferentiability of the thermodynamic functions, or through lack of clustering of the states, but no framework really existed for the definition and characterization of pure phases and mixed phases. The realization that the equilibrium states could in fact be identified as states over the appropriate C*-dynamical systems immediately provided this framework.