Related papers: The constraint algebra of quantum gravity in the l…

The notions of the Novikov deformation of a commutative associative algebra and the corresponding classical limit are introduced. We show such a classical limit belongs to a subclass of transposed Poisson algebras, and hence the Novikov…

We provide a full quantization of the vacuum Gowdy model with local rotational symmetry. We consider a redefinition of the constraints where the Hamiltonian Poisson-commutes with itself. We then apply the canonical quantization program of…

The operator algebraic framework plays an important role in mathematical physics. Many different operator algebras exist for example for a theory of quantum mechanics. In Loop Quantum Gravity only two algebras have been introduced until…

In this review, the fundamental structure of loop quantum gravity is presented pedagogically. Our main aim is to help non-experts to understand the motivations, basic structures, as well as general results. It may also be beneficial to…

In this article we examine a Hamiltonian constraint operator governing the dynamics of simple quantum states, whose graph consists of a single six-valent vertex, in quantum-reduced loop gravity. To this end, we first derive the action of…

In its canonical formulation, general relativity is subject to gauge transformations that are equivalent to space-time coordinate changes of general covariance only when the gauge generators, given by the Hamiltonian and diffeomorphism…

Spherically symmetric gravity in Ashtekar variables coupled to Yang-Mills theory in two dimensions and its relation to dilaton gravity and Poisson sigma models are discussed. After introducing its loop quantization, quantum corrections for…

The Lie-Rinehart algebra of a manifold M, defined by the Lie structure of the vector fields, their action and their module structure on the infinitely differentiable functions on M, is a common, diffeomorphism invariant, algebra for both…

It is shown that if a representation of a *-algebra on a vector space $V$ is an irreducible *-representation with respect to some inner product on $V$ then under appropriate technical conditions this property determines the inner product…

In this paper we define a quantum version of the ``fusion'' tensor product of two representations of an affine Kac-Moody algebra.It is replaced by what we call fusion action of the category of finite-dimensional representations of quantum…

We show that the non-Abelian nature of geometric fluxes---the corner-stone in the definition of quantum geometry in the framework of loop quantum gravity (LQG)---follows directly form the continuum canonical commutations relations of…

Quantization relates Poisson algebras to $C^*$-algebras. The analysis of local gauge symmetries in algebraic quantum field theory is approached through the quantization of classical gauge theories, regarded as constrained dynamical systems.…

Using self dual Ashtekar variables, we investigate (at the effective level) the spherically symmetry reduced model of loop quantum gravity, both in vacuum and when coupled to a scalar field. Within the real Ashtekar-Barbero formulation, the…

Generators and relations are given for the subalgebra of cocommutative elements in the quantized coordinate rings of the classical groups, where the deformation parameter q is transcendental. This is a ring theoretic formulation of the well…

We introduce a new class of algebras called Poisson orders. This class includes the symplectic reflection algebras of Etingof and Ginzburg, many quantum groups at roots of unity, and enveloping algebras of restricted Lie algebras in…

We introduce a new operator representing the three-dimensional scalar curvature in loop quantum gravity. The operator is constructed by writing the Ricci scalar classically as a function of the Ashtekar variables and regularizing the…

Spherically symmetric models of loop quantum gravity have been studied recently by different methods that aim to deal with structure functions in the usual constraint algebra of gravitational systems. As noticed by Gambini and Pullin, a…

We formulate Yang-Mills theory in terms of the large-N limit, viewed as a classical limit, of gauge-invariant dynamical variables, which are closely related to Wilson loops, via deformation quantization. We obtain a Poisson algebra of these…

Some aspects of the algebraic quantization programme proposed by Ashtekar are revisited in this article. It is proved that, for systems with first-class constraints, the involution introduced on the algebra of quantum operators via reality…

Pure (2+1)-dimensional Einstein gravity is analysed in the Ashtekar formulation, when the spatial manifold is a torus. We have found a set of globally defined observables, forming a closed algebra. This allowed us to solve the quantum…