Editing Quantum gravity (section)

== Candidate theories ==
There are a number of proposed quantum gravity theories.<ref>A timeline and overview can be found in {{Cite arXiv
|last=Rovelli
|first=Carlo
|author-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
|last=Ashtekar |first=Abhay
|author-link=Abhay Ashtekar
|date=2007
|title=Loop Quantum Gravity: Four Recent Advances and a Dozen Frequently Asked Questions
|conference=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
|isbn=978-981-283-426-3
|s2cid=119663169
}}</ref><ref>{{cite journal
|last=Schwarz
|first=John H.
|title=String Theory: Progress and Problems
|journal=[[Progress of Theoretical Physics Supplement]]
|volume=170
|pages=214–226
|date=2007
|arxiv=hep-th/0702219
|doi=10.1143/PTPS.170.214
|bibcode=2007PThPS.170..214S
|s2cid=16762545
}}</ref>

=== String theory ===
{{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]]

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 one-dimensional extended objects: string theory.<ref>An accessible introduction at the undergraduate level can be found in {{Cite book
|last=Zwiebach
|first=Barton
|author-link= Barton Zwiebach
|title=A First Course in String Theory
|publisher=[[Cambridge University Press]]
|date=2004
|isbn=978-0-521-83143-7
}}, and more complete overviews in {{Cite book
|last=Polchinski
|first=Joseph
|author-link=Joseph Polchinski
|date=1998
|title=String Theory Vol. I: An Introduction to the Bosonic String
|publisher=[[Cambridge University Press]]
|isbn=978-0-521-63303-1
}} and {{Cite book
|last=Polchinski
|first=Joseph
|author-link=Joseph Polchinski
|date=1998b
|title=String Theory Vol. II: Superstring Theory and Beyond
|publisher=[[Cambridge University Press]]
|isbn=978-0-521-63304-8
}}</ref> At the energies reached in current experiments, these strings are indistinguishable from point-like particles, but, crucially, different [[Normal mode|modes]] of oscillation of one and the same type of fundamental string appear as particles with different ([[electric]] and other) [[charge (physics)|charges]]. In this way, string theory promises to be a [[theory of everything|unified description]] of all particles and interactions.<ref>{{Cite journal
|last=Ibanez
|first=L. E.
|title=The second string (phenomenology) revolution
|journal=[[Classical and Quantum Gravity]]
|volume=17
|issue=5
|date=2000
|pages=1117–1128
|arxiv=hep-ph/9911499
|doi=10.1088/0264-9381/17/5/321
|bibcode = 2000CQGra..17.1117I |s2cid=15707877
}}</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 is 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|2012|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
|last=Weinberg
|first=Steven
|chapter=Chapter 31
|author-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
|isbn=978-0-521-55002-4
|url-access=registration
|url=https://archive.org/details/quantumtheoryoff00stev
}}</ref> form part of a hypothesized eleven-dimensional model known as [[M-theory]], which would constitute a uniquely defined and consistent theory of quantum gravity.<ref>{{Cite journal
|last=Townsend
|first=Paul K.
|title=Four Lectures on M-Theory
|journal=High Energy Physics and Cosmology
|volume=13
|series=ICTP Series in Theoretical Physics
|page=385
|date=1996
|arxiv=hep-th/9612121
|bibcode=1997hepcbconf..385T
}}</ref><ref>{{Cite journal
|last=Duff
|first=Michael
|author-link=Michael Duff (physicist)
|title=M-Theory (the Theory Formerly Known as Strings)
|journal=[[International Journal of Modern Physics A]]
|volume=11 |issue=32
|date=1996
|pages=5623–5642
|doi=10.1142/S0217751X96002583
|arxiv=hep-th/9608117
|bibcode=1996IJMPA..11.5623D
|s2cid=17432791
}}</ref> As presently understood, however, string theory admits a very large number (10<sup>500</sup> 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|thumb|upright=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 that there is 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 spectra. 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
|author-link=Abhay Ashtekar
|title=New variables for classical and quantum gravity
|journal=[[Physical Review Letters]]
|volume=57
|pages=2244–2247
|date=1986
|doi=10.1103/PhysRevLett.57.2244
|pmid=10033673
|issue=18
|bibcode=1986PhRvL..57.2244A
}}</ref><ref>{{Cite journal
|last=Ashtekar
|first=Abhay
|author-link=Abhay Ashtekar
|title=New Hamiltonian formulation of general relativity
|journal=[[Physical Review D]]
|volume=36|issue=6|pages=1587–1602
|date=1987
|doi=10.1103/PhysRevD.36.1587
|pmid=9958340
|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=Rovelli
|first=Carlo
|author-link=Carlo Rovelli
|title=Loop Quantum Gravity
|journal=[[Living Reviews in Relativity]]
|volume=1
|date=1998
|issue=1
|page=1
|doi=10.12942/lrr-1998-1
|doi-access=free
|pmid=28937180
|pmc=5567241
|arxiv=gr-qc/9710008
|bibcode=1998LRR.....1....1R
}}</ref><ref>{{cite journal
 | last1=Ashtekar
 | first1=Abhay
 | author-link=Abhay Ashtekar
 | first2=Jerzy
 | last2=Lewandowski
 | title=Background Independent Quantum Gravity: A Status Report
 | journal=[[Classical and Quantum Gravity]]
 | volume=21
 | date=2004
 | issue=15
 | pages=R53–R152
 | 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
|chapter=Lectures on Loop Quantum Gravity
|date=2003
|volume=631
|pages=41–135
|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
|title=Quantum Gravity
|date=2004
|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.

