Quantum gravity

Last updated

Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics, and where quantum effects cannot be ignored, [1] such as in the vicinity of black holes or similar compact astrophysical objects where the effects of gravity are strong, such as neutron stars.


Three of the four fundamental forces of physics are described within the framework of quantum mechanics and quantum field theory. The current understanding of the fourth force, gravity, is based on Albert Einstein's general theory of relativity, which is formulated within the entirely different framework of classical physics. However, that description is incomplete: describing the gravitational field of a black hole in the general theory of relativity, physical quantities such as the spacetime curvature diverge at the center of the black hole.

This signals the breakdown of the general theory of relativity and the need for a theory that goes beyond general relativity into the quantum. At distances very close to the center of the black hole (closer than the Planck length), quantum fluctuations of spacetime are expected to play an important role. [2] To describe these quantum effects a theory of quantum gravity is needed. Such a theory should allow the description to be extended closer to the center and might even allow an understanding of physics at the center of a black hole. On more formal grounds, one can argue that a classical system cannot consistently be coupled to a quantum one. [3] [4] :1112

The field of quantum gravity is actively developing and theorists are exploring a variety of approaches to the problem of quantum gravity, the most popular approaches being M-theory and loop quantum gravity. [5] All of these approaches aim to describe the quantum behavior of the gravitational field. This does not necessarily include unifying all fundamental interactions into a single mathematical framework. However, many approaches to quantum gravity, such as string theory, try to develop a framework that describes all fundamental forces. Such theories are often referred to as a theory of everything . Others, such as loop quantum gravity, make no such attempt; instead, they make an effort to quantize the gravitational field while it is kept separate from the other forces.

One of the difficulties of formulating a quantum gravity theory is that quantum gravitational effects only appear at length scales near the Planck scale, around 10−35 meters, a scale far smaller, and hence only accessible with far higher energies, than those currently available in high energy particle accelerators. Therefore, physicists lack experimental data which could distinguish between the competing theories which have been proposed [n.b. 1] [n.b. 2] and thus thought experiment approaches are suggested as a testing tool for these theories. [6] [7] [8]


Question, Web Fundamentals.svgUnsolved 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 Quantum gravity.svg
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." [9] On the other hand, quantum field theory is typically formulated in the flat spacetime used in special relativity. No theory has yet proven successful in describing the general situation where the dynamics of matter, modeled with quantum mechanics, affect the curvature of spacetime. If one attempts to treat gravity as simply another quantum field, the resulting theory is not renormalizable. [10] Even in the simpler case where the curvature of spacetime is fixed a priori, developing quantum field theory becomes more mathematically challenging, and many ideas physicists use in quantum field theory on flat spacetime are no longer applicable. [11]

It is widely hoped that a theory of quantum gravity would allow us to understand problems of very high energy and very small dimensions of space, such as the behavior of black holes, and the origin of the universe. [1]

Quantum mechanics and general relativity

Gravity Probe B (GP-B) has measured spacetime curvature near Earth to test related models in application of Einstein's general theory of relativity. 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.


The observation that all fundamental forces except gravity have one or more known messenger particles leads researchers to believe that at least one must exist for gravity. This hypothetical particle is known as the graviton. These particles act as a force particle similar to the photon of the electromagnetic interaction. Under mild assumptions, the structure of general relativity requires them to follow the quantum mechanical description of interacting theoretical spin-2 massless particles. [12] [13] [14] [15] [16] Many of the accepted notions of a unified theory of physics since the 1970s assume, and to some degree depend upon, the existence of the graviton. The Weinberg–Witten theorem places some constraints on theories in which the graviton is a composite particle. [17] [18] While gravitons are an important theoretical step in a quantum mechanical description of gravity, they are generally believed to be indetectable because they interact too weakly. [19]

Nonrenormalizability of 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. [4] :xxxvixxxviii;211212 [20] For a quantum field theory to be well defined according to this understanding of the subject, it must be asymptotically free or asymptotically safe. The theory must be characterized by a choice of finitely many parameters, which could, in principle, be set by experiment. For example, in quantum electrodynamics these parameters are the charge and mass of the electron, as measured at a particular energy scale.

