Loop quantum gravity

Loop quantum gravity (LQG), also known as loop gravity, quantum geometry and canonical quantum general relativity, is a proposed quantum theory of spacetime which attempts to blend together the seemingly incompatible theories of quantum mechanics and general relativity. This theory is one of a family of theories called canonical quantum gravity. It was developed in parallel with loop quantization, a rigorous framework for nonperturbative quantization of diffeomorphisminvariant gauge theory. In plain english this is a quantum theory of gravity in which the very space that all other physics occurs in is quantized.
This is not the most popular theory of quantum gravity and many physicist have big philosophical problems with it. For one thing the critics of this theory cite that it does not predict the existence of extra dimensions, does not predict the masses or charges of particles, etc; such as in String theory. The rebuttal in general boils down to LQG being a theory of gravity and nothing more . In the view of those scientist who agree with LQG the facts that it does not predict any of those properties of particles are not a problem. There are many other theories of quantum gravity a list of them can be found on the Quantum gravity page. This very article has been subject of intense debate which can be found on the talk page Talk:Loop quantum gravity.
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Loop quantum gravity in general, and its ambitions
LQG in itself was initially less ambitious than string theory, purporting only to be a quantum theory of gravity. String theory, on the other hand, appears to predict not only gravity but also various kinds of matter and energy that lie inside spacetime. Many string theorists believe that it is not possible to quantize gravity in 3+1 dimensions without creating these artifacts. But this is not proven, and it is also not proven that the matter artifacts of string theory are exactly the same as observed matter. Should LQG succeed as a quantum theory of gravity, the known matter fields would have to be incorporated into the theory a posteriori. Lee Smolin, one of the fathers of LQG, has explored the possibility that string theory and LQG are two different approximations to the same ultimate theory.
The main claimed successes of loop quantum gravity are: (1) that it is a nonperturbative quantization of 3space geometry, with quantized area and volume operators; (2) that it includes a calculation of the entropy of black holes; and (3) that it is a viable gravityonly alternative to string theory. However, these claims are not universally accepted. While many of the core results are rigorous mathematical physics, their physical interpretations are speculative. LQG may or may not be viable as a refinement of either gravity or geometry; entropy is calculated for a kind of hole which may or may not be a black hole.
It should be noted that several quantum gravity alternatives, such as spin foam, and causal dynamic triangulation are also sometimes called "loop quantum gravity".
The incompatibility between quantum mechanics and general relativity
Main article: quantum gravity
Quantum field theory studied on curved (nonMinkowskian) backgrounds has shown that some of the core assumptions of quantum field theory cannot be carried over. In particular, the vacuum, when it exists, is shown to depend on the path of the observer through spacetime (see Unruh effect).
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 emergent. The other view is that backgroundindependence is fundamental, and quantum mechanics needs to be generalized to settings where there is no a priori specified time.
Loop quantum gravity is an effort to formulate a backgroundindependent quantum theory. Topological quantum field theory is a backgroundindependent quantum theory, but it lacks causallypropagating local degrees of freedom needed for 3 + 1 dimensional gravity.
History of LQG
Main article: history of loop quantum gravity
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 EinsteinCartan theory with a complex connection. He was able to quantize gravity using 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. Because the Ashtekar formulation was backgroundindependent, it was possible to use Wilson loops as the basis for a nonperturbative quantization of gravity. Explicit (spatial) diffeomorphism invariance of the vacuum state plays an essential role in the regularization of the Wilson loop states.
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.
The ingredients of loop quantum gravity
Loop quantization
At the core of loop quantum gravity is a framework for nonperturbative quantization of diffeomorphisminvariant gauge theories, which one might call loop quantization. While originally developed in order to quantize vacuum general relativity in 3+1 dimensions, the formalism can accommodate arbitrary spacetime dimensionalities, fermions (Baez and Krasnov), an arbitrary gauge group (or even quantum group), and supersymmetry (Smolin), and results in a quantization of the kinematics of the corresponding diffeomorphisminvariant gauge theory. Much work remains to be done on the dynamics, the classical limit and the correspondence principle, all of which are necessary in one way or another to make contact with experiment.
In a nutshell, loop quantization is the result of applying C*algebraic quantization to a noncanonical algebra of gaugeinvariant classical observables. Noncanonical means that the basic observables quantized are not generalized coordinates and their conjugate momenta. Instead, the algebra generated by spin network observables (built from holonomies) and field strength fluxes is used.
Loop quantization techniques are particularly successful in dealing with topological quantum field theories, where they give rise to statesum/spinfoam models such as the TuraevViro model of 2+1 dimensional general relativity. A much studied topological quantum field theory is the socalled BF theory in 3+1 dimensions, because classical general relativity can be formulated as a BF theory with constraints, and it is hoped that a consistent quantization of gravity may arise from perturbation theory of BF spinfoam models.
Lorentz invariance
For detailed discussion see the Lorentz covariance page.
LQG is a quantization of a classical Lagrangian field theory which is equivalent to the usual EinsteinCartan theory in that it leads to the same equations of motion describing general relativity with torsion. As such, it can be argued that LQG respects local Lorentz invariance. Global Lorentz invariance is broken in LQG just as in general relativity. A positive cosmological constant can be realized in LQG by replacing the Lorentz group with the corresponding quantum group.
Diffeomorphism invariance and background independence
General covariance (also known as diffeomorphism invariance) is the invariance of physical laws (for example, the equations of general relativity) under arbitrary coordinate transformations. This symmetry is one of the defining features of general relativity. LQG preserves this symmetry by requiring that the physical states must be invariant under the generators of diffeomorphisms. The interpretation of this condition is well understood for purely spatial diffemorphisms; however the understanding of diffeomorphisms involving time (the Hamiltonian constraint) is more subtle because it is related to dynamics and the socalled problem of time in general relativity, and a generally accepted calculational framework to account for this constraint is yet to be found.
