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Axioms 2013, 2(4), 477-489; doi:10.3390/axioms2040477
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; in revised form: 28 November 2013 / Accepted: 10 December 2013 / Published: 13 December 2013
(This article belongs to the Special Issue Quantum Statistical Inference)
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Abstract: 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.
Keywords: quantum logic; Piron’s representation theorem; foundations of quantum mechanics
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MDPI and ACS Style
Brunet, O. Orthogonality and Dimensionality. Axioms 2013, 2, 477-489.AMA Style
Brunet O. Orthogonality and Dimensionality. Axioms. 2013; 2(4):477-489.Chicago/Turabian Style
Brunet, Olivier. 2013. "Orthogonality and Dimensionality." Axioms 2, no. 4: 477-489.