Related papers: Orthogonality and Dimensionality
The basic notions of quantum mechanics are formulated in terms of separable infinite dimensional Hilbert space $\mathcal{H}$. In terms of the Hilbert lattice $\mathcal{L}$ of closed linear subspaces of $\mathcal{H}$ the notions of state and…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
The traditional display of elements in the periodic table is convenient for the study of chemistry and physics. However, the atomic number alone is insufficient for training statistical machine learning models to describe and extract…
The signaling dimension of a given physical system quantifies the minimum dimension of a classical system required to reproduce all input/output correlations of the given system. Thus, unlike other dimension measures - such as the dimension…
We present two geometric interpretations for complex multivectors and determinants: a little known one in terms of square roots of volumes, and a new one which uses fractions of volumes and allows graphical representations. The fraction…
We give a correspondence between simple matroids and a reconstruction of Alfred North Whitehead's theory of dimension, as developed in "On Mathematical Concepts of the Material World" (1906). In brief, if a geometrical system in the…
A periodic lattice in Euclidean space is the infinite set of all integer linear combinations of basis vectors. Any lattice can be generated by infinitely many different bases. This ambiguity was only partially resolved, but standard…
We present and study approximate notions of dimensional and margin complexity, which correspond to the minimal dimension or norm of an embedding required to approximate, rather then exactly represent, a given hypothesis class. We show that…
Let $\big(\mathscr{X}, \langle\cdot, \cdot\rangle\big)$ be a Hilbert $C^*$-module over a $C^*$-algebra $\mathscr{A}$ and let $\mathcal{S}(\mathscr{A})$ be the set of states on $\mathscr{A}$. In this paper, we first compute the norm…
We introduce Backstr\"om pairs and Backstr\"om rings, study their derived categories and construct for them a sort of categorical resolutions. For the latter we define the global dimension, construct a sort of semi-orthogonal decomposition…
This article introduces Hilbert $*$-categories: an abstraction of categories with similar algebraic and analytic properties to the categories of real, complex, and quaternionic Hilbert spaces and bounded linear maps. Other examples include…
A notion of general manifolds is introduced. It covers all usual manifolds in mathematics. Essentially, it is a way how to get a bigger 'fibration' over a site which locally coincides with a given one. An enrichment with generalized…
Taking several statistical examples, in particular one involving a choice of experiment, as points of departure, and making symmetry assumptions, the link towards quantum theory developed in Helland (2005a,b) is surveyed and clarified. The…
In this work, we attempt to define a notion of compositeness compatible with Quantum Field Theory. Considering the analytic properties of the S-matrix, we conclude that there is no satisfactory definition of compositeness compatible with…
We discuss correspondence between the predictions of quantum theories for rotation angle formulated in infinite and finite dimensional Hilbert spaces, taking as example, the calculation of matrix elements of phase-angular momentum…
In this paper we outline a Matrix Ansatz approach to some problems of combinatorial enumeration. The idea is that many interesting quantities can be expressed in terms of products of matrices, where the matrices obey certain relations. We…
For the first time it is shown that the logic of quantum mechanics can be derived from Classical Physics. An orthomodular lattice of propositions, characteristic of quantum logic, is constructed for manifolds in Einstein's theory of general…
In this talk we entertain the possibility that the synthesis of general covariance and quantum mechanics requires an extension of the basic kinematical setup of quantum mechanics. According to the holographic principle, regions of spacetime…
Formalizations of quantum information theory in category theory and type theory, for the design of verifiable quantum programming languages, need to express its two fundamental characteristics: (1) parameterized linearity and (2) metricity.…
In this article we prove the topological minimality of unions of several almost orthogonal planes of arbitrary dimensions. A particular case was proved in arXiv:1103.1468, where we proved the Almgren minimality (which is a weaker property…