Whether or not Lorentz invariance is broken in the lowenergy limit of LQG, the theory is formally background independent. The equations of LQG are not embedded in or presuppose space and time (except for its topology that cannot be changed), but rather they are expected to give rise to space and time at large distances compared to the Planck length. It has not been yet shown that LQG's description of spacetime at the Planckian scale has the right continuum limit described by general relativity with possible quantum corrections.
Problems
As of now there is not one experiment which verifies or refutes any aspect of LQG. This is a problem which plagues many current theories of quantum gravity. This problem is so persistent because LQG applies on a small scale to the weakest of the forces of nature. This poblem however cannot be minimized as it is the biggest problem any scientific theory can have. Theory without experiment is just faith. The second problem is that a crucial free parameter in the theory known as the Immirzi parameter is a logarithm of a Transcendental number. This has negative impications for the computation of the enthropy of a black hole useing LQG (Although to be fair the transcendental number is just the result of a calculation and not an experiment wihch are the only true test of scientific reality). Since Beckenstien and Hawking computed the enthropy of a black hole this computation has been a crucial litmus test for any theory of quantum gravity. Last and most profoundly LQG has failed to gain support in the physics community at large mainly because of its limited scope. An observation is that many scientist believe that we could formulate a theory of quantum gravity which is just for four dimensions and is unconcerned with other forces but why? Why do that when via String theory or M theory we are so close to a theory that takes account of everything we know and predicts so very much that we do not know? At this point Loop theorist disagree. They feel that a proper theory of quantum gravity is a prerequisite for any theory of everything. This philosophical problem could be the most fatal problem that LQG faces in the future. Only time and experimentation will tell the tale.
See also
 Heyting algebra
 mathematical category theory
 noncommutative geometry
 topos theory
 C*algebra
 Regge calculus
Bibliography
arXiv:grqc/03113V2 (http://www.arxiv.org/multi?archive=grqc&file=new+abstracts&year=%2705&month=03&args=0301113v2&%2Fabs=+Show+Abstract+&search_year=past+year&field_1=au&query_1=&subj_condmat=%3E+condmat+subject+classes&subj_physics=%3E+physics+subject+classes)
 Popular books:
 Magazine articles:
 Lee Smolin, "Atoms in Space and Time," Scientific American, January 2004
 Easier introductory/expository works:
 Abhay Ashtekar, Gravity and the quantum, eprint available as grqc/0410054 (http://arxiv.org/abs/grqc/0410054)
 John C. Baez and Javier Perez de Muniain, Gauge Fields, Knots and Quantum Gravity, World Scientific (1994)
 Carlo Rovelli, A Dialog on Quantum Gravity, eprint available as hepth/0310077 (http://arxiv.org/abs/hepth/0310077)
 More advanced introductory/expository works:
 Abhay Ashtekar, New Perspectives in Canonical Gravity, Bibliopolis (1988).
 Abhay Ashtekar, Lectures on NonPerturbative Canonical Gravity, World Scientific (1991)
 Abhay Ashtekar and Jerzy Lewandowski, Background independent quantum gravity: a status report, eprint available as grqc/0404018 (http://arxiv.org/abs/grqc/0404018)
 Rodolfo Gambini and Jorge Pullin, Loops, Knots, Gauge Theories and Quantum Gravity, Cambridge University Press (1996)
 Hermann Nicolai, Kasper Peeters, Marija Zamaklar, Loop quantum gravity: an outside view, eprint available as hepth/0501114 (http://arxiv.org/abs/hepth/0501114)
 Carlo Rovelli, Loop Quantum Gravity, Living Reviews in Relativity (http://relativity.livingreviews.org/) 1, (1998), 1, online article (http://www.livingreviews.org/lrr19981), 2001 15 August version.
 Carlo Rovelli, Quantum Gravity, Cambridge University Press (2004); draft available online (http://www.cpt.univmrs.fr/~rovelli/book.pdf)
 Thomas Thiemann, Introduction to modern canonical quantum general relativity, eprint available as grqc/0110034 (http://arxiv.org/abs/grqc/0110034)
 Thomas Thiemann, Lectures on loop quantum gravity, eprint available as grqc/0210094 (http://arxiv.org/abs/grqc/0210094)
 Conference proceedings:
 John C. Baez (ed.), Knots and Quantum Gravity
 Fundamental research papers:
 Abhay Ashtekar, New variables for classical and quantum gravity, Phys. Rev. Lett., 57, 22442247, 1986
 Abhay Ashtekar, New Hamiltonian formulation of general relativity, Phys. Rev. D36, 15871602, 1987
 Roger Penrose, Angular momentum: an approach to combinatorial spacetime in Quantum Theory and Beyond, ed. Ted Bastin, Cambridge University Press, 1971
 Carlo Rovelli and Lee Smolin, Loop space representation of quantum general relativity, Nuclear Physics B331 (1990) 80152
 Carlo Rovelli and Lee Smolin, Discreteness of area and volume in quantum gravity, Nucl. Phys., B442 (1995) 593622, eprint available as grqc/9411005 (http://xxx.lanl.gov/abs/grqc/9411005)
External links
 Quantum Gravity, Physics, and Philosophy: http://www.qgravity.org/
 Resources for LQG and spin foams: http://jdc.math.uwo.ca/spinfoams/
 Gammaray Large Area Space Telescope: http://glast.gsfc.nasa.gov/ca:Teoria de la xarxa d'espín
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