On the other hand, in quantizing gravity there are, in perturbation theory, infinitely many independent parameters (counterterm coefficients) needed to define the theory. For a given choice of those parameters, one could make sense of the theory, but since it is impossible to conduct infinite experiments to fix the values of every parameter, it has been argued that one does not, in perturbation theory, have a meaningful physical theory. At low energies, the logic of the renormalization group tells us that, despite the unknown choices of these infinitely many parameters, quantum gravity will reduce to the usual Einstein theory of general relativity. On the other hand, if we could probe very high energies where quantum effects take over, then every one of the infinitely many unknown parameters would begin to matter, and we could make no predictions at all. [21]

It is conceivable that, in the correct theory of quantum gravity, the infinitely many unknown parameters will reduce to a finite number that can then be measured. One possibility is that normal perturbation theory is not a reliable guide to the renormalizability of the theory, and that there really is a UV fixed point for gravity. Since this is a question of non-perturbative quantum field theory, finding a reliable answer is difficult, pursued in the asymptotic safety program. Another possibility is that there are new, undiscovered symmetry principles that constrain the parameters and reduce them to a finite set. This is the route taken by string theory, where all of the excitations of the string essentially manifest themselves as new symmetries. [22] [ better source needed ]

Quantum gravity as an effective field 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. [23] Furthermore, many theorists argue that the Standard Model should be regarded as an effective field theory itself, with "nonrenormalizable" interactions suppressed by large energy scales and whose effects have consequently not been observed experimentally. [24]

By treating general relativity as an effective field theory, one can actually make legitimate predictions for quantum gravity, at least for 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. [23]

Spacetime background dependence

A fundamental lesson of general relativity is that there is no fixed spacetime background, as found in Newtonian mechanics and special relativity; the spacetime geometry is dynamic. While easy to grasp in principle, this is the hardest idea to understand about general relativity, and its consequences are profound and not fully explored, even at the classical level. To a certain extent, general relativity can be seen to be a relational theory, [25] in which the only physically relevant information is the relationship between different events in 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 Point&string.png
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.[ citation needed ]

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

A conceptual difficulty in combining quantum mechanics with general relativity arises from the contrasting role of time within these two frameworks. In quantum theories time acts as an independent background through which states evolve, with the Hamiltonian operator acting as the generator of infinitesimal translations of quantum states through time. [26] In contrast, general relativity treats time as a dynamical variable which relates directly with matter and moreover requires the Hamiltonian constraint to vanish. [27] Because this variability of time has been observed macroscopically, it removes any possibility of employing a fixed notion of time, similar to the conception of time in quantum theory, at the macroscopic level.

Candidate theories

There are a number of proposed quantum gravity theories. [28] Currently, there is still no complete and consistent quantum theory of gravity, and the candidate models still need to overcome major formal and conceptual problems. They also face the common problem that, as yet, there is no way to put quantum gravity predictions to experimental tests, although there is hope for this to change as future data from cosmological observations and particle physics experiments becomes available. [29] [30]

String theory

Projection of a Calabi-Yau manifold, one of the ways of compactifying the extra dimensions posited by string theory Calabi-Yau.png
Projection of a Calabi–Yau manifold, one of the ways of 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. [31] 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. [32] The theory is successful in that one mode will always correspond to a graviton, the messenger particle of gravity; however, the price of this success are unusual features such as six extra dimensions of space in addition to the usual three for space and one for time. [33]

In what is called the second superstring revolution, it was conjectured that both string theory and a unification of general relativity and supersymmetry known as supergravity [34] form part of a hypothesized eleven-dimensional model known as M-theory, which would constitute a uniquely defined and consistent theory of quantum gravity. [35] [36] As presently understood, however, string theory admits a very large number (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

Simple spin network of the type used in loop quantum gravity Spin network.svg
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. [37] [38] In the quantum theory, space is represented by a network structure called a spin network, evolving over time in discrete steps. [39] [40] [41] [42]

The dynamics of the theory is today constructed in several versions. One version starts with the canonical quantization of general relativity. The analogue of the Schrödinger equation is a Wheeler–DeWitt equation, which can be defined within the theory. [43] In the covariant, or spinfoam formulation of the theory, the quantum dynamics is obtained via a sum over discrete versions of spacetime, called spinfoams. These represent histories of spin networks.

