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International Journal of Theoretical Physics

Category-Theoretic Interpretative Framework of the Complementarity Principle in Quantum Mechanics

Elias Zafiris, Vassilios Karakostas

Published Nov 13, 2019• 2 citations
Publisher
DOI
10.1007/s10773-019-04292-4

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Categorical quantum mechanics studies quantum theory in the framework of dagger-compact closed categories. Using this framework, we establish a tight relationship between two key quantum theoretical notions: non-locality and complementarity. In particular, we establish a direct connection between Mermin-type non-locality scenarios, which we generalise to an arbitrary number of parties, using systems of arbitrary dimension, and performing arbitrary measurements, and a new stronger notion of complementarity which we introduce here. Our derivation of the fact that strong complementarity is a necessary condition for a Mermin scenario provides a crisp operational interpretation for strong complementarity. We also provide a complete classification of strongly complementary observables for quantum theory, something which has not yet been achieved for ordinary complementarity. Since our main results are expressed in the (diagrammatic) language of dagger-compact categories, they can be applied outside of quantum theory, in any setting which supports the purely algebraic notion of strongly complementary observables. We have therefore introduced a method for discussing non-locality in a wide variety of models in addition to quantum theory. The diagrammatic calculus substantially simplifies (and sometimes even trivialises) many of the derivations, and provides new insights. In particular, the diagrammatic computation of correlations clearly shows how local measurements interact to yield a global overall effect. In other words, we depict non-locality.
This paper develops a structural framework connecting quantum mechanics, category theory, and arithmetic geometry through the language of non-semisimple categories. Within the Algorithmic Motives (AM) and Ramified Arithmetic Coordinates (RAC) frameworks, core phenomena of quantum mechanics—including superposition, measurement, the Born rule, complementarity, and entanglement—are interpreted as consequences of categorical structures such as extension groups, global dimension, and exact functors. Superposition is identified with non-split extensions classified by Ext¹, while measurement is modeled as a functor to a semisimple category, forcing the annihilation of extension classes and yielding wavefunction collapse as an algebraic necessity. The Born rule is derived as the unique probability assignment compatible with the output category’s semisimple structure. Complementarity arises from the algebraic invariance of squared amplitudes under phase transformations. The framework further introduces a geometric interpretation of entanglement via covering spaces and deck transformations, and provides a categorical explanation of the quantum eraser phenomenon. These constructions are extended to arithmetic geometry through a dictionary linking Betti and étale realizations to wave and particle descriptions. A central component is the Ramified Arithmetic Coordinates (RAC) framework, which interprets the critical point of L-functions as a ramification point analogous to a coordinate singularity in general relativity. The modified trace is proposed as a coordinate system resolving this singularity. Within this structure, the Generalized Riemann Hypothesis is reformulated as a faithfulness condition for a modified comparison functor, with precise separation between proved finite-level results and conjectural infinite-level extensions. All statements are explicitly classified as Theorem, Structural Correspondence, or Conjecture, ensuring clarity between established results and proposed extensions. The paper is intended as a research exposition introducing a unified algebraic framework rather than a claim of equivalence between physical and arithmetic systems.
In this writing, I reconstruct the Principle of Complementarity in Quantum Mechanics, elaborated by Niels Bohr, and the conception of objectivity implied in it (and in Bohr's writings).Then, I connect it with the epistemological pragmatism of the philosopher Richard Rorty.My aim is to reinterpret some issues of this quantum description that concern the category of objectivity from the Rortian perspective in a plausible way.This allows me to connect in a positive and a negative sense some quantum (and Bohrian) assumptions to the Rortian perspective, but mainly, it allows to reconsider the framework of scientific objects from the modern perspective to a new pragmatist and anti-essentialist point of view.