Quantum gravity
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics, and where quantum effects cannot be ignored,^{[1]} such as in the vicinity of black holes or similar compact astrophysical objects where the effects of gravity are strong.
Three of the four fundamental forces of physics are described within the framework of quantum mechanics and quantum field theory. The current understanding of the fourth force, gravity, is based on Albert Einstein's general theory of relativity, which is formulated within the entirely different framework of classical physics. However, that description is incomplete: describing the gravitational field of a black hole in the general theory of relativity, physical quantities such as the spacetime curvature diverge at the center of the black hole.
This signals the breakdown of the general theory of relativity and the need for a theory that goes beyond general relativity into the quantum. At distances very close to the center of the black hole (closer than the Planck length), quantum fluctuations of spacetime are expected to play an important role.^{[2]} To describe these quantum effects a theory of quantum gravity is needed. Such a theory should allow the description to be extended closer to the center and might even allow an understanding of physics at the center of a black hole. On more formal grounds one can argue that a classical system cannot consistently be coupled to a quantum one.^{[3]}^{[4]}^{:11–12}
The field of quantum gravity is actively developing and theorists are exploring a variety of approaches to the problem of quantum gravity, the most popular approaches being string theory and loop quantum gravity.^{[5]} All these approaches aim to describe the quantum behavior of the gravitational field. This does not necessarily include unifying all fundamental interactions into a single mathematical framework. However, many approaches to quantum gravity, such as string theory, try to develop a framework that describes all fundamental forces. Such theories are often referred to as a theory of everything. Others, such as loop quantum gravity, make no such attempt; instead, they make an effort to quantize the gravitational field while it is kept separate from the other forces.
One of the difficulties of formulating a quantum gravity theory is that quantum gravitational effects only appear at length scales near the Planck scale, around 10^{−35} meters, a scale far smaller, and hence only accessible with far higher energies, than those currently available in high energy particle accelerators. Therefore, physicists lack experimental data which could distinguish between the competing theories which have been proposed^{[n.b. 1]}^{[n.b. 2]} and thus thought experiment approaches are suggested as a testing tool for these theories.^{[6]}^{[7]}^{[8]}
OverviewEdit
Unsolved problem in physics: How can the theory of quantum mechanics be merged with the theory of general relativity / gravitational force and remain correct at microscopic length scales? What verifiable predictions does any theory of quantum gravity make? (more unsolved problems in physics)

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. General relativity models gravity as curvature of spacetime: in the slogan of John Archibald Wheeler, "Spacetime tells matter how to move; matter tells spacetime how to curve."^{[9]} On the other hand, quantum field theory is typically formulated in the flat spacetime used in special relativity. No theory has yet proven successful in describing the general situation where the dynamics of matter, modeled with quantum mechanics, affect the curvature of spacetime. If one attempts to treat gravity as simply another quantum field, the resulting theory is not renormalizable.^{[10]} Even in the simpler case where the curvature of spacetime is fixed a priori, developing quantum field theory becomes more mathematically challenging, and many ideas physicists use in quantum field theory on flat spacetime are no longer applicable.^{[11]}
It is widely hoped that a theory of quantum gravity would allow us to understand problems of very high energy and very small dimensions of space, such as the behavior of black holes, and the origin of the universe.^{[1]}
Quantum mechanics and general relativityEdit
GravitonEdit
The observation that all fundamental forces except gravity have one or more known messenger particles leads researchers to believe that at least one must exist for gravity. This hypothetical particle is known as the graviton. These particles act as a force particle similar to the photon of the electromagnetic interaction. Under mild assumptions the structure of general relativity requires them to follow the quantum mechanical description of interacting theoretical spin2 massless particles.^{[12]}^{[13]}^{[14]}^{[15]}^{[16]} Many of the accepted notions of a unified theory of physics since the 1970s assume, and to some degree depend upon, the existence of the graviton. The Weinberg–Witten theorem places some constraints on theories in which the graviton is a composite particle.^{[17]}^{[18]} While gravitons are an important theoretical step in a quantum mechanical description of gravity, they are generally believed to be indetectable because they interact too weakly.^{[19]}
Nonrenormalizability of gravityEdit
General relativity, like electromagnetism, is a classical field theory. One might expect that, as with electromagnetism, the gravitational force should also have a corresponding quantum field theory.
