Quantum gravity: Difference between revisions

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{{short description|FieldDescription of theoreticalgravity physicsusing discrete values}}

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'''Quantum gravity''' ('''QG''') is a field of [[theoretical physics]] that seeks to describe [[gravity]] according to the principles of [[quantum mechanics]],. It deals with environments in which neither andgravitational wherenor quantum effects cannotcan be ignored,<ref name="scholarpedia">{{cite journal | last1last = Rovelli | first1first = Carlo | authorlinkauthor-link = Carlo Rovelli | year = 2008| title = Quantum gravity | url = http://www.scholarpedia.org/article/Quantum_gravity | journal = [[Scholarpedia]] | volume = 3 | issue = 5| page = 7117 | doi = 10.4249/scholarpedia.7117 | bibcode = 2008SchpJ...3.7117R | doi-access = free }}</ref> such as nearin the vicinity of [[black holes]] or similar compact astrophysical objects, whereas well as in the effectsearly stages of gravitythe areuniverse strongmoments after the [[Big Bang]].<ref>{{Cite book |last=Kiefer |first=Claus |title=Quantum gravity |date=2012 |publisher=Oxford University Press |isbn=978-0-19-958520-5 |edition=3rd |series=International series of monographs on physics |___location=Oxford |pages=1–4 |language=en}}</ref>

Three of the four [[fundamental force]]s of nature are described within the framework of quantum mechanics and [[quantum field theory]]: the [[Electromagnetism|electromagnetic interaction]], the [[Strong interaction|strong force]], and the [[Weak interaction|weak force]]; this leaves gravity as the only interaction that has not been fully accommodated. The current understanding of gravity is based on [[Albert Einstein]]'s [[general theory of relativity]], which incorporates his theory of special relativity and deeply modifies the understanding of concepts like time and space. Although general relativity is highly regarded for its elegance and accuracy, it has limitations: the [[Gravitational singularity|gravitational singularities]] inside black holes, the ad hoc postulation of [[dark matter]], as well as [[dark energy]] and its relation to the [[cosmological constant]] are among the current unsolved mysteries regarding gravity,<ref>{{Cite journal |last=Mannheim |first=Philip |date=2006 |title=Alternatives to dark matter and dark energy |journal=Progress in Particle and Nuclear Physics |language=en |volume=56 |issue=2 |pages=340–445 |doi=10.1016/j.ppnp.2005.08.001|arxiv=astro-ph/0505266 |bibcode=2006PrPNP..56..340M |s2cid=14024934 }}</ref> all of which signal the collapse of the general theory of relativity at different scales and highlight the need for a gravitational theory that goes into the quantum realm. At distances close to the [[Planck length]], like those near the center of a black hole, [[quantum fluctuations]] of spacetime are expected to play an important role.<ref>{{cite web

The current understanding of [[gravity]] is based on [[Albert Einstein]]'s [[general theory of relativity]], which is formulated within the framework of [[classical physics]]. On the other hand, the other three [[fundamental force]]s of physics are described within the framework of [[quantum mechanics]] and [[quantum field theory]], radically different formalisms for describing physical phenomena.<ref>{{cite book |last=Griffiths |first=David J. |title=Introduction to Quantum Mechanics |date=2004 |publisher=Pearson Prentice Hall |oclc=803860989}}</ref> It is sometimes argued that a quantum mechanical description of gravity is necessary on the grounds that one cannot consistently couple a classical system to a quantum one.<ref>{{cite book |last=Wald |first=Robert M. |title=General Relativity |url=https://archive.org/details/generalrelativit0000wald |url-access=registration |date=1984 |publisher=University of Chicago Press |oclc=471881415 |page=[https://archive.org/details/generalrelativit0000wald/page/382 382]}}</ref><ref name="FLoG">{{Cite book|title=Feynman Lectures on Gravitation|last=Feynman|first=Richard P.|last2=Morinigo|first2=Fernando B.|last3=Wagner|first3=William G.|publisher=Addison-Wesley|year=1995|isbn=978-0201627343|___location=Reading, Mass.|oclc=32509962|author-link=Richard Feynman|url-access=registration|url=https://archive.org/details/feynmanlectureso0000feyn_g4q1}}</ref>{{rp|11&ndash;12}}

| url = https://www.quantamagazine.org/black-hole-singularities-are-as-inescapable-as-expected-20191202/

| title = Black Hole Singularities Are as Inescapable as Expected

| last = Nadis

| first = Steve

| date = 2 December 2019

| website = quantamagazine.org

| publisher = [[Quanta Magazine]]

| access-date = 22 April 2020

| archive-date = 14 April 2020

| archive-url = https://web.archive.org/web/20200414150244/https://www.quantamagazine.org/black-hole-singularities-are-as-inescapable-as-expected-20191202/

| url-status = live

}}</ref> Finally, the discrepancies between the predicted value for the [[vacuum energy]] and the observed values (which, depending on considerations, can be of 60 or 120 orders of magnitude)<ref>{{cite journal|last=Bousso |first=Raphael |title=The cosmological constant |journal=General Relativity and Gravitation |volume=40 |year=2008 |issue=2–3 |pages=607–637 |arxiv=0708.4231 |doi=10.1007/s10714-007-0557-5|bibcode=2008GReGr..40..607B }}</ref><ref>{{cite journal|doi=10.1088/2058-7058/34/03/32 |first=Rob |last=Lea |title=A new generation takes on the cosmological constant |journal=Physics World |volume=34 |number=3 |page=42 |year=2021|bibcode=2021PhyW...34c..42L }}</ref> highlight the necessity for a quantum theory of gravity.

WhileThe a quantum theory of [[gravity]] may be needed to reconcile general relativity with the principlesfield of quantum mechanics, difficulties arise when applying the usual prescriptions of [[quantum field theory]] to the force of gravity via hypothesised [[graviton]] bosons.<ref name=":1" /> The problem is that the theory one gets in thisactively way is not [[renormalizable]] — it predicts infinite values for some observable properties, such as the mass of particlesdeveloping, and thereforetheorists cannotare beexploring used to make meaningful physicala predictionsvariety of those properties. As a result, theorists have taken up more radical approaches to the problem of quantum gravity, the most popular approaches being [[string M-theory]] and [[loop quantum gravity]].<ref>{{cite book |last=Penrose |first=Roger |title=The road to reality : a complete guide to the laws of the universe |url=https://archive.org/details/roadtoreality00penr_319 |url-access=limited |date=2007 |publisher=Vintage |oclc=716437154 |page=[https://archive.org/details/roadtoreality00penr_319/page/n1045 1017]|isbn=9780679776314 }}</ref> AlthoughAll someof these approaches aim to describe the quantum gravitybehavior theoriesof the [[gravitational field]], which does not necessarily include [[Theory of everything|unifying all fundamental interactions]] into a single mathematical framework. However, many approaches to quantum gravity, such as [[string theory]], try to unifydevelop gravitya withframework thethat otherdescribes all fundamental forces. Such a theory is often referred to as a [[fundamentaltheory forceof everything]]s,. othersSome of the approaches, 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. Other lesser-known but no less important theories include [[causal dynamical triangulation]], [[Noncommutative geometry#Applications in mathematical physics|noncommutative geometry]], and [[twistor theory]].<ref>{{Cite arXiv |last=Rovelli |first=Carlo |date=2001 |title=Notes for a brief history of quantum gravity |eprint=gr-qc/0006061 }}</ref>

One of the difficulties of formulating a quantum gravity theory is that direct observation of quantum gravitational effects is thought to 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 accelerator]]s. Therefore, physicists lack experimental data which could distinguish between the competing theories which have been proposed.<ref group="n.b.">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.</ref><ref group="n.b.">On the quantization of the geometry of spacetime, see also in the article [[Planck length]], in the examples</ref>

Strictly speaking, the aim of quantum gravity is only to describe the quantum behavior of the gravitational field and should not be confused with the objective of [[Theory of everything|unifying all fundamental interactions]] into a single mathematical framework. A quantum field theory of gravity that is unified with a [[Grand Unified Theory|grand unified theory]] is sometimes referred to as a [[theory of everything]] (TOE). While any substantial improvement into the present understanding of gravity would aid further work towards unification, the study of quantum gravity is a field in its own right with various branches having different approaches to unification.

