Related papers: An Introduction to Topos Physics
All current approaches to quantum gravity employ essentially standard quantum theory including, in particular, continuum quantities such as the real or complex numbers. However, I wish to argue that this may be fundamentally wrong in so far…
Our starting point is a particular `canvas' aimed to `draw' theories of physics, which has symmetric monoidal categories as its mathematical backbone. In this paper we consider the conceptual foundations for this canvas, and how these can…
The classifying topos of a geometric theory is a topos such that geometric morphisms into it correspond to models of that theory. We study classifying toposes for different infinitary logics: first-order, sub-first-order (i.e. geometric…
A core level of basic information for physics is identified, based on an analysis of the characteristics of the parameters space, time, mass and charge. At this level, it is found that certain symmetries operate, which can be used to…
This is an introduction to type theory, synthetic topology, and homotopy type theory from a category-theoretic and topological point of view, written as a chapter for the book "New Spaces for Mathematics and Physics" (ed. Gabriel Catren and…
Group Theory has become an invaluable tool in the physics community. Despite numerous introductory books, the subject remains challenging for beginners. Mathematica has emerged as a popular tool for research and education, offering various…
An introduction is given to an algebraic formulation and generalisation of the consistent histories approach to quantum theory. The main technical tool in this theory is an orthoalgebra of history propositions that serves as a generalised…
These notes review a description of quantum mechanics in terms of the topology of spaces, basing on the axioms of Topological Quantum Field Theory and path integral formalism. In this description quantum states and operators are encoded by…
This paper deals with the foundations of quantum mechanics. We start by outlining the characterisation, due to Birkhoff and Von Neumann, of the logical structures of the theories of classical physics and quantum mechanics, as boolean and…
Ideas from quantum field theory and topology have proved remarkably fertile in suggesting new phenomena in the quantum physics of condensed matter. Here I'll supply some broad, unifying context, both conceptual and historical, for the…
We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. In this second part we introduce the fundamental concepts of topological spaces, convergence, and continuity, as…
Topoi are categories which have enough structure to interpret higher order logic. They admit two notions of morphism: logical morphisms which preserve all of the structure and therefore the interpretation of higher order logic, and…
In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology: namely, a linear operator behaves very much…
Quantum set theory (QST) and topos quantum theory (TQT) are two long running projects in the mathematical foundations of quantum mechanics that share a great deal of conceptual and technical affinity. Most pertinently, both approaches…
We explore the possibility of replacing point set topology by higher category theory and topos theory as the foundation for quantum general relativity. We discuss the BC model and problems of its interpretation, and connect with the…
We present an abstract unifying framework for interpreting Stone-type dualities; several known dualities are seen to be instances of just one topos-theoretic phenomenon, and new dualities are introduced. In fact, infinitely many new…
In this paper we present a new categorical approach which attempts to provide an original understanding of QM. Our logos categorical approach attempts to consider the main features of the quantum formalism as the standpoint to develop a…
In this thesis I present a short review of ideas in quantum information theory. The first chapter contains introductory material, sketching the central ideas of probability and information theory. Quantum mechanics is presented at the level…
Topology forms a cornerstone in modern condensed matter and statistical physics, offering a new framework to classify the phases and phase transitions beyond the traditional Landau paradigm. However, it is widely believed that topological…
In a previous paper, we have proposed assigning as the value of a physical quantity in quantum theory, a certain kind of set (a sieve) of quantities that are functions of the given quantity. The motivation was in part physical---such a…