Introduction to Loop Quantum Gravity and Spin Foams

TL;DR

This work presents a thorough introduction to non-perturbative, background-independent quantum gravity via Loop Quantum Gravity (LQG) and its covariant spin-foam formulation. It first develops the Ashtekar-Barbero variables and the kinematical Hilbert space built from holonomies and fluxes, yielding a discrete quantum geometry evidenced by area and volume operators with eigenvalues tied to spins. It then outlines the Dirac program for quantizing GR, the treatment of Gauss and diffeomorphism constraints, and the remaining scalar constraint, whose quantization leads to a rich structure of dressed spin networks and an arena for black hole entropy computations. The spin-foam section connects the canonical theory to a covariant path integral, detailing 3d exactly solvable models and four-dimensional constrained BF theories, including Barrett-Crane-type models and group-field-theory approaches. Together, these sections illuminate how LQG aims to reproduce GR in the semiclassical limit while predicting Planck-scale discreteness and offering a covariant dynamics framework, albeit with ongoing challenges in the continuum limit and semi-classical correspondence.

Abstract

These notes are a didactic overview of the non perturbative and background independent approach to a quantum theory of gravity known as loop quantum gravity. The definition of real connection variables for general relativity, used as a starting point in the program, is described in a simple manner. The main ideas leading to the definition of the quantum theory are naturally introduced and the basic mathematics involved is described. The main predictions of the theory such as the discovery of Planck scale discreteness of geometry and the computation of black hole entropy are reviewed. The quantization and solution of the constraints is explained by drawing analogies with simpler systems. Difficulties associated with the quantization of the scalar constraint are discussed.In a second part of the notes, the basic ideas behind the spin foam approach are presented in detail for the simple solvable case of 2+1 gravity. Some results and ideas for four dimensional spin foams are reviewed.

Figures (20)

• Figure 1: The larger cone represents the light-cone at a point according to the ad hoc background $\eta_{ab}$. The smaller cones are a cartoon representation of the fluctuations of the true gravitational field represented by $g_{ab}$.
• Figure 2: An example of three different graphs on which the Wilson loop function (\ref{['wl']}) can be defined. The distinction must be physically irrelevant.
• Figure 3: Examples of spin networks.
• Figure 4: Schematic representation of the construction of a spin network. To each node we associate an invariant vector in the tensor product of irreducible representations converging at the node. In this case we take $\iota^{n_1n_2n_3n_4}\in j \otimes k \otimes p \otimes s$, and the relevant piece of spin network function is $\stackrel{ j}{\Pi}(h_{e_1}[A])_{m_1n_1} \stackrel{ k}{\Pi}(h_{e_2}[A])_{ m_2n_2} \stackrel{ p}{\Pi}(h_{e_3}[A])_{ m_3n_3} \stackrel{ s}{\Pi}(h_{e_4}[A])_{ m_4n_4}\iota^{ n_1n_2n_3n_4}$.
• Figure 5: On the left: any invariant vector can be decomposed in terms of the (unique up to normalization) three valent ones. At each three node the standard rules of addition of angular momentum must be satisfied for a non vanishing intertwiner to exist. On the right: an example of spin network with the explicit decomposition of intertwiners.

Theorems & Definitions (1)

• Theorem 1