Abstract
In the framework of quantum theory, events are identified as measurement outcomes referent to corresponding observables. The theory then provides the means of correlating these events. In this respect, the conceptual complexity of any ontological interpretation of quantum theory stems from two factors: [1] The existence of an event referring to a quantum system can be inferred only probabilistically, and only relative to a particular measurement context of a selected observable—a local context whereby the universe, represented by a global state vector, is decomposed into “system,” “measuring apparatus,” and “environment” with their respective state vectors. The actualization of probability-valuated, potential quantum events can be affirmed only after a measured result has been registered by the corresponding measuring apparatus. [2] The totality of events related to the behavior of a quantum system cannot be actualized within the same local measurement context due to the property of non-commutativity of quantum observables. These two factors together necessitate a thorough rethinking of our conceptual and mathematical representation of the notion of a physical continuum suited to the quantum domain of discourse. To this end, we propose a mereotopological, category theoretic interpretation of quantum mechanics that captures the relationship between a global description in terms of non-commutative algebras of quantum observables, and a description in terms of local Boolean algebras associated with particular measurement contexts. In this relational realist framework, which integrates key aspects of Whiteheadian philosophy and Grothendieck topology, the measuring apparatus and environment are thus understood not as external agents that causally affect a measured system, but rather as constitutive of an internal-relational mereotopological localizing scheme at the macroscopic level with respect to a global algebra of quantum observables.