Retrospective and Outlook

Abstract

The virtues of the algebraic approach are

• 1) Naturalness of the conceptual structure; directness and unambiguity of the physical interpretation.

• 2) The sometimes amazing harmony between physical questions and mathematical structures. The theory of von Neumann algebras and C*-algebras offers an adequate language for the concise formulation of the principle of locality in special relativistic quantum physics and tools for further development of the physical theory. But more striking are cases where the discussion of physical questions and the development of a mathematical structure proceed parallel, ignorant of each other and motivated by completely disjoint objectives, till at some time the close ties between them are noticed and a mutually fruitful interaction between physicists and mathematicians sets in. One example in our context is the formulation of equilibrium quantum statistical mechanics in the algebraic setting and the inception of the Tomita-Takesaki theory of modular automorphisms. Another one is the correspondence between the statistics parameter and the Jones index.

• 3) The fact that a few principles stake out a vast territory in a qualitatively correct manner. The essential principles are locality in the sense of special relativity, stability (positivity of the energy), nuclearity. The consequences range from the emergence of the particle picture with all its ramifications (statistics, charge structure, collision theory) to a natural characterization of equilibrium states and the second law of thermodynamics.

• 4) A rather detailed understanding of the anatomy of the theory with the recognition of those points where further structural elements are needed and where modifications are demanded if one aspires to a synthesis with the principles of general relativity.