The Loop representation in gauge theories and quantum gravity (original) (raw)

Extended loops: A new arena for nonperturbative quantum gravity

Physical Review Letters, 1994

We propose a new representation for gauge theories and quantum gravity. It can be viewed as a generalization of the loop representation. We make use of a recently introduced extension of the group of loops into a Lie Group. This extension allows the use of functional methods to solve the constraint equations. It puts in a precise framework the regularization problems of the

A review on Loop Quantum Gravity

arXiv: General Relativity and Quantum Cosmology, 2018

The aim of this dissertation is to review `Loop Quantum Gravity', explaining the main structure of the theory and indicating its main open issues. We will develop the two main lines of research for the theory: the canonical quantization (first two chapters) and spin foams (third). The final chapter will be devoted to studying some of the problems of the theory and what things remain to be developed. In chapter 3 we will also include an example of a simple calculation done in the frame of LQG: Schwarzschild black hole entropy.

Quantum-reduced loop gravity: Relation with the full theory

Physical Review D, 2013

The quantum-reduced loop-gravity technique has been introduced for dealing with cosmological models. We show that it can be applied rather generically: anytime the spatial metric can be gauge-fixed to a diagonal form. The technique selects states based on reduced graphs with Livine-Speziale coherent intertwiners and could simplify the analysis of the dynamics in the full theory.

Lattice knot theory and quantum gravity in the loop representation

Physical Review D, 1997

We present an implementation of the loop representation of quantum gravity on a square lattice. Instead of starting from a classical lattice theory, quantizing and introducing loops, we proceed backwards, setting up constraints in the lattice loop representation and showing that they have appropriate (singular) continuum limits and algebras. The diffeomorphism constraint reproduces the classical algebra in the continuum and has as solutions lattice analogues of usual knot invariants. We discuss some of the invariants stemming from Chern--Simons theory in the lattice context, including the issue of framing. We also present a regularization of the Hamiltonian constraint. We show that two knot invariants from Chern--Simons theory are annihilated by the Hamiltonian constraint through the use of their skein relations, including intersections. We also discuss the issue of intersections with kinks. This paper is the first step towards setting up the loop representation in a rigorous, computable setting.

Unified model of loop quantum gravity and matter

General Relativity and Gravitation, 2006

We reconsider the unified model of gravitation and Yang-Mills interactions proposed by Chakraborty and Peldán, in the light of recent formal developments in loop quantum gravity. In particular, we show that one can promote the Hamiltonian constraint of the unified model to a well defined anomaly-free quantum operator using the techniques introduced by Thiemann, at least for the Euclidean theory. The Lorentzian version of the model can be consistently constructed, but at the moment appears to yield a correct weak field theory only under restrictive assumptions, and its quantization appears problematic.

Regularization ambiguities in loop quantum gravity

Physical Review D, 2006

One of the main achievements of loop quantum gravity is the consistent quantization of the analog of the Wheeler-DeWitt equation which is free of ultra-violet divergences. However, ambiguities associated to the intermediate regularization procedure lead to an apparently infinite set of possible theories. The absence of an UV problem-the existence of well behaved regularization of the constraints-is intimately linked with the ambiguities arising in the quantum theory. Among these ambiguities there is the one associated to the SU (2) unitary representation used in the diffeomorphism covariant "point-splitting" regularization of the non linear functionals of the connection. This ambiguity is labelled by a half-integer m and, here, it is referred to as the m-ambiguity. The aim of this paper is to investigate the important implications of this ambiguity.

A short review of loop quantum gravity

Reports on Progress in Physics, 2021

An outstanding open issue in our quest for physics beyond Einstein is the unification of general relativity (GR) and quantum physics. Loop quantum gravity (LQG) is a leading approach toward this goal. At its heart is the central lesson of GR: Gravity is a manifestation of spacetime geometry. Thus, the approach emphasizes the quantum nature of geometry and focuses on its implications in extreme regimes -near the big bang and inside black holes-where Einstein's smooth continuum breaks down. We present a brief overview of the main ideas underlying LQG and highlight a few recent advances. This report is addressed to non-experts.

The kinematical frame of Loop Quantum Gravity I

In loop quantum gravity in the connection representation, the quantum configuration space A/G, which is a compact space, is much larger than the classical configuration space A/G of connections modulo gauge transformations. One finds that A/G is homeomorphic to the space Hom(L * , G))/Ad. We give a new, natural proof of this result, suggesting the extension of the hoop group L * to a larger, compact group M(L * ) that contains L * as a dense subset. This construction is based on almost periodic functions. We introduce the Hilbert algebra L 2 (M(L * )) of M(L * ) with respect to the Haar measure ξ on M(L * ). The measure ξ is shown to be invariant under 3-diffeomorphisms. This is the first step in a proof that L 2 (M(L * )) is the appropriate Hilbert space for loop quantum gravity in the loop representation. In a subsequent paper, we will reinforce this claim by defining an extended loop transform and its inverse.

Quantum-reduced loop gravity: Cosmology

Physical Review D, 2013

We introduce a new framework for loop quantum gravity: mimicking the spinfoam quantization procedure we propose to study the symmetric sectors of the theory imposing the reduction weakly on the full kinematical Hilbert space of the canonical theory. As a first application of Quantum-Reduced Loop Gravity we study the inhomogeneous Bianchi I model. The emerging quantum cosmological model represents a simplified arena on which the complete canonical quantization program can be tested. The achievements of this analysis could elucidate the relationship between Loop Quantum Cosmology and the full theory.

The Constraint Algebra of Quantum Gravity in the Loop Representation

International Journal of Modern Physics D, 1995

We study the algebra of constraints of quantum gravity in the loop representation based on Ashtekar’s new variables. We show by direct computation that the quantum commutator algebra reproduces the classical Poisson bracket one, in the limit in which regulators are removed. The calculation illustrates the use of several computational techniques for the loop representation.