The Role of Type III Factors in Quantum Field Theory

TL;DR

This paper reviews the significance of type III factors in quantum field theory, highlighting their physical implications and the unique hyperfinite type III_1 structure of local algebras in relativistic quantum systems.

Contribution

It summarizes the physical consequences of type III factors in RQFT and discusses the established uniqueness of their hyperfinite type III_1 classification.

Findings

Type III factors naturally occur in RQFT.

Local algebras are isomorphic to the unique hyperfinite type III_1 factor.

The net structure characterizes specific quantum field theories.

Abstract

One of von Neumann's motivations for developing the theory of operator algebras and his and Murray's 1936 classification of factors was the question of possible decompositions of quantum systems into independent parts. For quantum systems with a finite number of degrees of freedom the simplest possibility, i.e., factors of type I in the terminology of Murray and von Neumann, are perfectly adequate. In relativistic quantum field theory (RQFT), on the other hand, factors of type III occur naturally. The same holds true in quantum statistical mechanics of infinite systems. In this brief review some physical consequences of the type III property of the von Neumann algebras corresponding to localized observables in RQFT and their difference from the type I case will be discussed. The cumulative effort of many people over more than 30 years has established a remarkable uniqueness result: The local algebras in RQFT are generically isomorphic to the unique, hyperfinite type ${\rm III}_{1}$ factor in Connes' classification of 1973. Specific theories are characterized by the net structure of the collection of these isomorphic algebras for different space-time regions, i.e., the way they are embedded into each other.