However, gravity is perturbatively nonrenormalizable.^{[4]}^{:xxxvi–xxxviii;211–212}^{[20]} 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, in perturbation theory, 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 it is impossible to conduct infinite experiments to fix the values of every parameter, it has been argued that one does not, in perturbation theory, 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.^{[21]}
It is conceivable that, in the correct theory of quantum gravity, the infinitely many unknown parameters will reduce 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 nonperturbative quantum field theory, finding a reliable answer is difficult, pursued in the asymptotic safety program. Another possibility is that there are new, undiscovered 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.^{[22]}^{[better source needed]}
Quantum gravity as an effective field theoryEdit
In an effective field theory, all but the first few of the infinite set of parameters in a nonrenormalizable theory are suppressed by huge energy scales and hence can be neglected when computing lowenergy effects. Thus, at least in the lowenergy regime, the model is a predictive quantum field theory.^{[23]} Furthermore, many theorists argue that the Standard Model should be regarded as an effective field theory itself, with "nonrenormalizable" interactions suppressed by large energy scales and whose effects have consequently not been observed experimentally.^{[24]}
By treating general relativity as an effective field theory, one can actually make legitimate predictions for quantum gravity, at least for lowenergy phenomena. An example is the wellknown calculation of the tiny firstorder quantummechanical correction to the classical Newtonian gravitational potential between two masses.^{[23]}
Spacetime background dependenceEdit
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,^{[25]} in which the only physically relevant information is the relationship between different events in spacetime.
On the other hand, quantum mechanics has depended since its inception on a fixed background (nondynamic) 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 theoryEdit
String theory can be seen as a generalization of quantum field theory where instead of point particles, stringlike objects propagate in a fixed spacetime background, although the interactions among closed strings give rise to spacetime 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.
Background independent theoriesEdit
Loop quantum gravity is the fruit of an effort to formulate a backgroundindependent quantum theory.
Topological quantum field theory provided an example of backgroundindependent 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.
Semiclassical quantum gravityEdit
Quantum field theory on curved (nonMinkowskian) 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 the least energy (and may or may not contain particles). See Quantum field theory in curved spacetime for a more complete discussion.
Problem of timeEdit
A conceptual difficulty in combining quantum mechanics with general relativity arises from the contrasting role of time within these two frameworks. In quantum theories time acts as an independent background through which states evolve, with the Hamiltonian operator acting as the generator of infinitesimal translations of quantum states through time.^{[26]} In contrast, general relativity treats time as a dynamical variable which interacts directly with matter and moreover requires the Hamiltonian constraint to vanish,^{[27]} removing any possibility of employing a notion of time similar to that in quantum theory.
Candidate theoriesEdit
There are a number of proposed quantum gravity theories.^{[28]} 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.^{[29]}^{[30]}
String theoryEdit
The central idea of string theory is to replace the classical concept of a point particle in quantum field theory, with a quantum theory of onedimensional extended objects: string theory.^{[31]} At the energies reached in current experiments, these strings are indistinguishable from pointlike 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.^{[32]} The theory is successful in that one mode will always correspond to a graviton, the messenger particle of gravity; however, the price of this success are unusual features such as six extra dimensions of space in addition to the usual three for space and one for time.^{[33]}
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^{[34]} form part of a hypothesized elevendimensional model known as Mtheory, which would constitute a uniquely defined and consistent theory of quantum gravity.^{[35]}^{[36]} As presently understood, however, string theory admits a very large number (10^{500} by some estimates) of consistent vacua, comprising the socalled "string landscape". Sorting through this large family of solutions remains a major challenge.
Loop quantum gravityEdit
Loop quantum gravity seriously considers general relativity's insight that spacetime is a dynamical field and is therefore a quantum object. Its second idea is that the quantum discreteness that determines the particlelike behavior of other field theories (for instance, the photons of the electromagnetic field) also affects the structure of space.
