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. It was developed in parallel with loop quantization, a rigorous framework for nonperturbative quantization of diffeomorphism-invariant gauge theories.

Theories about quantum gravity

LQG is controversial. Naturally, its practitioners consider it the main competitor of string theory as a theory of quantum gravity. LQG is, however, not even considered a contender by string theorists. See objections to the theory of loop quantum gravity for a more thorough discussion of some criticisms. String theory is itself also subject to various kinds of critical sniping, but has been a dominant presence since the mid-1980s in the quantum gravity area. In numerical terms, stringy people outnumber loopy people by a factor of roughly 10 and stringy papers outnumber loopy papers by a factor of roughly 50. The 'minority status' of LQG is not of course of any weight scientifically.

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, automatically accommodates matter particles, gauge vector bosons and the graviton, which suggested early in its development that strings might be able to model all known fundamental physics. Should LQG succeed as a quantum theory of gravity, however, the known matter fields would have to be incorporated into the theory using the broader formalism. 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 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 candidate for a quantum theory of gravity. Many of the core results in LQG are established at the level of rigour of mathematical physics, and rely heavily on algebraic quantum field theory. Its main shortcomings are: not yet having a picture of dynamics but only of kinematics; not yet able to perform particle physics calculations; not yet able to recover the classical limit. These difficulties may all be related.

The incompatibility between quantum mechanics and general relativity

Main article: quantum gravity

Quantum field theory studied on curved (non-Minkowskian) 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 space-time (see Unruh effect).

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 emergent. The other view is that background-independence is fundamental, and quantum mechanics needs to be generalized to settings where there is no a priori specified time.

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 lacked the causally-propagating local degrees of freedom needed for a description of 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 Einstein-Cartan 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 background-independent, 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 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 (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 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, because classical general relativity can be formulated as a BF theory with constraints, and it is hoped that a consistend quantization of gravity may arise from perturbation theory of BF spin-foam 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 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.

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 so-called 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 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 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.

Open problems

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 where the metric tensor carries no physical degrees of freedom. 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.

Time

Additionally, in LQG, time is not continuous but discrete and quantized, just as space is: there is a minimum moment of time, Planck time, which is on the order of 10⁻⁴³ seconds, and shorter intervals of time have no physical meaning. This carries the physical implication that relativity's prediction of time dilation due to accelerating speed or gravitational field, must be quantized, and must consist of multiples of Planck time units. (This helps resolve the time zero singularity problem: see subsection "The big bang")

Particle physics

While classical particle physics posit particles traveling through space and time that is continuous 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 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.

LQG does not constrain the spectrum of non-gravitational forces and elementary particles. Unlike the situation in string theory, all of them must be added to LQG by hand. It has proved difficult to incorporate elementary scalar fields, Higgs mechanism, and CP-violation into the framework of LQG.

Quantum field theory

Quantum field theory is 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 apparently no physical "room" for the ultraviolet infinities to occur. However, this argument may be compromised if LQG does not admit a limit of smooth geometry at long distance scales. It should bear in mind that LQG is constructed as an alternative to perturbative quantum field theory on a fixed background. In its present form, it does not allow a perturbative calculation of graviton scattering or other processes and it is not clear whether it ever will.

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 in its theoretical framework. As of today, the appearance of smooth space and gravitons in LQG has not been demonstrated, and henceforth the questions about graviton scattering cannot be answered in LQG.

Loop quantum gravity's implications

Space atoms

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.

Kinematics

Kinematics in loop quantum gravity is the physics of space and time at the Planck scale. It is expressed in terms of area and volume operators, and spin foam formalism.

Area and volume operators

One of the key results of loop quantum gravity is quantization of areas: according to several related derivations based on loop quantum gravity, the operator of the area <math>A<math> of a two-dimensional surface <math>\Sigma<math> should have discrete spectrum. Every spin network is an eigenstate of each such operator, and the area eigenvalue equals

<math>A_{\Sigma} = 8\pi G_{\mathrm{Newton}} \gamma \sum_i \sqrt{j_i(j_i+1)}<math>

where the sum goes over all intersections <math>i<math> of <math>\Sigma<math> with the spin network. In this formula, <math>G_{\mathrm{Newton}}<math> is the gravitational constant, <math>\gamma<math> is the Immirzi parameter and <math>j_i=0,0.5,1,1.5,\dots<math> is the spin associated with the link <math>i<math> of the spin network. The two-dimensional area is therefore "concentrated" in the intersections with the spin network.

