Related papers: Modular Theory and Geometry
The main purpose of this paper is to present a new approach to logic or what we will call superlogic. This approach constitutes a new way of looking at the connection between quantum mechanics and logic. It is a {\it geometrisation} of the…
The aim of these notes is to elucidate some aspects of quantum field theory in curved spacetime, especially those relating to the notion of particles. A selection of issues relevant to wave-particle duality is given. The case of a generic…
The scope of this review is to give a pedagogical introduction to some new calculations and methods developed by the author in the context of quantum groups and their applications. The review is self- contained and serves as a "first aid…
We develop a mathematical theory of quantization of multidimensional variational principles, and compare it with traditional constructions of quantum field theory. We conjecture that mathematical realization of quantum field theory axioms,…
Some explanations and implications of the underlying theory approach for quantum theories (QM or QFT) are discussed and suggested. This simple idea seems to have significantly nontrivial effects for our understanding of the quantum…
This is an overview of higher structural constructions in physics. The main motivations of our current attempt are as follows: (i) to provide a brief introduction to derived algebraic geometry, (ii) to understand how derived objects…
Generalisations of geometry have emerged in various forms in the study of field theory and quantization. This mini-review focuses on the role of higher geometry in three selected physical applications. After motivating and describing some…
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.
We comment on the present status, the concepts and their limitations, and the successes and open problems of the various approaches to a relativistic quantum theory of elementary particles, with a hindsight to questions concerning quantum…
We present a generally covariant approach to quantum mechanics in which generalized positions, momenta and time variables are treated as coordinates on a fundamental "phase-spacetime." We show that this covariant starting point makes…
The two essential ideas in this paper are, on the one hand, that a considerable amount of the power of quantum computation may be obtained by adding to a classical computer a few specialized quantum modules and, on the other hand, that such…
We review the generalization of field theory to space-time with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and…
A mathematically rigorous Hamiltonian formulation for classical and quantum field theories is given. New results include clarifications of the structure of linear fields, and a plausible formulation for nonlinear fields. Many mathematical…
We describe the elements of a novel structural approach to classical field theory, inspired by recent developments in perturbative algebraic quantum field theory. This approach is local and focuses mainly on the observables over field…
The present text surveys some relevant situations and results where basic Module Theory interacts with computational aspects of operator algebras. We tried to keep a balance between constructive and algebraic aspects.
Using tangent bundle geometry we construct an equivalent reformulation of classical field theory on flat spacetimes which simultaneously encodes the perspectives of multiple observers. Its generalization to curved spacetimes realizes a new…
We give a rough description of the 'categories' formed by quantum field theories. A few recent mathematical conjectures derived from quantum field theories, some of which are now proven theorems, will be presented in this language.
Recent critiques of the semantic conception of scientific theories suggest that a theory is not best formulated as a collection of models satisfying some set of kinematical or dynamical conditions. Thus it has been argued that additional…
The algebraic approach to QFT, which for several decades has enriched QFT with structural theorems, has recently shown its utility in various constructions of actual interest. In these lecture notes I explain how AQFT (in particular the…
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…