Interpreting Probabilities in Quantum Field Theoryand Quantum Statistical Mechanics

In ordinary nonrelativistic quantum mechanics (QM), the observables pertaining to a system typically form the self-adjoint part of the algebra B(H) of bounded operators acting on a Hilbert space H.1 B(H) is an algebra of the genus von Neumann and the species Type I factor.2 Here we consider quantum systems whose observable-algebras belong to the same genus, but correspond to more exotic species. Settings in which the exotic species occur include relativistic quantum field theory (QFT) and the thermodynamic limit of quantum statistical mechanics (QSM), reached by letting the number of systems one considers and the volume they occupy go to infinity while keeping their density finite. The aim of this essay is to articulate the impact the non-Type-I von Neumann algebras have on the interpretation of quantum probability. We proceed as follows. Section 2 sets the stage for the rest of our discussion by highlighting key elements of the formalism and interpretation of quantum probability in the familiar setting of Type-I von Neumann algebras. Key elements of the formalism include Gleason’s Theorem and Lüders’ Rule of Conditionalization; key elements of its standard interpretation include the use of minimal projection operators to characterize not only the preparation of quantum states but also the results of quantum measurements, as well as the manner in which the former assign probabilities to the latter. Section 3 motivates the significance of non-Type-I algebras by describing some physical situations that give rise to them. It also reviews some of the novel features of these algebras, including features that might seem to be impediments to