Quantum gravity (QG) is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics (describing three of the four known fundamental interactions) with general relativity (describing the fourth, gravity). It is hoped that development of such a theory would unify into a single mathematical framework all fundamental interactions and to describe all known observable interactions in the universe, at both subatomic and cosmological scales.
Such a theory of quantum gravity would yield the same experimental results as ordinary quantum mechanics in conditions of weak gravity (gravitational potentials much less than c2) and the same results as Einsteinian general relativity in phenomena at scales much larger than individual molecules (action much larger than reduced Planck's constant), but moreover be able to predict the outcome of situations where both quantum effects and strong-field gravity are important (at the Planck scale, unless large extra dimension conjectures are correct).
If the theory of quantum gravity also achieves a grand unification of the other known interactions, it is referred to as a theory of everything (TOE).
Motivation for quantizing gravity comes from the remarkable success of the quantum theories of the other three fundamental interactions, and from experimental evidence suggesting that gravity can be made to show quantum effects.[1][2][3] Although some quantum gravity theories such as string theory and other unified field theories (or 'theories of everything') attempt to unify gravity with the other fundamental forces, others such as loop quantum gravity make no such attempt; they simply quantize the gravitational field while keeping it separate from the other forces.
Observed physical phenomena can be described well by quantum mechanics or general relativity, without needing both. This can be thought of as due to an extreme separation of mass scales at which they are important. Quantum effects are usually important only for the "very small", that is, for objects no larger than typical molecules. General relativistic effects, on the other hand, show up mainly for the "very large" bodies such as collapsed stars. (Planets' gravitational fields, as of 2011, are well-described by linearized gravity except for Mercury's perihelion precession; so strong-field effects—any effects of gravity beyond lowest nonvanishing order in φ/c2—have not been observed even in the gravitational fields of planets and main sequence stars). There is a lack of experimental evidence relating to quantum gravity, and classical physics adequately describes the observed effects of gravity over a range of 50 orders of magnitude of mass, i.e., for masses of objects from about 10−23 to 1030 kg.
Diagram showing where quantum gravity sits in the hierarchy of physics theories
Much of the difficulty in meshing these theories at all energy scales comes from the different assumptions that these theories make on how the universe works. Quantum field theory depends on particle fields embedded in the flat space-time of special relativity. General relativity models gravity as a curvature within space-time that changes as a gravitational mass moves. Historically, the most obvious way of combining the two (such as treating gravity as simply another particle field) ran quickly into what is known as the renormalization problem. In the old-fashioned understanding of renormalization, gravity particles would attract each other and adding together all of the interactions results in many infinite values which cannot easily be cancelled out mathematically to yield sensible, finite results. This is in contrast with quantum electrodynamics where, while the series still do not converge, the interactions sometimes evaluate to infinite results, but those are few enough in number to be removable via renormalization.
Quantum gravity can be treated as an effective field theory. Effective quantum field theories come with some high-energy cutoff, beyond which we do not expect that the theory provides a good description of nature. The "infinities" then become large but finite quantities proportional to this finite cutoff scale, and correspond to processes that involve very high energies near the fundamental cutoff. These quantities can then be absorbed into an infinite collection of coupling constants, and at energies well below the fundamental cutoff of the theory, to any desired precision; only a finite number of these coupling constants need to be measured in order to make legitimate quantum-mechanical predictions. This same logic works just as well for the highly successful theory of low-energy pions as for quantum gravity. Indeed, the first quantum-mechanical corrections to graviton-scattering and Newton's law of gravitation have been explicitly computed[4] (although they are so astronomically small that we may never be able to measure them). In fact, gravity is in many ways a much better quantum field theory than the Standard Model, since it appears to be valid all the way up to its cutoff at the Planck scale. (By comparison, the Standard Model is expected to start to break down above its cutoff at the much smaller scale of around 1000 GeV.[citation needed])
While confirming that quantum mechanics and gravity are indeed consistent at reasonable energies, it is clear that near or above the fundamental cutoff of our effective quantum theory of gravity (the cutoff is generally assumed to be of the order of the Planck scale), a new model of nature will be needed. Specifically, the problem of combining quantum mechanics and gravity becomes an issue only at very high energies, and may well require a totally new kind of model.
The general approach to deriving a quantum gravity theory that is valid at even the highest energy scales is to assume that such a theory will be simple and elegant and, accordingly, to study symmetries and other clues offered by current theories that might suggest ways to combine them into a comprehensive, unified theory. One problem with this approach is that it is unknown whether quantum gravity will actually conform to a simple and elegant theory, as it should resolve the dual conundrums of special relativity with regard to the uniformity of acceleration and gravity, and general relativity with regard to spacetime curvature.
