Related papers: Categorified Algebra and Quantum Mechanics
This paper describes how the structure of the state space of the quantum harmonic oscillator can be described by an adjunction of categories, that encodes the raising and lowering operators into a commutative comonoid. The formulation is an…
We review the problem of finding a general framework within which one can construct quantum theories of non-standard models for space, or space-time. The starting point is the observation that entities of this type can typically be regarded…
A model category is called combinatorial if it is cofibrantly generated and its underlying category is locally presentable. As shown in recent years, homotopy categories of combinatorial model categories share useful properties, such as…
This thesis provides an introduction to the various category theory ideas employed in topological quantum field theory. These theories are viewed as symmetric monoidal functors from topological cobordism categories into the category of…
Using the fact that any linear representation of a group can be embedded into permutations, we propose a constructive description of quantum behavior that provides, in particular, a natural explanation of the appearance of complex numbers…
`Categorification' is the process of replacing equations by isomorphisms. We describe some of the ways a thoroughgoing emphasis on categorification can simplify and unify mathematics. We begin with elementary arithmetic, where the category…
Quantum contextuality, a fundamental feature distinguishing quantum theory from classical models, is investigated via algebraic and topological structures inherent in modular tensor categories. This work rigorously demonstrates that braid…
Quantum physics has revealed many interesting formal properties associated with the algebra of two operators, A and B, satisfying the partial commutation relation AB-BA=1. This study surveys the relationships between classical combinatorial…
Machine Learning classification models learn the relation between input as features and output as a class in order to predict the class for the new given input. Quantum Mechanics (QM) has already shown its effectiveness in many fields and…
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…
Quantum categories were introduced in [4] as generalizations of both bi(co)algebroids and small categories. We clarify details of that work. In particular, we show explicitly how the monadic definition of a quantum category unpacks to a set…
A quantitative model of concurrent interaction is introduced. The basic objects are linear combinations of partial order relations, acted upon by a group of permutations that represents potential non-determinism in synchronisation. This…
Classical block designs are important combinatorial structures with a wide range of applications in Computer Science and Statistics. Here we give a new abstract description of block designs based on the arrow category construction. We show…
We formulate quantum mechanics in spacetimes with real-order fractional geometry and more general factorizable measures. In spacetimes where coordinates and momenta span the whole real line, Heisenberg's principle is proven and the…
The combinatorial theory of species developed by Joyal provides a foundation for enumerative combinatorics of objects constructed from finite sets. In this paper we develop an analogous theory for the enumerative combinatorics of objects…
The regular objects in various categories, such as maps, hypermaps or covering spaces, can be identified with the normal subgroups N of a given group \Gamma, with quotient group isomorphic to \Gamma/N. It is shown how to enumerate such…
We construct a colored operad whose category of algebras is the category of algebraic quantum field theories. This is achieved by a construction that depends on the choice of a category, whose objects provide the operad colors, equipped…
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…
We show that observables in QED-type theories can be realized in terms of a combinatorial structure called chord diagrams. One advantage of this combinatorial representation is that it simplifies the study of the asymptotic behavior of…
Quantum algebras are a mathematical tool which provides us with a class of symmetries wider than that of Lie algebras, which are contained in the former as a special case. After a self-contained introduction to the necessary mathematical…