[[Thought experiment]] approaches have been suggested as a testing tool for quantum gravity theories.<ref>

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 meter, a scale far smaller, and equivalently far larger in energy, than those currently accessible by high energy [[particle accelerator]]s. Therefore, physicists lack experimental data which could distinguish between the competing theories which have been proposed<ref group="n.b.">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.</ref><ref group="n.b.">On the quantization of the geometry of spacetime, see also in the article [[Planck length]], in the examples</ref> and thus thought experiment approaches are suggested as a testing tool for these theories.<ref>{{cite journal

{{cite journal

|last = Bose

|firstlast1 = S.Lindner

|first1 = Nethanel H.

|display-authors=etal

|last2 = Peres

|title = Spin Entanglement Witness for Quantum Gravity

|datefirst2 = 2017Asher

|title = Testing quantum superpositions of the gravitational field with Bose-Einstein condensates

|journal = [[Physical Review Letters]]

|volumedate = 1192005

|journal = [[Physical Review A]]

|issue = 4

|pagesvolume = 24040171

|issue = 2

|doi = 10.1103/PhysRevLett.119.240401

|pmidpages = 29286711024101

|doi = 10.1103/PhysRevA.71.024101

|arxiv = 1707.06050

|arxiv = gr-qc/0410030

}}</ref><ref>{{cite journal

|bibcode = 2005PhRvA..71b4101L

|last = Marletto

}}</ref><ref>

|first = C.

{{cite arXiv

|last2 = Vedral

|first2last1 = V.Kafri

|first1 = Dvir

|title = Gravitationally Induced Entanglement between Two Massive Particles is Sufficient Evidence of Quantum Effects in Gravity

|datelast2 = 2017Taylor

|first2 = Jacob M

|journal = [[Physical Review Letters]]

|title = A noise inequality for classical forces

|volume = 119

|issuedate = 242013

|pagesclass = 240402 quant-ph

|eprint = 1311.4558

|doi = 10.1103/PhysRevLett.119.240402

}}</ref> In the field of quantum gravity there are several open questions – e.g., it is not known how spin of elementary particles sources gravity, and thought experiments could provide a pathway to explore possible resolutions to these questions,<ref name="Spin-Spacetime Censorship AdP">{{cite journal|journal=Annalen der Physik|first1=J.|last1=Nemirovsky|first2=E.|last2=Cohen|title=Spin Spacetime Censorship|volume = 534|issue = 1|doi = 10.1002/andp.202100348|date=5 November 2021|last3=Kaminer|first3=I.|arxiv=1812.11450|s2cid=119342861}}</ref> even in the absence of lab experiments or physical observations.

|pmid = 29286752

|arxiv = 1707.06036

In the early 21st century, new experiment designs and technologies have arisen which suggest that indirect approaches to testing quantum gravity may be feasible over the next few decades.<ref name="nautilus">{{cite web |last1=Hossenfelder |first1=Sabine |title=What Quantum Gravity Needs Is More Experiments |url=http://nautil.us/issue/45/power/what-quantum-gravity-needs-is-more-experiments |website=Nautilus |accessdate=21 September 2020 |date=2 February 2017 |archive-date=28 January 2018 |archive-url=https://web.archive.org/web/20180128021051/http://nautil.us/issue/45/power/what-quantum-gravity-needs-is-more-experiments |url-status=dead }}</ref><ref name="springer1">{{cite book |title=Experimental search for quantum gravity |date=2017 |publisher=Springer |___location=Cham |isbn=9783319645360}}</ref><ref name="tabletop">{{cite journal |last1=Carney |first1=Daniel |last2=Stamp |first2=Philip C. E. |last3=Taylor |first3=Jacob M. |title=Tabletop experiments for quantum gravity: a user's manual |journal=Classical and Quantum Gravity |pages=034001 |doi=10.1088/1361-6382/aaf9ca |date=7 February 2019 |volume=36 |issue=3 |arxiv=1807.11494 |bibcode=2019CQGra..36c4001C |s2cid=119073215 }}</ref><ref>{{Cite journal |last1=Danielson |first1=Daine L. |last2=Satishchandran |first2=Gautam |last3=Wald |first3=Robert M. |date=2022-04-05 |title=Gravitationally mediated entanglement: Newtonian field versus gravitons |url=https://link.aps.org/doi/10.1103/PhysRevD.105.086001 |journal=Physical Review D |volume=105 |issue=8 |pages=086001 |arxiv=2112.10798 |doi=10.1103/PhysRevD.105.086001 |bibcode=2022PhRvD.105h6001D |s2cid=245353748 |access-date=2022-12-11 |archive-date=2023-01-22 |archive-url=https://web.archive.org/web/20230122174555/https://journals.aps.org/prd/abstract/10.1103/PhysRevD.105.086001 |url-status=live }}</ref> This field of study is called [[phenomenological quantum gravity]].

}}</ref><ref name="Spin Spacetime Censorship">{{Cite arXiv|eprint=1812.11450v2|class=gr-qc|first1=J.|last1=Nemirovsky|first2=E.|last2=Cohen|title=Spin Spacetime Censorship|date=30 Dec 2018|last3=Kaminer|first3=I.|ref=harv}}</ref>

== Overview ==

{{unsolved|physics|How can the theory of [[quantum mechanics]] be merged with the theory of [[general relativity]] / [[gravitation]]al force and remain correct at microscopic length scales? What verifiable predictions does any theory of quantum gravity make?}}

[[File:Quantum gravity.svg|right|thumb|480pxupright=1.4|Diagram showing the place of quantum gravity 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. 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."<ref>{{cite book|first=John Archibald|last=Wheeler|title=Geons, Black Holes, and Quantum Foam: A Life in Physics|year=2010|publisher=[[W. W. Norton & Company]]|isbn=9780393079487|pages=235}}</ref> 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 [[renormalization|renormalizable]].<ref name=":1">{{cite book|title=Quantum Field Theory in a Nutshell|last=Zee|first=Anthony|publisher=[[Princeton University Press]]|year=2010|isbn=978-0-691-14034-6|edition=second|___location=|pages=[https://archive.org/details/isbn_9780691140346/page/172 172,434&ndash;435]|oclc=659549695|author-link=Anthony Zee|title-link=Quantum Field Theory in a Nutshell}}</ref> 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.<ref name="Wald 1994">{{cite book |last=Wald |first=Robert M. |title= Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics |date=1994 |publisher=University of Chicago Press |isbn=978-0-226-87027-4}}</ref>

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 hole]]s, and the [[Big Bang|origin of the universe]].<ref name="scholarpedia"/>

One major obstacle is that for [[quantum field theory in curved spacetime]] with a fixed metric, [[Boson|bosonic]]/[[Fermion|fermionic]] operator fields [[Lie superalgebra|supercommute]] for [[Causal structure|spacelike separated points]]. (This is a way of imposing a [[principle of locality#Relativistic quantum mechanics|principle of locality]].) However, in quantum gravity, the metric is dynamical, so that whether two points are spacelike separated depends on the state. In fact, they can be in a [[quantum superposition]] of being spacelike and not spacelike separated.{{cn|date=June 2024}}

== Quantum mechanics and general relativity ==

[[File:Gravity Probe B.jpg|thumb|right|150px|[[Gravity Probe B]] (GP-B) has measured spacetime curvature near Earth to test related models in application of Einstein's general theory of relativity.]]