Such a theory is required in order to understand problems involving the combination of very high energy and very small dimensions of space, such as the behavior of black holes, and the origin of the universe.
Gravity Probe B (GP-B) has measured spacetime curvature near Earth to test related models in application of Einstein's general theory of relativity.
At present, one of the deepest problems 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 atomic scale. This problem must be put in the proper context, however. In particular, contrary to the popular claim that quantum mechanics and general relativity are fundamentally incompatible, one can demonstrate that the structure of general relativity essentially follows inevitably from the quantum mechanics of interacting theoretical spin-2 massless particles [5][6][7][8][9] (called gravitons).
While there is no concrete proof of the existence of gravitons, quantized theories of matter may necessitate their existence.[citation needed] Supporting this theory is the observation that all fundamental forces except gravity have one or more known messenger particles, leading researchers to believe that at least one most likely does exist; they have dubbed these hypothetical particles gravitons. Many of the accepted notions of a unified theory of physics since the 1970s, including string theory, superstring theory, M-theory, loop quantum gravity, all assume, and to some degree depend upon, the existence of the graviton. Many researchers[who?] view the detection of the graviton as vital to validating their work.
The dilaton made its first appearance in Kaluza–Klein theory, a five-dimensional theory that combined gravitation and electromagnetism. Generally, it appears in string theory. More recently, it has appeared in the lower-dimensional many-bodied gravity problem[10] based on the field theoretic approach of Roman Jackiw. The impetus arose from the fact that complete analytical solutions for the metric of a covariant N-body system have proven elusive in General Relativity. To simplify the problem, the number of dimensions was lowered to (1+1) namely one spatial dimension and one temporal dimension. This model problem, known as R=T theory[11] (as opposed to the general G=T theory) was amenable to exact solutions in terms of a generalization of the Lambert W function. It was also found that the field equation governing the dilaton (derived from differential geometry) was the Schrödinger equation and consequently amenable to quantization.[12] Thus, one had a theory which combined gravity, quantization and even the electromagnetic interaction, promising ingredients of a fundamental physical theory. It is worth noting that the outcome revealed a previously unknown and already existing natural link between general relativity and quantum mechanics. However, this theory needs to be generalized in (2+1) or (3+1) dimensions although, in principle, the field equations are amenable to such generalization as shown with the inclusion of a one-graviton process[13] and yielding the correct Newtonian limit in d dimensions if a dilaton is included. However, it is not yet clear what the full field equation will govern the dilaton in higher dimensions. This is further complicated by the fact that gravitons can propagate in (3+1) dimensions and consequently that would imply gravitons and dilatons exist in the real world. Moreover, detection of the dilaton is expected to be even more elusive than the graviton. However, since this approach allows for the combination of gravitational, electromagnetic and quantum effects, their coupling could potentially lead to a means of vindicating the theory, through cosmology and perhaps even experimentally.
General relativity, like electromagnetism, is a classical field theory. One might expect that, as with electromagnetism, there should be a corresponding quantum field theory.
However, gravity is perturbatively nonrenormalizable.[14] For a quantum field theory to be well-defined according to this understanding of the subject, it must be asymptotically free or asymptotically safe. The theory must be characterized by a choice of finitely many parameters, which could, in principle, be set by experiment. For example, in quantum electrodynamics, these parameters are the charge and mass of the electron, as measured at a particular energy scale.
On the other hand, in quantizing gravity, there are infinitely many independent parameters (counterterm coefficients) needed to define the theory. For a given choice of those parameters, one could make sense of the theory, but since we can never do infinitely many experiments to fix the values of every parameter, we do not have a meaningful physical theory:
- At low energies, the logic of the renormalization group tells us that, despite the unknown choices of these infinitely many parameters, quantum gravity will reduce to the usual Einstein theory of general relativity.
- On the other hand, if we could probe very high energies where quantum effects take over, then every one of the infinitely many unknown parameters would begin to matter, and we could make no predictions at all.
As explained below, there is a way around this problem by treating QG as an effective field theory.
Any meaningful theory of quantum gravity that makes sense and is predictive at all energy scales must have some deep principle that reduces the infinitely many unknown parameters to a finite number that can then be measured.
- One possibility is that normal perturbation theory is not a reliable guide to the renormalizability of the theory, and that there really is a UV fixed point for gravity. Since this is a question of non-perturbative quantum field theory, it is difficult to find a reliable answer, but some people still pursue this option.
