Loop quantum gravity

Not to be confused with the path integral formulation of LQG, see spin foam. This article is about LQG in its Canonical formulation..

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Loop quantum gravity (LQG), also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity. Loop quantum gravity suggests 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 called spin foam, which should not be confused with quantum foam. A major quantum gravity contender with string theory, loop quantum gravity incorporates general relativity without requiring string theory's 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. Roughly, LQG tries to establish a quantum theory of gravity in which the very space itself, where all other physical phenomena occur, becomes quantized.

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

Broad Overview of LQG

The problem with quantum gravity is that native mechanisms to quantize gravity (which have been applied successfully to other fields) fail for gravity. That means that something fundamental has to be changed for quantum gravity.

There are different approaches to solve these problems, e.g., string theory, asymptotic safety, and loop quantum gravity (LQG).

Essentially LQG does the following: it introduces new variables which replace the (in general relativity (GR)) well-known metric that describes spacetime and curvature. What happens is that these new variables are rather close to fields that we know from gauge theory like QED and QCD. Indeed in a certain sense gravity looks rather similar to QCD, but there is one additional property of gravity that allows one to apply a second mathematical technique which essentially replaces the fundamental fields with something like "fluxes through surfaces" or "fluxes along circles". These surfaces and circles are embedded into spacetime.

The next step is again rather technical and it becomes possible due to so-called diffeomorphism invariance (which does not exist in other field theories): one can get rid of the embedding of circles and surfaces into spacetime. Instead one replaces these entities with a so-called spin network, i.e. a graph with nodes and links between nodes where each link and each node carries some numbers which represent abstract entities from which certain properties of spacetime can be reconstructed. You can think about spacetime as made of cells; each cell has 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.

The problem with this picture is that one might think about these cells as sitting in spacetime - but this is fundamentally wrong: this picture is only due to the construction of the spin network, but basically there is no spacetime anymore; all there are are nodes and links (and certain numbers attributed to nodes and links). Spacetime is no longer fundamental but becomes an entity emerging from the more fundamental graphs with their nodes and links. The graphs are called spin networks because the numbers they are carrying have properties well-known from quantum mechanical spins. But this is a mathematical property only, it does not mean that there are real spinning objects.

Compare this emerging spacetime to a water surface of a lake. We know that it consists of atoms, and as soon as we get this picture it is clear that there is no water between the atoms; the surface is only an emerging phenomenon, the true fundamental objects are the atoms. In the same sense the spin networks are the fundamental entities from which spacetime, surfaces etc. and their properties like volume, area, curvature etc. can be constructed. Dynamics of spacetime (which was curvature, gravitational waves etc. in GR) is 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 "is" spacetime. Instead quantized spacetime is a superposition of (infinitely many) spin networks. This is well-known in quantum mechanics; there is no reason why an atom should be in a certain state; we can achieve that via preparation or measurement, but in principle a single atom can be in an arbitrary complex quantum state which is a superposition of "an atom sitting here, an atom moving in a certain direction over there, an atom moving in this or that direction, ...".

So classical spacetime is recovered by two averaging process: first there seems to be a regime where this 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; second from this single spin network one can reconstruct spacetime in the same sense as one can reconstruct the water surface from the individual atoms. But there may be different regimes (e.g. in black holes or close to the big bang) where this classical picture and this averaging does no longer work. It may be that in these regimes all there is are spin networks without any classical property like smooth spacetime, areas, volume etc. It's like looking at a single atom, or gas: there is no water surface anymore; and there seems to be no smooth spacetime near a singularity of a black hole or the big bang. In order to understand these new non-classical regimes of spacetime a fundamentally new picture is required. This is what LQG (and other approaches) are aiming for: construct a new mathematical model from which well-known classical spacetime (like in GR) can be reconstructed, but which does not break down in certain regimes but remains well-defined and consistent.

History of LQG

Main article: History of loop quantum gravity

In 1986, Abhay Ashtekar reformulated Einstein's 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 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

Main article: 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 space-time. 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) space-time. 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 (cf. the section on General relativity#Evolution equations), the result is an analogue of the Schrödinger equation called the Wheeler-deWitt equation, which some argue is ill-defined.^[1] A major break-through 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.

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 contact 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

Main article: Lorentz covariance

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 are 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 diffemorphisms. 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 is 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 includes 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 includes spinfoam models ^[14] and entropic gravity.^[15]

See also

• Loop quantum cosmology
• Heyting algebra
• Category theory
• Noncommutative geometry
• Topos theory
• C*-algebra
• Regge calculus
• Double special relativity
• Lorentz invariance in loop quantum gravity
• Immirzi parameter
• Invariance mechanics
• spin foam
• group field theory
• string-net
• Kodama state
• supersymmetry

Notes

1. ^ Cf. section 3 in Kuchař 1973.
2. ^ See Ashtekar 1986, Ashtekar 1987.
3. ^ For a review, see Thiemann 2006; more extensive accounts can be found in Rovelli 1998, Ashtekar & Lewandowski 2004 as well as in the lecture notes Thiemann 2003.
4. ^ See List of loop quantum gravity researchers
5. ^ Baez, John; Krasnov, Kirill (1997). Quantization of Diffeomorphism-Invariant Theories with Fermions. arXiv:hep-th/9703112. Bibcode 1998JMP....39.1251B. doi:10.1063/1.532400. 
6. ^ Ling, Yi; Smolin, Lee (1999). "Supersymmetric Spin Networks and Quantum Supergravity". arXiv:hep-th/9904016 [hep-th].  }}
7. ^ See, e.g., Stuart Kauffman and Lee Smolin "A Possible Solution For The Problem Of Time In Quantum Cosmology" (1997). [1]
8. ^ See, e.g., Lee Smolin, "The Case for Background Independence", in Dean Rickles, et al. (eds.) The Structural Foundations of Quantum Gravity (2006), p 196 ff.
9. ^ For a highly technical explanation, see, e.g., Carlo Rovelli (2004). Quantum Gravity, p 13 ff.
10. ^ Hermann Nicolai; Kasper Peeters; Marija Zamaklar (2005), "Loop quantum gravity: an outside view", Classical and Quantum Gravity 22 (19): R193–R247, arXiv:hep-th/0501114, Bibcode 2005CQGra..22R.193N, doi:10.1088/0264-9381/22/19/R01. 
11. ^ Hermann Nicolai; Kasper Peeters; Marija Zamaklar (2005), "Loop quantum gravity: an outside view", Classical and Quantum Gravity 22 (19): R193–R247, arXiv:hep-th/0501114, Bibcode 2005CQGra..22R.193N, doi:10.1088/0264-9381/22/19/R01. 
12. ^ Fermi GBM/LAT Collaborations (2009). "Testing Einstein's special relativity with Fermi's short hard gamma-ray burst GRB090510". arXiv:0908.1832 [astro-ph.HE]. 
13. ^ Goswami et al.; Joshi, Pankaj S.; Singh, Parampreet (2006), "Quantum evaporation of a naked singularity", Physical Review Letters 96 (3): 31302, arXiv:gr-qc/0506129, Bibcode 2006PhRvL..96c1302G, doi:10.1103/PhysRevLett.96.031302. 
14. ^ http://math.ucr.edu/home/baez/week280.html
15. ^ Lee Smolin (2010). "Newtonian gravity in loop quantum gravity". arXiv:1001.3668 [gr-qc]. 

References

• Topical Reviews
  • 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)