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# History of loop quantum gravity

General relativity is the theory of gravitation published by Albert Einstein in 1915. According to it, the force of gravity is a manifestation of the local geometry of spacetime. Mathematically, the theory is modelled after Bernhard Riemann's metric geometry, but the Lorentz group of spacetime symmetries (an essential ingredient of Einstein's own theory of special relativity) replaces the group of rotational symmetries of space. Loop quantum gravity inherits this geometric interpretation of gravity, and posits that a quantum theory of gravity is fundamentally a quantum theory of spacetime.

In the 1920s, the French mathematician Elie Cartan formulated Einstein's theory in the language of bundles and connections, a generalization of Riemann's geometry to which Cartan made important contributions. The so-called Einstein-Cartan theory of gravity not only reformulated but also generalized general relativity, and allowed spacetimes with torsion as well as curvature. In Cartan's geometry of bundles, the concept of parallel transport is more fundamental than that of distance, the centerpiece of Riemannian geometry. A similar conceptual shift occurs between the invariant interval of Einstein's general relativity and the parallel transport of Einstein-Cartan theory.

In the 1960s, physicist Roger Penrose explored the idea of space arising from a quantum combinatorial structure. His investigations resulted in the development of spin networks. Because this was a quantum theory of the rotational group and not the Lorentz group, Penrose went on to develop twistors.

In 1986, physicist Abhay Ashtekar reformulated Einstein's field equations of general relativity using what have come to be known as Ashtekar variables, a particular flavor of Einstein-Cartan theory with a complex connection. Using this reformulation, he was able to quantize gravity using well-known techniques from quantum gauge field theory. In the Ashtekar formulation, the fundamental objects are a rule for parallel transport (technically, a connection) and a coordinate frame (called a vierbein) at each point.

The quantization of gravity in the Ashtekar formulation was based on Wilson loops, a technique developed in the 1970s to study the strong-interaction regime of quantum chromodynamics. It is interesting in this connection that Wilson loops were known to be ill-behaved in the case of standard quantum field theory on (flat) Minkowski space, and so did not provide a nonperturbative quantization of QCD. However, because the Ashtekar formulation was background-independent, it was possible to use Wilson loops as the basis for nonperturbative quantization of gravity.

Ashtekar's work resulted, for the first time, in a setting where the Wheeler-DeWitt equation could be written in terms of a well-defined Hamiltonian operator on a well-defined Hilbert space, and led to construction of the first known exact solution, the so-called Chern-Simons form or Kodama state. The physical interpretation of this state remains obscure.

Around 1990, Carlo Rovelli and Lee Smolin obtained an explicit basis of states of quantum geometry, which turned out to be labelled by Penrose's spin networks. In this context, spin networks arose as a generalization of Wilson loops necessary to deal with mutually intersecting loops. Mathematically, spin networks are related to group representation theory and can be used to construct knot invariants such as the Jones polynomial. Being closely related to topological quantum field theory and group representation theory, LQG is mostly established at the level of rigour of mathematical physics, as compared to string theory, which is established at the level of rigour of physics.

After the spin network basis was described, progress was made on the analysis of the spectra of various operators resulting in a predicted spectrum for area and volume (see below). Work on the semiclassical limit, the continuum limit, and dynamics was intense after this, but progress slower.

On the semiclassical limit front, the goal is to obtain and study analogues of the harmonic oscillator coherent states (candidates are known as weave states).

LQG was initially formulated as a quantization of the Hamiltonian ADM formalism, according to which the Einstein equations are a collection of constraints (Gauss, Diffeomorphism and Hamiltonian). The kinematics are encoded in the Gauss and Diffeomorphism constraints, whose solution is the space spanned by the spin network basis. The problem is to define the Hamiltonian constraint as a self-adjoint operator on the kinematical state space. The most promising work in this direction is Thomas Thiemann's Phoenix program.

