Loop quantum gravity

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Loop quantum gravity (LQG), also known as loop gravity and quantum geometry, is a proposed quantum hypothesis of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity.

Loop quantum gravity postulates that space can be viewed as an extremely fine fabric or network "woven" of finite quantised loops of excited gravitational fields called spin networks. When viewed over time, these spin networks are referred to as "spin foam" (which should not be confused with quantum foam). The theory of LQG is considered a major quantum gravity contender, along with string theory, but has the perceived advantage of consistently incorporating general relativity without requiring resorting to the use of "higher dimensions".

LQG preserves many of the important features of general relativity while simultaneously employing quantization of both space and time at the Planck scale in the tradition of quantum mechanics. The technique of loop quantization was developed for the nonperturbative quantization of diffeomorphism-invariant gauge theory. LQG tries to establish a quantum theory of gravity in which space itself - where all other physical phenomena occur - becomes quantized.

LQG is one of a family of theories called canonical quantum gravity. LQG also includes matter and forces, but does not address the problem of the unification of all physical forces in the way some other quantum gravity theories (such as string theory) do.

Broad overview of LQG

The native mechanisms in quantum gravity (successfully applied to other fields) fail to quantize gravity - indicating that something fundamental in the theory is missing, incomplete, or has to be changed. Several different approaches attempt to solve this problem, including string theory, asymptotic safety, and loop quantum gravity (LQG).

Essentially, LQG introduces new variables which replace the well-known metric in general relativity (GR) that describes spacetime and curvature. These new variables are rather close to fields that are known from gauge theory, like QED and QCD. In a certain sense gravity appears similar to QCD, but gravity has an additional property that allows the application of a second mathematical technique which replaces the fundamental fields with "fluxes through surfaces" or "fluxes along circles". These surfaces and circles are embedded into spacetime.

It thus becomes possible to eliminate the embedding of circles and surfaces into spacetime through so-called diffeomorphism invariance (which does not exist in other field theories). This is accomplished by replacing these entities with a so-called "spin network" - a graph with nodes and links among them, where each link and node carry numerical values which represent abstract entities from which certain properties of spacetime can be reconstructed. Conceptually, spacetime can be thought of as cells, each with a certain volume carried by a node. Each cell has certain surfaces, and the link between different nodes (sitting inside these cells) carry the areas of the surfaces.

It would not be correct to envision these cells as sitting in spacetime, since there is no "spacetime" anymore - only nodes and links, and certain numerical values associated with these nodes and links. Spacetime, therefore, is no longer fundamental, but is more accurately described as an entity emerging from the more fundamental graphs and their associated nodes and links. The graphs are called spin networks because the numerical values with which they are associated have properties well-known from quantum mechanical spins. This is a mathematical property only, and does not indicate that there are actual spinning objects.

It is possible to compare this emerging spacetime to the surface of a lake. We know that the water surface consists of atoms, and that there is no water between these atoms. Consequently, the surface is only an emerging phenomenon, the true fundamental objects are the atoms. In the same sense, spin networks are fundamental entities from which spacetime, surfaces and their properties (such as volume, area and curvature) can be constructed. Dynamics of spacetime (such as curvature and gravitational waves in General Relativity) are replaced by dynamics of spin networks: within a given graph new nodes with new links can appear (or disappear) according to specific mathematical rules.

A spin network is not a mechanical object which comprises spacetime. Instead, quantized spacetime is a superposition of an infinite number of spin networks. This is very well known in quantum mechanics: there is no reason why an atom should be in a "certain state". In principle, a single atom can be in an arbitrary complex quantum state, a phenomenon which has been described as a superposition of "an atom sitting here, an atom moving in a certain direction over there, an atom moving in this or that direction, ...".

Therefore, classical spacetime is recovered by two averaging processes:

• First, there seems to be a regime where the superposition of spin networks is peaked around a single classical spacetime (i.e. where one single spin network dominates the superposition of infinitely many spin networks)
• Secondly, from this single spin network, one can reconstruct spacetime in the same sense as one can reconstruct the "water surface" from the individual atoms

However, there may be different regimes (i.e. black holes, Big Bang singularities, events around the Planck scales) in which this classical picture and averaging no longer yield correct results. Potentially, in these regimes only, spin networks exist without classical properties (such as "smooth" spacetime, area or volume). Analogous to the individual atoms comprising a lake surface, there is no water surface anymore, and these theories suggest that there seems to be no smooth spacetime near Big Bang or black hole singularities, or at Planck scale lengths. To understand these new "non-classical" regimes of spacetime, a fundamentally new picture is required. LQG and certain other approaches represent new mathematical models from which the well-known "classical" General Relativity spacetime can be reconstructed, but which also remain well-defined and consistent in more "extreme" regimes.

