Related papers: The loop derivative as a curvature
It is shown that every quantum principal bundle with a compact structure group is a Hopf-Galois extension. This property naturally extends to the level of general differential structures, so that every differential calculus over a quantum…
For a Lie groupoid $G$, the differential forms on its nerve comprise a double complex. A natural question is if this statement extends to forms with values in a representation $V$ of $G$. In this paper, we research two types of covariant…
Motivated by extending the functional stochastic calculus, to important functionals to which it does not apply, a notion of functional derivative along a curve is introduced. This new setting is developed by incorporating path-dependent…
In this paper we introduce a notion of parallel transport for principal bundles with connections over differentiable stacks. We show that principal bundles with connections over stacks can be recovered from their parallel transport thereby…
We outline the recent classification of differential structures for all main classes of quantum groups. We also outline the algebraic notion of `quantum manifold' and `quantum Riemannian manifold' based on quantum group principal bundles, a…
"Quantum Topology" deals with the general quantum theory as the theory of the functional quantum space; space time and energy momentum forms form a connected manifold; a functional quantum space on the quantum level. The general quantum…
Dynamical evolution is described as a parallel section on an infinite dimensional Hilbert bundle over the base manifold of all frames of reference. The parallel section is defined by an operator-valued connection whose components are the…
Measuring comodules are defined and shown to provide a useful generalization of the set of maps between modules with a broad range of applications. Three applications are described. Connections on bundles are described in terms of measuring…
We present an extension to arbitrary dimensions of a worldline path integral approach to one-loop quantum gravity, which was previously formulated in four spacetime dimensions. By utilizing this method, we recalculate gauge invariant…
A model is proposed to demonstrate that classical general relativity can emerge from loop quantum gravity, in a relational description of gravitational field in terms of the coordinates given by matter. Local Dirac observables and coherent…
The recent developments of the ``connection'' and ``loop'' representations have given the possibility to show that the two representation are equivalent and that it is possible to transform any result from one representation into the other.…
Aspects of the full theory of loop quantum gravity can be studied in a simpler context by reducing to symmetric models like cosmological ones. This leads to several applications where loop effects play a significant role when one is…
Loop quantum gravity is an approach to quantum gravity that starts from the Hamiltonian formulation in terms of a connection and its canonical conjugate. Quantization proceeds in the spirit of Dirac: First one defines an algebra of basic…
The gauge symmetry of classical general relativity under space-time diffeomorphisms implies that any path integral quantization which can be interpreted as a sum over space-time geometries, gives rise to a formal invariant of smooth…
We present an interpretation of loop quantization in the framework of lattice gauge theory. Within this context the lack of appropriate notions of effective theories and renormalization group flow exhibit loop quantization as an incomplete…
The approaches to quantum field theories based in the so called loop representation deserved much attention recently. In it, closed curves and holonomies around them play a central role. In this framework the group of loops and the group of…
We introduce a category of noncommutative bundles. To establish geometry in this category we construct suitable noncommutative differential calculi on these bundles and study their basic properties. Furthermore we define the notion of a…
We define a categorical action of the shifted quantum loop group of $\mathfrak{sl}_2$ on the derived categories of Quot schemes of finite length quotient sheaves on a smooth projective curve. As an application, we obtain a semi-orthogonal…
We continue an analysis of representations of cylindrical functions and fluxes which are commonly used as elementary variables of Loop Quantum Gravity. We consider an arbitrary principal bundle of a compact connected structure group and…
We consider the variational complex on infinite jet space and the complex of variational derivatives for Lagrangians of multidimensional paths and study relations between them. The discussion of the variational (bi)complex is set up in…