- Another possibility is that there are new symmetry principles that constrain the parameters and reduce them to a finite set. This is the route taken by string theory, where all of the excitations of the string essentially manifest themselves as new symmetries.
In an effective field theory, all but the first few of the infinite set of parameters in a non-renormalizable theory are suppressed by huge energy scales and hence can be neglected when computing low-energy effects. Thus, at least in the low-energy regime, the model is indeed a predictive quantum field theory.[4] (A very similar situation occurs for the very similar effective field theory of low-energy pions.) Furthermore, many theorists agree that even the Standard Model should really be regarded as an effective field theory as well, with "nonrenormalizable" interactions suppressed by large energy scales and whose effects have consequently not been observed experimentally.
Recent work[4] has shown that by treating general relativity as an effective field theory, one can actually make legitimate predictions for quantum gravity, at least for low-energy phenomena. An example is the well-known calculation of the tiny first-order quantum-mechanical correction to the classical Newtonian gravitational potential between two masses.
A fundamental lesson of general relativity is that there is no fixed spacetime background, as found in Newtonian mechanics and special relativity; the spacetime geometry is dynamic. 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 relational theory,[15] 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 inception on a fixed background (non-dynamic) structure. In the case of quantum mechanics, it is time that is given and not dynamic, 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.
String theory can be seen as a generalization of quantum field theory where instead of point particles, string-like objects propagate in a fixed spacetime background, although the interactions among closed strings give rise to space-time in a dynamical way. Although string theory had its origins in the study of quark confinement and not of quantum gravity, it was soon discovered that the string spectrum contains the graviton, and that "condensation" of certain vibration modes of strings is equivalent to a modification of the original background. In this sense, string perturbation theory exhibits exactly the features one would expect of a perturbation theory that may exhibit a strong dependence on asymptotics (as seen, for example, in the AdS/CFT correspondence) which is a weak form of background dependence.
Loop quantum gravity is the fruit of an 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 in 3+1 dimensions which has local degrees of freedom according to general relativity. In 2+1 dimensions, however, gravity is a topological field theory, and it has been successfully quantized in several different ways, including spin networks.
Quantum field theory on curved (non-Minkowskian) backgrounds, while not a full quantum theory of gravity, has shown many promising early results. In an analogous way to the development of quantum electrodynamics in the early part of the 20th century (when physicists considered quantum mechanics in classical electromagnetic fields), the consideration of quantum field theory on a curved background has led to predictions such as black hole radiation.
Phenomena such as the Unruh effect, in which particles exist in certain accelerating frames but not in stationary ones, do not pose any difficulty when considered on a curved background (the Unruh effect occurs even in flat Minkowskian backgrounds). The vacuum state is the state with least energy (and may or may not contain particles). See Quantum field theory in curved spacetime for a more complete discussion.
There are other points of tension between quantum mechanics and general relativity.
- First, classical general relativity breaks down at singularities, and quantum mechanics becomes inconsistent with general relativity in the neighborhood of singularities (however, no one is certain that classical general relativity applies near singularities in the first place).
- Second, it is not clear how to determine the gravitational field of a particle, since under the Heisenberg uncertainty principle of quantum mechanics its location and velocity cannot be known with certainty. The resolution of these points may come from a better understanding of general relativity.[16]
- Third, there is the Problem of Time in quantum gravity. Time has a different meaning in quantum mechanics and general relativity and hence there are subtle issues to resolve when trying to formulate a theory which combines the two.[17]
There are a number of proposed quantum gravity theories.[18] Currently, there is still no complete and consistent quantum theory of gravity, and the candidate models still need to overcome major formal and conceptual problems. They also face the common problem that, as yet, there is no way to put quantum gravity predictions to experimental tests, although there is hope for this to change as future data from cosmological observations and particle physics experiments becomes available.[19][20]
Main article:
String theoryOne suggested starting point is ordinary quantum field theories which, after all, are successful in describing the other three basic fundamental forces in the context of the standard model of elementary particle physics. However, while this leads to an acceptable effective (quantum) field theory of gravity at low energies,[21] gravity turns out to be much more problematic at higher energies. Where, for ordinary field theories such as quantum electrodynamics, a technique known as renormalization is an integral part of deriving predictions which take into account higher-energy contributions,[22] gravity turns out to be nonrenormalizable: at high energies, applying the recipes of ordinary quantum field theory yields models that are devoid of all predictive power.[23]
One attempt to overcome these limitations is to replace ordinary quantum field theory, which is based on the classical concept of a point particle, with a quantum theory of one-dimensional extended objects: string theory.[24] At the energies reached in current experiments, these strings are indistinguishable from point-like particles, but, crucially, different modes of oscillation of one and the same type of fundamental string appear as particles with different (electric and other) charges. In this way, string theory promises to be a unified description of all particles and interactions.[25] The theory is successful in that one mode will always correspond to a graviton, the messenger particle of gravity; however, the price to pay are unusual features such as six extra dimensions of space in addition to the usual three for space and one for time.[26]
In what is called the second superstring revolution, it was conjectured that both string theory and a unification of general relativity and supersymmetry known as supergravity[27] form part of a hypothesized eleven-dimensional model known as M-theory, which would constitute a uniquely defined and consistent theory of quantum gravity.[28][29] As presently understood, however, string theory admits a very large number (10500 by some estimates) of consistent vacua, comprising the so-called "string landscape". Sorting through this large family of solutions remains one of the major challenges.
