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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]].
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"/>{{rp|xxxvi&ndash;xxxviii;211&ndash;212}}<ref>{{ cite book | last= Hamber | first= H. W. | title= Quantum Gravitation – The Feynman Path Integral Approach | publisher = Springer Publishing | 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.
However, gravity is perturbatively [[nonrenormalizable]].<ref name="FLoG"/>{{rp|xxxvi&ndash;xxxviii;211&ndash;212}}<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.{{citation needed|date=February 2019}}
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.{{citation needed|date=February 2019}}
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* {{cite book | author=Herbert W. Hamber | title= Quantum Gravitation | publisher = Springer Publishing | date=2009 | doi=10.1007/978-3-540-85293-3 | isbn=978-3-540-85292-6| url= http://cds.cern.ch/record/1233211 }}
* {{cite book | author=Herbert W. Hamber | title= Quantum Gravitation | publisher = Springer Nature | date=2009 | doi=10.1007/978-3-540-85293-3 | isbn=978-3-540-85292-6| url= http://cds.cern.ch/record/1233211 }}
* {{Cite book
* {{Cite book
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Revision as of 02:34, 21 February 2019

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 near compact astrophysical objects where the effects of gravity are strong.

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 forces of physics are described within the framework of quantum mechanics and quantum field theory, radically different formalisms for describing physical phenomena.[2] 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.[3][4]: 11–12 

While a quantum theory of gravity may be needed to reconcile general relativity with the principles of quantum mechanics, difficulties arise when applying the usual prescriptions of quantum field theory to the force of gravity via graviton bosons.[5] The problem is that the theory one gets in this way is not renormalizable (it predicts infinite values for some observable properties such as the mass of particles) and therefore cannot be used to make meaningful physical predictions. As a result, theorists have taken up more radical approaches to the problem of quantum gravity, the most popular approaches being string theory and loop quantum gravity.[6] Although some quantum gravity theories, such as string theory, try to unify gravity with the other fundamental forces, 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.

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 unifying all fundamental interactions into a single mathematical framework. A quantum field theory of gravity that is unified with a 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.

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 accelerators. Therefore physicists lack experimental data which could distinguish between the competing theories which have been proposed[7][8] and thus gedanken experimental approaches are suggested as a testing tool for these theories.[9][10] [11]

Overview

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)
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."[12] 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.[5] 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.[13]

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 relativity

File:Gravity Probe B.jpg
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 article: Graviton

At present, one of the deepest problems in theoretical physics is harmonizing the theory of general relativity, which describes gravitation, and applications to large-scale structures (stars, planets, galaxies), with quantum mechanics, which describes the other three fundamental forces acting on the atomic scale. This problem must be put in the proper context, however. In particular, contrary to the popular claim that quantum mechanics and general relativity are fundamentally incompatible, one can demonstrate that the structure of general relativity essentially follows inevitably from the quantum mechanics of interacting theoretical spin-2 massless particles (called gravitons).[14][15][16][17][18]

No concrete proof of gravitons exists, but quantized theories of matter may necessitate their existence.[19] 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.

The Weinberg–Witten theorem places some constraints on theories in which the graviton is a composite particle.[20][21]

Dilaton

Main article: Dilaton

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[22] 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 theory,[23] 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.[24]

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[25] that is seen in condensed matter physics and superfluids. The field equations are amenable to such a generalization, as shown with the inclusion of a one-graviton process,[26] 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.[27] 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.

Nonrenormalizability of gravity

Further information: Renormalization

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 [28] 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.[citation needed]

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 find a reliable answer, but some people still pursue this option. 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.[29][better source needed]

Quantum gravity as an effective field theory

Main article: Effective field theory

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 low-energy effects. Thus, at least in the low-energy regime, the model is a predictive quantum field theory.[30] 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.[31]

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.[30]

Spacetime background dependence

Main article: 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 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,[32] in which the only physically relevant information is the relationship between different events in space-time.

