Related papers: Is there a Jordan geometry underlying quantum phys…
We characterize the space of restrictions of real rational functions to certain algebraic Jordan curves in the plane via the Dirichlet-to-Neumann map associated to the domain in the complex plane bounded by the curve and its Bergman kernel.…
In this scientific preface to the first issue of International Journal of Geometric Methods in Modern Physics, we briefly survey some peculiarities of geometric techniques in quantum models.
Two novel frameworks for handling mathematical and physical problems are introduced. The first, the emerging Jordan form, generalizes the concept of the Jordan canonical form, a well-established tool of linear algebra. The second, dual…
The aim of this paper is to offer an overview of the most important applications of Jordan structures inside mathematics and also to physics, up-dated references being included. For a more detailed treatment of this topic see - especially -…
Quantum geometry, i.e., the quantum theory of intrinsic and extrinsic spatial geometry, is a cornerstone of loop quantum gravity. Recently, there have been many new ideas in this field, and I will review some of them. In particular, after a…
We define symmetric spaces in arbitrary dimension and over arbitrary non-discrete topological fields $\K$, and we construct manifolds and symmetric spaces associated to topological continuous quasi-inverse Jordan pairs and -triple systems.…
The issue of whether some manifestations of gravitation in the quantum domain, are indicative or not of a non-geometrical aspect in gravitation is discussed. We examine some examples that have been considered in this context, providing a…
A conjecture for the dimension and the character of the homogenous components of the free Jordan algebras is proposed. As a support of the conjecture, some numerical evidences are generated by a computer and some new theoretical results are…
While describing the results of our recent work on exceptional Lie and Jordan algebras, so tightly intertwined in their connection with elementary particles, we will try to stimulate a critical discussion on the nature of spacetime and…
In this paper, we investigate metric Jordan algebras, and follow the lines of the paper (J. Milnor: Curvatures of left invariant metrics on Lie groups. Adv. Math. (1976)). Firstly, we define the Jordan-Levi-Civita connection, then we show…
Any canonical quantum theory can be understood to arise from the compatibility of the statistical geometry of distinguishable observations with the canonical Poisson structure of Hamiltonian dynamics. This geometric perspective offers a…
We continue the study undertaken in \cite{DV} of the exceptional Jordan algebra $J = J_3^8$ as (part of) the finite-dimensional quantum algebra in an almost classical space-time approach to particle physics. Along with reviewing known…
Over the past six years, a detailed framework has been constructed to unravel the quantum nature of the Riemannian geometry of physical space. A review of these developments is presented at a level which should be accessible to graduate…
Physical theories can be characterized in terms of their state spaces and their evolutive equations. The kinematical structure and the dynamical structure of finite dimensional quantum theory are, in light of the Choi-Jamio{\l}kowski…
A representation of finite-dimensional probabilistic models in terms of formally real Jordan algebras is obtained, in a strikingly easy way, from simple assumptions. This provides a framework in which real, complex and quaternionic quantum…
The application of geometry to physics has provided us with new insightful information about many physical theories such as classical mechanics, general relativity, and quantum geometry (quantum gravity). The geometry also plays an…
We discuss the basic properties of Lie groupoids, Lie algebroids and Lie pseudo-groups in view of applying these techniques to the analysis of Jordan-H\"older resolutions and, subsequently, to the integration of partial differential…
We formulate quantum group Riemannian geometry as a gauge theory of quantum differential forms. We first develop (and slightly generalise) classical Riemannian geometry in a self-dual manner as a principal bundle frame resolution and a dual…
We discuss new approaches to fundamental problems of mathematics and mathematical physics such as mathematical foundation of quantum field theory, the Riemann hypothesis, and construction of noncommutative algebraic geometry.
In 2003 Peter Cameron introduced the concept of a Jordan scheme and asked whether there exist Jordan schemes which are not symmetrisations of coherent configurations (proper Jordan schemes). The question was answered affirmatively by the…