The main result of loop quantum gravity is the derivation of a granular structure of space at the Planck length. This is derived from following considerations: In the case of electromagnetism, the quantum operator representing the energy of each frequency of the field has a discrete spectrum. Thus the energy of each frequency is quantized, and the quanta are the photons. In the case of gravity, the operators representing the area and the volume of each surface or space region likewise have discrete spectrum. Thus area and volume of any portion of space are also quantized, where the quanta are elementary quanta of space. It follows, then, that spacetime has an elementary quantum granular structure at the Planck scale, which cuts off the ultraviolet infinities of quantum field theory.
The quantum state of spacetime is described in the theory by means of a mathematical structure called spin networks. Spin networks were initially introduced by Roger Penrose in abstract form, and later shown by Carlo Rovelli and Lee Smolin to derive naturally from a nonperturbative quantization of general relativity. Spin networks do not represent quantum states of a field in spacetime: they represent directly quantum states of spacetime.
The theory is based on the reformulation of general relativity known as Ashtekar variables, which represent geometric gravity using mathematical analogues of electric and magnetic fields.^{[37]}^{[38]} In the quantum theory, space is represented by a network structure called a spin network, evolving over time in discrete steps.^{[39]}^{[40]}^{[41]}^{[42]}
The dynamics of the theory is today constructed in several versions. One version starts with the canonical quantization of general relativity. The analogue of the Schrödinger equation is a Wheeler–DeWitt equation, which can be defined within the theory.^{[43]} In the covariant, or spinfoam formulation of the theory, the quantum dynamics is obtained via a sum over discrete versions of spacetime, called spinfoams. These represent histories of spin networks.
Other approachesEdit
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.^{[44]}^{[45]} Examples include:
 Asymptotic safety in quantum gravity
 Euclidean quantum gravity
 Causal dynamical triangulation^{[46]}
 Causal fermion systems
 Causal Set Theory
 Covariant Feynman path integral approach
 Dilatonic quantum gravity
 Group field theory
 Wheeler–DeWitt equation
 Geometrodynamics
 Hořava–Lifshitz gravity
 Integral method^{[47]}
 MacDowell–Mansouri action
 Noncommutative geometry
 Pathintegral based models of quantum cosmology^{[48]}
 Regge calculus
 Scale relativity
 Shape Dynamics
 Spontaneous Quantum Gravity^{[49]}^{[50]}^{[51]}
 Stringnets and quantum graphity
 Superfluid vacuum theory a.k.a. theory of BEC vacuum
 Supergravity
 Twistor theory^{[52]}
 Canonical quantum gravity
 Quantum holonomy theory^{[53]}
Experimental testsEdit
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 theoretical development has been slow, the field of phenomenological quantum gravity, which studies the possibility of experimental tests, has obtained increased attention.^{[54]}
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 spacetime foam.
ESA's INTEGRAL satellite measured polarization of photons of different wavelengths and was able to place a limit in the granularity of space ^{[55]} that is less than 10⁻⁴⁸m or 13 orders of magnitude below the Planck scale .
The BICEP2 experiment detected what was initially thought to be primordial Bmode polarization caused by gravitational waves in the early universe. Had the signal in fact been primordial in origin, it could have been an indication of quantum gravitational effects, but it soon transpired that the polarization was due to interstellar dust interference.^{[56]}
Thought experimentsEdit
As explained above, quantum gravitational effects are extremely weak and therefore difficult to test. For this reason, thought experiments are becoming an important theoretical tool. An important aspect of quantum gravity relates to the question of coupling of spin and spacetime. While spin and spacetime are expected to be coupled,^{[57]} the precise nature of this coupling is currently unknown. In particular and most importantly, it is not known how quantum spin sources gravity and what is the correct characterization of the spacetime of a single spinhalf particle. To analyze this question, thought experiments in the context of quantum information, have been suggested.^{[8]} This work shows that, in order to avoid violation of relativistic causality, the measurable spacetime around a spinhalf particle's (rest frame) must be spherically symmetric  i.e., either spacetime is spherically symmetric, or somehow measurements of the spacetime (e.g., timedilation measurements) should create some sort of back action that affects and changes the quantum spin.