=== Graviton ===

{{Main|Graviton}}

AtThe present,observation onethat of the deepest problems in theoretical physics is harmonizing the theory ofall [[generalfundamental relativityforces]], whichexcept describesgravity gravitationhave andone appliesor tomore large-scaleknown structures ([[star]]s,messenger [[planetsparticles]], [[galaxies]]),leads withresearchers [[quantumto mechanics]],believe whichthat describesat theleast otherone threemust [[fundamentalexist forces]] acting on the [[atom]]icfor scalegravity. This problemhypothetical mustparticle beis putknown inas the proper''graviton''. context,These however.particles Inact particular,as contrarya to[[force theparticle]] popularsimilar claimto thatthe quantum[[photon]] mechanicsof andthe generalelectromagnetic relativityinteraction. areUnder fundamentallymild incompatibleassumptions, one can demonstrate that the structure of general relativity essentiallyrequires followsthem inevitablyto fromfollow the quantum mechanicsmechanical description of interacting theoretical spin-2 massless particles (called [[graviton]]s).<ref name=Kraichnan1955>{{cite journal

|last = Kraichnan

|first = R. H.

|authorlinkauthor-link = Robert Kraichnan

|title = Special-Relativistic Derivation of Generally Covariant Gravitation Theory

|date = 1955

Line 63 ⟶ 83:

|pages = 1118–1122

|doi = 10.1103/PhysRev.98.1118

|ref = harv

|bibcode = 1955PhRv...98.1118K }}</ref><ref name=Gupta1954>{{cite journal

|last = Gupta

|first = S. N.

|authorlinkauthor-link = Suraj N. Gupta

|title = Gravitation and Electromagnetism

|date = 1954

Line 75 ⟶ 94:

|pages = 1683–1685

|doi = 10.1103/PhysRev.96.1683

|ref = harv

|bibcode = 1954PhRv...96.1683G }}</ref><ref name=Gupta1957>{{cite journal

|last = Gupta

|first = S. N.

|authorlinkauthor-link = Suraj N. Gupta

|title = Einstein's and Other Theories of Gravitation

|date = 1957

Line 87 ⟶ 105:

|pages = 334–336

|doi = 10.1103/RevModPhys.29.334

|ref = harv

|bibcode=1957RvMP...29..334G

}}</ref><ref name=Gupta1962>{{cite book

Line 101 ⟶ 118:

|last = Deser

|first = S.

|authorlinkauthor-link = Stanley Deser

|title = Self-Interaction and Gauge Invariance

|date = 1970

|journal = [[General Relativity and Gravitation]]

|volume = 1

|issue = 1

|pages = 9–18

|doi = 10.1007/BF00759198

|ref = harv

|bibcode=1970GReGr...1....9D

|arxiv = gr-qc/0411023 }}</ref>|s2cid = 14295121

}}</ref> 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 [[composite gravity|the graviton is a composite particle]].<ref>{{cite journal|first1=Steven|last1=Weinberg|first2=Edward|last2=Witten|author-link1=Steven Weinberg|author-link2=Edward Witten|title=Limits on massless particles|journal=[[Physics Letters B]]|volume=96|issue=1–2|year=1980|pages=59&ndash;62|doi=10.1016/0370-2693(80)90212-9|bibcode=1980PhLB...96...59W}}</ref><ref>{{cite book|first1=Gary T.|last1=Horowitz|first2=Joseph|last2=Polchinski|author-link2=Joseph Polchinski|chapter=Gauge/gravity duality|title=Approaches to Quantum Gravity|publisher=[[Cambridge University Press]]|editor-last=Oriti|editor-first=Daniele|isbn=9780511575549|oclc=873715753|arxiv=gr-qc/0602037|bibcode=2006gr.qc.....2037H|year=2006}}</ref> While gravitons are an important theoretical step in a quantum mechanical description of gravity, they are generally believed to be undetectable because they interact too weakly.<ref>{{cite journal

|last1 = Rothman

No concrete proof of gravitons exists, but quantized theories of matter may necessitate their existence.<ref>{{cite book|author=Charles Ginenthal|url=https://books.google.com/?id=M8RACwAAQBAJ&printsec=frontcover#v=onepage&q&f=false

|first1 = Tony

|title=Newton, Einstein, and Velikovsky|isbn=9781329742567

|last2 = Boughn

|date=2015-12-07

|first2 = Stephen

}}</ref> 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. This hypothetical particle is known as the ''graviton''. The predicted find would result in the classification of the graviton as a [[force particle]] similar to the [[photon]] of the electromagnetic interaction. 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. These include [[string theory]], [[superstring theory]], and [[M-theory]]. Detection of gravitons would validate these various lines of research to unify quantum mechanics and relativity theory.

|date = 2006

|title = Can Gravitons be Detected?

The [[Weinberg–Witten theorem]] places some constraints on theories in which [[composite gravity|the graviton is a composite particle]].<ref>{{cite journal|first1=Steven|last1=Weinberg|first2=Edward|last2=Witten|author-link1=Steven Weinberg|author-link2=Edward Witten|title=Limits on massless particles|journal=[[Physics Letters B]]|volume=96|issue=1–2|year=1980|pages=59&ndash;62|doi=10.1016/0370-2693(80)90212-9|bibcode=1980PhLB...96...59W}}</ref><ref>{{cite book|first1=Gary T.|last1=Horowitz|first2=Joseph|last2=Polchinski|author-link2=Joseph Polchinski|chapter=Gauge/gravity duality|title=Approaches to Quantum Gravity|publisher=[[Cambridge University Press]]|editor-last=Oriti|editor-first=Daniele|isbn=9780511575549|oclc=873715753|arxiv=gr-qc/0602037|bibcode=2006gr.qc.....2037H|year=2006}}</ref>

|url = https://link.springer.com/article/10.1007/s10701-006-9081-9

|journal = Foundations of Physics

=== Dilaton ===

|volume = 36

{{Main|Dilaton}}

|issue = 12

The [[dilaton]] made its first appearance in [[Kaluza–Klein theory]], a five-dimensional theory that combined [[gravitation]] and [[electromagnetism]]. It appears in [[string theory]]. However, it's become central to the lower-dimensional many-bodied gravity problem<ref>{{cite journal|last1=Ohta|first1=Tadayuki|last2=Mann|first2=Robert|title=Canonical reduction of two-dimensional gravity for particle dynamics|journal=[[Classical and Quantum Gravity]]|volume=13|issue=9|pages=2585–2602|date=1996|doi=10.1088/0264-9381/13/9/022|arxiv=gr-qc/9605004|ref=harv|bibcode = 1996CQGra..13.2585O }}</ref> 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'' - one spatial dimension and one temporal dimension. This model problem, known as [[R=T model|''R=T'' theory]],<ref>{{cite journal|last1=Sikkema|first1=A E|last2=Mann|first2=R B|title=Gravitation and cosmology in (1+1) dimensions|journal=[[Classical and Quantum Gravity]]|volume=8|issue=1|pages=219–235|date=1991|doi=10.1088/0264-9381/8/1/022|ref=harv|bibcode = 1991CQGra...8..219S }}</ref> as opposed to the general ''G=T'' theory, was amenable to exact solutions in terms of a generalization of the [[Lambert W function]]. Also, the field equation governing the dilaton, derived from [[differential geometry]], as the [[Schrödinger equation]] could be amenable to quantization.<ref>{{cite journal|author1=Farrugia|author2=Mann|author3=Scott|doi=10.1088/0264-9381/24/18/006|journal=[[Classical and Quantum Gravity]] |volume=24|title=N-body Gravity and the Schroedinger Equation|issue=18|pages=4647–4659|date=2007|arxiv=gr-qc/0611144|bibcode = 2007CQGra..24.4647F }}</ref>