=== Other theories ===
There are a number of other approaches to quantum gravity. The theories 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
|author-link=Christopher Isham
|contribution=Prima facie questions in quantum gravity
|editor-last=Ehlers
|editor-first=Jürgen
|editor2-last=Friedrich
|editor2-first=Helmut
|volume=434
|pages=1–21
|date=1994
|publisher=Springer
|arxiv=gr-qc/9310031
|isbn=978-3-540-58339-4
|bibcode=1994LNP...434....1I
|doi=10.1007/3-540-58339-4_13
|series=Lecture Notes in Physics
|s2cid=119364176
}}</ref><ref>{{Cite journal
|last=Sorkin
|first=Rafael D.
|author-link= Rafael Sorkin
|title=Forks in the Road, on the Way to Quantum Gravity
|arxiv=gr-qc/9706002
|journal=[[International Journal of Theoretical Physics]]
|volume=36
|date=1997
|issue=12
|pages=2759–2781
|doi=10.1007/BF02435709
|bibcode = 1997IJTP...36.2759S |s2cid=4803804
}}</ref> Such theories 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
|first=Renate
|title=Discrete Approaches to Quantum Gravity in Four Dimensions
|journal=[[Living Reviews in Relativity]]
|volume=1
|issue=1
|page=13
|date=1998
|bibcode=1998LRR.....1...13L
|arxiv = gr-qc/9805049 |doi = 10.12942/lrr-1998-13 |doi-access=free
|pmid=28191826
 |pmc=5253799
}}</ref>
* [[Causal fermion system]]s
* [[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]]
* [[Geometrodynamics]]
* [[Hořava–Lifshitz gravity]]
* [[MacDowell–Mansouri action]]
* [[noncommutative standard model|Noncommutative geometry]]
* [[path integral formulation|Path-integral]] based models of [[quantum cosmology]]<ref>{{Cite book
|last=Hawking
|first=Stephen W.
|author-link=Stephen Hawking
|contribution=Quantum cosmology
|pages =631–651
|editor2-last=Israel
|editor2-first=Werner
|editor1-last=Hawking
|editor1-first=Stephen W.
|title=300 Years of Gravitation
|publisher=Cambridge University Press
|date=1987
|isbn=978-0-521-37976-2
}}</ref>
* [[Regge calculus]]
* [[Shape dynamics|Shape Dynamics]]
* [[String-net]]s and [[quantum graphity]]
* [[Supergravity]]
* [[Twistor theory]]<ref>See ch. 33 in {{Harvnb|Penrose|2005}} and references therein.</ref>
* [[Canonical quantum gravity]]
}}