See alsoEdit
 Abraham–Lorentz force
 Beyond black holes
 Black hole electron
 Centauro event
 De Sitter relativity
 Dilaton
 Doubly special relativity
 Event symmetry
 Fock–Lorentz symmetry
 Gravitomagnetism
 Hawking radiation
 List of quantum gravity researchers
 Macrocosm and microcosm
 Orders of magnitude (length)
 Penrose interpretation
 Planck epoch
 Planck units
 Quantum realm
 Swampland (physics)
 Weak Gravity Conjecture
NotesEdit
 ^ Quantum effects in the early universe might have an observable effect on the structure of the present universe, for example, or gravity might play a role in the unification of the other forces. Cf. the text by Wald cited above.
 ^ On the quantization of the geometry of spacetime, see also in the article Planck length, in the examples
ReferencesEdit
 ^ ^{a} ^{b} Rovelli, Carlo (2008). "Quantum gravity". Scholarpedia. 3 (5): 7117. Bibcode:2008SchpJ...3.7117R. doi:10.4249/scholarpedia.7117.
 ^ Nadis, Steve (2 December 2019). "Black Hole Singularities Are as Inescapable as Expected". quantamagazine.org. Quanta Magazine. Retrieved 22 April 2020.
 ^ Wald, Robert M. (1984). General Relativity. University of Chicago Press. p. 382. OCLC 471881415.
 ^ ^{a} ^{b} Feynman, Richard P.; Morinigo, Fernando B.; Wagner, William G. (1995). Feynman Lectures on Gravitation. Reading, Mass.: AddisonWesley. ISBN 9780201627343. OCLC 32509962.
 ^ Penrose, Roger (2007). The road to reality : a complete guide to the laws of the universe. Vintage. p. 1017. OCLC 716437154.
 ^ Bose, S.; et al. (2017). "Spin Entanglement Witness for Quantum Gravity". Physical Review Letters. 119 (4): 240401. arXiv:1707.06050. doi:10.1103/PhysRevLett.119.240401. PMID 29286711.
 ^ Marletto, C.; Vedral, V. (2017). "Gravitationally Induced Entanglement between Two Massive Particles is Sufficient Evidence of Quantum Effects in Gravity". Physical Review Letters. 119 (24): 240402. arXiv:1707.06036. doi:10.1103/PhysRevLett.119.240402. PMID 29286752.
 ^ ^{a} ^{b} Nemirovsky, J.; Cohen, E.; Kaminer, I. (30 Dec 2018). "Spin Spacetime Censorship". arXiv:1812.11450v2 [grqc].CS1 maint: ref=harv (link)
 ^ Wheeler, John Archibald (2010). Geons, Black Holes, and Quantum Foam: A Life in Physics. W. W. Norton & Company. p. 235. ISBN 9780393079487.
 ^ Zee, Anthony (2010). Quantum Field Theory in a Nutshell (second ed.). Princeton University Press. pp. 172, 434–435. ISBN 9780691140346. OCLC 659549695.
 ^ Wald, Robert M. (1994). Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. University of Chicago Press. ISBN 9780226870274.
 ^ Kraichnan, R. H. (1955). "SpecialRelativistic Derivation of Generally Covariant Gravitation Theory". Physical Review. 98 (4): 1118–1122. Bibcode:1955PhRv...98.1118K. doi:10.1103/PhysRev.98.1118.CS1 maint: ref=harv (link)
 ^ Gupta, S. N. (1954). "Gravitation and Electromagnetism". Physical Review. 96 (6): 1683–1685. Bibcode:1954PhRv...96.1683G. doi:10.1103/PhysRev.96.1683.CS1 maint: ref=harv (link)
 ^ 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.CS1 maint: ref=harv (link)
 ^ Gupta, S. N. (1962). "Quantum Theory of Gravitation". Recent Developments in General Relativity. Pergamon Press. pp. 251–258.
