Related papers: Lie algebra of Ashtekar-Barbero connection operato…
In this paper we establish the existence of the non-perturbative theory of quantum gravity known as quantum holonomy theory by showing that a Hilbert space representation of the QHD(M) algebra, which is an algebra generated by…
Lie algebroids provide a natural medium to discuss classical systems, however, quantum systems have not been considered. In aim of this paper is to attempt to rectify this situation. Lie algebroids are reviewed and their use in classical…
The mathematical structure of homogeneous loop quantum cosmology is analyzed, starting with and taking into account the general classification of homogeneous connections not restricted to be Abelian. As a first consequence, it is seen that…
The machinery of noncommutative geometry is applied to a space of connections. A noncommutative function algebra of loops closely related to holonomy loops is investigated. The space of connections is identified as a projective limit of…
Covariant loop gravity comes out of the canonical analysis of the Palatini action and the use of the Dirac brackets arising from dealing with the second class constraints (``simplicity'' constraints). Within this framework, we underline a…
In [arXiv:1411.3592] an extension of the Ashtekar-Lewandowski (AL) state space of Loop Quantum Gravity was set up with the help a projective formalism introduced by Kijowski [Kijowski 1977; see also: arXiv:1304.6330, arXiv:1411.3590]. The…
Within the context of the Ashtekar variables, the Hamiltonian constraint of four-dimensional pure General Relativity with cosmological constant, $\Lambda$, is reexpressed as an affine algebra with the commutator of the imaginary part of the…
This paper introduces Lie groups in degenerate geometric (Clifford) algebras that preserve four fundamental subspaces determined by the grade involution and reversion under the adjoint and twisted adjoint representations. We prove that…
A real Lie algebra with a compatible Hilbert space structure (in the sense that the scalar product is invariant) is called a Hilbert-Lie algebra. Such Lie algebras are natural infinite-dimensional analogues of the compact Lie algebras; in…
This paper is concerned with the holonomy of a class of spaces which includes Landsberg spaces of Finsler geometry. The methods used are those of Lie groupoids and algebroids as developed by Mackenzie. We prove a version of the…
This paper develops a geometric framework for virtual constraints on Lie groups, with emphasis on mechanical systems modeled as affine connection systems. Virtual holonomic and virtual nonholonomic constraints, including linear and affine…
Lie groups and quantum algebras are connected through their common universal enveloping algebra. The adjoint action of Lie group on its algebra is naturally extended to related q-algebra and q-coalgebra. In such a way, quantum structure can…
The canonical quantization of diffeomorphism invariant theories of connections in terms of loop variables is revisited. Such theories include general relativity described in terms of Ashtekar-Barbero variables and extension to Yang-Mills…
From the Holst action in terms of complex valued Ashtekar variables additional reality conditions mimicking the linear simplicity constraints of spin foam gravity are found. In quantum theory with the results of You and Rovelli we are able…
It is shown that the description of certain class of representations of the holonomy Lie algebra associated to hyperplane arrangement $\Delta$ is essentially equivalent to the classification of $\vee$-systems associated to $\Delta.$ The…
Recently the $\text{SU}(2)$ spin-network states in loop quantum gravity is generalized to those of the corresponding affine Lie algebra. We show that if one literally starts from the full $\text{SL}(2,\mathbb{C})$ group, this procedure…
Inspired by Bohr's dictum that "physical phenomena are observed relative to different experimental setups", this article investigates the notion of relativity in Bohr's sense, starting from a set of binary elements. The most general form of…
In this paper, we study the controllability properties and the Lie algebra structure of networks of particles with spin immersed in an electro-magnetic field. We relate the Lie algebra structure to the properties of a graph whose nodes…
We present a new formulation of quantum holonomy theory, which is a candidate for a non-perturbative and background independent theory of quantum gravity coupled to matter and gauge degrees of freedom. The new formulation is based on a…
Quantum theory can be formulated as a theory of operations, more specific, of complex represented operations from real Lie groups. Hilbert space eigenvectors of acting Lie operations are used as states or particles. The simplest simple Lie…