Other 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. [44] [45] 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. [54]

The most widely pursued possibilities for quantum gravity phenomenology include violations of Lorentz invariance, imprints of quantum gravitational effects in the cosmic microwave background (in particular its polarization), and decoherence induced by fluctuations [55] [56] [57] in the space-time foam. [58]

ESA's INTEGRAL satellite measured polarization of photons of different wavelengths and was able to place a limit in the granularity of space [59] that is less than 10⁻⁴⁸m or 13 orders of magnitude below the Planck scale .

The BICEP2 experiment detected what was initially thought to be primordial 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]

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 the coupling of spin and spacetime. While spin and spacetime are expected to be coupled, [61] 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. [8] 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


  1. 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.
  2. On the quantization of the geometry of spacetime, see also in the article Planck length, in the examples

Related Research Articles

General relativity Einsteins theory of gravitation as curved spacetime

General relativity, also known as the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time or four-dimensional spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.

In theories of quantum gravity, the graviton is the hypothetical quantum of gravity, an elementary particle that mediates the force of gravity. There is no complete quantum field theory of gravitons due to an outstanding mathematical problem with renormalization in general relativity. In string theory, believed to be a consistent theory of quantum gravity, the graviton is a massless state of a fundamental string.

Wormhole Hypothetical topological feature of spacetime

A wormhole is a speculative structure linking disparate points in spacetime, and is based on a special solution of the Einstein field equations.

Alcubierre drive Hypothetical mode of transportation by warping space

The Alcubierre drive, Alcubierre warp drive, or Alcubierre metric is a speculative idea based on a solution of Einstein's field equations in general relativity as proposed by Mexican theoretical physicist Miguel Alcubierre, by which a spacecraft could achieve apparent faster-than-light travel if a configurable energy-density field lower than that of vacuum could be created.

Gravitational singularity Location in spacetime where the mass and gravitational field of a celestial body is predicted to become infinite

A gravitational singularity, spacetime singularity or simply singularity is a location in spacetime where the mass and gravitational field of a celestial body is predicted to become infinite by general relativity in a way that does not depend on the coordinate system. The quantities used to measure gravitational field strength are the scalar invariant curvatures of spacetime, which includes a measure of the density of matter. Since such quantities become infinite at the singularity point, the laws of normal spacetime break down.

Loop quantum gravity Theory of quantum gravity, merging quantum mechanics and general relativity

Loop quantum gravity (LQG) is a theory of quantum gravity, which aims to merge quantum mechanics and general relativity, incorporating matter of the Standard Model into the framework established for the pure quantum gravity case. As a candidate for quantum gravity, LQG competes with string theory.

Scalar field Assignment of numbers to points in space

In mathematics and physics, a scalar field associates a scalar value to every point in a space – possibly physical space. The scalar may either be a (dimensionless) mathematical number or a physical quantity. In a physical context, scalar fields are required to be independent of the choice of reference frame, meaning that any two observers using the same units will agree on the value of the scalar field at the same absolute point in space regardless of their respective points of origin. Examples used in physics include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields, such as the Higgs field. These fields are the subject of scalar field theory.

No-hair theorem All black hole solutions of the Einstein–Maxwell equations can be characterized by mass, electric charge, and angular momentum

The no-hair theorem states that all black hole solutions of the Einstein–Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterized by only three externally observable classical parameters: mass, electric charge, and angular momentum. All other information about the matter that formed a black hole or is falling into it "disappears" behind the black-hole event horizon and is therefore permanently inaccessible to external observers. Physicist John Archibald Wheeler expressed this idea with the phrase "black holes have no hair," which was the origin of the name. In a later interview, Wheeler said that Jacob Bekenstein coined this phrase.

Doubly special relativity (DSR) – also called deformed special relativity or, by some, extra-special relativity – is a modified theory of special relativity in which there is not only an observer-independent maximum velocity, but also, an observer-independent maximum energy scale and/or a minimum length scale. This contrasts with other Lorentz-violating theories, such as the Standard-Model Extension, where Lorentz invariance is instead broken by the presence of a preferred frame. The main motivation for this theory is that the Planck energy should be the scale where as yet unknown quantum gravity effects become important and, due to invariance of physical laws, this scale should remain fixed in all inertial frames.