 ^ Deser, S. (1970). "SelfInteraction and Gauge Invariance". General Relativity and Gravitation. 1 (1): 9–18. arXiv:grqc/0411023. Bibcode:1970GReGr...1....9D. doi:10.1007/BF00759198.CS1 maint: ref=harv (link)
 ^ Weinberg, Steven; Witten, Edward (1980). "Limits on massless particles". Physics Letters B. 96 (1–2): 59–62. Bibcode:1980PhLB...96...59W. doi:10.1016/03702693(80)902129.
 ^ Horowitz, Gary T.; Polchinski, Joseph (2006). "Gauge/gravity duality". In Oriti, Daniele (ed.). Approaches to Quantum Gravity. Cambridge University Press. arXiv:grqc/0602037. Bibcode:2006gr.qc.....2037H. ISBN 9780511575549. OCLC 873715753.
 ^ Rothman, Tony; Boughn, Stephen (2006). "Can Gravitons be Detected?". Foundations of Physics. 36 (12): 1801–1825. arXiv:grqc/0601043. doi:10.1007/s1070100690819.
 ^ Hamber, H. W. (2009). Quantum Gravitation – The Feynman Path Integral Approach. Springer Nature. ISBN 9783540852926.
 ^ Goroff, Marc H.; Sagnotti, Augusto; Sagnotti, Augusto (1985). "Quantum gravity at two loops". Physics Letters B. 160 (1–3): 81–86. Bibcode:1985PhLB..160...81G. doi:10.1016/03702693(85)914704.CS1 maint: ref=harv (link)
 ^ Distler, Jacques (20050901). "Motivation". golem.ph.utexas.edu. Retrieved 20180224.
 ^ ^{a} ^{b} Donoghue, John F. (editor) (1995). "Introduction to the Effective Field Theory Description of Gravity". In Cornet, Fernando (ed.). Effective Theories: Proceedings of the Advanced School, Almunecar, Spain, 26 June–1 July 1995. Singapore: World Scientific. arXiv:grqc/9512024. Bibcode:1995gr.qc....12024D. ISBN 9789810229085.CS1 maint: extra text: authors list (link)
 ^ ZinnJustin, Jean (2007). Phase transitions and renormalization group. Oxford: Oxford University Press. ISBN 9780199665167. OCLC 255563633.
 ^ Smolin, Lee (2001). Three Roads to Quantum Gravity. Basic Books. pp. 20–25. ISBN 9780465078356. Pages 220–226 are annotated references and guide for further reading.
 ^ Sakurai, J. J.; Napolitano, Jim J. (20100714). Modern Quantum Mechanics (2 ed.). Pearson. p. 68. ISBN 9780805382914.
 ^ Novello, Mario; Bergliaffa, Santiago E. (20030611). Cosmology and Gravitation: Xth Brazilian School of Cosmology and Gravitation; 25th Anniversary (1977–2002), Mangaratiba, Rio de Janeiro, Brazil. Springer Science & Business Media. p. 95. ISBN 9780735401310.
 ^ A timeline and overview can be found in Rovelli, Carlo (2000). "Notes for a brief history of quantum gravity". arXiv:grqc/0006061.CS1 maint: ref=harv (link) (verify against ISBN 9789812777386)
 ^ 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. The Eleventh 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. ISBN 9789812834263.CS1 maint: ref=harv (link)
 ^ Schwarz, John H. (2007). "String Theory: Progress and Problems". Progress of Theoretical Physics Supplement. 170: 214–226. arXiv:hepth/0702219. Bibcode:2007PThPS.170..214S. doi:10.1143/PTPS.170.214.CS1 maint: ref=harv (link)
 ^ An accessible introduction at the undergraduate level can be found in Zwiebach, Barton (2004). A First Course in String Theory. Cambridge University Press. ISBN 9780521831437.CS1 maint: ref=harv (link), and more complete overviews in Polchinski, Joseph (1998). String Theory Vol. I: An Introduction to the Bosonic String. Cambridge University Press. ISBN 9780521633031. and Polchinski, Joseph (1998b). String Theory Vol. II: Superstring Theory and Beyond. Cambridge University Press. ISBN 9780521633048.