Simple
spin network of the type used in loop quantum gravity
Another approach to quantum gravity starts with the canonical quantization procedures of quantum theory. Starting with the initial-value-formulation of general relativity (cf. the section on evolution equations, above), the result is an analogue of the Schrödinger equation: the Wheeler–DeWitt equation, which some argue is ill-defined.[30] 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.[31][32] 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.[33][34][35][36]
There are a number of other approaches to quantum gravity. The approaches differ depending on which features of general relativity and quantum theory are accepted unchanged, and which features are modified.[37][38] Examples include:
In quantum field theory, the Weinberg–Witten theorem places some constraints on theories of composite gravity/emergent gravity. However, recent developments attempt to show that if locality is only approximate and the holographic principle is correct, the Weinberg–Witten theorem would not be valid[citation needed].
As was emphasized above, quantum gravitational effects are extremely weak and therefore difficult to test. For this reason, the possibility of experimentally testing quantum gravity had not received much attention prior to the late 1990s. However, in the past decade, physicists have realized that evidence for quantum gravitational effects can guide the development of the theory. Since the theoretical development has been slow, the phenomenology of quantum gravity which studies the possibility of experimental tests, has obtained increased attention.[45][46]
There is presently no confirmed experimental signature of quantum gravitational effects. The most widely pursued possibilities for quantum gravity phenomenology include violations of Lorentz invariance, imprints of quantum gravitational effects in the Cosmic Microwave Background (in particular its polarization), and decoherence induced by fluctuations in the space-time foam.
- ^ Nesvizhevsky, Nesvizhevsky et al. (2002-01-17). "Quantum states of neutrons in the Earth's gravitational field". Nature 415 (6869): 297–299. Bibcode 2002Natur.415..297N. DOI:10.1038/415297a. http://www.nature.com/nature/journal/v415/n6869/abs/415297a.html. Retrieved 2011-04-21.
- ^ Jenke, Geltenbort, Lemmel & Abele, Tobias; Geltenbort, Peter; Lemmel, Hartmut; Abele, Hartmut (2011-04-17). "Realization of a gravity-resonance-spectroscopy technique". Nature 7 (6): 468–472. Bibcode 2011NatPh...7..468J. DOI:10.1038/nphys1970. http://www.nature.com/nphys/journal/vaop/ncurrent/full/nphys1970.html. Retrieved 2011-04-21.
- ^ Palmer, Jason (2011-04-18). "Neutrons could test Newton's gravity and string theory". BBC News. http://www.bbc.co.uk/news/science-environment-13097370. Retrieved 2011-04-21.
- ^ a b c Donoghue (1995). "Introduction to the Effective Field Theory Description of Gravity". arXiv:gr-qc/9512024 [gr-qc].
- ^ Kraichnan, R. H. (1955). "Special-Relativistic Derivation of Generally Covariant Gravitation Theory". Physical Review 98 (4): 1118–1122. Bibcode 1955PhRv...98.1118K. DOI:10.1103/PhysRev.98.1118.
- ^ Gupta, S. N. (1954). "Gravitation and Electromagnetism". Physical Review 96 (6): 1683–1685. Bibcode 1954PhRv...96.1683G. DOI:10.1103/PhysRev.96.1683.
- ^ Gupta, S. N. (1957). "Einstein's and Other Theories of Gravitation". Reviews of Modern Physics 29 (3): 334–336. Bibcode 1957RvMP...29..334G. DOI:10.1103/RevModPhys.29.334.