On the other hand, quantum mechanics has depended since its inception on a fixed background (non-dynamic) structure. In the case of quantum mechanics, it is time that is given and not dynamic, just as in Newtonian classical mechanics. In relativistic quantum field theory, just as in classical field theory, Minkowski spacetime is the fixed background of the theory.

String theory

Interaction in the subatomic world: world lines of point-like particles in the Standard Model or a world sheet swept up by closed 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 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 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 networks.

Semi-classical quantum 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.

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 article: 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 operator acting as the generator of infinitesimal translations of quantum states through time.[33] In contrast, general relativity treats time as a dynamical variable which interacts directly with matter and moreover requires the Hamiltonian constraint to vanish,[34] removing any possibility of employing a notion of time similar to that in quantum theory.

Candidate theories

There are a number of proposed quantum gravity theories.[35] 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.[36][37]

String theory

Main article: String theory
Projection of a Calabi–Yau manifold, one of the ways of 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 (quantum) field theory of gravity at low energies,[30] 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,[38] 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.[39]

One attempt to overcome these limitations is to replace ordinary quantum field theory, which is based on the classical concept of a point particle, with a quantum theory of one-dimensional extended objects: string theory.[40] At the energies reached in current experiments, these strings are indistinguishable from point-like particles, but, crucially, different modes of oscillation of one and the same type of fundamental string appear as particles with different (electric and other) charges. In this way, string theory promises to be a unified description of all particles and interactions.[41] 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.[42]

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[43] form part of a hypothesized eleven-dimensional model known as M-theory, which would constitute a uniquely defined and consistent theory of quantum gravity.[44][45] 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 article: Loop quantum gravity
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 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 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 and magnetic fields.[46][47] In the quantum theory, space is represented by a network structure called a spin network, evolving over time in discrete steps.[48][49][50][51]

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.[52] 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 approaches

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.[53][54] Examples include:

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, 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.[59]

The most widely pursued possibilities for quantum gravity phenomenology include violations of Lorentz invariance, imprints of quantum gravitational effects in the cosmic microwave background (in particular its polarization), and decoherence induced by fluctuations in the space-time foam.

The BICEP2 experiment detected what was initially thought to be primordial B-mode 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.[60]

