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Axioms 2013, 2(4), 477-489;

Orthogonality and Dimensionality

CAPP, Laboratoire d'Informatique de Grenoble, Bâtiment IMAG C, 220, rue de la Chimie, 38400 Saint Martin d'Hères, France
Received: 26 October 2013 / Revised: 28 November 2013 / Accepted: 10 December 2013 / Published: 13 December 2013
(This article belongs to the Special Issue Quantum Statistical Inference)
Full-Text   |   PDF [178 KB, uploaded 13 December 2013]


In this article, we present what we believe to be a simple way to motivate the use of Hilbert spaces in quantum mechanics. To achieve this, we study the way the notion of dimension can, at a very primitive level, be defined as the cardinality of a maximal collection of mutually orthogonal elements (which, for instance, can be seen as spatial directions). Following this idea, we develop a formalism based on two basic ingredients, namely an orthogonality relation and matroids which are a very generic algebraic structure permitting to define a notion of dimension. Having obtained what we call orthomatroids, we then show that, in high enough dimension, the basic constituants of orthomatroids (more precisely the simple and irreducible ones) are isomorphic to generalized Hilbert lattices, so that their presence is a direct consequence of an orthogonality-based characterization of dimension. View Full-Text
Keywords: quantum logic; Piron’s representation theorem; foundations of quantum mechanics quantum logic; Piron’s representation theorem; foundations of quantum mechanics
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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Brunet, O. Orthogonality and Dimensionality. Axioms 2013, 2, 477-489.

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