Related papers: Categorical Operator Algebraic Foundations of Rela…
Many insights into the quantum world can be found by studying it from amongst more general operational theories of physics. In this thesis, we develop an approach to the study of such theories purely in terms of the behaviour of their…
This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita-Takesaki modular theory and A. Connes non-commutative geometry aiming at the reconstruction of…
The standard formulation of quantum theory assumes a predefined notion of time. This is a major obstacle in the search for a quantum theory of gravity, where the causal structure of space-time is expected to be dynamical and fundamentally…
We reconsider differential geometry from the point of view of the quantum theory of non-relativistic spinning particles, which provides examples of supersymmetric quantum mechanics. This enables us to encode geometrical structure in…
A central theme in current work in quantum information and quantum foundations is to see quantum mechanics as occupying one point in a space of possible theories, and to use this perspective to understand the special features and properties…
We use tools from non-standard analysis to formulate the building blocks of quantum field theory within the framework of categorical quantum mechanics. Building upon previous work, we construct an object of *Hilb having quantum fields as…
An algebraic formulation of general relativity is proposed. The formulation is applicable to quantum gravity and noncommutative space. To investigate quantum gravity we develop the canonical formalism of operator geometry, after…
We propose a mathematical structure, based on a noncommutative geometry, which combines essential aspects of general relativity and quantum mechanics, and leads to correct "limiting cases" of both these theories. We quantize a groupoid…
The model of the position-dependent noncommutativety in quantum mechanics is proposed. We start with a given commutation relations between the operators of coordinates [x^{i},x^{j}]=\omega^{ij}(x), and construct the complete algebra of…
Motivated by the sharp contrast between classical and quantum physics as probability theories, in these lecture notes I introduce the basic notions of operator algebras that are relevant for the algebraic approach to quantum physics.…
We present a model unifying general relativity and quantum mechanics. The model is based on the (noncommutative) algebra \mbox{{\cal A}} on the groupoid \Gamma = E \times G where E is the total space of the frame bundle over spacetime, and…
Several algebro-geometric properties of commutative rings of partial differential operators as well as several geometric constructions are investigated. In particular, we show how to associate a geometric data by a commutative ring of…
Observable properties of a classical physical system can be modelled deterministically as functions from the space of pure states to outcomes; dually, states can be modelled as functions from the algebra of observables to outcomes. The…
Based on the assumption that time evolves only in one direction and mechanical systems can be described by Lagrangeans, a dynamical C*-algebra is presented for non-relativistic particles at atomic scales. Without presupposing any…
We investigate the quantum structure of spacetime at fundamental scales via a novel, Lorentz-invariant noncommutative coordinate framework. Building on insights from noncommutative geometry, spectral theory, and algebraic quantum field…
This paper charts a very direct path between the categorical approach to quantum mechanics, due to Abramsky and Coecke, and the older convex-operational approach based on ordered vector spaces (recently reincarnated as "generalized…
This paper presents a research program aimed at establishing relational foundations for relativistic quantum physics. Although the formalism is still under development, we believe it has matured enough to be shared with the broader…
This article develops a practical technique for studying representations of $\Bbbk$-linear categories arising in the categorification of quantum groups. We work in terms of locally unital algebras which are $\mathbb{Z}$-graded with graded…
In my Montreal lecture notes of 1988, it was suggested that the theory of linear quantum groups can be presented in the framework of the category of {\it quadratic algebras} (imagined as algebras of functions on "quantum linear spaces"),…
The operator algebras of a new family of relativistic geometric models of the relativistic oscillator are studied. It is shown that, generally, the operator of number of quanta and the pair of the shift operators of each model are the…