Loop quantum gravity
From Wikinfo
Loop quantum gravity (LQG) is a proposed quantum theory of spacetime which blends together the seemingly incompatible theories of quantum mechanics and general relativity (see below). 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 3-space 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.
The incompatibility between quantum mechanics and general relativity
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, a situation which led Richard Feynman (who himself had made important attempts at understanding quantum gravity) to write, in desperation, "Remind me not to come to any more gravity conferences" in a letter to his wife in the early 1960's. 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.
The history of LQG
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. Mathematically, the theory is modelled after 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. LQG 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 rotation 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 flavour 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 1970's to study the strongly-interacting 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.
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.
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Area and volume operators
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.
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.
Loop quantum gravity's implications
The classical limit
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 dimensions. To date, it has not been shown that LQG reproduces classical gravity in 3+1 dimensions. Thus, it remains unclear whether LQG successfully merges quantum mechanics with General Relativity.
Special relativity and Lorentz invariance
In order to account for the structure of space and time at planck scale, LQG breaks lorentz invariance and posits that certain well known effects of special relativity such as length contraction and time dilation cannot occur below the threshold of the planck scale.
General relativity and diffeomorphism invariance
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Background independence
LQG is background independent. The equations of LQG are not embedded or presuppose space and time, but rather they give rise to and create space and time at the planck and trans-plancking distance. This reflects a philosophical view that gravity is the very geometric fabric of space and time, and that a quantum theory of gravity must be a quantum theory of space and time. Hence any theory of quantum gravity must give an account of space and time. Unfortunately, it has not been yet shown that LQG's description of spacetime at the plancking and transplancking scales can lead to spacetime as described by General Relativity.
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 vacuum. There have been proposals to include a positive cosmological constant in LQG involving a state referred to as the Kodama state after Hideo Kodama. Some have argued by analogy with other theories that this state is unphysical. This issue is still unresolved.
Black hole thermodynamics
While experimental tests for LQG maybe years in the future, one conceptual test any candidate for QG must pass is that it must derive the correct formulua Hawking derived for the entropy of a blackhole.
With the proper Immirzi parameter, LQG can calculate and reproduce the Hawking formula for all blackholes. While String/M-theory does not need the Immirzi parameter, it can only derive the correct Hawking formula for extremal black holes -- black holes with a net electric charge, which are unlikely to exist physically. To date, the Immirzi cannot be derived from more fundamental principles, and is an unavoidable artifact of quantization of general relativity's field equations.
LQG's interpretation of black hole entropy is that the spacetime fabric that make up the black hole horizon is quantized per plank area, and the Bekenstein/Hawking entropy represents the degrees of freedom present in each planck quanta.
Additionally, the spectrum of radiation of particles emanating from the event horizon of a black hole has been calculated from LQG's theoretical framework and precisely predicted.
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The Big Bang
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.
Particle physics
While classical particle physics posit particles traveling through space and time that is infinitely continous and therefore infinitely divisible, LQG predicts that space-time is quantized or granular. The two different models of space and time affects the way ultra high energy cosmic rays (UHECR) interacts with the background, with quantized spacetime predicting that the threshold for allowable energies for such high energy particles be raised. Such particles have been observed, however, alternative explanations have not been ruled out.
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Quantum field theory
Quantum field theory is perturbative and background dependent. One problem LQG may be able to address in QFT is the ultraviolet catastrophe.
The term ultraviolet catastrophe has also been applied to similar situations in quantum electrodynamics in which summing across all energies results in an infinite value because the higher energy terms do not decrease quickly enough to create finite values.
In LQG, the background quantum field theory depends on is quantized, and hence, there is no physical "room" for the ultraviolet infinities to occur.
Graviton
In quantum field theories, the graviton is a hypothetical elementary particle that transmits the force of gravity in most quantum gravity systems. In order to do this gravitons have to be always-attractive (gravity never pushes), work over any distance (gravity is universal) and come in unlimited numbers (to provide high strengths near stars). In quantum theory, this defines an even-spin (spin 2 in this case) boson with a rest mass of zero.