See also

References

  1. ^ a b Rovelli, Carlo (2008). "Quantum gravity". Scholarpedia. 3 (5): 7117. Bibcode:2008SchpJ...3.7117R. doi:10.4249/scholarpedia.7117.{{cite journal}}: CS1 maint: unflagged free DOI (link)
  2. ^ Griffiths, David J. (2004). Introduction to Quantum Mechanics. Pearson Prentice Hall. OCLC 803860989.
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  5. ^ a b Zee, Anthony (2010). Quantum Field Theory in a Nutshell (second ed.). Princeton University Press. pp. 172, 434–435. ISBN 978-0-691-14034-6. OCLC 659549695.
  6. ^ Penrose, Roger (2007). The road to reality : a complete guide to the laws of the universe. Vintage. p. 1017. OCLC 716437154.
  7. ^ 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.
  8. ^ On the quantization of the geometry of spacetime, see also in the article Planck length, in the examples
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  11. ^ Nemirovsky, J.; Cohen, E.; Kaminer, I. (30 Dec 2018). "Spin Spacetime Censorship". arXiv:1812.11450v1 [gr-qc]. {{cite arXiv}}: Invalid |ref=harv (help)
  12. ^ Wheeler, John Archibald (2010). Geons, Black Holes, and Quantum Foam: A Life in Physics. W. W. Norton & Company. p. 235. ISBN 9780393079487.
  13. ^ Wald, Robert M. (1994). Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. University of Chicago Press. ISBN 978-0-226-87027-4.
  14. ^ Kraichnan, R. H. (1955). "Special-Relativistic Derivation of Generally Covariant Gravitation Theory". Physical Review. 98 (4): 1118–1122. Bibcode:1955PhRv...98.1118K. doi:10.1103/PhysRev.98.1118. {{cite journal}}: Invalid |ref=harv (help)
  15. ^ Gupta, S. N. (1954). "Gravitation and Electromagnetism". Physical Review. 96 (6): 1683–1685. Bibcode:1954PhRv...96.1683G. doi:10.1103/PhysRev.96.1683. {{cite journal}}: Invalid |ref=harv (help)
  16. ^ 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. {{cite journal}}: Invalid |ref=harv (help)
  17. ^ Gupta, S. N. (1962). "Quantum Theory of Gravitation". Recent Developments in General Relativity. Pergamon Press. pp. 251–258.
  18. ^ Deser, S. (1970). "Self-Interaction and Gauge Invariance". General Relativity and Gravitation. 1: 9–18. arXiv:gr-qc/0411023. Bibcode:1970GReGr...1....9D. doi:10.1007/BF00759198. {{cite journal}}: Invalid |ref=harv (help)
  19. ^ Charles Ginenthal (2015-12-07). Newton, Einstein, and Velikovsky. ISBN 9781329742567.
  20. ^ Weinberg, Steven; Witten, Edward (1980). "Limits on massless particles". Physics Letters B. 96 (1–2): 59–62. Bibcode:1980PhLB...96...59W. doi:10.1016/0370-2693(80)90212-9.
  21. ^ Horowitz, Gary T.; Polchinski, Joseph (2006). "Gauge/gravity duality". In Oriti, Daniele (ed.). Approaches to Quantum Gravity. Cambridge University Press. arXiv:gr-qc/0602037. Bibcode:2006gr.qc.....2037H. ISBN 9780511575549. OCLC 873715753.
  22. ^ Ohta, Tadayuki; Mann, Robert (1996). "Canonical reduction of two-dimensional gravity for particle dynamics". Classical and Quantum Gravity. 13 (9): 2585–2602. arXiv:gr-qc/9605004. Bibcode:1996CQGra..13.2585O. doi:10.1088/0264-9381/13/9/022. {{cite journal}}: Invalid |ref=harv (help)
  23. ^ Sikkema, A E; Mann, R B (1991). "Gravitation and cosmology in (1+1) dimensions". Classical and Quantum Gravity. 8 (1): 219–235. Bibcode:1991CQGra...8..219S. doi:10.1088/0264-9381/8/1/022. {{cite journal}}: Invalid |ref=harv (help)
  24. ^ Farrugia; Mann; Scott (2007). "N-body Gravity and the Schroedinger Equation". Classical and Quantum Gravity. 24 (18): 4647–4659. arXiv:gr-qc/0611144. Bibcode:2007CQGra..24.4647F. doi:10.1088/0264-9381/24/18/006.