In particle physics, the hypothetical dilaton particle is a particle of a scalar field that appears in theories with extra dimensions when the volume of the compactified dimensions varies. It appears as a radion in Kaluza–Klein theory's compactifications of extra dimensions. In Brans–Dicke theory of gravity, Newton's constant is not presumed to be constant but instead 1/G is replaced by a scalar field and the associated particle is the dilaton.

In theoretical physics, the Einstein–Cartan theory, also known as the Einstein–Cartan–Sciama–Kibble theory, is a classical theory of gravitation similar to general relativity. The theory was first proposed by Élie Cartan in 1922. Einstein–Cartan theory is the simplest Poincaré gauge theory.

The chronology protection conjecture is a hypothesis first proposed by Stephen Hawking that the laws of physics prevent time travel on all but microscopic scales. The permissibility of time travel is represented mathematically by the existence of closed timelike curves in some solutions to the field equations of general relativity. The chronology protection conjecture should be distinguished from chronological censorship under which every closed timelike curve passes through an event horizon, which might prevent an observer from detecting the causal violation.

Micro black holes, also called quantum mechanical black holes or mini black holes, are hypothetical tiny black holes, for which quantum mechanical effects play an important role. The concept that black holes may exist that are smaller than stellar mass was introduced in 1971 by Stephen Hawking.

Numerical relativity is one of the branches of general relativity that uses numerical methods and algorithms to solve and analyze problems. To this end, supercomputers are often employed to study black holes, gravitational waves, neutron stars and many other phenomena governed by Einstein's theory of general relativity. A currently active field of research in numerical relativity is the simulation of relativistic binaries and their associated gravitational waves. Other branches are also active.

A ring singularity or ringularity is the gravitational singularity of a rotating black hole, or a Kerr black hole, that is shaped like a ring.

Causal sets

The causal sets program is an approach to quantum gravity. Its founding principles are that spacetime is fundamentally discrete and that spacetime events are related by a partial order. This partial order has the physical meaning of the causality relations between spacetime events.

In classical theories of gravitation, the changes in a gravitational field propagate. A change in the distribution of energy and momentum of matter results in subsequent alteration, at a distance, of the gravitational field which it produces. In the relativistic sense, the "speed of gravity" refers to the speed of a gravitational wave, which, as predicted by general relativity and confirmed by observation of the GW170817 neutron star merger, is the same speed as the speed of light (c).

The world crystal is a theoretical model in cosmology which provides an alternative understanding of gravity proposed by Hagen Kleinert.

In mathematical physics, de Sitter invariant special relativity is the speculative idea that the fundamental symmetry group of spacetime is the indefinite orthogonal group SO(4,1), that of de Sitter space. In the standard theory of general relativity, de Sitter space is a highly symmetrical special vacuum solution, which requires a cosmological constant or the stress–energy of a constant scalar field to sustain.


Gravitoelectromagnetism, abbreviated GEM, refers to a set of formal analogies between the equations for electromagnetism and relativistic gravitation; specifically: between Maxwell's field equations and an approximation, valid under certain conditions, to the Einstein field equations for general relativity. Gravitomagnetism is a widely used term referring specifically to the kinetic effects of gravity, in analogy to the magnetic effects of moving electric charge. The most common version of GEM is valid only far from isolated sources, and for slowly moving test particles.