|pages = 1801–1825

|doi = 10.1007/s10701-006-9081-9

This combines gravity, quantization, and even the electromagnetic interaction, promising ingredients of a fundamental physical theory. This outcome revealed a previously unknown and already existing natural link between general relativity and quantum mechanics. There lacks clarity in the generalization of this theory to ''3+1'' dimensions. However, a recent derivation in ''3+1'' dimensions under the right coordinate conditions yields a formulation similar to the earlier ''1+1'', a dilaton field governed by the [[logarithmic Schrödinger equation]]<ref>{{cite journal|last1=Scott|first1=T.C.|last2=Zhang|first2=Xiangdong|last3=Mann|first3=Robert|last4=Fee|first4=G.J.|title=Canonical reduction for dilatonic gravity in 3 + 1 dimensions|journal=[[Physical Review D]]|volume=93|issue=8|pages=084017|date=2016|doi=10.1103/PhysRevD.93.084017|bibcode = 2016PhRvD..93h4017S|arxiv=1605.03431}}</ref> that is seen in [[condensed matter physics]] and [[Superfluidity|superfluids]]. The field equations are amenable to such a generalization, as shown with the inclusion of a one-graviton process,<ref>{{cite journal|last1=Mann|first1=R B|last2=Ohta|first2=T|title=Exact solution for the metric and the motion of two bodies in (1+1)-dimensional gravity|journal=[[Physical Review|Phys. Rev. D]]|volume=55|issue=8|pages=4723–4747|date=1997|doi=10.1103/PhysRevD.55.4723|arxiv=gr-qc/9611008|ref=harv|bibcode=1997PhRvD..55.4723M}}</ref> and yield the correct Newtonian limit in ''d'' dimensions, but only with a dilaton. Furthermore, some speculate on the view of the apparent resemblance between the dilaton and the [[Higgs boson]].<ref>{{cite journal|last1=Bellazzini|first1=B.|last2=Csaki|first2=C.|last3=Hubisz|first3=J.|last4=Serra|first4=J.|last5=Terning|first5=J.|title=A higgs-like dilaton|journal=Eur. Phys. J. C|volume=73|number=2|pages=2333|date=2013|doi=10.1140/epjc/s10052-013-2333-x|bibcode=2013EPJC...73.2333B|arxiv=1209.3299}}</ref> However, there needs more experimentation to resolve the relationship between these two particles. Finally, since this theory can combine gravitational, electromagnetic, and quantum effects, their coupling could potentially lead to a means of testing the theory through cosmology and experimentation.

|arxiv = gr-qc/0601043

|bibcode = 2006FoPh...36.1801R

|s2cid = 14008778

|access-date = 2020-05-15

|archive-date = 2020-08-06

|archive-url = https://web.archive.org/web/20200806234929/https://link.springer.com/article/10.1007/s10701-006-9081-9

|url-status = live

}}</ref>

=== Nonrenormalizability of gravity ===

{{Further|Renormalization|Asymptotic safety in quantum gravity}}

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]].<ref name="FLoG"/>{{rpCite book |xxxvi&ndash;xxxviii;211&ndash;212last=Feynman |first=Richard P. |title=Feynman Lectures on Gravitation |publisher=Addison-Wesley |year=1995 |isbn=978-0201627343 |___location=Reading, Massachusetts |pages=xxxvi–xxxviii, 211–212 |language=en-us}}</ref><ref>{{ cite book | last= Hamber | first= H. W. | title= Quantum Gravitation – The Feynman Path Integral Approach | publisher = Springer Nature | date=2009 | isbn=978-3-540-85292-6 }}</ref> For a quantum field theory to be well defined according to this understanding of the subject, it must be [[asymptotic freedom|asymptotically free]] or [[asymptotic safety|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.<ref>{{citationCite neededjournal|doi=10.1016/0370-2693(85)91470-4|last1=Goroff|first1=Marc H.|last2=Sagnotti|first2=Augusto|last3=Sagnotti|first3=Augusto|title=Quantum gravity at two loops|date=February1985|journal=[[Physics Letters B]]|volume=160|issue=1–3|pages=81–86|bibcode = 1985PhLB..160...81G 2019}}</ref>

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 [[non-perturbative]] quantum field theory, it is difficult to findfinding a reliable answer is difficult, butpursued somein peoplethe still[[Asymptotic pursuesafety thisin optionquantum gravity|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.<ref>{{Cite web|url=https://golem.ph.utexas.edu/~distler/blog/archives/000639.html|title=Motivation|last=Distler|first=Jacques|authorlinkauthor-link=Jacques Distler|date=2005-09-01|website=golem.ph.utexas.edu|language=en|access-date=2018-02-24|archive-date=2019-02-11|archive-url=https://web.archive.org/web/20190211070351/https://golem.ph.utexas.edu/~distler/blog/archives/000639.html|url-status=live}}</ref>{{Better source needed|date=February 2019}}

=== Quantum gravity as an effective field theory ===

{{Main|Effective field theory}}

In an [[effective field theory]], not 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 low-energy effects. Thus, at least in the low-energy regime, the model is a predictive quantum field theory.<ref name=":0" /> 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.<ref>{{Citecite book|title=Phase transitions and renormalization group|last=Zinn-Justin|first=Jean|date=2007|publisher=[[Oxford University Press]]|isbn=9780199665167|___location=Oxford|pages=|oclc=255563633|author-link=Jean Zinn-Justin}}</ref>

|author1-link=John Francis Donoghue (physicist)

|last = Donoghue

|first=John F.

|contribution=Introduction to the Effective Field Theory Description of Gravity

|date=1995

|arxiv=gr-qc/9512024

|editor-last=Cornet

|editor-first=Fernando

|title=Effective Theories: Proceedings of the Advanced School, Almunecar, Spain, 26 June–1 July 1995

|isbn=978-981-02-2908-5

|publisher = [[World Scientific]]

|___location = Singapore

|bibcode=1995gr.qc....12024D

}}</ref> 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.<ref>{{Cite book|title=Phase transitions and renormalization group|last=Zinn-Justin|first=Jean|date=2007|publisher=[[Oxford University Press]]|isbn=9780199665167|___location=Oxford|oclc=255563633|author-link=Jean Zinn-Justin}}</ref>

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.<ref name=":0" /> Another example is the calculation of the corrections to the Bekenstein-Hawking entropy formula.<ref>{{cite journal |last1=Calmet |last2=Kuipers |first1=Xavier |first2=Folkert |title=Quantum gravitational corrections to the entropy of a Schwarzschild black hole |journal=Phys. Rev. D|year=2021 |volume=104 |issue=6 |page=6 |doi=10.1103/PhysRevD.104.066012 |arxiv=2108.06824 |bibcode=2021PhRvD.104f6012C |s2cid=237091145 }}</ref><ref>{{cite journal |last1=Campos Delgado|first1=Ruben |title=Quantum gravitational corrections to the entropy of a Reissner-Nordström black hole |journal=Eur. Phys. J. C|year=2022 |volume=82 |issue=3 |page=272 |doi=10.1140/epjc/s10052-022-10232-0|arxiv=2201.08293 |bibcode=2022EPJC...82..272C |s2cid=247824137 |doi-access=free }}</ref>

=== Spacetime background dependence ===

{{Main|Background independence}}

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 easysimple to grasp in principle, this is thea hardestcomplex 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]],<ref>{{cite book

|last = Smolin

|first = Lee

|authorlinkauthor-link= Lee Smolin

|title = Three Roads to Quantum Gravity

|publisher = [[Basic Books]]

Line 152 ⟶ 191:

|pages = [https://archive.org/details/threeroadstoquan00smol_0/page/20 20–25]

|isbn = 978-0-465-07835-6|title-link = Three Roads to Quantum Gravity

}} Pages 220–226 are annotated references and guide for further reading.</ref> in which the only physically relevant information is the relationship between different events in space-timespacetime.

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 ====

[[File:Point&string.png|right|thumb|class=skin-invert-image|Interaction in the subatomic world: [[world line]]s of point-like [[Subatomic particle|particles]] in the [[Standard Model]] or a [[world sheet]] swept up by closed [[string (physics)|strings]] in string 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 dynamicaldynamic 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 independence|background dependence]].

==== Background independent theories ====

[[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 network]]s.{{Citation needed|date=September 2020}}

=== Semi-classical quantum gravity ===

{{Main article|Quantum field theory in curved spacetime|Semiclassical gravity}}

[[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.

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 the least energy (and may or may not contain particles).

See [[Quantum field theory in curved spacetime]] for a more complete discussion.