  25. ^ Scott, T.C.; Zhang, Xiangdong; Mann, Robert; Fee, G.J. (2016). "Canonical reduction for dilatonic gravity in 3 + 1 dimensions". Physical Review D. 93 (8): 084017. arXiv:1605.03431. Bibcode:2016PhRvD..93h4017S. doi:10.1103/PhysRevD.93.084017.
  26. ^ Mann, R B; Ohta, T (1997). "Exact solution for the metric and the motion of two bodies in (1+1)-dimensional gravity". Phys. Rev. D. 55 (8): 4723–4747. arXiv:gr-qc/9611008. Bibcode:1997PhRvD..55.4723M. doi:10.1103/PhysRevD.55.4723. {{cite journal}}: Invalid |ref=harv (help)
  27. ^ Bellazzini, B.; Csaki, C.; Hubisz, J.; Serra, J.; Terning, J. (2013). "A higgs-like dilaton". Eur. Phys. J. C. 73 (2): 2333. arXiv:1209.3299. Bibcode:2013EPJC...73.2333B. doi:10.1140/epjc/s10052-013-2333-x.
  28. ^ Hamber, H. W. (2009). Quantum Gravitation – The Feynman Path Integral Approach. Springer Nature. ISBN 978-3-540-85292-6.
  29. ^ Distler, Jacques (2005-09-01). "Motivation". golem.ph.utexas.edu. Retrieved 2018-02-24.
  30. ^ a b c 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:gr-qc/9512024. Bibcode:1995gr.qc....12024D. ISBN 978-981-02-2908-5. {{cite book}}: |first= has generic name (help)
  31. ^ Zinn-Justin, Jean (2007). Phase transitions and renormalization group. Oxford: Oxford University Press. ISBN 9780199665167. OCLC 255563633.
  32. ^ Smolin, Lee (2001). Three Roads to Quantum Gravity. Basic Books. pp. 20–25. ISBN 978-0-465-07835-6. Pages 220–226 are annotated references and guide for further reading.
  33. ^ Sakurai, J. J.; Napolitano, Jim J. (2010-07-14). Modern Quantum Mechanics (2 ed.). Pearson. p. 68. ISBN 978-0-8053-8291-4.
  34. ^ Novello, Mario; Bergliaffa, Santiago E. (2003-06-11). 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 978-0-7354-0131-0.
  35. ^ A timeline and overview can be found in Rovelli, Carlo (2000). "Notes for a brief history of quantum gravity". arXiv:gr-qc/0006061. {{cite arXiv}}: Invalid |ref=harv (help) (verify against ISBN 9789812777386)
  36. ^ Ashtekar, Abhay (2007). "Loop Quantum Gravity: Four Recent Advances and a Dozen Frequently Asked Questions". 11th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity. p. 126. arXiv:0705.2222. Bibcode:2008mgm..conf..126A. doi:10.1142/9789812834300_0008. ISBN 978-981-283-426-3. {{cite book}}: |journal= ignored (help); Invalid |ref=harv (help)
  37. ^ Schwarz, John H. (2007). "String Theory: Progress and Problems". Progress of Theoretical Physics Supplement. 170: 214–226. arXiv:hep-th/0702219. Bibcode:2007PThPS.170..214S. doi:10.1143/PTPS.170.214. {{cite journal}}: Invalid |ref=harv (help)
  38. ^ Weinberg, Steven (1996). "Chapters 17–18". The Quantum Theory of Fields II: Modern Applications. Cambridge University Press. ISBN 978-0-521-55002-4.
  39. ^ 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:10.1016/0370-2693(85)91470-4. {{cite journal}}: Invalid |ref=harv (help)
  40. ^ An accessible introduction at the undergraduate level can be found in Zwiebach, Barton (2004). A First Course in String Theory. Cambridge University Press. ISBN 978-0-521-83143-7. {{cite book}}: Invalid |ref=harv (help), and more complete overviews in Polchinski, Joseph (1998). String Theory Vol. I: An Introduction to the Bosonic String. Cambridge University Press. ISBN 978-0-521-63303-1. and Polchinski, Joseph (1998b). String Theory Vol. II: Superstring Theory and Beyond. Cambridge University Press. ISBN 978-0-521-63304-8.
  41. ^ Ibanez, L. E. (2000). "The second string (phenomenology) revolution". Classical and Quantum Gravity. 17 (5): 1117–1128. arXiv:hep-ph/9911499. Bibcode:2000CQGra..17.1117I. doi:10.1088/0264-9381/17/5/321. {{cite journal}}: Invalid |ref=harv (help)