History of LQG

In 1986, Abhay Ashtekar reformulated the Einstein field equations of general relativity. 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 background-independent, Carlo Rovelli and Lee Smolin understood that 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, Rovelli and Smolin obtained an explicit basis of states of quantum geometry, which turned out to be labelled by Roger Penrose's spin networks, and showed that the geometry is quantized - that is, the (non-gauge-invariant) quantum operators representing area and volume have a discrete spectrum. In this context, spin networks arose as a generalization of Wilson loops that was 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.

Key concepts of loop quantum gravity

In the framework of quantum field theory, and using the standard techniques of perturbative calculations, one finds that gravitation is non-renormalizable (in contrast to the electroweak and strong interactions of the Standard Model of particle physics. This implies that there are infinitely many free parameters in the theory and, thus, that it cannot be predictive.

In general relativity, the Einstein field equations assign a geometry (via a metric) to spacetime. Before this, there is no physical notion of distance or time measurements. In this sense, general relativity is said to be background independent. An immediate conceptual issue that arises is that the usual framework of quantum mechanics - including quantum field theory - relies on a reference (background) spacetime. Therefore, one approach to finding a quantum theory of gravity is to understand how to do quantum mechanics without relying on such a background; this is the approach of the canonical quantization/loop quantum gravity/spin foam approaches.

Simple spin network of the type used in loop quantum gravity

Starting with the initial-value-formulation of general relativity, the result is an analogue of the Schrödinger equation, called the Wheeler-deWitt equation, which some argue is ill-defined.^[1] A major breakthrough came with the introduction of what are now known as Ashtekar variables, which represent geometric gravity using mathematical analogues of electric and magnetic fields.^[2] The resulting candidate for a theory of quantum gravity is Loop Quantum Gravity, in which space is represented by a network structure called a spin network, evolving over time in discrete steps.^[3]

Though not proven, it may be impossible to quantize gravity in 3+1 dimensions without creating matter and energy "artifacts". Should LQG succeed as a quantum theory of gravity, the known matter fields will have to be incorporated into the theory a posteriori. Many of the approaches now being actively pursued (by Renate Loll, Jan Ambjørn, Lee Smolin, Sundance Bilson-Thompson, Laurent Freidel, Mark B. Wise and others^[4]) combine matter with geometry.

To date, the main successes of loop quantum gravity are:

1. It is a nonperturbative quantization of 3-space geometry, with quantized area and volume operators
2. It includes a calculation of the entropy of black holes
3. It replaces the Big Bang spacetime singularity with a Big Bounce

These claims are not universally accepted among physicists, who are presently divided between different approaches to the problem of quantum gravity. LQG may possibly be viable as a refinement of either gravity or geometry. Many of the core results are rigorous mathematical physics; their physical interpretations remain speculative. Three speculative physical interpretations of LQG's core mathematical results are loop quantization, Lorentz invariance, General covariance, and background independence (discussed below). Another physical test for LQG is to reproduce the physics of general relativity coupled with quantum field theory (discussed under "Problems").

Loop quantization

At the core of loop quantum gravity is a framework for nonperturbative quantization of diffeomorphism-invariant 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,^[5], an arbitrary gauge group (or even quantum group), and supersymmetry,^[6] and results in a quantization of the kinematics of the corresponding diffeomorphism-invariant 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 the theory strictly testable and consistent with experiment.

In a nutshell, loop quantization is the result of applying C*-algebraic quantization to a non-canonical algebra of gauge-invariant classical observables. Non-canonical 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 state-sum/spin-foam models such as the Turaev-Viro model of 2+1 dimensional general relativity. A much-studied topological quantum field theory is the so-called BF theory in 3+1 dimensions. Since classical general relativity can be formulated as a BF theory with constraints, scientists hope that a consistent quantization of gravity may arise from the perturbation theory of BF spin foam models.

Lorentz invariance

LQG is a quantization of a classical Lagrangian field theory which is equivalent to the usual Einstein-Cartan 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.

