H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics

A certain class of Frobenius algebras has been used to characterize orthonormal bases and observables on finite-dimensional Hilbert spaces. The presence of units in these algebras means that they can only be realized finite-dimensionally. We seek a suitable generalization, which will allow arbitrary bases and observables to be described within categorical axiomatizations of quantum mechanics. We develop a definition of H*-algebra that can be interpreted in any symmetric monoidal dagger category, reduces to the classical notion from functional analysis in the category of (possibly infinite-dimensional) Hilbert spaces, and hence provides a categorical way to speak about orthonormal bases and quantum observables in arbitrary dimension. Moreover, these algebras reduce to the usual notion of Frobenius algebra in compact categories. We then investigate the relations between nonunital Frobenius algebras and H*-algebras. We give a number of equivalent conditions to characterize when they coincide in the category of Hilbert spaces. We also show that they always coincide in categories of generalized relations and positive matrices.Comment: 29 pages. Final versio

The Kinetic Basis of Self-Organized Pattern Formation

In his seminal paper on morphogenesis (1952), Alan Turing demonstrated that different spatio-temporal patterns can arise due to instability of the homogeneous state in reaction-diffusion systems, but at least two species are necessary to produce even the simplest stationary patterns. This paper is aimed to propose a novel model of the analog (continuous state) kinetic automaton and to show that stationary and dynamic patterns can arise in one-component networks of kinetic automata. Possible applicability of kinetic networks to modeling of real-world phenomena is also discussed.Comment: 8 pages, submitted to the 14th International Conference on the Synthesis and Simulation of Living Systems (Alife 14) on 23.03.2014, accepted 09.05.201

Logic is to the quantum as geometry is to gravity

I will propose that the reality to which the quantum formalism implicitly refers is a kind of generalized history, the word history having here the same meaning as in the phrase sum-over-histories. This proposal confers a certain independence on the concept of event, and it modifies the rules of inference concerning events in order to resolve a contradiction between the idea of reality as a single history and the principle that events of zero measure cannot happen (the Kochen-Specker paradox being a classic expression of this contradiction). The so-called measurement problem is then solved if macroscopic events satisfy classical rules of inference, and this can in principle be decided by a calculation. The resulting conception of reality involves neither multiple worlds nor external observers. It is therefore suitable for quantum gravity in general and causal sets in particular.Comment: plainTeX, 26 pages, 2 figures. To appear in G.F.R. Ellis, J. Murugan and A. Weltman (eds), {\it Foundations of Space and Time} (Cambridge University Press). Most current version is available at http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/ (or wherever my home-page may be)

Three Puzzles on Mathematics, Computation, and Games

In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure

Theoretical Engineering and Satellite Comlink of a PTVD-SHAM System

This paper focuses on super helical memory system's design, 'Engineering, Architectural and Satellite Communications' as a theoretical approach of an invention-model to 'store time-data'. The current release entails three concepts: 1- an in-depth theoretical physics engineering of the chip including its, 2- architectural concept based on VLSI methods, and 3- the time-data versus data-time algorithm. The 'Parallel Time Varying & Data Super-helical Access Memory' (PTVD-SHAM), possesses a waterfall effect in its architecture dealing with the process of voltage output-switch into diverse logic and quantum states described as 'Boolean logic & image-logic', respectively. Quantum dot computational methods are explained by utilizing coiled carbon nanotubes (CCNTs) and CNT field effect transistors (CNFETs) in the chip's architecture. Quantum confinement, categorized quantum well substrate, and B-field flux involvements are discussed in theory. Multi-access of coherent sequences of 'qubit addressing' in any magnitude, gained as pre-defined, here e.g., the 'big O notation' asymptotically confined into singularity while possessing a magnitude of 'infinity' for the orientation of array displacement. Gaussian curvature of k(k<0) is debated in aim of specifying the 2D electron gas characteristics, data storage system for defining short and long time cycles for different CCNT diameters where space-time continuum is folded by chance for the particle. Precise pre/post data timing for, e.g., seismic waves before earthquake mantle-reach event occurrence, including time varying self-clocking devices in diverse geographic locations for radar systems is illustrated in the Subsections of the paper. The theoretical fabrication process, electromigration between chip's components is discussed as well.Comment: 50 pages, 10 figures (3 multi-figures), 2 tables. v.1: 1 postulate entailing hypothetical ideas, design and model on future technological advances of PTVD-SHAM. The results of the previous paper [arXiv:0707.1151v6], are extended in order to prove some introductory conjectures in theoretical engineering advanced to architectural analysi