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Loop Quantum Gravity
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Loop quantum gravity
Loop quantum gravity (LQG) is a proposed quantum theory of spacetime which blends together the seemingly incompatible theories (see below) of quantum mechanics and general relativity. As a theory of quantum gravity, it is the main competitor of string theory, although stringy people outnumber loopy people by a factor of roughly 10:1. The main successes of Loop Quantum Gravity are: a nonperturbative quantization of 3space geometry, with quantized area and volume operators; a calculation of the entropy of physical black holes; and a proof by example that it is not necessary to have a theory of everything in order to have a sensible candidate for a quantum theory of gravity. Its main shortcomings are: not yet having a picture of dynamics but only of kinematics; not yet able to incorporate particle physics; not yet able to recover the classical limit.
Table of contents 
The incompatibility between quantum mechanics and general relativity
The fundamental lesson of general relativity is that there is no fixed spacetime background, as found in Newtonian mechanics and special relativity. While easy to grasp in principle, this is the hardest idea to understand about General Relativity, and its consequences are profound and not fully understood, even at the classical level. To a certain extent, general relativity can be seen to be a completely relational theory, in which the only physically relevant information is the relationship between different events in spacetime.
On the other hand, quantum mechanics since its invention has depended on a fixed background (nondynamical) structure. In the case of quantum mechanics, it is time that is given and not dynamical, just as in Newtonian classical mechanics. In relativistic quantum field theory, just as in classical field theory, Minkowski spacetime is the fixed background of the theory. Finally, string theory started out as a generalization of quantum field theory where instead of point particles, stringlike objects propagate in a fixed spacetime background. No attempt will be made to describe string theory/Mtheory in more depth in this article, since it wouldn't be possible to do it justice.
Quantum field theory on curved (nonMinkowskian) backgrounds, while not a quantum theory of gravity, has shown that some of the core assumptions of quantum field theory cannot be carried over to curved spacetime, let alone to fullblown quantum gravity. In particular, the vacuum, when it exists, is shown to depend on the path of the observer though spacetime. Also, the field concept is seen to be fundamental over the particle concept (which arises as a convenient way to describe localized interactions).
Historically, there have been two reactions to the apparent inconsistency of quantum theories with the necessary backgroundindependence of general relativity. The first is that the geometric interpretation of General relativity is not fundamental, but just an emergent quality of some backgrounddependent theory. This is explicitly stated, for example, in Steven Weinberg's classic Gravitation and Cosmology textbook. The opposing view is that backgroundindependence is fundamental, and quantum mechanics needs to be generalized to settings where there is no apriori specified time. The geometric point of view is expounded in the classic text Gravitation, by Misner, Wheeler and Thorne. It is interesting that two books by giants of theoretical physics expressing completely opposite views of the meaning of gravitation were published almost simultaneously in the early 1970's. The reason was that an impasse had been reached. Since then, though, progress was rapid on both fronts, leading ultimately to String Theory and Loop Quantum Gravity.
Loop quantum gravity is the fruit of the effort to formulate a backgroundindependent quantum theory. Topological quantum field theory provided an example of backgroundindependent quantum theory, but with no local degrees of freedom, and only finitely many degrees of freedom globally. This is inadequate to describe gravity, which even in vacuum has local degrees of freedom according to general relativity.
Wilson loops and spin networks
In LQG, the fabric of spacetime is a foamy network of interacting loops mathematically described by spin networks. These loops are about 10^{35} meters in size, called the Planck scale. The loops knot together forming edges, surfaces, and vertices, much as do soap bubbles joined together. In other words, spacetime itself is quantized. Any attempt to divide a loop would, if successful, cause it to divide into two loops each with the original size. In LQG, spin networks represent the quantum states of the geometry of relative spacetime. Looked at another way, Einstein's theory of general relativity is (as Einstein predicted) a classical approximation of a quantized geometry.
LQG and quantum cosmology
An important principle in quantum cosmology that LQG adheres to is that there are no observers outside the universe. All observers must be a part of the universe they are observing. However, because light cones limit the information that is available to any observer, the Platonic idea of absolute truths does not exist in a LQG universe. Instead, there exists a consistency of truths in that every observer will report consistent (not necessarily the same) results if truthful.
Another important principle is the issue of the cosmological constant, which is the energy density inherent in a vaccuum. Because string theory/mtheory makes use of supersymmetry, the physics implies a negative or a zero comsological constant. This is in apparent contradiction to observation, which observes a positive, but very close to zero, cosmological constant. However, the ground state in LQG is positive, although very small; LQG, unlike its rival string theory/mtheory, apparently incorporates a positive cosmological constant in agreement with observation.
Experimental tests of LQG?
Unlike string theory and Mtheory, LQG makes experimentally testable hypotheses.
The path taken by a photon through a discrete spacetime geometry would be different from the path taken by the same photon through continuous spacetime. Normally, such differences should be insignificant, but Giovanni AmelinoCamelia points out that photons which have travelled from distant galaxies may reveal the structure of spacetime. LQG predicts that more energetic photons should travel ever so slightly faster than less energetic photons. This effect would be too small to observe within our galaxy. However, light reaching us from gamma ray bursts in other galaxies should manifest a varying spectral shift over time. In other words, distant gamma ray bursts should appear to start off more bluish and end more reddish. LQG physcists anxiously await results from spacebased gammaray spectrometry experiments  a mission set to launch in September, 2006.
The recent result that gravity propagates at the speed of light is consistent with LQG. However, the result significantly constrains string theory and probably Mtheory because large numbers of dimensions would allow gravity to propagate along extra dimensions. This result does not by itself rule out all forms of string theory.
People in LQG and related areas
Loop quantum gravity theorists:

Abhay Ashtekar

John Baez

Julian Barbour

Louis Crane

Laurent Freidel

Rodolfo Gambini

Christopher Isham

Fotini MarkopoulouKalamara

Carlo Rovelli

Lee Smolin

Thomas Thiemann
Bibliography

Popular books:

Julian Barbour, The End of Time.

Lee Smolin, Three Roads to Quantum Gravity


Introductory/expository works:

John Baez and Javier Perez de Muniain, Gauge Fields, Knots and Quantum Gravity, World Scientific (1994), ISBN 9810220340

Carlo Rovelli, A Dialog on Quantum Gravity, preprint available as hepth/0310077


Advanced books, reports, conference proceedings:

Robert M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, Chicago University Press (1994), ISBN 0226870278

Robert M. Wald, General Relativity, Chicago University Press, ISBN 0226870332

Steven Weinberg, Gravitation and Cosmology: principles and applications of the general theory of relativity, Wiley (1972), ISBN 0471925675

Misner, Thorne and Wheeler, Gravitation, Freeman, (1973), ISBN 0716703440

A. Ashtekar, Lectures on NonPerturbative Canonical Gravity, World Scientific (1991)

Rodolfo Gambini and Jorge Pullin, Loops, Knots, Gauge Theories and Quantum Gravity

John Baez (ed.), Knots and Quantum Gravity

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