It remains open to debate whether loop quantum gravity requires, or does not require, the graviton, or whether the graviton can be accounted for its theoretical framework.
Parsomonious approach to quantum gravity
To date, the supersymmetry hypothesized/required/predicted by String theory has not been observed experimentally nor is required by particle physics. If both LQG and String Theory succeed as quantum theories of gravity, LQG is more parsimonious because it does not require the introduction of unobserved spontaneosly-broken supersymmetry.
To date, there is no experimental evidence for supersymmetry and no experimental evidence for spatial dimensions beyond the 3 large spatial dimensions required by string theory and M-theory.
LQG in its current formulation does not require supersymmetry and additional spatial dimensions. Hence, LQG is formulated in 3 spatial dimensions and one dimension of time; 3 + 1.
If experimental evidence confirms supersymmetry in the form of supersymmetric particles such as the neutralino, what is believed to be the lightest-massed superpartner of the neutrino quite possibly as early as 2007 when Europe's Large Hadron Collider will be in operation with sufficient energies to produce such particles, it is possible to modify LQG's spin networks to accommodate these discoveries by requiring the spin networks to carry more quantum numbers.
Chaos theory and classical physics
Chaos theory sensitivity on the initial conditions means that two such systems with however small a difference in their initial state eventually will end up with a finite difference between their states (however, two deterministic systems with identical initial conditions will remain identical). Since most chaos theory is formulated in partial differential equations, it is thought that for two systems to have the same exact initial conditions, they must be reproduced with infinite accuracy.
Loop quantum gravity suggests that the planck scale represents the physical cut-off allowed for such sensitivity -- a chaotic's system's sensitivity to initial conditions need only be reproduced at the Planck scale.
Similiary, many of the equations in classical physics, such as the equations for motion, must been seen as a classical approximation to quantized geometry, and hence, a particle traveling through space is "jumping" from one space-time quantum to the next, much as the electron orbits of a hydrogen atom are quantized.
Mathematics
As with any theory of quantum gravity, esearch into LQG employs a variety of techniques and fields of mathematics. A partial list would include
- Heyting algebra
- mathematical category theory
- noncommutative geometry
- topos theory
- C* Algebra
- Regge calculus
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Philosophy
LQG takes certain philosophical positions, and itself influences philosophy, especially philosophy of science.
One of the long standing debates in a branch of philosophy known as metaphysics concerns the "ultimate" nature of space and time. LQG preserves Einstein's position on spacetime by preserving diffomorphism invariance, while breaking lorentz invariance. On the philosophy of space and time, two well developed positions have been developed, one, that space and time is absolute, and the other, space and time is relational. LQG takes the latter position. While philosophy and metaphysics offers no clear cut method of determining which of the two positions is more plausible, LQG offers predictions that could, if not yet technologically feasible at least in principle, be empirically tested or experimentally falsified.
An underlying philosophy of LQG is that fundamental physics should provide an account for space and time.
Certain predictions of LQG, such as the holographic principle, and its parsimonious methodology, has resulted in fruitful debate in the philosophy of science and physics community.
Research and progress in Quantum Gravity has stimulated philosophy of science and mathematics, including LQG and String theory.
Differences between LQG and string/M-theory
String theory and LQG are the products of different communities. String theory emerged from the particle physics community and was originally formulated as a theory that depends on a background spacetime, flat or curved, which obeys Einstein's equations. This is now known to be just an approximation to a mysterious and not well-formulated underlying theory which may or may not be background independent.
In contrast, LQG was formulated with background independence in mind. However, it has been difficult to show that classical gravity can be recovered from the theory. Thus, LQG and string theory seem somewhat complementary. String theory easily recovers classical gravity, but lacks a fundamental, perhaps background independent, description. LQG is a background independent theory of something, but the classical limit has yet not proven tractable. This has led some people to conjecture that LQG and string theory may both be aspects of some new theory, or that, perhaps there is some synthesis of the techniques of each that will lead to a complete theory of quantum gravity. For now, this is mostly a fond hope with little evidence.