 ^ Ibanez, L. E. (2000). "The second string (phenomenology) revolution". Classical and Quantum Gravity. 17 (5): 1117–1128. arXiv:hepph/9911499. Bibcode:2000CQGra..17.1117I. doi:10.1088/02649381/17/5/321.CS1 maint: ref=harv (link)
 ^ 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). "Chapter 31". The Quantum Theory of Fields II: Modern Applications. Cambridge University Press. ISBN 9780521550024.
 ^ Townsend, Paul K. (1996). "Four Lectures on MTheory". High Energy Physics and Cosmology. ICTP Series in Theoretical Physics. 13: 385. arXiv:hepth/9612121. Bibcode:1997hepcbconf..385T.CS1 maint: ref=harv (link)
 ^ Duff, Michael (1996). "MTheory (the Theory Formerly Known as Strings)". International Journal of Modern Physics A. 11 (32): 5623–5642. arXiv:hepth/9608117. Bibcode:1996IJMPA..11.5623D. doi:10.1142/S0217751X96002583.CS1 maint: ref=harv (link)
 ^ 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.CS1 maint: ref=harv (link)
 ^ 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. PMID 9958340.CS1 maint: ref=harv (link)
 ^ Thiemann, Thomas (2007). Loop Quantum Gravity: An Inside View. Approaches to Fundamental Physics. Lecture Notes in Physics. 721. pp. 185–263. arXiv:hepth/0608210. Bibcode:2007LNP...721..185T. doi:10.1007/9783540711179_10. ISBN 9783540711155.CS1 maint: ref=harv (link)
 ^ Rovelli, Carlo (1998). "Loop Quantum Gravity". Living Reviews in Relativity. 1. Retrieved 20080313.CS1 maint: ref=harv (link)
 ^ Ashtekar, Abhay; Lewandowski, Jerzy (2004). "Background Independent Quantum Gravity: A Status Report". Classical and Quantum Gravity. 21 (15): R53–R152. arXiv:grqc/0404018. Bibcode:2004CQGra..21R..53A. doi:10.1088/02649381/21/15/R01.
 ^ Thiemann, Thomas (2003). "Lectures on Loop Quantum Gravity". Quantum Gravity. Lecture Notes in Physics. 631. pp. 41–135. arXiv:grqc/0210094. Bibcode:2003LNP...631...41T. doi:10.1007/9783540452300_3. ISBN 9783540408109.CS1 maint: ref=harv (link)
 ^ Rovelli, Carlo (2004). Quantum Gravity. Cambridge University Press. ISBN 9780521715966.
 ^ Isham, Christopher J. (1994). "Prima facie questions in quantum gravity". In Ehlers, Jürgen; Friedrich, Helmut (eds.). Canonical Gravity: From Classical to Quantum. Canonical Gravity: From Classical to Quantum. Lecture Notes in Physics. 434. Springer. pp. 1–21. arXiv:grqc/9310031. Bibcode:1994LNP...434....1I. doi:10.1007/3540583394_13. ISBN 9783540583394.
 ^ Sorkin, Rafael D. (1997). "Forks in the Road, on the Way to Quantum Gravity". International Journal of Theoretical Physics. 36 (12): 2759–2781. arXiv:grqc/9706002. Bibcode:1997IJTP...36.2759S. doi:10.1007/BF02435709.CS1 maint: ref=harv (link)
 ^ Loll, Renate (1998). "Discrete Approaches to Quantum Gravity in Four Dimensions". Living Reviews in Relativity. 1 (1): 13. arXiv:grqc/9805049. Bibcode:1998LRR.....1...13L. doi:10.12942/lrr199813. PMC 5253799. PMID 28191826.CS1 maint: ref=harv (link)
 ^ Klimets AP, Philosophy Documentation Center, Western UniversityCanada, 2017, pp.2530
 ^ Hawking, Stephen W. (1987). "Quantum cosmology". In Hawking, Stephen W.; Israel, Werner (eds.). 300 Years of Gravitation. Cambridge University Press. pp. 631–651. ISBN 9780521379762.