  42. ^ 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.
  43. ^ Weinberg, Steven (2000). "Chapter 31". The Quantum Theory of Fields II: Modern Applications. Cambridge University Press. ISBN 978-0-521-55002-4.
  44. ^ Townsend, Paul K. (1996). "Four Lectures on M-Theory". High Energy Physics and Cosmology. ICTP Series in Theoretical Physics. 13: 385. arXiv:hep-th/9612121. Bibcode:1997hepcbconf..385T. {{cite journal}}: Invalid |ref=harv (help); Unknown parameter |booktitle= ignored (help)
  45. ^ Duff, Michael (1996). "M-Theory (the Theory Formerly Known as Strings)". International Journal of Modern Physics A. 11 (32): 5623–5642. arXiv:hep-th/9608117. Bibcode:1996IJMPA..11.5623D. doi:10.1142/S0217751X96002583. {{cite journal}}: Invalid |ref=harv (help)
  46. ^ 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. {{cite journal}}: Invalid |ref=harv (help)
  47. ^ 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. {{cite journal}}: Invalid |ref=harv (help)
  48. ^ Thiemann, Thomas (2007). Loop Quantum Gravity: An Inside View. Lecture Notes in Physics. Vol. 721. pp. 185–263. arXiv:hep-th/0608210. Bibcode:2007LNP...721..185T. doi:10.1007/978-3-540-71117-9_10. ISBN 978-3-540-71115-5. {{cite book}}: |journal= ignored (help); Invalid |ref=harv (help)
  49. ^ Rovelli, Carlo (1998). "Loop Quantum Gravity". Living Reviews in Relativity. 1. Retrieved 2008-03-13. {{cite journal}}: Invalid |ref=harv (help)
  50. ^ Ashtekar, Abhay; Lewandowski, Jerzy (2004). "Background Independent Quantum Gravity: A Status Report". Classical and Quantum Gravity. 21 (15): R53–R152. arXiv:gr-qc/0404018. Bibcode:2004CQGra..21R..53A. doi:10.1088/0264-9381/21/15/R01.
  51. ^ Thiemann, Thomas (2003). Lectures on Loop Quantum Gravity. Lecture Notes in Physics. Vol. 631. pp. 41–135. arXiv:gr-qc/0210094. Bibcode:2003LNP...631...41T. doi:10.1007/978-3-540-45230-0_3. ISBN 978-3-540-40810-9. {{cite book}}: Invalid |ref=harv (help)
  52. ^ Rovelli, Carlo (2004). Quantum Gravity. Cambridge University Press. ISBN 978-0-521-71596-6.
  53. ^ Isham, Christopher J. (1994). "Prima facie questions in quantum gravity". In Ehlers, Jürgen; Friedrich, Helmut (eds.). Canonical Gravity: From Classical to Quantum. Lecture Notes in Physics. Vol. 434. Springer. pp. 1–21. arXiv:gr-qc/9310031. Bibcode:1994LNP...434....1I. doi:10.1007/3-540-58339-4_13. ISBN 978-3-540-58339-4. {{cite book}}: |journal= ignored (help)
  54. ^ Sorkin, Rafael D. (1997). "Forks in the Road, on the Way to Quantum Gravity". International Journal of Theoretical Physics. 36 (12): 2759–2781. arXiv:gr-qc/9706002. Bibcode:1997IJTP...36.2759S. doi:10.1007/BF02435709. {{cite journal}}: Invalid |ref=harv (help)
  55. ^ Loll, Renate (1998). "Discrete Approaches to Quantum Gravity in Four Dimensions". Living Reviews in Relativity. 1: 13. arXiv:gr-qc/9805049. Bibcode:1998LRR.....1...13L. doi:10.12942/lrr-1998-13. PMID 28191826. Retrieved 2008-03-09. {{cite journal}}: Invalid |ref=harv (help)CS1 maint: unflagged free DOI (link)
  56. ^ Hawking, Stephen W. (1987). "Quantum cosmology". In Hawking, Stephen W.; Israel, Werner (eds.). 300 Years of Gravitation. Cambridge University Press. pp. 631–651. ISBN 978-0-521-37976-2.
  57. ^ See ch. 33 in Penrose 2004 and references therein.
  58. ^ 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.
  59. ^ 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 978-1-61122-957-8.{{cite book}}: CS1 maint: location (link)
  60. ^ Cowen, Ron (30 January 2015). "Gravitational waves discovery now officially dead". Nature. doi:10.1038/nature.2015.16830.

Further reading

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