Spin foams are new framework intended to tackle the problem of dynamics and the continuum limit simultaneously. Heuristically, it would be expected that evolution between spin network states might be described by discrete combinatorial operations on the spin networks, which would then trace a two-dimensional skeleton of spacetime. This approach is related to state-sum models of statistical mechanics and topological quantum field theory such as the Turaeev-Viro model of 3D quantum gravity, and also to the Regge calculus approach to calculate the Feynman path integral of general relativity by discretizing spacetime.

Some radical approaches to spin foams include the work on causal sets by Fotini Markopoulou and Rafael Sorkin, among others.

Bibliography

* Topical Reviews

o Carlo Rovelli, Loop Quantum Gravity, Living Reviews in Relativity 1, (1998), 1, online article, 2001 15 August version.

o Thomas Thiemann, Lectures on loop quantum gravity, e-print available as gr-qc/0210094

o Abhay Ashtekar and Jerzy Lewandowski, Background independent quantum gravity: a status report, e-print available as gr-qc/0404018

o Carlo Rovelli and Marcus Gaul, Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance, e-print available as gr-qc/9910079.

o Lee Smolin, The case for background independence, e-print available as hep-th/0507235.

* Popular books:

o Julian Barbour, The End of Time

o Lee Smolin, Three Roads to Quantum Gravity

o Carlo Rovelli, Che cos'è il tempo? Che cos'è lo spazio?, Di Renzo Editore, Roma, 2004. French translation: Qu'est ce que le temps? Qu'est ce que l'espace?, Bernard Gilson ed, Brussel, 2006. English translation: What is Time? What is space?, Di Renzo Editore, Roma, 2006.

* Magazine articles:

o Lee Smolin, "Atoms in Space and Time," Scientific American, January 2004

* Easier introductory, expository or critical works:

o Abhay Ashtekar, Gravity and the quantum, e-print available as gr-qc/0410054

o John C. Baez and Javier Perez de Muniain, Gauge Fields, Knots and Quantum Gravity, World Scientific (1994)

o Carlo Rovelli, A Dialog on Quantum Gravity, e-print available as hep-th/0310077

o Carlo Rovelli, Quantum Gravity, Cambridge University Press (2004); draft available online

o Thomas Thiemann, Introduction to modern canonical quantum general relativity, e-print available as gr-qc/0110034

o Abhay Ashtekar, New Perspectives in Canonical Gravity, Bibliopolis (1988).

o Abhay Ashtekar, Lectures on Non-Perturbative Canonical Gravity, World Scientific (1991)

o Rodolfo Gambini and Jorge Pullin, Loops, Knots, Gauge Theories and Quantum Gravity, Cambridge University Press (1996)

o Hermann Nicolai, Kasper Peeters, Marija Zamaklar, Loop quantum gravity: an outside view, e-print available as hep-th/0501114

o "Loop and Spin Foam Quantum Gravity: A Brief Guide for beginners arXiv:hep-th/0601129 H. Nicolai and K. Peeters.

o Edward Witten, Quantum Background Independence In String Theory, e-print available as hep-th/9306122.

* Conference proceedings:

o John C. Baez (ed.), Knots and Quantum Gravity

* Fundamental research papers:

o Abhay Ashtekar, New variables for classical and quantum gravity, Phys. Rev. Lett., 57, 2244-2247, 1986

o Abhay Ashtekar, New Hamiltonian formulation of general relativity, Phys. Rev. D36, 1587-1602, 1987

o Roger Penrose, Angular momentum: an approach to combinatorial space-time in Quantum Theory and Beyond, ed. Ted Bastin, Cambridge University Press, 1971

o Carlo Rovelli and Lee Smolin, Knot theory and quantum gravity, Phys. Rev. Lett., 61 (1988) 1155

o Carlo Rovelli and Lee Smolin, Loop space representation of quantum general relativity, Nuclear Physics B331 (1990) 80-152