  1. 1 2 Rovelli, Carlo (2008). "Quantum gravity". Scholarpedia . 3 (5): 7117. Bibcode:2008SchpJ...3.7117R. doi: 10.4249/scholarpedia.7117 .
  2. Nadis, Steve (2 December 2019). "Black Hole Singularities Are as Inescapable as Expected". quantamagazine.org. Quanta Magazine . Retrieved 22 April 2020.
  3. Wald, Robert M. (1984). General Relativity . University of Chicago Press. p.  382. OCLC   471881415.
  4. 1 2 Feynman, Richard P.; Morinigo, Fernando B.; Wagner, William G. (1995). Feynman Lectures on Gravitation . Reading, Mass.: Addison-Wesley. ISBN   978-0201627343. OCLC   32509962.
  5. Penrose, Roger (2007). The road to reality : a complete guide to the laws of the universe . Vintage. p.  1017. OCLC   716437154.
  6. Bose, S.; et al. (2017). "Spin Entanglement Witness for Quantum Gravity". Physical Review Letters . 119 (4): 240401. arXiv: 1707.06050 . Bibcode:2017PhRvL.119x0401B. doi:10.1103/PhysRevLett.119.240401. PMID   29286711. S2CID   2684909.
  7. Marletto, C.; Vedral, V. (2017). "Gravitationally Induced Entanglement between Two Massive Particles is Sufficient Evidence of Quantum Effects in Gravity". Physical Review Letters . 119 (24): 240402. arXiv: 1707.06036 . Bibcode:2017PhRvL.119x0402M. doi:10.1103/PhysRevLett.119.240402. PMID   29286752. S2CID   5163793.
  8. 1 2 Nemirovsky, J.; Cohen, E.; Kaminer, I. (30 Dec 2018). "Spin Spacetime Censorship". arXiv: 1812.11450v2 [gr-qc].
  9. Wheeler, John Archibald (2010). Geons, Black Holes, and Quantum Foam: A Life in Physics. W. W. Norton & Company. p. 235. ISBN   9780393079487.
  10. 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.
  11. Wald, Robert M. (1994). Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. University of Chicago Press. ISBN   978-0-226-87027-4.
  12. 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.
  13. Gupta, S. N. (1954). "Gravitation and Electromagnetism". Physical Review . 96 (6): 1683–1685. Bibcode:1954PhRv...96.1683G. doi:10.1103/PhysRev.96.1683.
  14. 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.
  15. Gupta, S. N. (1962). "Quantum Theory of Gravitation". Recent Developments in General Relativity. Pergamon Press. pp. 251–258.
  16. Deser, S. (1970). "Self-Interaction and Gauge Invariance". General Relativity and Gravitation . 1 (1): 9–18. arXiv: gr-qc/0411023 . Bibcode:1970GReGr...1....9D. doi:10.1007/BF00759198. S2CID   14295121.
  17. 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.
  18. 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.
  19. Rothman, Tony; Boughn, Stephen (2006). "Can Gravitons be Detected?". Foundations of Physics. 36 (12): 1801–1825. arXiv: gr-qc/0601043 . Bibcode:2006FoPh...36.1801R. doi:10.1007/s10701-006-9081-9. S2CID   14008778.
  20. Hamber, H. W. (2009). Quantum Gravitation – The Feynman Path Integral Approach. Springer Nature. ISBN   978-3-540-85292-6.
  21. 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.
  22. Distler, Jacques (2005-09-01). "Motivation". golem.ph.utexas.edu. Retrieved 2018-02-24.
  23. 1 2 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.CS1 maint: extra text: authors list (link)
  24. Zinn-Justin, Jean (2007). Phase transitions and renormalization group. Oxford: Oxford University Press. ISBN   9780199665167. OCLC   255563633.
  25. 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.
  26. Sakurai, J. J.; Napolitano, Jim J. (2010-07-14). Modern Quantum Mechanics (2 ed.). Pearson. p. 68. ISBN   978-0-8053-8291-4.
  27. 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.
  28. A timeline and overview can be found in Rovelli, Carlo (2000). "Notes for a brief history of quantum gravity". arXiv: gr-qc/0006061 . (verify against ISBN   9789812777386)
  29. Ashtekar, Abhay (2007). "Loop Quantum Gravity: Four Recent Advances and a Dozen Frequently Asked Questions". 11th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity. The Eleventh Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity. p. 126. arXiv: 0705.2222 . Bibcode:2008mgm..conf..126A. doi:10.1142/9789812834300_0008. ISBN   978-981-283-426-3. S2CID   119663169.
  30. 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. S2CID   16762545.
  31. 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., 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.
  32. 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. S2CID   15707877.
