Orthogonality and Dimensionality
Résumé
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 \emph{orthomatroids}, we then show that, in high enough dimension, the basic ingredients of orthomatroids (more precisely the simple and irreducible ones) are isomorphic to generalized Hilbert lattices, so that the latter are a direct consequence of an orthogonality-based characterization of dimension.
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