Related papers: Orthogonality and Dimensionality
How do symmetries induce natural and useful quantum structures? This question is investigated in the context of models of three interacting particles in one-dimension. Such models display a wide spectrum of possibilities for dynamical…
An orthogonality space is a set equipped with a symmetric, irreflexive relation called orthogonality. Every orthogonality space has an associated complete ortholattice, called the logic of the orthogonality space. To every poset, we…
In a finite dimensional Hilbert space, each normalized vector (state) can be chosen as a member of an orthonormal basis of the space. We give a proof of this statement in a manner that seems to be more comprehensible for physics students…
We introduce the notion of order projections using the order unit property of a positive element in an order unit space and characterize them in terms of (geometric) orthogonality. We describe order projections of the order unit space…
Coordination geometries describe how the neighbours of a central particle are arranged around it. Such geometries can be thought to lie in an abstract topological space; a model of this space could provide a mathematical basis for…
A matroid is a machine capturing linearity of mathematical objects and producing combinatorial structures. Matroid structure arises everywhere since linearity is a ubiquitous concept. One natural way to obtain matroids is by considering…
Motivated by Quantum Bayesianism I give background for a general epistemic approach to quantum mechanics, where complementarity and symmetry are the only essential features. A general definition of a symmetric epistemic setting is…
We show that a partition of the unity (or POVM) on a Hilbert space that is almost orthogonal is close to an orthogonal POVM in the same von Neumann algebra. This generalizes to infinite dimension previous results in matrix algebras by…
In a general triangulated category, the finiteness of the finitistic dimension serves as a prerequisite for a categorical obstruction, via the singularity category, to the existence of bounded $t$-structures. In this paper, we investigate…
We study the Hadamard product of the linear forms defining a hyperplane arrangement with those of its dual, which we view as generating an ideal in a certain polynomial ring. We use this ideal, which we call the ideal of pairs, to study…
To the best of our current understanding, quantum mechanics is part of the most fundamental picture of the universe. It is natural to ask how pure and minimal this fundamental quantum description can be. The simplest quantum ontology is…
A description of physical reality in which wholeness is the foundation is discussed along with the motivation for such an attempt. As a possible mathematical framework within which a physical theory based on wholeness may be expressed,…
We invent the notion of a {\it dimension of a variety} $V$ as the cardinality of all its proper {\it derived} subvarieties (of the same type). The dimensions of varieties of lattices, varieties of regular bands and other general algebraic…
What is the "right way" to define dimension? Mathematicians working in the early and middle $20$th-century formalized three intuitive definitions of dimension that all turned out to be equivalent on separable metric spaces. But were these…
This article is an introductory work to a larger research project devoted to pure, applied and philosophical aspects of dimension theory. It concerns a novel approach toward an alternate dimension theory foundation: the point-dimension…
Quantum logic aims to capture essential quantum mechanical structure in order-theoretic terms. The Achilles' heel of quantum logic is the absence of a canonical description of composite systems, given descriptions of their components. We…
The Hilbert space dimension of a quantum system is the most basic quantifier of its information content. Lower bounds on the dimension can be certified in a device-independent way, based only on observed statistics. We highlight that some…
In this short article we show an orthogonal decomposition of a Hilbert space as a sum of null solutions of the first derivative and the first derivative of a traceless higher order Hilbert/Sobolev space. We define orthogonal projections and…
This paper addresses the question why quantum mechanics is formulated in a unitary Hilbert space, i.e. in a manifestly complex setting. Investigating the linear dynamics of real quantum theory in a finite-dimensional Euclidean Hilbert space…
In considering the nature of the basic mathematical structures appropriate for describing the fundamental elements of particle physics a significant role for the octonions, as an extension from the complex numbers and uniquely the largest…