  33. 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.
  34. Weinberg, Steven (2000). "Chapter 31". The Quantum Theory of Fields II: Modern Applications . Cambridge University Press. ISBN   978-0-521-55002-4.
  35. 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.
  36. 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. S2CID   17432791.
  37. 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.
  38. Ashtekar, Abhay (1987). "New Hamiltonian formulation of general relativity". Physical Review D . 36 (6): 1587–1602. Bibcode:1987PhRvD..36.1587A. doi:10.1103/PhysRevD.36.1587. PMID   9958340.
  39. Thiemann, Thomas (2007). "Loop Quantum Gravity: An Inside View". Approaches to Fundamental Physics . Lecture Notes in Physics. 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. S2CID   119572847.Missing or empty |title= (help)
  40. Rovelli, Carlo (1998). "Loop Quantum Gravity". Living Reviews in Relativity . 1 (1): 1. arXiv: gr-qc/9710008 . Bibcode:1998LRR.....1....1R. doi:10.12942/lrr-1998-1. PMC   5567241 . PMID   28937180 . Retrieved 2008-03-13.
  41. 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. S2CID   119175535.
  42. Thiemann, Thomas (2003). "Lectures on Loop Quantum Gravity". Quantum Gravity. Lecture Notes in Physics. 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. S2CID   119151491.
  43. Rovelli, Carlo (2004). Quantum Gravity. Cambridge University Press. ISBN   978-0-521-71596-6.
  44. Isham, Christopher J. (1994). "Prima facie questions in quantum gravity". In Ehlers, Jürgen; Friedrich, Helmut (eds.). Canonical Gravity: From Classical to Quantum. Canonical Gravity: From Classical to Quantum. Lecture Notes in Physics. 434. Springer. pp. 1–21. arXiv: gr-qc/9310031 . Bibcode:1994LNP...434....1I. doi:10.1007/3-540-58339-4_13. ISBN   978-3-540-58339-4. S2CID   119364176.
  45. 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. S2CID   4803804.
  46. Loll, Renate (1998). "Discrete Approaches to Quantum Gravity in Four Dimensions". Living Reviews in Relativity . 1 (1): 13. arXiv: gr-qc/9805049 . Bibcode:1998LRR.....1...13L. doi:10.12942/lrr-1998-13. PMC   5253799 . PMID   28191826.
  47. Klimets AP, Philosophy Documentation Center, Western University-Canada, 2017, pp.25-30
  48. 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.
  49. Spontaneous Quantum Gravity , retrieved 2020-01-19
  50. Maithresh, Palemkota; Singh, Tejinder P. (2020). "Proposal for a new quantum theory of gravity III: Equations for quantum gravity, and the origin of spontaneous localisation". Zeitschrift für Naturforschung A. 0 (2): 143–154. arXiv: 1908.04309 . Bibcode:2019arXiv190804309M. doi:10.1515/zna-2019-0267. ISSN   1865-7109. S2CID   204924253.
  51. Singh, Tejinder P. (2019-12-05). "Spontaneous quantum gravity". arXiv: 1912.03266 [physics.pop-ph].
  52. See ch. 33 in Penrose 2004 and references therein.
  53. 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. S2CID   84118515.
  54. 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.CS1 maint: location (link)
  55. Oniga, Teodora; Wang, Charles H.-T. (2016-02-09). "Quantum gravitational decoherence of light and matter". Physical Review D. 93 (4): 044027. doi:10.1103/PhysRevD.93.044027. hdl: 2164/5830 .
  56. Oniga, Teodora; Wang, Charles H.-T. (2017-10-05). "Quantum coherence, radiance, and resistance of gravitational systems". Physical Review D. 96 (8): 084014. doi:10.1103/PhysRevD.96.084014. hdl: 2164/9320 .
  57. Quiñones, D. A.; Oniga, T.; Varcoe, B. T. H.; Wang, C. H.-T. (2017-08-15). "Quantum principle of sensing gravitational waves: From the zero-point fluctuations to the cosmological stochastic background of spacetime". Physical Review D. 96 (4): 044018. doi:10.1103/PhysRevD.96.044018. hdl: 2164/9150 .
  58. Oniga, Teodora; Wang, Charles H.-T. (2016-09-19). "Spacetime foam induced collective bundling of intense fields". Physical Review D. 94 (6): 061501. doi:10.1103/PhysRevD.94.061501. hdl: 2164/7434 .
  59. https://www.esa.int/Science_Exploration/Space_Science/Integral_challenges_physics_beyond_Einstein
  60. Cowen, Ron (30 January 2015). "Gravitational waves discovery now officially dead". Nature . doi:10.1038/nature.2015.16830. S2CID   124938210.
  61. Yuri.N., Obukhov, "Spin, gravity, and inertia", Physical review letters 86.2 (2001): 192.arXiv : 0012102v1

Further reading