Related papers: Quantum Principal Bundles and Their Characteristic…
A new realist interpretation of quantum mechanics is introduced. Quantum systems are shown to have two kinds of properties: the usual ones described by values of quantum observables, which are called extrinsic, and those that can be…
We introduce some equivalent notions of homomorphisms between quantum groups that behave well with respect to duality of quantum groups. Our equivalent definitions are based on bicharacters, coactions, and universal quantum groups,…
We define the notion of characteristic classes for supermanifolds endowed with a homological vector field $Q$. These take values in the cohomology of the Lie derivative operator $L_Q$ acting on arbitrary tensor fields. We formulate a…
This paper considers a generalization of the notion of quantum observables in ontological models of quantum mechanics. Within this framework it is possible to construct physical models where quantum noncommutativity can arise dynamically.…
The natural generalization of the notion of bundle in quantum geometry is that of bimodule. If the base space has quantum group symmetries one is particularly interested in bimodules covariant (equivariant) under these symmetries. Most…
After about a century since the first attempts by Bohr, the interpretation of quantum theory is still a field with many open questions. In this article a new interpretation of quantum theory is suggested, motivated by philosophical…
The Gamma-class is a characteristic class for complex manifolds with transcendental coefficients. It defines an integral structure of quantum cohomology, or more precisely, an integral lattice in the space of flat sections of the quantum…
In this paper and a companion paper, we show how the framework of information geometry, a geometry of discrete probability distributions, can form the basis of a derivation of the quantum formalism. The derivation rests upon a few…
We describe a cohomological framework for measurement based quantum computation, in which symmetry plays a central role. Therein, the essential information about the computational output is contained in topological invariants, namely…
We define and study quantum permutations of infinite sets. This leads to discrete quantum groups which can be viewed as infinite variants of the quantum permutation groups introduced by Wang. More precisely, the resulting quantum groups…
We introduce the definition of the typical irreducible modules of the generalized quantum groups, and prove the Weyl-Kac-type formulas of their characters. As a by-product, we obtain the Weyl-Kac-type character formulas of the typical…
From a geometrical point of view it is, so far, not sufficiently well understood what should be a "noncommutative principal bundle". Still, there is a well-developed abstract algebraic approach using the theory of Hopf algebras. An…
We introduce and examine the notion of principal $\mathbb{Z}_2^n$-bundles, i.e., principal bundles in the category of $\mathbb{Z}_2^n$-manifolds. The latter are higher graded extensions of supermanifolds in which a $\mathbb{Z}_2^n$-grading…
A quantum principal bundle is constructed for every Coxeter group acting on a finite-dimensional Euclidean space $E$, and then a connection is also defined on this bundle. The covariant derivatives associated to this connection are the…
Over the quantum weighted 1-dimensional complex projective spaces, called quantum teardrops, the quantum line bundles associated with the quantum principal U(1)-bundles introduced and studied by Brzezinski and Fairfax are explicitly…
The properties which give quantum mechanics its unique character - unitarity, complementarity, non-commutativity, uncertainty, nonlocality - derive from the algebraic structure of Hermitian operators acting on the wavefunction in complex…
The main goal of this article is to construct some geometric invariants for the topology of the set $\mathcal{F}$ of flat connections on a principal $G$-bundle $P\,\longrightarrow\, M$. Although the characteristic classes of principal…
We formulate the notion of quantum channels in the framework of quantum tomography and address there the issue of whether such maps can be regarded as classical stochastic maps. In particular kernels of maps acting on probability…
The concept of quantum commutativity with respect to an action or coaction of a given Hopf algebra is used for the algebraic description of a system of particles and their interaction with certain quantum field. Graded commutativity and…
This article provides a basic introduction to some concepts of non-commutative geometry. The importance of quantum groups and quantum spaces is stressed. Canonical non-commutativity is understood as an approximation to the quantum group…