$\mathrm{Op}^*$ and $\mathrm C^*$ dynamical systems~I.

A. V. Voronin ; S. S. Horuzhy

Teoretičeskaâ i matematičeskaâ fizika, Tome 82 (1990) no. 2, pp. 163-177

Résumé

The concept of an $\mathrm{Op}^*$ dynamical system is introduced and provides the basis of a systematic study of the problem of describing the vacuum structure of quantum field theory, formulated as a problem of the decomposition of operators and states for an algebra of unbounded operators ($\mathrm{Op}^*$ algebra) with a group of automorphisms. The following result makes it possible to develop a new solution of this problem, namely, it is found (Theorem 1) that for $\mathrm{Op}^*$ algebras Araki's theorem, which states that the commutant of a quasilocal $\mathrm C^*$ algebra with cyclic vacuum is Abelian, is true and can be very easily proved. Introducing the concept of an orthogonal measure on an $\mathrm{Op}^*$ algebra, and generalizing Tomita's theorem on orthogonal measures on $\mathrm C^*$ algebras, we obtain for $\mathrm{Op}^*$ algebras a connection between the spatial decomposition and the decomposition of states. The key Theorem 5 solves the decomposition problem for $\mathrm{Op}^*$ dynamical systems and completely reveals their structural similarity with the wellstudied $\mathrm C^*$ dynamical systems. The physical consequences of this solution are analyzed, and also the properties of Lorentz invariance of an $\mathrm{Op}^*$ system.