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

Extended loop representation of quantum gravity

Physical Review D, 1995

A new representation of Quantum Gravity is developed. This formulation is based on an extension of the group of loops. The enlarged group, that we call the Extended Loop Group, behaves locally as an infinite dimensional Lie group. Quantum Gravity can be realized on the state space of extended loop dependent wavefunctions. The extended representation generalizes the loop representation and contains this representation as a particular case. The resulting diffeomorphism and hamiltonian constraints take a very simple form and allow to apply functional methods and simplify the loop calculus. In particular we show that the constraints are linear in the momenta. The nondegenerate solutions known in the loop representation are also solutions of the constraints in the new representation. The practical calculation advantages allows to find a new solution to the Wheeler-DeWitt equation. Moreover, the extended representation puts in a precise framework some of the regularization problems of the loop representation. We show that the solutions are generalized knot invariants, smooth in the extended variables, and any framing is unnecessary.

Knot polynomial states of quantum gravity in terms of loops and extended loops: Some remarks

Journal of Mathematical Physics, 1995

In this paper we review the status of several solutions to all the constraints of quantum gravity that have been proposed in terms of loops and extended loops, based on knot polynomials. We discuss pitfalls of several of the results, and in particular the issues of covariance and regularization of the constraints in terms of extended loops. We also propose a formalism for ‘‘thickened out loops,’’ which does not face the covariance problems of extended loops and may allow to regularize expressions in a consistent manner.

Loop quantum gravity and quanta of space: a primer

We present a straightforward and self-contained introduction to the basics of the loop approach to quantum gravity, and a derivation of what is arguably its key result, namely the spectral analysis of the area operator. We also discuss the arguments supporting the physical prediction following this result: that physical geometrical quantities are quantized in a non-trivial, computable, fashion. These results are not new; we present them here in a simple form that avoids the many non-essential complications of the first derivations.

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.

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.

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.