 ^ Spontaneous Quantum Gravity, retrieved 20200119
 ^ Maithresh, Palemkota; Singh, Tejinder P. (20191005). "Proposal for a new quantum theory of gravity III: Equations for quantum gravity, and the origin of spontaneous localisation". Zeitschrift für Naturforschung A. 0 (2): 143–154. arXiv:1908.04309. Bibcode:2019arXiv190804309M. doi:10.1515/zna20190267. ISSN 18657109.
 ^ Singh, Tejinder P. (20191205). "Spontaneous quantum gravity". arXiv:1912.03266 [physics.popph].
 ^ See ch. 33 in Penrose 2004 and references therein.
 ^ Aastrup, J.; Grimstrup, J. M. (27 Apr 2015). "Quantum Holonomy Theory". Fortschritte der Physik. 64 (10): 783. arXiv:1504.07100. Bibcode:2016ForPh..64..783A. doi:10.1002/prop.201600073.
 ^ Hossenfelder, Sabine (2011). "Experimental Search for Quantum Gravity". In V. R. Frignanni (ed.). Classical and Quantum Gravity: Theory, Analysis and Applications. Chapter 5: Nova Publishers. ISBN 9781611229578.CS1 maint: location (link)
 ^ https://www.esa.int/Science_Exploration/Space_Science/Integral_challenges_physics_beyond_Einstein
 ^ Cowen, Ron (30 January 2015). "Gravitational waves discovery now officially dead". Nature. doi:10.1038/nature.2015.16830.
 ^ Yuri.N., Obukhov, "Spin, gravity, and inertia", Physical review letters 86.2 (2001): 192.arXiv:0012102v1
Further readingEdit
 Ahluwalia, D. V. (2002). "Interface of Gravitational and Quantum Realms". Modern Physics Letters A. 17 (15–17): 1135–1145. arXiv:grqc/0205121. Bibcode:2002MPLA...17.1135A. doi:10.1142/S021773230200765X.CS1 maint: ref=harv (link)
 Ashtekar, Abhay (2005). The winding road to quantum gravity (PDF). Current Science. 89. pp. 2064–2074. Bibcode:2007laec.book...69A. CiteSeerX 10.1.1.616.8952. doi:10.1142/9789812772718_0005. ISBN 9789812700490.CS1 maint: ref=harv (link)
 Carlip, Steven (2001). "Quantum Gravity: a Progress Report". Reports on Progress in Physics. 64 (8): 885–942. arXiv:grqc/0108040. Bibcode:2001RPPh...64..885C. doi:10.1088/00344885/64/8/301.CS1 maint: ref=harv (link)
 Herbert W. Hamber (2009). Quantum Gravitation. Springer Nature. doi:10.1007/9783540852933. ISBN 9783540852926.
 Kiefer, Claus (2007). Quantum Gravity. Oxford University Press. ISBN 9780199212521.CS1 maint: ref=harv (link)
 Kiefer, Claus (2005). "Quantum Gravity: General Introduction and Recent Developments". Annalen der Physik. 15 (1): 129–148. arXiv:grqc/0508120. Bibcode:2006AnP...518..129K. doi:10.1002/andp.200510175.CS1 maint: ref=harv (link)
 Lämmerzahl, Claus, ed. (2003). Quantum Gravity: From Theory to Experimental Search. Lecture Notes in Physics. Springer. ISBN 9783540408109.CS1 maint: ref=harv (link)
 Rovelli, Carlo (2004). Quantum Gravity. Cambridge University Press. ISBN 9780521837330.CS1 maint: ref=harv (link)
 Trifonov, Vladimir (2008). "GRfriendly description of quantum systems". International Journal of Theoretical Physics. 47 (2): 492–510. arXiv:mathph/0702095. Bibcode:2008IJTP...47..492T. doi:10.1007/s1077300794743.CS1 maint: ref=harv (link)
External linksEdit
Wikiquote has quotations related to: Quantum gravity 
 "Planck Era" and "Planck Time" (up to 10^{−43} seconds after birth of Universe) (University of Oregon).
 "Quantum Gravity", BBC Radio 4 discussion with John Gribbin, Lee Smolin and Janna Levin (In Our Time, Feb. 22, 2001)