- ^ Gupta, S. N. (1962). "Quantum Theory of Gravitation". Recent Developments in General Relativity. Pergamon Press. pp. 251–258.
- ^ Deser, S. (1970). "Self-Interaction and Gauge Invariance". General Relativity and Gravitation 1: 9–18. arXiv:gr-qc/0411023. Bibcode 1970GReGr...1....9D. DOI:10.1007/BF00759198.
- ^ Ohta, Tadayuki; Mann, Robert (1996). "Canonical reduction of two-dimensional gravity for particle dynamics". Classical and Quantum Gravity 13 (9): 2585–2602. arXiv:gr-qc/9605004. Bibcode 1996CQGra..13.2585O. DOI:10.1088/0264-9381/13/9/022.
- ^ Sikkema, A E; Mann, R B (1991). "Gravitation and cosmology in (1+1) dimensions". Classical and Quantum Gravity 8: 219–235. Bibcode 1991CQGra...8..219S. DOI:10.1088/0264-9381/8/1/022.
- ^ Farrugia; Mann; Scott (2007). "N-body Gravity and the Schroedinger Equation". Classical and Quantum Gravity 24 (18): 4647–4659. arXiv:gr-qc/0611144. Bibcode 2007CQGra..24.4647F. DOI:10.1088/0264-9381/24/18/006.
- ^ Mann, R B; Ohta, T (1997). "Exact solution for the metric and the motion of two bodies in (1+1)-dimensional gravity". Phys. Rev. D. 55 (8): 4723–4747. arXiv:gr-qc/9611008. Bibcode 1997PhRvD..55.4723M. DOI:10.1103/PhysRevD.55.4723.
- ^ Feynman, R. P.; Morinigo, F. B., Wagner, W. G., & Hatfield, B. (1995). Feynman lectures on gravitation. Addison-Wesley. ISBN 0-201-62734-5.
- ^ Smolin, Lee (2001). Three Roads to Quantum Gravity. Basic Books. pp. 20–25. ISBN 0-465-07835-4. Pages 220-226 are annotated references and guide for further reading.
- ^ Hunter Monroe (2005). "Singularity-Free Collapse through Local Inflation". arXiv:astro-ph/0506506 [astro-ph].
- ^ Edward Anderson (2010). "The Problem of Time in Quantum Gravity". arXiv:1009.2157 [gr-qc].
- ^ A timeline and overview can be found in Rovelli, Carlo (2000). "Notes for a brief history of quantum gravity". arXiv:gr-qc/0006061 [gr-qc].
- ^ Ashtekar, Abhay (2007). "Loop Quantum Gravity: Four Recent Advances and a Dozen Frequently Asked Questions". 11th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity. p. 126. arXiv:0705.2222. Bibcode 2008mgm..conf..126A. DOI:10.1142/9789812834300_0008.
- ^ Schwarz, John H. (2007). "String Theory: Progress and Problems". Progress of Theoretical Physics Supplement 170: 214–226. arXiv:hep-th/0702219. Bibcode 2007PThPS.170..214S. DOI:10.1143/PTPS.170.214.
- ^ Donoghue, John F.(editor), (1995). "Introduction to the Effective Field Theory Description of Gravity". In Cornet, Fernando. Effective Theories: Proceedings of the Advanced School, Almunecar, Spain, 26 June–1 July 1995. Singapore: World Scientific. arXiv:gr-qc/9512024. ISBN 981-02-2908-9.
- ^ Weinberg, Steven (1996). "17-18". The Quantum Theory of Fields II: Modern Applications. Cambridge University Press. ISBN 0-521-55002-5.
- ^ Goroff, Marc H.; Sagnotti, Augusto (1985). "Quantum gravity at two loops". Physics Letters B 160: 81–86. Bibcode 1985PhLB..160...81G. DOI:10.1016/0370-2693(85)91470-4.
- ^ An accessible introduction at the undergraduate level can be found in Zwiebach, Barton (2004). A First Course in String Theory. Cambridge University Press. ISBN 0-521-83143-1. , and more complete overviews in Polchinski, Joseph (1998). String Theory Vol. I: An Introduction to the Bosonic String. Cambridge University Press. ISBN 0-521-63303-6. and Polchinski, Joseph (1998b). String Theory Vol. II: Superstring Theory and Beyond. Cambridge University Press. ISBN 0-521-63304-4.
