Related papers: Symmetry, Self-Duality and the Jordan Structure of…
This note adds to the recent spate of derivations of the probabilistic apparatus of finite-dimensional quantum theory from various axiomatic packages. We offer two different axiomatic packages that lead easily to the Jordan algebraic…
Among the many important geometric properties of quantum state space are: transitivity of the group of symmetries of the cone of unnormalized states on its interior (homogeneity), identification of this cone with its dual cone of effects…
The famous Koecher-Vinberg theorem characterises the finite dimensional formally real Jordan algebras among the finite dimensional order unit spaces as the ones that have a symmetric cone. An alternative characterisation of symmetric cones…
A celebrated result of Koecher and Vinberg asserts the one-one correspondence between the finite dimensional formally real Jordan algebras and Euclidean symmetric cones. We extend this result to the infinite dimensional setting.
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…
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…
Using a result of H. Hanche-Olsen, we show that (subject to fairly natural constraints on what constitutes a system, and on what constitutes a composite system), orthodox finite-dimensional complex quantum mechanics with superselection…
We show that finite-dimensional order unit spaces equipped with a continuous sequential product as defined by Gudder and Greechie are homogeneous and self-dual. As a consequence of the Koecher-Vinberg theorem these spaces therefore…
There have been several propositions for a geometric and essentially non-linear formulation of quantum mechanics. From a purely mathematical point of view, the point of view of Jordan algebra theory might give new strength to such…
Jordan algebras were first introduced in an effort to restructure quantum mechanics purely in terms of physical observables. In this paper we explain why, if one attempts to reformulate the internal structure of the standard model of…
Symmetry postulates play a crucial role in various approaches to reconstruct quantum theory from a few basic principles. Discrete and continuous symmetries are under consideration. The continuous case better matches the physical needs for…
Jordan algebras arise naturally in (quantum) information geometry, and we want to understand their role and their structure within that framework. Inspired by Kirillov's discussion of the symplectic structure on coadjoint orbits, we provide…
The goal of this note is to show that Jordan algebras and superalgebras provide an elegant and concise language for formulating quantum mechanical problems with inherent (super)conformal symmetry. The superconformal symmetries of the…
The famous Koecher-Vinberg theorem characterizes the Euclidean Jordan algebras among the finite dimensional order unit spaces as the ones that have a symmetric cone. Recently Walsh gave an alternative characterization of the Euclidean…
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 Jordan structure of finite-dimensional quantum theory is derived, in a conspicuously easy way, from a few simple postulates concerning abstract probabilistic models (each defined by a set of basic measurements and a convex set of…
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 interplay between the algebraic structure (operator algebras) for the quantum observables and the convex structure of the state space has been explored for a long time and most advanced results are due to Alfsen and Shultz. Here we…
Based on an interpretation of the quark-lepton symmetry in terms of the unimodularity of the color group $SU(3)$ and on the existence of 3 generations, we develop an argumentation suggesting that the "finite quantum space" corresponding to…
Quantum theory's Hilbert space apparatus in its finite-dimensional version is nearly reconstructed from four simple and quantum-mechanically motivated postulates for a quantum logic. The reconstruction process is not complete, since it…