=== Problem of time ===

{{Main|Problem of time}}

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 (quantum mechanics)|Hamiltonian operator]] acting as the [[Translation operator (quantum mechanics)|generator of infinitesimal translations]] of quantum states through time.<ref>{{Cite book|title=Modern Quantum Mechanics|lastlast1=Sakurai|firstfirst1=J. J.|last2=Napolitano|first2=Jim J.|date=2010-07-14|publisher=Pearson|isbn=978-0-8053-8291-4|edition=2|page=68|language=Englishen}}</ref> In contrast, general relativity [[Einstein field equations|treats time as a dynamical variable]] which interactsrelates directly with matter and moreover requires the Hamiltonian constraint to vanish,.<ref>{{Cite book|url=https://books.google.com/books?id=vDWvUBiNgNkC|title=Cosmology and Gravitation: Xth Brazilian School of Cosmology and Gravitation; 25th Anniversary (1977–2002), Mangaratiba, Rio de Janeiro, Brazil|lastlast1=Novello|firstfirst1=Mario|last2=Bergliaffa|first2=Santiago E.|date=2003-06-11|publisher=Springer Science & Business Media|isbn=978-0-7354-0131-0|page=95|language=en}}</ref> removingBecause this variability of time has been [[Gravitational time dilation#Experimental confirmation|observed macroscopically]], it removes any possibility of employing a fixed notion of time, similar to thatthe conception of time in quantum theory, at the macroscopic level.

== Candidate theories ==

Line 180 ⟶ 219:

|last=Rovelli

|first=Carlo

|authorlinkauthor-link=Carlo Rovelli

|date=2000

|title=Notes for a brief history of quantum gravity

|eprint=gr-qc/0006061

}} (verify against {{ISBN|9789812777386}})</ref> 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 become available.<ref>{{cite conference

|ref=harv

|class=

}} (verify against {{ISBN|9789812777386}})</ref> 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.<ref>{{cite book

|last=Ashtekar |first=Abhay

|authorlinkauthor-link=Abhay Ashtekar

|date=2007

|chaptertitle=Loop Quantum Gravity: Four Recent Advances and a Dozen Frequently Asked Questions

|titleconference=11thThe Eleventh Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity

|journal=The Eleventh Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity

|page=126

|arxiv=0705.2222

|doi=10.1142/9789812834300_0008

|bibcode=2008mgm..conf..126A

|ref=harv

|isbn=978-981-283-426-3

|s2cid=119663169

}}</ref><ref>{{cite journal

|last=Schwarz

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|doi=10.1143/PTPS.170.214

|bibcode=2007PThPS.170..214S

|s2cid=16762545

|ref=harv

}}</ref>

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{{Main|String theory}}

[[File:Calabi-Yau.png|thumb|Projection of a [[Calabi–Yau manifold]], one of the ways of [[Compactification (physics)|compactifying]] the extra dimensions posited by string theory]]

One suggested starting point is ordinary quantum field theories which 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 field theory|effective (quantum) field theory]] of gravity at low energies,<ref name=":0">{{cite book

|last = Donoghue

|first=John F. (editor)

|contribution=Introduction to the Effective Field Theory Description of Gravity

|date=1995

|arxiv=gr-qc/9512024

|editor-last=Cornet

|editor-first=Fernando

|title=Effective Theories: Proceedings of the Advanced School, Almunecar, Spain, 26 June–1 July 1995

|isbn=978-981-02-2908-5

|publisher = [[World Scientific]]

|___location = Singapore

|bibcode=1995gr.qc....12024D

}}</ref> gravity turns out to be much more problematic at higher energies. 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,<ref>{{Cite book

|last=Weinberg

|first=Steven

|chapter=Chapters 17–18

|authorlink=Steven Weinberg

|title=The Quantum Theory of Fields II: Modern Applications

|publisher=[[Cambridge University Press]]

|date=1996

|isbn=978-0-521-55002-4

|url-access=registration

|url=https://archive.org/details/quantumtheoryoff00stev

}}</ref> but 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.<ref>{{Cite journal

|doi=10.1016/0370-2693(85)91470-4

|last=Goroff|first=Marc H.

|last2=Sagnotti|first2=Augusto

|last3=Sagnotti|first3=Augusto|title=Quantum gravity at two loops

|date=1985

|journal=[[Physics Letters B]]

|volume=160

|issue=1–3|pages=81–86

|ref=harv

|bibcode = 1985PhLB..160...81G }}</ref>

OneThe attemptcentral toidea overcomeof thesestring limitationstheory is to replace ordinary [[quantum field theory]], which is based on the classical concept of a [[point particle]], in quantum field theory with a quantum theory of one-dimensional extended objects: [[string theory]].<ref>An accessible introduction at the undergraduate level can be found in {{Cite book

|last=Zwiebach

|first=Barton

|authorlinkauthor-link= Barton Zwiebach

|title=A First Course in String Theory

|publisher=[[Cambridge University Press]]

|date=2004

|isbn=978-0-521-83143-7

|ref=harv

}}, and more complete overviews in {{Cite book

|last=Polchinski

|first=Joseph

|authorlinkauthor-link=Joseph Polchinski

|date=1998

|title=String Theory Vol. I: An Introduction to the Bosonic String

Line 272:

|last=Polchinski

|first=Joseph

|authorlinkauthor-link=Joseph Polchinski

|date=1998b

|title=String Theory Vol. II: Superstring Theory and Beyond

Line 288:

|arxiv=hep-ph/9911499

|doi=10.1088/0264-9381/17/5/321

|bibcode = 2000CQGra..17.1117I |s2cid=15707877

|ref=harv

|bibcode = 2000CQGra..17.1117I }}</ref> 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 areis unusual features such as six extra dimensions of space in addition to the usual three for space and one for time.<ref>For the graviton as part of the string spectrum, e.g. {{Harvnb|Green|Schwarz|Witten|19872012|loc=sec. 2.3 and 5.3}}; for the extra dimensions, ibid sec. 4.2.</ref>

In what is called the [[History of string theory#1984–1989: first superstring revolution|second superstring revolution]], it was conjectured that both string theory and a unification of general relativity and [[supersymmetry]] known as [[supergravity]]<ref>{{Cite book

Line 295:

|first=Steven

|chapter=Chapter 31

|authorlinkauthor-link=Steven Weinberg

|title=The Quantum Theory of Fields II: Modern Applications

|publisher=Cambridge University Press

|date=2000

|chapter-url=https://books.google.com/books?id=aYDDRKqODpUC&printsec=frontcover

|isbn=978-0-521-55002-4

|url-access=registration

Line 309:

|journal=High Energy Physics and Cosmology

|volume=13

|booktitle=1996 Summer School in High Energy Physics and Cosmology

|series=ICTP Series in Theoretical Physics

|page=385

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|arxiv=hep-th/9612121

|bibcode=1997hepcbconf..385T

|ref=harv

}}</ref><ref>{{Cite journal

|last=Duff

|first=Michael

|authorlinkauthor-link=Michael Duff (physicist)

|title=M-Theory (the Theory Formerly Known as Strings)

|journal=[[International Journal of Modern Physics A]]

Line 328 ⟶ 326:

|arxiv=hep-th/9608117

|bibcode=1996IJMPA..11.5623D

|s2cid=17432791

|ref=harv

}}</ref> 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 a major challenge.

