Observables and Operators

Roderich Tumulka

Atsushi Okamoto

Operational decomposition was introduced in the preceding paper as the process through which the latent structure is operationally assigned to the observable and hidden domains under a given interaction. However, the condition under which this operational assignment becomes established has not yet been clarified. In this paper, we propose that observation induces operational decomposition. Observation is treated as the operational interaction through which the assignment of the latent structure to the observable and hidden domains becomes established, rather than as a process of information acquisition. From the observational perspective, this principle corresponds to the previously introduced statement that measurement grants accessibility. These two principles are interpreted as complementary descriptions of the same operational interaction, viewed respectively from structural and observational perspectives. The present work thereby clarifies the conceptual role of observation in establishing operational decomposition while intentionally leaving its detailed dynamical mechanism outside the scope of the paper.

Davide Sangiorgi

Juha-Pekka Pellonpää

We represent quantum phase observables as phase shift covariant normalized positive operator measures. The phase operators are the first moment operators of the phase observables. A phase operator determines the associated phase observable uniquely. We show that the Cahill–Glauber s -ordered phase operators are determined by phase shift covariant generalized operator measures, which are ordinary operator measures whenever Re s < 0 and phase observables when s ≤ −1. The Wigner–Weyl quantized phase operator is not determined by any phase observable. We investigate the classical limit of covariant (generalized) operator measures in coherent states.

Atsushi Okamoto

The Intrinsic Observability Framework (IOF) distinguishes structural decomposition from operational decomposition as complementary levels of organization. Structural decomposition establishes the observable and hidden domains as the architecture of observability, whereas operational decomposition assigns the latent structure to these domains under a given observational interaction. The present work extends this framework by introducing two complementary forms of totality: structural totality and operational totality. Structural totality represents the completeness of the structural architecture established through structural decomposition, consisting of the observable and hidden domains. Operational totality, in contrast, represents the completeness of the latent structure as operationally assigned across these domains through operational decomposition. The distinction therefore concerns different objects of completeness: structural totality refers to the architecture within which assignment is possible, whereas operational totality refers to the content completely assigned within that architecture. Operational totality is consequently neither an incomplete portion nor a degraded representation of structural totality. Different observational interactions may establish different operational totalities within a common structural totality. Such observer dependence does not imply incompleteness or mutually isolated operational worlds. Each operational totality is complete with respect to its own operational assignment, while the admissible relations among independently established operational totalities are constrained by the Global Consistency Rule (GCR). The resulting framework therefore distinguishes structural completeness, operational completeness, and global consistency, providing an organizational interpretation of how distinct observer-dependent totalities can coexist while collectively supporting a shared and coherent operational reality without modifying the mathematical formalism or empirical predictions of quantum mechanics.

J. Rak, Michael J. Tannenbaum