- ^ Ibanez, L. E. (2000). "The second string (phenomenology) revolution". Classical & Quantum Gravity 17 (5): 1117–1128. arXiv:hep-ph/9911499. Bibcode 2000CQGra..17.1117I. DOI:10.1088/0264-9381/17/5/321.
- ^ For the graviton as part of the string spectrum, e.g. Green, Schwarz & Witten 1987, sec. 2.3 and 5.3; for the extra dimensions, ibid sec. 4.2.
- ^ Weinberg, Steven (2000). "31". The Quantum Theory of Fields II: Modern Applications. Cambridge University Press. ISBN 0-521-55002-5. http://books.google.com/books?id=aYDDRKqODpUC&printsec=frontcover.
- ^ Townsend, Paul K. (1996). Four Lectures on M-Theory. ICTP Series in Theoretical Physics. p. 385. arXiv:hep-th/9612121. Bibcode 1997hepcbconf..385T.
- ^ Duff, Michael (1996). "M-Theory (the Theory Formerly Known as Strings)". International Journal of Modern Physics A 11 (32): 5623–5642. arXiv:hep-th/9608117. Bibcode 1996IJMPA..11.5623D. DOI:10.1142/S0217751X96002583.
- ^ Kuchař, Karel (1973). "Canonical Quantization of Gravity". In Israel, Werner. Relativity, Astrophysics and Cosmology. D. Reidel. pp. 237–288 (section 3). ISBN 90-277-0369-8.
- ^ Ashtekar, Abhay (1986). "New variables for classical and quantum gravity". Physical Review 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.
- ^ Thiemann, Thomas (2006). "Loop Quantum Gravity: An Inside View". Approaches to Fundamental Physics 721: 185. arXiv:hep-th/0608210. Bibcode 2007LNP...721..185T.
- ^ Rovelli, Carlo (1998). "Loop Quantum Gravity". Living Reviews in Relativity 1. http://www.livingreviews.org/lrr-1998-1. Retrieved 2008-03-13.
- ^ Ashtekar, Abhay; Lewandowski, Jerzy (2004). "Background Independent Quantum Gravity: A Status Report". Classical & Quantum Gravity 21 (15): R53–R152. arXiv:gr-qc/0404018. Bibcode 2004CQGra..21R..53A. DOI:10.1088/0264-9381/21/15/R01.
- ^ Thiemann, Thomas (2003). "Lectures on Loop Quantum Gravity". Lecture Notes in Physics 631: 41–135. arXiv:gr-qc/0210094. Bibcode 2003LNP...631...41T.
- ^ Isham, Christopher J. (1994). "Prima facie questions in quantum gravity". In Ehlers, Jürgen; Friedrich, Helmut. Canonical Gravity: From Classical to Quantum. Springer. arXiv:gr-qc/9310031. ISBN 3-540-58339-4.
- ^ Sorkin, Rafael D. (1997). "Forks in the Road, on the Way to Quantum Gravity". International Journal of Theoretical Physics 36 (12): 2759–2781. arXiv:gr-qc/9706002. Bibcode 1997IJTP...36.2759S. DOI:10.1007/BF02435709.
- ^ Loll, Renate (1998). "Discrete Approaches to Quantum Gravity in Four Dimensions". Living Reviews in Relativity 1: 13. arXiv:gr-qc/9805049. Bibcode 1998LRR.....1...13L. http://www.livingreviews.org/lrr-1998-13. Retrieved 2008-03-09.
- ^ Sorkin, Rafael D. (2005). "Causal Sets: Discrete Gravity". In Gomberoff, Andres; Marolf, Donald. Lectures on Quantum Gravity. Springer. arXiv:gr-qc/0309009. ISBN 0-387-23995-2.
- ^ See Daniele Oriti and references therein.
- ^ Hawking, Stephen W. (1987). "Quantum cosmology". In Hawking, Stephen W.; Israel, Werner. 300 Years of Gravitation. Cambridge University Press. pp. 631–651. ISBN 0-521-37976-8.
- ^ Wen 2006
- ^ See ch. 33 in Penrose 2004 and references therein.
- ^ Hossenfelder, Sabine (2011). "Experimental Search for Quantum Gravity". In V. R. Frignanni. Classical and Quantum Gravity: Theory, Analysis and Applications. Chapter 5: Nova Publishers. ISBN 978-1-61122-957-8. https://www.novapublishers.com/catalog/product_info.php?products_id=15903.
- ^ "1010.3420] Experimental Search for Quantum Gravity". Arxiv.org. 2010-10-17. http://arxiv.org/abs/1010.3420. Retrieved 2012-04-08.