=== Loop quantum gravity ===

{{Main|Loop quantum gravity}}

[[File:Spin network.svg|right|thumb|200pxupright=1|Simple [[spin network]] of the type used in loop quantum gravity]]

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 particle-like 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 thethat derivationthere ofis a granular structure of space at the Planck length. This is derived from the 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 spectrumspectra. 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 network]]s. 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 non-perturbative 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 field|electric]] and [[magnetic field]]s.<ref>{{Cite journal

|last=Ashtekar

|first=Abhay

|authorlinkauthor-link=Abhay Ashtekar

|title=New variables for classical and quantum gravity

|journal=[[Physical Review Letters]]

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|pmid=10033673

|issue=18

|ref=harv

|bibcode=1986PhRvL..57.2244A

}}</ref><ref>{{Cite journal

|last=Ashtekar

|first=Abhay

|authorlinkauthor-link=Abhay Ashtekar

|title=New Hamiltonian formulation of general relativity

|journal=[[Physical Review D]]

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|date=1987

|doi=10.1103/PhysRevD.36.1587

|pmid=9958340

|ref=harv|bibcode = 1987PhRvD..36.1587A }}</ref>

|bibcode = 1987PhRvD..36.1587A }}</ref> In the quantum theory, space is represented by a network structure called a [[spin network]], evolving over time in discrete steps.<ref>{{Cite book|last=Thiemann|first=Thomas|title=Approaches to Fundamental Physics|year=2007|isbn=978-3-540-71115-5|series=Lecture Notes in Physics|volume=721|pages=185–263|chapter=Loop Quantum Gravity: An Inside View|bibcode=2007LNP...721..185T|doi=10.1007/978-3-540-71117-9_10|arxiv=hep-th/0608210|s2cid=119572847}}</ref><ref>{{cite journal

|last=Thiemann

|first=Thomas

|title=Loop Quantum Gravity: An Inside View

|journal=[[Approaches to Fundamental Physics]]

|volume=721

|pages=185–263

|year=2007

|arxiv=hep-th/0608210

|bibcode=2007LNP...721..185T

|ref=harv

|doi=10.1007/978-3-540-71117-9_10

|series=Lecture Notes in Physics

|isbn=978-3-540-71115-5

}}</ref><ref>{{cite journal

|last=Rovelli

|first=Carlo

|authorlinkauthor-link=Carlo Rovelli

|title=Loop Quantum Gravity

|journal=[[Living Reviews in Relativity]]

|volume=1

|date=1998

|issue=1

|url=http://www.livingreviews.org/lrr-1998-1

|page=1

|accessdate=2008-03-13

|doi=10.12942/lrr-1998-1

|ref=harv

|doi-access=free

|pmid=28937180

|pmc=5567241

|arxiv=gr-qc/9710008

|bibcode=1998LRR.....1....1R

}}</ref><ref>{{cite journal

| lastlast1=Ashtekar

| firstfirst1=Abhay

| authorlinkauthor-link=Abhay Ashtekar

| first2=Jerzy

| last2=Lewandowski

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| arxiv=gr-qc/0404018

| doi=10.1088/0264-9381/21/15/R01

|bibcode = 2004CQGra..21R..53A | s2cid=119175535
}}</ref><ref>{{Cite book

|last=Thiemann

|first=Thomas

|titlechapter=Lectures on Loop Quantum Gravity

|date=2003

|volume=631

|pages=41–135

|ref=harv

|arxiv=gr-qc/0210094

|bibcode=2003LNP...631...41T

|doi = 10.1007/978-3-540-45230-0_3 |series=Lecture Notes in Physics

|isbn=978-3-540-40810-9

|title=Quantum Gravity

|s2cid=119151491

}}</ref>

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.<ref>{{Cite book

|last=Rovelli

|first=Carlo

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|publisher=Cambridge University Press

|isbn=978-0-521-71596-6

}}</ref> 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.

}}</ref>

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 approachesproposals ===

There are a number of other approaches to quantum gravity. The approachestheories differ depending on which features of general relativity and quantum theory are accepted unchanged, and which features are modified.<ref>{{Cite book

|last=Isham

|first=Christopher J.

|title=Canonical Gravity: From Classical to Quantum

|authorlink=Christopher Isham

|author-link=Christopher Isham

|contribution=Prima facie questions in quantum gravity

|editor-last=Ehlers

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|editor2-last=Friedrich

|editor2-first=Helmut

|title=Canonical Gravity: From Classical to Quantum

|journal=Canonical Gravity: From Classical to Quantum

|volume=434

|pages=1–21

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|doi=10.1007/3-540-58339-4_13

|series=Lecture Notes in Physics

|s2cid=119364176

}}</ref><ref>{{Cite journal

|last=Sorkin

|first=Rafael D.

|authorlinkauthor-link= Rafael Sorkin

|title=Forks in the Road, on the Way to Quantum Gravity

|arxiv=gr-qc/9706002

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|pages=2759–2781

|doi=10.1007/BF02435709

|ref=harv|bibcode = 1997IJTP...36.2759S }}</ref> Examples include:|s2cid=4803804

}}</ref> Such approaches include:

{{columns-list|colwidth=17em|

* [[Asymptotic safety in quantum gravity]]

* [[Euclidean quantum gravity]]

* [[Virtual black hole|Integral method]]<ref>{{Cite web |last=Klimets |first=A. P. |date=2017 |title=Philosophy Documentation Center, Western University – Canada |url=https://philpapers.org/archive/ALXOTF.pdf |url-status=live |archive-url=https://web.archive.org/web/20190701011840/https://philpapers.org/archive/ALXOTF.pdf |archive-date=2019-07-01 |access-date=2020-04-24 |publisher=Philosophy Documentation Center, Western University – Canada |pages=25–32}}</ref>

* [[Causal dynamical triangulation]]<ref>{{Cite journal

|last=Loll

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|journal=[[Living Reviews in Relativity]]

|volume=1

|issue=1

|page=13

|date=1998

|url=http://www.livingreviews.org/lrr-1998-13

|bibcode=1998LRR.....1...13L

|arxiv = gr-qc/9805049 |doi = 10.12942/lrr-1998-13 |doi-access=free

|accessdate=2008-03-09

|pmid=28191826

|ref=harv

|arxiv = gr-qc/9805049 |doi = 10.12942/lrr-1998-13 |pmid=28191826

|pmc=5253799

}}</ref>

Line 481 ⟶ 472:

* [[causal sets|Causal Set Theory]]

* Covariant Feynman [[path integral formulation|path integral]] approach

* [[Dilaton#The dilaton in quantum gravity|Dilatonic quantum gravity]]

*[[Double copy theory]]

* [[Group field theory]]

* [[Wheeler–DeWitt equation]]

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|last=Hawking

|first=Stephen W.

|authorlinkauthor-link=Stephen Hawking

|contribution=Quantum cosmology

|pages =631–651

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}}</ref>

* [[Regge calculus]]

* [[Scale relativity]]

* [[Shape dynamics|Shape Dynamics]]

* [[String-net]]s and [[quantum graphity]]

* [[Superfluid vacuum theory]] a.k.a. theory of [[BEC vacuum]]

* [[Supergravity]]

* [[Twistor theory]]<ref>See ch. 33 in {{Harvnb|Penrose|20042005}} and references therein.</ref>

* [[Canonical quantum gravity]]

*[[An Exceptionally Simple Theory of Everything]] (Garret Lisi's E8 model)

* Quantum [[holonomy]] theory<ref>{{Cite journal|title = Quantum Holonomy Theory |journal=Fortschritte der Physik |volume=64 |issue=10 |pages=783 |author1=Aastrup, J. |author2=Grimstrup, J. M. |date = 27 Apr 2015|arxiv= 1504.07100|doi=10.1002/prop.201600073 |bibcode=2016ForPh..64..783A }}</ref>

* Vipremigini model

}}

== Experimental tests ==

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, insince the past decade2000s, 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.<ref>{{cite book |last=Hossenfelder |first=Sabine |title=Classical and Quantum Gravity: Theory, Analysis and Applications |date=2011 |publisher=Nova Publishers|___location=Chapter 5|isbn=978-1-61122-957-8 |editor=Frignanni |editor-first=V. R. |chapter=Experimental Search for Quantum Gravity |access-date=2012-04-01 |chapter-url=https://www.novapublishers.com/catalog/product_info.php?products_id=15903|chapter=Experimental Search for Quantum Gravity|editorarchive-url=Vhttps://web. Rarchive.org/web/20170701040647/https://www.novapublishers.com/catalog/product_info.php?products_id=15903 Frignanni|archive-date=2017-07-01 |url-status=dead}}</ref>

The most widely pursued possibilities for quantum gravity phenomenology include violationsgravitationally ofmediated [[Lorentz covariance|Lorentz invariance]]entanglement, imprints of quantum gravitational effects in the [[cosmic microwave background]] (in particular its polarization), and decoherence induced by fluctuations in the [[space-time foam]].<ref>

{{cite journal

|last1 = Lindner

|first1 = Nethanel H.