Experimental tests of LQG
LQG may make hypotheses that can be experimentally testable in the near future.
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. Alternatively, highly energetic photons from gamma ray bursts should arrive a split second sooner than less energetic photons. LQG physicists eagerly await results from space-based gamma-ray spectrometry experiments -- a mission set to launch in February, 2007.
LQG and the sociology and politics of science
To date, there is no experimental evidence to support any theory of quantum gravity, whether String/M-theory or Loop quantum gravity. As previously noted, however, the string community outnumbers the loop community by at least a factor of 10:1. Furthermore, string theory has achieved far more success than loop quantum gravity in attracting graduate and doctoral students, professors, research positions, and funding, and more recognition in popular culture. How this imbalance has come about, and the nature, contraversies, and interactions between the two communities has provided facinating material in the sociology, economics, and politics of science, academia, and higher learning.
Recently, Joao Magueijo wrote in his book "Faster Than the Speed of Light: The Story of a Scientific Speculation" of the intense rivalry, even hostility between these opposing camps of quantum gravity theorists. The hostility stems in part from funding: academic positions open to PhDs for string theorists are unavailable to Loop Quantum Gravity theorists. It also stems from the fact that string/m-theory and LQG in their present forms, makes different physical predictions, and appear to be incompatible, and that it is psychologically difficult for anyone to accept he or she may have dedicated their graduate studies and academic careers on the wrong theory.
On the physics newsgroup sci.physics.research, vigorous debates used to take place between string theorists, notably Lubos Motl and loop quantum gravity theorist and (now former) s.p.r. moderator John Baez, with general relativists such as and Steve Carlip taking sides "against" strings. As a result Lubos Motl spearheaded the creation of a new newsgroup, sci.physics.strings. These heated debates did not preclude scientific collaboration of sorts. After Baez announced in his column This Week's Finds that Olaf Dreyer had found a connection between the Immirzi Parameter and the asymptotic behaviour of black hole quasinormal modes (in numerical general relativity), Lubos Motl wasted no time proving that the exact asymptotic behaviour was as predicted numerically.
Recent research directions
- Hamiltonian loop quantum gravity
- Spin foam formalism
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People in LQG and related areas
Loop quantum gravity theorists:
- Abhay Ashtekar
- John Baez
- Julian Barbour
- John Barrett
- Martin Bojowald
- Louis Crane
- Laurent Freidel
- Rodolfo Gambini
- Giorgio Immirzi
- Christopher Isham
- Kirill Krasnov
- Jerzy Lewandowski
- Renate Loll
- Fotini Markopoulou-Kalamara
- Donald Marolf
- Hugo Morales
- Alejandro Perez
- Jorge Pullin
- Michael Reisenberger
- Carlo Rovelli
- Lee Smolin
- Thomas Thiemann
- [[Jos� Antonio Zapata]]
Bibliography
- Popular books:
- 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 hep-th/0310077
- Advanced books, reports, conference proceedings:
- Robert M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, Chicago University Press (1994), ISBN 0226-87027-8
- Robert M. Wald, General Relativity, Chicago University Press, ISBN 0-226-87033-2
- Steven Weinberg, Gravitation and Cosmology: principles and applications of the general theory of relativity, Wiley (1972), ISBN 0-471-92567-5
- Misner, Thorne and Wheeler, Gravitation, Freeman, (1973), ISBN 0-7167-0344-0
- A. Ashtekar, Lectures on Non-Perturbative Canonical Gravity, World Scientific (1991)
- Rodolfo Gambini and Jorge Pullin, Loops, Knots, Gauge Theories and Quantum Gravity
- John Baez (ed.), Knots and Quantum Gravity
External links
- Quantum Gravity, Physics, and Philosophy: http://www.qgravity.org/
- Gamma-ray Large Area Space Telescope: http://glast.gsfc.nasa.gov/ scientific american article January 2004
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References
- Adapted from the Wikipedia article, "Loop_quantum_gravity" http://en.wikipedia.org/wiki/Loop_quantum_gravity, used under the GNU Free Documentation License
