Conflicts and the Basic Incompatibility

Kenneth Searcy

When closure criteria conflict, projection and duality become unavoidable. This paper derives projection structure and duality from purely pre-physical constraints on describability. Building on prior results establishing irreversible transition, ordered asymmetry, forced quantitative measure, and thresholded evaluation, we introduce plural closure criteria as jointly necessary but not globally compatible. We show that no single global evaluation (even if multi-valued) can preserve all required distinctions once repair requirements conflict. The minimal structural resolution is criterion-indexed stabilization: projection operators Pi characterized by viability, idempotence, and fixing of already-viable states. Duality arises as the existence of two necessary stabilized descriptions of one underlying state, and complementarity appears as non-commutation of projections under mutual disruption. The paper remains prior to probability, Hilbert space structure, and physical interpretation, and closes by identifying the next forced step: a minimal theory of selection and interaction (interference-like effects) between incompatible projections.

Gilad Gour, Teiko Heinosaari, Robert W. Spekkens

Oľga Nánásiová, Andrei Yu. Khrennikov

Mariana Storrer, Patrick Lima, Ana C. S. Costa +2 more

The concept of compatibility originally emerged as a synonym for the commutativity of observables and later evolved into the notion of measurement compatibility. In any case, however, it has remained predominantly algebraic in nature, tied to the formalism of quantum mechanics. Recently, still within the quantum domain, the concept of context incompatibility has been proposed as a resource for detecting eavesdropping in quantum communication channels. Here, we propose a significant generalization of this concept by introducing the notion of theory-independent context compatibility, a concept that is trivially satisfied by classical statistical theory but is found in conflict with quantum mechanics. Moreover, we propose a figure of merit capable of quantifying the degree of violation of theory-independent context incompatibility, and we experimentally demonstrate, using a quantum optics platform, that quantum systems can exhibit pronounced degrees of violation. Besides yielding a concept that extends to generic probabilistic theories and retrieving the notion of measurement incompatibility in the quantum domain, our results offer a promising perspective on evaluating the role of incompatibility in the manifestation of non-local correlations.

Noriaki Horiuchi

Hitoshi Kitada, Lancelot R. Fletcher

The notions of time in the theories of Newton and Einstein are reviewed so that certain of their assumptions are clarified. These assumptions will be seen as the causes of the incompatibility between the two different ways of understanding time, and seen to be philosophical hypotheses, rather than purely scientific ones. The conflict between quantum mechanics and (general) relativity is shown to be a consequence of retaining the Newtonian conception of time in the context of quantum mechanics. As a remedy for this conflict, an alternative definition of time -- earlier presented in Kitada 1994a and 1994b -- is reviewed with less mathematics and more emphasis on its philosophical aspects. Based on this revised understanding of time it is shown that quantum mechanics and general relativity are reconciled while preserving the current mathematical formulations of both theories.