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At present, the deepest problem in theoretical physics is harmonizing the theory of general relativity, which describes gravitation and applies to large-scale structures (stars, planets, galaxies), with quantum mechanics, which describes the other three fundamental forces acting on the microscopic scale.
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 explored, 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 space-time.
On the other hand, quantum mechanics has depended since its invention on a fixed background (non-dynamical) 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, string-like objects propagate in a fixed spacetime background. No attempt will be made to describe string theory/M-theory in more depth in this article, since it wouldn't be possible to do it justice.
Quantum field theory on curved (non-Minkowskian) 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 full-blown quantum gravity. In particular, the vacuum, when it exists, is shown to depend on the path of the observer through space-time. 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 background-independence of general relativity. The first is that the geometric interpretation of General Relativity is not fundamental, but just an emergent quality of some background-dependent theory. This is explicitly stated, for example, in Steven Weinberg's classic Gravitation and Cosmology textbook. The opposing view is that background-independence is fundamental, and quantum mechanics needs to be generalized to settings where there is no a-priori 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 1970s. 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 background-independent quantum theory. Topological quantum field theory provided an example of background-independent 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.
General relativity is the theory of gravitation published by Albert Einstein in 1915. According to general relativity, the force of gravity is a manifestation of the local geometry of spacetime. LQG inherits this geometric interpretation of gravity, and posits that a quantum theory of gravity is also a quantum theory of spacetime.
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 rotation group and not the lorentz group, Penrose went on to develop twistors.
In 1986 physcist Abhay Ashtekar reformulated Einstein's field equations of General Relativity using what have come to be known as Ashtekar variables. Using this reformulation, he was able to quantize gravity.
Around 1990, Carlo Rovelli and Lee Smolin obtained an explicit basis of states of quantum geometry, labelled by Penrose's spin networks.
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Area and volume operators
Spin foams
The most obvious ways of combining the two (such as treating gravity as simply another particle field) run quickly into what is known as the renormalization problem. Gravity particles would attract each other and if you add together all of the interactions you end up with many infinite results which can not easily be cancelled out. This is in contrast with quantum electrodynamics where the interactions do result in some infinite results, but those are few enough in number to be removable via renormalization.
Thus the development of a quantum theory of gravity must come about by different means than were used for the other forces.
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.
Any successful theory of quantum gravity must provide physical predictions that closely match known observation, and reproduce the results of quantum field theory and gravity. To date Einstein's theory of General Relativity is the most successful theory of gravity. It has been shown that quantizing the field equations of General Relativity will not necessarily recover those equations in the classical limit. It remains unclear whether LQG yields results that match General Relativity in the domain of low-energy, macroscopic and astronomical realm. To date, LQG has been shown to yield results that match General Relativity in 1+1 and 2+1, and certain conformations of 3+1 spacetime, such as a flat, smooth spacetime, it has not been show to yield General Relativity in all conformations of 3+1 spacetime. Thus, it remains unclear whether LQG successfully merges quantum mechanics with General Relativity.
Another important principle is the issue of the "cosmological constant", which is the energy density inherent in a vacuum. Because string theory/m-theory 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; LQG, unlike its rival string theory/m-theory, apparently incorporates a positive cosmological constant in agreement with observation. Specifically, a positive cosmological constant would suggest that the expansion of the universe is accelerating, whereas a zero cosmological constant would predict that gravity would slow down the expansion of the universe.
The research into loop quantum gravity with the positive cosmological constant was done by a LQG physcist named Hideo Kodama. Recently, famed superstring/M-theorist Edward Witten wrote a highly influential paper "Kodama state is unphysical" in which he argued that LQG does not incorporate a positive cosmological constant in a way that agrees with physical reality. His article has been hotly debated by both the string community and LQG community. LQG theorists plan to publish papers addressing Witten's challenge.
String theorists have suggested that gravity, mediated by gravitons, can escape into higher dimensions, which would explain the apparent accelerated expansion of the universe. Gravity is leaking out of our universe, and so the rate at which the universe expands would accelerate.
Several LQG physcists have shown that LQG can get rid of the infinities and singularities present when General Relativity is applied to the Big Bang.
While standard physics tools break down, LQG have provided internally self-consistent models of a Big Bounce in the time preceding the Big Bang.
As a rough approximation, string theory/M-theory follows quantum mechanics, and posits that gravity is mediated by gravitons embedded in flat space-time. LQG follows Einstein's theory of general relativity and models gravity as a curvature within spacetime. Much of the radical differences between these theories of quantum gravity comes from the radically different assumptions that these theories have on how the universe works.
Unlike its rival string/M-theory,which purports to be a theory of everything, and explains matter and energy, and the unification of the four forces, LQG is primarily interested in gravity, and space-time. While string/M-theory posits gravity is a field through spacetime, LQG sees gravity as spacetime itself.
The current formulation of LQG posits that spacetime has 3 spatial dimensions and 1 dimension for time.
string/M-theory posits that space consists of 10 spatial dimensions and 1 dimension of time. The additional dimensions have not yet been experimentally observed.
LQG does not make use of supersymmetry, while String/M-theory not ony requires supersymmetry, it also predicts supersymetric particles. Supersymmetry and supersymetric particles have not yet been experimentally observed.
LQG predicts spacetime is discrete and quantized; String/M-theory allows for spacetime to be infinitely continous.
String/M-theories subsumes grand unified theories which attempt to unite the strong and electroweak forces. Many of these theories predict proton decay and magnetic monopoles which has not been observed.
LQG does not attempt to unify gravity with strong and electro-weak forces.
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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 Amelino-Camelia points out that photons which have traveled 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 physicists eagerly await results from space-based gamma-ray spectrometry experiments -- a mission set to launch in February, 2007.
The recent result that gravity propagates at the speed of light is consistent with LQG. However, the result significantly constrains string theory and probably M-theory 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.
The incompatibility between quantum mechanics and general relativity
The History of LQG
Wilson loops and spin networks
The development of a quantum field theory of a force invariably results in infinite (and therefore useless) answers. Physicists have developed mathematical techniques (renormalization) to eliminate these infinities which work for the electromagnetic, strong nuclear and weak nuclear forces, but not gravity. LQG and the classical limit
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.LQG and the Big Bang
Differences between LQG and string/M-theory
Einstein's theory of general relativity models gravity
as a curvature within space-time that changes as mass moves.
Quantum mechanics depends on particle fields embedded in the flat
space-time of either Newtonian mechanics or special relativity. Experimental tests of LQG?
Unlike string theory and M-theory, LQG makes hypotheses that may be experimentally testable in the near future.People in LQG and related areas
Loop quantum gravity theorists:
Bibliography
External links
scientific american article January 2004