|last2 = Peres

|first2 = Asher

|title = Testing quantum superpositions of the gravitational field with Bose-Einstein condensates

|date = 2005

|journal = [[Physical Review A]]

|volume = 71

|issue = 2

|pages = 024101

|doi = 10.1103/PhysRevA.71.024101

|arxiv = gr-qc/0410030

|bibcode = 2005PhRvA..71b4101L

}}</ref><ref>

{{cite arXiv

|last1 = Kafri

|first1 = Dvir

|last2 = Taylor

|first2 = Jacob M

|title = A noise inequality for classical forces

|date = 2013

|class = quant-ph

|eprint = 1311.4558

}}</ref>

violations of [[Lorentz covariance|Lorentz invariance]], imprints of quantum gravitational effects in the [[cosmic microwave background]] (in particular its polarization), and decoherence induced by fluctuations<ref>{{Cite journal|last1=Oniga|first1=Teodora|last2=Wang|first2=Charles H.-T.|date=2016-02-09|title=Quantum gravitational decoherence of light and matter|url=https://link.aps.org/doi/10.1103/PhysRevD.93.044027|journal=Physical Review D|volume=93|issue=4|pages=044027|doi=10.1103/PhysRevD.93.044027|arxiv=1511.06678|bibcode=2016PhRvD..93d4027O|hdl=2164/5830|s2cid=119210226|hdl-access=free|access-date=2021-01-01|archive-date=2023-01-22|archive-url=https://web.archive.org/web/20230122174605/https://journals.aps.org/prd/abstract/10.1103/PhysRevD.93.044027|url-status=live}}</ref><ref>{{Cite journal|last1=Oniga|first1=Teodora|last2=Wang|first2=Charles H.-T.|date=2017-10-05|title=Quantum coherence, radiance, and resistance of gravitational systems|url=https://link.aps.org/doi/10.1103/PhysRevD.96.084014|journal=Physical Review D|volume=96|issue=8|pages=084014|doi=10.1103/PhysRevD.96.084014|arxiv=1701.04122|bibcode=2017PhRvD..96h4014O|hdl=2164/9320|s2cid=54777871|hdl-access=free|access-date=2021-01-01|archive-date=2023-01-22|archive-url=https://web.archive.org/web/20230122174600/https://journals.aps.org/prd/abstract/10.1103/PhysRevD.96.084014|url-status=live}}</ref><ref>{{Cite journal|last1=Quiñones|first1=D. A.|last2=Oniga|first2=T.|last3=Varcoe|first3=B. T. H.|last4=Wang|first4=C. H.-T.|date=2017-08-15|title=Quantum principle of sensing gravitational waves: From the zero-point fluctuations to the cosmological stochastic background of spacetime|url=https://link.aps.org/doi/10.1103/PhysRevD.96.044018|journal=Physical Review D|volume=96|issue=4|pages=044018|doi=10.1103/PhysRevD.96.044018|arxiv=1702.03905|bibcode=2017PhRvD..96d4018Q|hdl=2164/9150|s2cid=55056264|hdl-access=free|access-date=2021-01-02|archive-date=2023-01-22|archive-url=https://web.archive.org/web/20230122174603/https://journals.aps.org/prd/abstract/10.1103/PhysRevD.96.044018|url-status=live}}</ref> in the [[space-time foam]].<ref>{{Cite journal|last1=Oniga|first1=Teodora|last2=Wang|first2=Charles H.-T.|date=2016-09-19|title=Spacetime foam induced collective bundling of intense fields|url=https://link.aps.org/doi/10.1103/PhysRevD.94.061501|journal=Physical Review D|volume=94|issue=6|pages=061501|doi=10.1103/PhysRevD.94.061501|arxiv=1603.09193|bibcode=2016PhRvD..94f1501O|hdl=2164/7434|s2cid=54872718|hdl-access=free|access-date=2021-01-02|archive-date=2023-01-22|archive-url=https://web.archive.org/web/20230122174605/https://journals.aps.org/prd/abstract/10.1103/PhysRevD.94.061501|url-status=live}}</ref> The latter scenario has been searched for in light from [[gamma-ray burst]]s and both astrophysical and atmospheric [[neutrino]]s, placing limits on phenomenological quantum gravity parameters.<ref>{{Cite journal |last1=Vasileiou |first1=Vlasios |last2=Granot |first2=Jonathan |last3=Piran |first3=Tsvi |last4=Amelino-Camelia |first4=Giovanni |date=2015-03-16 |title=A Planck-scale limit on spacetime fuzziness and stochastic Lorentz invariance violation |journal=Nature Physics |volume=11 |issue=4 |pages=344–346 |doi=10.1038/nphys3270 |bibcode=2015NatPh..11..344V |s2cid=54727053 |issn=1745-2473|doi-access=free }}</ref><ref>{{Cite journal |last1=The IceCube Collaboration |last2=Abbasi |first2=R. |last3=Ackermann |first3=M. |last4=Adams |first4=J. |last5=Aguilar |first5=J. A. |last6=Ahlers |first6=M. |last7=Ahrens |first7=M. |last8=Alameddine |first8=J. M. |last9=Alispach |first9=C. |last10=Alves Jr |first10=A. A. |last11=Amin |first11=N. M. |last12=Andeen |first12=K. |last13=Anderson |first13=T. |last14=Anton |first14=G. |last15=Argüelles |first15=C. |date=2022-11-01 |title=Search for quantum gravity using astrophysical neutrino flavour with IceCube |url=https://www.nature.com/articles/s41567-022-01762-1 |journal=Nature Physics |language=en |volume=18 |issue=11 |pages=1287–1292 |doi=10.1038/s41567-022-01762-1 |bibcode=2022NatPh..18.1287I |s2cid=243848123 |issn=1745-2473|arxiv=2111.04654 }}</ref><ref>{{cite journal | arxiv=2308.00105 | doi=10.1038/s41567-024-02436-w | title=Search for decoherence from quantum gravity with atmospheric neutrinos | date=2024 | last1=Abbasi | first1=R. | last2=Ackermann | first2=M. | last3=Adams | first3=J. | last4=Agarwalla | first4=S. K. | last5=Aguilar | first5=J. A. | last6=Ahlers | first6=M. | last7=Alameddine | first7=J. M. | last8=Amin | first8=N. M. | last9=Andeen | first9=K. | last10=Anton | first10=G. | last11=Argüelles | first11=C. | last12=Ashida | first12=Y. | last13=Athanasiadou | first13=S. | last14=Ausborm | first14=L. | last15=Axani | first15=S. N. | last16=Bai | first16=X. | last17=Balagopal v | first17=A. | last18=Baricevic | first18=M. | last19=Barwick | first19=S. W. | last20=Basu | first20=V. | last21=Bay | first21=R. | last22=Beatty | first22=J. J. | last23=Tjus | first23=J. Becker | last24=Beise | first24=J. | last25=Bellenghi | first25=C. | last26=Benning | first26=C. | last27=Benzvi | first27=S. | last28=Berley | first28=D. | last29=Bernardini | first29=E. | last30=Besson | first30=D. Z. | journal=Nature Physics | volume=20 | issue=6 | pages=913–920 | bibcode=2024NatPh..20..913T | display-authors=1 }}</ref>

[[ESA]]'s [[INTEGRAL]] satellite measured polarization of photons of different wavelengths and was able to place a limit in the granularity of space that is less than 10⁻⁴⁸&nbsp;m, or 13 orders of magnitude below the Planck scale.<ref>{{Cite web|date=2011-06-30|title=Integral challenges physics beyond Einstein|url=https://www.esa.int/Science_Exploration/Space_Science/Integral_challenges_physics_beyond_Einstein|url-status=live|access-date=2021-11-06|website=European Space Agency|archive-date=2021-11-13|archive-url=https://web.archive.org/web/20211113230038/https://www.esa.int/Science_Exploration/Space_Science/Integral_challenges_physics_beyond_Einstein}}</ref><ref>{{Cite journal|last1=Laurent|first1=P.|last2=Götz|first2=D.|last3=Binétruy|first3=P.|last4=Covino|first4=S.|last5=Fernandez-Soto|first5=A.|date=2011-06-28|title=Constraints on Lorentz Invariance Violation using integral/IBIS observations of GRB041219A|url=https://link.aps.org/doi/10.1103/PhysRevD.83.121301|journal=Physical Review D|language=en|volume=83|issue=12|pages=121301|doi=10.1103/PhysRevD.83.121301|arxiv=1106.1068|bibcode=2011PhRvD..83l1301L|hdl=10261/37661 |s2cid=53603505|issn=1550-7998|access-date=2021-11-06|archive-date=2023-01-22|archive-url=https://web.archive.org/web/20230122175047/https://journals.aps.org/prd/abstract/10.1103/PhysRevD.83.121301|url-status=live}}</ref>{{better source needed|date=September 2024}}

[[ESA]]'s [[INTEGRAL]] satellite measured polarization of photons of different wavelengths and was able to place a limit in the granularity of space <ref>https://www.esa.int/Science_Exploration/Space_Science/Integral_challenges_physics_beyond_Einstein</ref> that is less than 10⁻⁴⁸m or 13 orders of magnitude below the Planck scale .

