On Inequivalent Representations

Question

Kronz and Lupher in their article, "Unitarily Inequivalent Representations in Algebraic Quantum Theory" say:

"Fock representations accommodate systems having infinite degrees of freedom; but they do not accommodate systems having an infinite number of particles (or subsystems)."

Then, they add:

"If the system is a finite-particle system, then all representations are unitarily equivalent—i.e., for any pair there is a unitary operator that transforms one into the other."

It contradicts, it seems, with writings of other physicists in which the inequivalence of some Fock spaces is shown. For example, Umezawa's book, Thermo Fields Dynamics and Condensed States (Ch. 2.4), and Blasone et al's book, Quantum Field Theory and its Macroscopic Manifestations (Ch. 2).

Where is the problem?

Kronz and Lupher's paper: https://link.springer.com/article/10.1007/s10773-005-4683-0

Answer 1

1. The first quotation states something incorrect. It is correct that Fock representations accommodate systems having infinite degrees of freedom. Moreover, normal states of the algebra are convex combinations of finite-particle vector states: normal states are interpreted as the states with a finite number of particles. However, the class of physical states is not identical with the class of normal states. The condition suggested by a Haag and other people is local normality, i.e. the restriction of a physically admissible state to any local algebra should be normal. Hence, local subsystems have a finite number of particles. As Bratteli and Robinson explain, however, a locally normal state can be used to describe an infinite number of particles for which the overall density is finite (Operator Algebras and Quantum Statistical Mechanics 2 p.26)

2. As for the second quotation it refers to systems of finite degrees of freedom. The Stone- von Neumann theorem identifies all regular states of the CCR-algebra with the locally normal states in the Fock representation of that algebra. This fact leads to the unitary equivalence of all regular representations of the CCR-algebra.