General covariance and background independence

General covariance, also known as "diffeomorphism invariance", is the invariance of physical laws under arbitrary coordinate transformations. An example of this is the equations of general relativity, where this symmetry is one of the defining features of the theory. LQG preserves this symmetry by requiring that the physical states remain invariant under the generators of diffeomorphisms. The interpretation of this condition is well understood for purely spatial diffeomorphisms. However, the understanding of diffeomorphisms involving time (the Hamiltonian constraint) is more subtle because it is related to dynamics and the so-called "problem of time" in general relativity.^[7] A generally accepted calculational framework to account for this constraint has yet to be found.^[8][9]

Whether or not Lorentz invariance is broken in the low-energy 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 invariant topology). Instead, they are expected to give rise to space and time at distances which are large compared to the Planck length.

Problems

Presently, no semiclassical limit recovering general relativity has been shown to exist. This means it remains unproven that LQG's description of spacetime at the Planck scale has the right continuum limit (described by general relativity with possible quantum corrections). It is thus unclear if the theory is in agreement with any experiment ever made. Specifically, the dynamics of the theory is encoded in the Hamiltonian constraint, but there is no candidate Hamiltonian.^[10] Other technical problems include finding off-shell closure of the constraint algebra and physical inner product vector space, coupling to matter fields of Quantum field theory, fate of the renormalization of the graviton in perturbation theory that lead to ultraviolet divergence beyond 2-loops (see One-loop Feynman diagram in Feynman diagram).^[11] The fate of Lorentz invariance in loop quantum gravity remains an open problem.^[12]

While there has been a recent proposal relating to observation of naked singularities,^[13] and doubly special relativity, as a part of a program called loop quantum cosmology, as of now there is no experimental observation for which loop quantum gravity makes a prediction not made by the Standard Model or general relativity (a problem that plagues all current theories of quantum gravity). Because of the above mentioned lack of a semiclassical limit, LQG cannot even reproduce the predictions made by the Standard Model.

Making predictions from the theory of LQG has been extremely difficult computationally, also a recurring problem with modern theories in physics.

Another problem is that a crucial free parameter in the theory, known as the Immirzi parameter, can only be computed by demanding agreement with Bekenstein and Hawking's calculation of the black hole entropy. Loop quantum gravity predicts that the entropy of a black hole is proportional to the area of the event horizon, but does not obtain the Bekenstein-Hawking formula S = A/4 unless the Immirzi parameter is chosen to give this value. A prediction directly from theory would be preferable.

Current LQG research directions attempt to address these known problems, and include spinfoam models ^[14] and entropic gravity.^[15]

See also

Notes

References

• Topical Reviews
  • Rovelli, Carlo (2011), Zakopane lectures on loop gravity, arXiv:gr-qc/1102.3660 
  • Rovelli, Carlo (1998), "Loop Quantum Gravity", Living Reviews in Relativity 1, http://www.livingreviews.org/lrr-1998-1, retrieved 2008-03-13 
  • Thiemann, Thomas (2003), "Lectures on Loop Quantum Gravity", Lectures Notes in Physics 631: 41–135, arXiv:gr-qc/0210094, Bibcode 2003LNP...631...41T 
  • Ashtekar, Abhay; Lewandowski, Jerzy (2004), "Background Independent Quantum Gravity: A Status Report", Classical and Quantum Gravity 21 (15): R53–R152, arXiv:gr-qc/0404018, Bibcode 2004CQGra..21R..53A, doi:10.1088/0264-9381/21/15/R01 
  • Carlo Rovelli and Marcus Gaul, Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance, e-print available as gr-qc/9910079.
  • Lee Smolin, The case for background independence, e-print available as hep-th/0507235.
  • Alejandro Corichi, Loop Quantum Geometry: A primer, e-print available as [2].
  • Alejandro Perez, Introduction to loop quantum gravity and spin foams, e-print available as [3].
  • Hermann Nicolai and Kasper Peeters Loop and spin foam quantum gravity: A Brief guide for beginners., e-print available as [4].
• Popular books:
  • Lee Smolin, Three Roads to Quantum Gravity
  • 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.
  • Julian Barbour, The End of Time: The Next Revolution in Our Understanding of the Universe
  • Musser, George (2008), "The Complete Idiot's Guide to String Theory", The Physics Teacher (Indianapolis: Alpha) 47 (2): 368, Bibcode 2009PhTea..47Q.128H, doi:10.1119/1.3072469, ISBN 978-1-59-257702-6  – Focuses on string theory but has an extended discussion of loop gravity as well.
• Magazine articles:
  • Lee Smolin, "Atoms of Space and Time," Scientific American, January 2004
  • Martin Bojowald, "Following the Bouncing Universe," Scientific American, October 2008
• Easier introductory, expository or critical works:
  • Abhay Ashtekar, Gravity and the quantum, e-print available as gr-qc/0410054 (2004)
  • John C. Baez and Javier Perez de Muniain, Gauge Fields, Knots and Quantum Gravity, World Scientific (1994)
  • Carlo Rovelli, A Dialog on Quantum Gravity, e-print available as hep-th/0310077 (2003)
• More advanced introductory/expository works:
  • Carlo Rovelli, Quantum Gravity, Cambridge University Press (2004); draft available online
  • Thomas Thiemann, Introduction to modern canonical quantum general relativity, e-print available as gr-qc/0110034
  • Thomas Thiemann, Introduction to Modern Canonical Quantum General Relativity, Cambridge University Press (2007)
  • Abhay Ashtekar, New Perspectives in Canonical Gravity, Bibliopolis (1988).
  • Abhay Ashtekar, Lectures on Non-Perturbative Canonical Gravity, World Scientific (1991)
  • 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, e-print available as hep-th/0501114
  • H. Nicolai and K. Peeters, Loop and Spin Foam Quantum Gravity: A Brief Guide for Beginners, e-print available as hep-th/0601129
  • T. Thiemann The LQG – String: Loop Quantum Gravity Quantization of String Theory (2004)