The [[BICEP and Keck Array|BICEP2 experiment]] detected what was initially thought to be primordial [[B-modes|B-mode polarization]] caused by [[gravitational wave]]s 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 [[cosmic dust|interstellar dust]] interference.<ref name="nature-20150130">

Line 527 ⟶ 546:

|journal=[[Nature (journal)|Nature]]

|doi=10.1038/nature.2015.16830

|s2cid=124938210

}}</ref>

}}</ref>

== Thought experiments ==

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,<ref>Yuri.N., Obukhov, "Spin, gravity, and inertia", Physical review letters 86.2 (2001): 192.{{arXiv|0012102v1}}</ref> 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 spin-half particle.

To analyze this question, thought experiments in the context of quantum information, have been suggested.<ref name="Spin Spacetime Censorship"/>

This work shows that, in order to avoid violation of relativistic causality, the measurable spacetime around a spin-half particle's (rest frame) must be spherically symmetric - i.e., either spacetime is spherically symmetric, or somehow measurements of the spacetime (e.g., time-dilation measurements) should create some sort of back action that affects and changes the quantum spin.

== See also ==

{{cols|colwidth=13em16em}}

* [[Abraham–Lorentz force]]

* [[Black hole thermodynamics#Beyond black holes|Beyond black holes]]

* [[Black hole electron]]

* [[Centauro event]]

* [[De Sitter relativity]]

* [[Dilaton]]

* [[Doubly special relativity]]

* [[EventGravitational symmetrydecoherence]]

* [[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)]]

* [[Virtual black hole]]

* [[Weak Gravity Conjecture]]

{{colend}}

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== References ==

{{Reflist|30em}}

== Sources ==

*{{Cite book |last1=Green |first1=Michael B. |title=Superstring theory. 1: Introduction |last2=Schwarz |first2=John H. |last3=Witten |first3=Edward |date=2012 |publisher=Cambridge University Press |isbn=978-1-107-02911-8 |edition=25th Anniversary |volume=l |orig-date=1987 |author-link1=Michael Green (physicist) |author-link2=John Henry Schwarz |author-link3=Edward Witten}}

* {{Cite book |last=Penrose |first=Roger |author-link=Roger Penrose |url=https://archive.org/details/roadtorealitycom00penr_0 |title=The road to reality: a complete guide to the laws of the universe |date=2005 |publisher=Knopf |isbn=978-0-679-45443-4 |___location=New York}}

== Further reading ==

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|bibcode=2002MPLA...17.1135A

|pages=1135–1145

|s2cid=119358167

|ref=harv

}}

* {{Cite bookjournal

|last=Ashtekar

|first=Abhay

|authorlinkauthor-link=Abhay Ashtekar

|title=The winding road to quantum gravity

|url= http://pubman.mpdl.mpg.de/pubman/item/escidoc:150927:1/component/escidoc:150926/2064.pdf

Line 590 ⟶ 602:

|pages=2064–2074

|date=2005

|issue=12

|ref=harv|bibcode=2007laec.book...69A

|jstor=24111069

|doi=10.1142/9789812772718_0005

|isbn=978-981-270-049-0

|citeseerx=10.1.1.616.8952

}}

* {{Cite journal

|last=Carlip

|first=Steven

|authorlinkauthor-link= Steve Carlip

|title=Quantum Gravity: a Progress Report

|journal=[[Reports on Progress in Physics]]

Line 607 ⟶ 617:

|arxiv=gr-qc/0108040

|doi=10.1088/0034-4885/64/8/301

|bibcode = 2001RPPh...64..885C |s2cid=118923209

|ref=harv

}}

|bibcode = 2001RPPh...64..885C }}

* {{citeCite book |last=Hamber author|first=H. W. |editor-first1=Herbert W. |editor-last1=Hamber |url=https://cds.cern.ch/record/1233211 |title= Quantum Gravitationgravitation: |the publisherFeynman =path Springerintegral Natureapproach | date=2009 |publisher=Springer |isbn=978-3-540-85292-6 |___location=Berlin |doi=10.1007/978-3-540-85293-3 | isbnhdl=97811858/00-3001M-5400000-852920013-6471D-A | urloclc= http://cds.cern.ch/record/1233211 248994165}}

* {{Cite book

|last=Kiefer

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|publisher=Oxford University Press

|isbn=978-0-19-921252-1

}}

|ref=harv}}

* {{Cite journal

|last=Kiefer

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|doi=10.1002/andp.200510175

|pages=129–148

|bibcode = 2006AnP...518..129K |s2cid=12984346

|ref=harv

}}

|bibcode = 2006AnP...518..129K }}

* {{Cite book

|editor-last=Lämmerzahl

Line 639 ⟶ 649:

|publisher= Springer

|isbn=978-3-540-40810-9

}}

|ref=harv}}

* {{Cite book

|last=Rovelli

|first=Carlo

|authorlinkauthor-link=Carlo Rovelli

|title=Quantum Gravity

|date=2004

|publisher=Cambridge University Press

|isbn=978-0-521-83733-0

}}

|ref=harv}}

* {{Cite journal

|last=Trifonov

|first=Vladimir

|title=GR-friendly description of quantum systems

|date=2008

|journal=[[International Journal of Theoretical Physics]]

|volume=47|issue=2|pages=492–510

|doi=10.1007/s10773-007-9474-3

|arxiv=math-ph/0702095

|ref=harv

|bibcode = 2008IJTP...47..492T }}

== External links ==

{{Wikiquote}}

* {{cite SEP|url-id=quantum-gravity|title=Quantum Gravity|author-last1=Weinstein|author-first1=Steven|author-last2=Rickles|first2=Dean}}

* [http://abyss.uoregon.edu/~js/cosmo/lectures/lec20.html "Planck Era" and "Planck Time"] (up to 10⁻⁴³ seconds after [[Big Bang|birth]] of [[Universe]]) ([[University of Oregon]]).

* [http://abyss.uoregon.edu/~js/cosmo/lectures/lec20.html "Planck Era" and "Planck Time"] {{Webarchive|url=https://web.archive.org/web/20181128045313/http://abyss.uoregon.edu/~js/cosmo/lectures/lec20.html |date=2018-11-28 }} (up to 10⁻⁴³ seconds after [[Big Bang|birth]] of [[Universe]]) ([[University of Oregon]]).

* [http://www.bbc.co.uk/programmes/p00547c4 "Quantum Gravity"], BBC Radio 4 discussion with John Gribbin, Lee Smolin and Janna Levin (''In Our Time'', Feb. 22, 2001)

* [http://www.bbc.co.uk/programmes/p00547c4 "Quantum Gravity"], BBC Radio 4 discussion with John Gribbin, Lee Smolin and Janna Levin (''In Our Time'', February 22, 2001)

{{Quantum gravity}}

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{{portal bar|Physics|Science}}

[[Category:Quantum gravity| ]]

[[Category:General relativity]]

[[Category:Physics beyond the Standard Model]]

[[Category:QuantumTheories of gravity|*]]

[[Category:Theories of gravitation]]