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Axioms 2013, 2(4), 477-489; doi:10.3390/axioms2040477
Article

Orthogonality and Dimensionality

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)
Download PDF [178 KB, uploaded 13 December 2013]
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 quantum logic; Piron’s representation theorem; foundations of quantum mechanics
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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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.

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