Similar quantization applies to the volume operators but the mathematics behind these derivations is less convincing.

Spin foams

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. Cosmologists working on the assumption of zero cosmological constant predicted that gravity would slow the rate at which the universe is expanding following the big bang. However, astronomical observations of the magnitude and cosmological redshift of Type I supernovae in remote galaxies implies that the rate at which the universe is expanding is actually increasing. General relativity has a constant, Lambda, to account for this, and the observations, recently supported by independent data on the cosmic microwave background, appear to require a positive cosmological constant. In string theory, there are many vacua with broken supersymmetry which have positive cosmological constant, but generically their value of Lambda is much larger than the observed value. In LQG, there have been proposals to include a positive cosmological constant, involving a state referred to as the Kodama state after Hideo Kodama, a state described by a Chern-Simons wave function. Some physicists, for example Edward Witten, have argued by analogy with other theories that this state is unphysical. This issue is considered unresolved by other physicists.

Standard quantum field theory and supersymmetric string theories make a prediction based on calculation of the vacuum energy density that differs from what has actually been observed by 120 orders of magnitude. To date, this remains an unsolved mystery that a successful quantum theory of gravity would hopefully avoid

Black hole thermodynamics

While experimental tests for LQG may be years in the future, one conceptual test any candidate for QG must pass is that it must derive the correct formula Hawking derived for the black hole entropy.

With the proper Immirzi parameter, LQG can calculate and reproduce the Hawking formula for all black holes. While string/M-theory does not need the Immirzi parameter, it can as yet only derive the Hawking formula for extremal black holes and near-extremal black holes -- black holes with a net electric charge, which differ from the nearly neutral black holes formed from the collapse of electrically neutral matter such as neutron stars. To date, the Immirzi parameter cannot be derived from more fundamental principles, and is an unavoidable artefact of quantization of general relativity's field equations.

LQG's interpretation of black hole entropy is that the spacetime fabric that makes up the black hole horizon is quantized per Planck area, and the Bekenstein-Hawking entropy represents the degrees of freedom present in each Planck quantum. LQG does not offer an explanation why the interior of the black hole carries no volume-extensive entropy. Instead, it assumes that the interior does not contribute. The spacetime is truncated at the event horizon, and consistency requires to add Chern-Simons theory at the event horizon. A calculation within Chern-Simons theory leads to the desired result for the entropy, proportional to the horizon area.

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. This prediction disagrees with Hawking's semiclassical calculation, but the use of a semiclassical calculation that is so far unconfirmed by experiment as a benchmark for an exact nonperturbative fully quantum calculation may be problematic. Modulo the Immirzi parameter, which is the only free parameter of LQG, it matches it on average, and additionally predicts a fine structure to it, which is experimentally testable and potentially an improvement.

The big bang

Several LQG physicists have shown that LQG can, at least formally, 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.

Theory of everything: unification of the four forces

Grand unification theory refers to a theory in particle physics that unifies the strong interaction and electroweak interactions. A so-called theory of everything (TOE) is a putative theory that unifies the four fundamental forces of nature: gravity, the strong nuclear force, the weak nuclear force, and electromagnetism. Since the strong and electroweak interactions are described by quantum field theory, such a theory would require gravity also to be quantized, bringing with it the inconsistencies noted above.

One candidate for a consistent quantum gravity is string theory, which in addition to gravity contains gauge vector bosons and matter particles reminiscent of those experimentally observed. This has led to attempts (so far unsuccessful) to construct TOE's within its framework. In contrast, LQG is just a theory of one part of the Universe, namely quantum gravity.

Unification in field theory or string theory is difficult or impossible to test directly, due to the extremely large energy (greater than 10¹⁶ GeV) at which unification is manifest. However, indirect tests exist, such as proton decay and the convergence of the coupling constants when extrapolated to high energy through the renormalization group. The simplest unified models (without supersymmetry) have failed such tests, but many models are still viable. Incorporating the correct strength of gravity in string unification is particularly challenging. While unified theories have greater explanatory and predictive power, it may be that nature does not favour them.

