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The Obvious -The Incredible: Continua, Fixed Points, and Boundaries of Canonicity in Quantum Mechanics and 2-Relational Geometry

Vladimir Veselov

Published Aug 13, 2026• 0 citations
Publisher
DOI
10.13140/rg.2.2.12394.07361

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Abstract We explore the relation of Van Vleck-Primas perturbation theory of quantum mechanics with the Lie-series-based perturbation theory of Hamiltonian systems in classical mechanics. In contrast to previous works on the relation of quantum and classical perturbation theories, our approach is not based on the conceptual similarities between the two methods. Instead, we show that for quantum systems with a finite-dimensional Hilbert space, the Van Vleck-Primas procedure can be recast exactly into a classical perturbation problem. As a non-obvious consequence, this approach gives a new way of calculating the geometric phase of quantum systems using tools from the theory of classical canonical transformations.
Unlike relativity quantum mechanics was not developed originally as a spacetime theory. Noncommutative geometry radically changed the situation. The present work builds upon noncommutative geometry and Cantorian spacetime theories to give a mathematical explanation coupled to a general and obvious physical reasoning for the quantum contra-intuitive results associated with particle wave duality, state vector reduction and related aspects at measurement. In particular we stipulate that a quantum particle is represented by a Cantor point via the dimensions of a Cantorian zero set given by dim (P) = (0,)φ while the quantum wave is represented by the emptiness between disjointed Cantorian points, i.e. a complementary empty Cantorian set of dim (W) = ( 1,)φ − where φ = ( 5 − 1)/2. Measurement interferes with the empty set and renders it non-empty and that accounts in various ways for the wave collapse and the disappearance of the interference fringes.