• Conference proceedings:
  • John C. Baez (ed.), Knots and Quantum Gravity
• Fundamental research papers:
  • Ashtekar, Abhay (1986), "New variables for classical and quantum gravity", Physical Reviews Letters 57 (18): 2244–2247, Bibcode 1986PhRvL..57.2244A, doi:10.1103/PhysRevLett.57.2244, PMID 10033673 
  • Ashtekar, Abhay (1987), "New Hamiltonian formulation of general relativity", Physical Review D 36 (6): 1587–1602, Bibcode 1987PhRvD..36.1587A, doi:10.1103/PhysRevD.36.1587 
  • Roger Penrose, Angular momentum: an approach to combinatorial space-time in Quantum Theory and Beyond, ed. Ted Bastin, Cambridge University Press, 1971
  • Rovelli, Carlo; Smolin, Lee (1988), "Knot theory and quantum gravity", Physical Review Letters 61 (10): 1155, Bibcode 1988PhRvL..61.1155R, doi:10.1103/PhysRevLett.61.1155. 
  • Rovelli, Carlo; Smolin, Lee (1990), "Loop space representation of quantum general relativity", Nuclear Physics B331: 80–152. 
  • Carlo Rovelli and Lee Smolin, Discreteness of area and volume in quantum gravity, Nucl. Phys., B442 (1995) 593-622, e-print available as gr-qc/9411005
  • Kuchař, Karel (1973), "Canonical Quantization of Gravity", in Israel, Werner, Relativity, Astrophysics and Cosmology, D. Reidel, pp. 237–288, ISBN 90-277-0369-8 
  • Thiemann, Thomas (2006), "Loop Quantum Gravity: An Inside View", Approaches to Fundamental Physics 721: 185, arXiv:hep-th/0608210, Bibcode 2007LNP...721..185T 

External links

• "Loop Quantum Gravity" by Carlo Rovelli Physics World, November 2003
• Quantum Foam and Loop Quantum Gravity
• Abhay Ashtekar: Semi-Popular Articles . Some excellent popular articles suitable for beginners about space, time, GR, and LQG.
• Loop Quantum Gravity: Lee Smolin.
• Loop Quantum Gravity on arxiv.org
• A list of LQG references catered to fresh graduates
• Loop Quantum Gravity Lectures Online by Lee Smolin
• Spin networks, spin foams and loop quantum gravity
• Wired magazine, News: Moving Beyond String Theory
• April 2006 Scientific American Special Issue, A Matter of Time, has Lee Smolin LQG Article Atoms of Space and Time
• September 2006, The Economist, article Looping the loop
• Gamma-ray Large Area Space Telescope: http://glast.gsfc.nasa.gov/
• Zeno meets modern science. Article from Acta Physica Polonica B by Z.K. Silagadze.
• Did pre-big bang universe leave its mark on the sky? - According to a model based on "loop quantum gravity" theory, a parent universe that existed before ours may have left an imprint (New Scientist, 10 April 2008)