Supersymmetry and extra dimensions

See supersymmetry for detailed discussion

LQG in its current formulation predicts no additional spatial dimensions, nor anything else about particle physics. Lee Smolin, one of the originators of LQG, has proposed that loop quantum gravity incorporating either supersymmetry or extra dimensions, or both, be called loop quantum gravity II, in light of experimental evidence.

Chaos theory and classical physics

Sensitivity on initial conditions, in the light of chaos theory means that two nonlinear systems with however small a difference in their initial state eventually will end up with a finite difference between their states. Loop quantum gravity suggests that the Planck scale represents the physical cut-off allowed for such sensitivity.

Differences between LQG and string/M-theory

Main article: objections to the theory of loop quantum gravity

String theory and LQG are the products of different communities within theoretical physics. It is not generally agreed whether they are in any sense compatible, and their differences have sometimes been represented as different ways of doing physics. This is a sharp debate, or at times presented as such: in other words matters are currently subject to dialectic rather than experimental test.

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 so far it lacks a universal, 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 in the near future

LQG may make predictions 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 (GLAST).

2007 will see the launch of GLAST, and (hopefully) the completion and operation of LHC. The results of these experiments will profoundly develop the course of QG. These experiments may establish spontaneously broken supersymmetry, Higgs boson and the Higgs field, extra spatial dimensions, and/or violations of Lorentz invariance.

If GLAST detects violations of Lorentz invariance in the form of energy-dependent photon velocity, in agreement with theoretical calculations, such observations would strongly support LQG. However, string theory would not necessarily be disfavoured, since although it predicts an underlying exact Lorentz symmetry, it is possible that this may be spontaneously broken through a nonzero expectation value of tensor fields.

Other topics where observation may affect the future theoretical development of quantum gravity are dark matter and dark energy.

Research in LQG and related areas

Active research directions

• Spin foam models
  • 2+1 and 3+1 theories
  • Barrett-Crane model
  • relation to the canonical approach
  • the Barbero-Immirzi parameter
  • canonical and spin foam geometries
  • the continuum limit
  • renormalization group flows
• the Hamiltonian constraint
  • 2+1 and 3+1 theories
  • spin-foam and canonical approach
  • quantum cosmology
  • Semi-classical corrections to Einstein equations
  • factor ordering
  • finding solutions and physical inner product
  • Thiemann's phoenix project.
• Semi-classical issues
  • kinematical and dynamical semi-classical states
  • quantum field theory on quantum geometry
  • quantum cosmology
  • Minkowski coherent state and Minkowski spin foam
• Loop quantum phenomenology
  • Lorentz invariance
  • Doubly-special relativity
  • quantum cosmology
  • Kodama state and de Sitter background
• Conceptual issues
  • observables through matter coupling
  • string theory in polymer representation
  • matter couplings on semi-classical states
  • the problem of time
  • spin foam histories
  • quantum groups in LQG

People

See list of loop quantum gravity researchers

Places

• New Brunswick
• Mississippi
• Louisiana
• Uruguay
• UC Riverside
• University of Western Ontario
• Kansas
• Penn State
• Hamilton
• Mexico
• Perimeter Institute for Theoretical Physics

See also

• Objections to the theory of loop quantum gravity
• Heyting algebra
• mathematical category theory
• noncommutative geometry
• topos theory
• C* Algebra
• Regge calculus

Bibliography

• Popular books:
  • Julian Barbour, The End of Time.
  • Lee Smolin, Three Roads to Quantum Gravity
• Magazine Articles
  • Lee Smolin, "Atoms in Space and Time," Scientific American, January 2004
• 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 (http://arxiv.org/abs/hep-th/0310077)
• Advanced books, reports, conference proceedings:
  • Carlo Rovelli, Quantum Gravity, Cambridge University Press (2004), ISBN 0521837332; draft available online (http://www.cpt.univ-mrs.fr/~rovelli/book.pdf)
  • 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/
• Resources for LQG and spin foams: http://jdc.math.uwo.ca/spin-foams/
• Gamma-ray Large Area Space Telescope: http://glast.gsfc.nasa.gov/

ca:Teoria de la xarxa d'espín de:Loop-Quantengravitation es:Gravedad cuántica de bucles it:gravitazione quantistica a loop