Related papers: Geometry of Quantum Principal Bundles II-Extended …
The phenomenon of quantum entanglement is thoroughly investigated, focussing especially on geometrical aspects and on bipartite systems. After introducing the formalism and discussing general aspects, some of the most important separability…
Algebras of functions on quantum weighted projective spaces are introduced, and the structure of quantum weighted projective lines or quantum teardrops are described in detail. In particular the presentation of the coordinate algebra of the…
The geometry of Quantum Mechanics in the context of uncertainty and complementarity, and probability is explored. We extend the discussion of geometry of uncertainty relations in wider perspective. Also, we discuss the geometry of…
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…
In this letter we investigate some aspects of the noncommutative differential geometry based on derivations of the algebra of endomorphisms of an oriented complex hermitian vector bundle. We relate it, in a natural way, to the geometry of…
This review paper is concerned with the generalizations to field theory of the tangent and cotangent structures and bundles that play fundamental roles in the Lagrangian and Hamiltonian formulations of classical mechanics. The paper…
The framework of generalized probabilistic theories is a powerful tool for studying the foundations of quantum physics. It provides the basis for a variety of recent findings that significantly improve our understanding of the rich physical…
We show that it is possible to represent various descriptions of Quantum Mechanics in geometrical terms. In particular we start with the space of observables and use the momentum map associated with the unitary group to provide an unified…
Perhaps the most important contribution of gauge theory to general mathematics is to point out the importance of association functors. Emphasizing category theory we characterize association functors by two of their natural properties and…
A gauge theory is associated with a principal bundle endowed with a connection permitting to define horizontal lifts of paths. The horizontal lifts of surfaces cannot be defined into a principal bundle structure. An higher gauge theory is…
The usual quantization of a classical space-time field does not touch the non-geometrical character of quantum mechanics. We believe that the deep problems of unification of general relativity and quantum mechanics are rooted in this poor…
The algebraic formulation of the quantum group covariant noncommutative geometry in the framework of the $R$-matrix approach to the theory of quantum groups is given. We consider structure groups taking values in the quantum groups and…
The leitmotiv of this review is noncommutative principal U(1)-bundles and associated line bundles. In the first part I give a brief introduction to Hopf-Galois theory and its applications, from field extensions to principal group actions. I…
Motivated by the computations done in \cite{C1}, where I introduced and discussed what I called the groupoid of generalized gauge transformations, viewed as a groupoid over the objects of the category $\mathsf{Bun}_{G,M}$ of principal…
These lecture notes cover 13 sessions and are presented as an e-print, intended to evolve over time. Quantum invariants do more than distinguish topological objects; they build bridges between topology, algebra, number theory and quantum…
We investigate the differential geometry of bipartite quantum states. In particular the manifold structures of pure bipartite states are studied in detail. The manifolds with respect to all normalized pure states of arbitrarily given…
We define a quantum generalization of the algebra of functions over an associated vector bundle of a principal bundle. Here the role of a quantum principal bundle is played by a Hopf-Galois extension. Smash products of an algebra times a…
We construct quantum group-valued canonical connections on quantum homogeneous spaces, including a q-deformed Dirac monopole on the quantum sphere of Podles quantum differential coming from the 3-D calculus of Woronowicz on $SU_q(2)$ . The…
The geometric properties of quantum states are crucial for understanding many physical phenomena in quantum mechanics, condensed matter physics, and optics. The central object describing these properties is the quantum geometric tensor,…
We outline how Drinfeld twist deformation techniques can be applied to the deformation quantization of principal bundles into noncommutative principal bundles, and more in general to the deformation of Hopf-Galois extensions. First we twist…