Related papers: Differential Geometry on the Space of Connections …
(This short article is a continuation of a longer, review work, in the same volume of Proceedings, by Ashtekar, Marolf and Mour\~ao [gr-qc/9403042]. All the details and other results are to be found in joint papers of the author with Abhay…
It is often noted that many of the basic concepts of differential geometry, such as the definition of connection, are purely algebraic in nature. Here, we review and extend existing work on fully algebraic formulations of differential…
The differential geometry on a Hopf algebra is constructed, by using the basic axioms of Hopf algebras and noncommutative differential geometry. The space of generalized derivations on a Hopf algebra of functions is presented via the smash…
In the present paper, we outline and expound the fundamental and novel qualitative-cum-philosophical premises, principles, ideas, concepts, constructions and results that originate from our ongoing research project of applying the…
We give detailed exposition of modern differential geometry from global coordinate independent point of view as well as local coordinate description suited for actual computations. In introduction, we consider Euclidean spaces and different…
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 ``geometry'', in the sense of the classical differential geometry of smooth manifolds (CDG), is put under scrutiny from the point of view of Abstract Differential Geometry (ADG), along with resulting, thereby, potential physical…
We construct a sheaf-theoretic representation of quantum observables algebras over a base category equipped with a Grothendieck topology, consisting of epimorphic families of commutative observables algebras, playing the role of local…
The topic of this thesis is the development of a versatile and geometrically motivated differential calculus on non-commutative or quantum spaces, providing powerful but easy-to-use mathematical tools for applications in physics and related…
Differential forms is a highly geometric formalism for physics used from field theories to General Relativity (GR) which has been a great upgrade over vector calculus with the advantages of being coordinate-free and carrying a high degree…
Many basis sets for electronic structure calculations evolve with varying external parameters, such as moving atoms in dynamic simulations, giving rise to extra derivative terms in the dynamical equations. Here we revisit these derivatives…
The fact that quantum theory is non-differentiable, while general relativity is built on the assumption of differentiability sources an incompatibility between quantum theory and gravity. Higher order geometry addresses this issue directly…
The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed in recent work (collaboration with H.…
In this brief note (written as a lengthy letter), we describe the construction of a representation for the Weyl-algebra underlying Loop Quantum Geometry constructed from a diffeomorphism variant state, which corresponds to a ''condensate''…
Quantum geometry on a discrete set means a directed graph with a weight associated to each arrow defining the quantum metric. However, these `lattice spacing' weights do not have to be independent of the direction of the arrow. We use this…
Quantum gauge theory in the connection representation uses functions of holonomies as configuration observables. Physical observables (gauge and diffeomorphism invariant) are represented in the Hilbert space of physical states; physical…
Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic…
A new approach is suggested to quantum differential calculus on certain quantum varieties. It consists in replacing quantum de Rham complexes with differentials satisfying Leibniz rule by those which are in a sense close to Koszul complexes…
Differential geometric formulation of quantum gauge theory of gravity is studied in this paper. The quantum gauge theory of gravity which is proposed in the references hep-th/0109145 and hep-th/0112062 is formulated completely in the…
Considering homogeneous four-dimensional space-time geometries within real projective geometry provides a mathematically well-defined framework to discuss their deformations and limits without the appearance of coordinate singularities. On…