Observables on σ-Frame effect Algebras as Upper Semicontinuous Functions

Jiří Janda, Yongming Li

Oscar E. Lanford, Derek W. Robinson

The equilibrium states for an infinite system of quantum mechanics may be represented by states over suitably chosen C* algebras. We consider the problem of associating an entropy with these states and finding its properties, such as positivity, subadditivity, etc. For the states of a quantum-spin system, a mean entropy is defined and it is shown that this entropy is affine and upper semicontinuous.

Anatolij Dvurečenskij

Gabriel Masarin Aguiar

This paper introduces the Relativity of Information Frames (RIF), a foundational framework that precedes and constrains dynamical theories. RIF treats information frames—σ-algebras of distinguishable events—as the fundamental objects of physical description, and formulates interaction as the joint realization of such frames subject to a symmetry principle: no information frame may be privileged over another. From this principle alone, the framework derives a canonical restriction on joint event structures. We prove the existence of a distinguished σ-algebra—the pointer algebra—which is the unique maximal event algebra compatible with perspectival symmetry and the existence of consistent probability assignments. Any empirical model admits a global probability measure on this algebra, while any strict extension generically fails to do so, reintroducing contextuality. These results impose non-trivial structural constraints that any admissible dynamics must respect. The framework is applied to quantum theory, not by postulating dynamics, but by showing how its core structures arise necessarily from informational consistency. Observables appear as perspectival maps, state update corresponds to interaction-induced coarse-graining, the Born rule follows from a martingale structure on the induced σ-algebra filtrations, Wigner-type scenarios are consistent, and unitary evolution is reinterpreted as gauge freedom in joint descriptions rather than ontic time evolution. RIF does not replace dynamical theories; it delimits them. It identifies which event structures, probability assignments, and symmetries are admissible prior to dynamics, and therefore clarifies what features of quantum theory are structural necessities rather than contingent dynamical postulates. The framework further implies intrinsic irreversibility, a relational notion of locality, and constraints on emergent geometric structure, which any subsequent dynamical formulation must accommodate.

Anatolij Dvurečenskij

Anna Jenčová, S. Pulmannová, E. Vinceková