Superposition, transition probabilities and primitive observables in infinite quantum systems

TL;DR

This paper extends superposition and transition probability concepts to infinite quantum systems, defining primitive observables that enable a meaningful superposition of excitations in algebraic quantum field theory contexts.

Contribution

It introduces a framework for superposition, transition probabilities, and primitive observables in infinite quantum systems, generalizing pure state concepts beyond type I factors.

Findings

States can be coherently superimposed regardless of global state type

Transition probabilities and orthogonal state families are constructed

A one-to-one correspondence between states and minimal projections is established

Abstract

The concepts of superposition and of transition probability, familiar from pure states in quantum physics, are extended to locally normal states on funnels of type I$_\infty$ factors. Such funnels are used in the description of infinite systems, appearing for example in quantum field theory or in quantum statistical mechanics; their respective constituents are interpreted as algebras of observables localized in an increasing family of nested spacetime regions. Given a generic reference state (expectation functional) on a funnel, e.g, a ground state or a thermal equilibrium state, it is shown that irrespective of the global type of this state all of its excitations, generated by the adjoint action of elements of the funnel, can coherently be superimposed in a meaningful manner. Moreover, these states are the extreme points of their convex hull and as such are analogues of pure states. As further support of this analogy, transition probabilities are defined, complete families of orthogonal states are exhibited and a one-to-one correspondence between the states and families of minimal projections on a Hilbert space is established. The physical interpretation of these quantities relies on a concept of primitive observables. It extends the familiar framework of observable algebras and avoids some counter intuitive features of that setting. Primitive observables admit a consistent statistical interpretation of corresponding measurements and their impact on states is described by a variant of the von Neumann-L\"uders projection postulate.