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Quantum gravity

Topic: Physics

 From HandWiki - Reading time: 24 min
Short description: Description of gravity using discrete values
A depiction of the cGh cube

Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored,[1] such as in the vicinity of black holes or similar compact astrophysical objects, such as neutron stars[2][3] as well as in the early stages of the universe moments after the Big Bang.[4]

Three of the four fundamental forces of nature are described within the framework of quantum mechanics and quantum field theory: the electromagnetic interaction, the strong force, and the weak force; this leaves gravity as the only interaction that has not been fully accommodated. The current understanding of gravity is based on Albert Einstein's general theory of relativity, which incorporates his theory of special relativity and deeply modifies the understanding of concepts like time and space. Although general relativity is highly regarded for its elegance and accuracy it has limitations: the gravitational singularities inside of black holes, the ad hoc postulation of dark matter, as well as dark energy and its relation to the cosmological constant are among the current unsolved mysteries regarding gravity;[5] all of which signal the collapse of the general theory of relativity at different scales and highlight the need for a gravitational theory that goes into the quantum realm. At distances close to the Planck length, like those near the center of the black hole, quantum fluctuations of spacetime are expected to play an important role.[6] The breakdown of general relativity at galactic and cosmological scales also points out the necessity for a more robust theory. Finally the discrepancies between the predicted value for the vacuum energy and the observed values (which, depending on the considerations, can be of 60 or 120 orders of magnitude)[7] highlight the necessity for a quantum theory of gravity.

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 being M-theory and loop quantum gravity.[8] All of these approaches aim to describe the quantum behavior of the gravitational field, which 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 a theory is often referred to as a theory of everything. Some of the approaches, such as loop quantum gravity, make no such attempt; instead, they make an effort to quantize the gravitational field while it is kept separate from the other forces. Other lesser-known but no less important theories include Causal dynamical triangulation, Noncommutative geometry, and Twistor theory.[9]

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

Thought experiment approaches have been suggested as a testing tool for quantum gravity theories.[10][11] In the field of quantum gravity there are several open questions – e.g., it is not known how spin of elementary particles sources gravity, and thought experiments could provide a pathway to explore possible resolutions to these questions,[12] even in the absence of lab experiments or physical observations.

In the early 21st century, new experiment designs and technologies have arisen which suggest that indirect approaches to testing quantum gravity may be feasible over the next few decades.[13][14][15][16] This field of study is called phenomenological quantum gravity.

Overview

Question, Web Fundamentals.svg 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."[17] 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.[18] 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.[19]

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

Graviton

Main page: Physics:Graviton

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.[20][21][22][23][24] 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.[25][26] While gravitons are an important theoretical step in a quantum mechanical description of gravity, they are generally believed to be undetectable because they interact too weakly.[27]

Nonrenormalizability of gravity

Further information: Physics:Renormalization and Physics:Asymptotic safety in quantum gravity

General relativity, like electromagnetism, is a classical field theory. One might expect that, as with electromagnetism, the gravitational force should also have a corresponding quantum field theory.

However, gravity is perturbatively nonrenormalizable.[28][29] 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.[30]

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.[31][better source needed]

Quantum gravity as an effective field theory

Main page: Physics: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.[32] 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.[33] Works pioneered by Barvinsky and Vilkovisky [34][35][36][37] suggest as a starting point up to second order in curvature the following action, consisting of local and non-local terms:

[math]\displaystyle{ \begin{align} \Gamma&=\int d^4x\, \sqrt{-g}\,\bigg(\frac{R}{16\pi G}+c_1(\mu)R^2 +c_2(\mu)R_{\mu\nu}R^{\mu\nu} +c_3(\mu)R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\bigg)\\&-\int d^4 x \sqrt{-g}\bigg[\alpha R\ln\left(\frac{\Box}{\mu^2}\right)R +\beta R_{\mu\nu}\ln\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu} + \gamma R_{\mu\nu\rho\sigma}\ln\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu\rho\sigma}\bigg], \end{align} }[/math]

where [math]\displaystyle{ \mu }[/math] is an energy scale. The exact values of the coefficients [math]\displaystyle{ c_1,c_2,c_3 }[/math] are unknown, as they depend on the nature of the ultra-violet theory of quantum gravity. [math]\displaystyle{ \ln\left(\Box/\mu^2\right) }[/math] is an operator with the integral representation

[math]\displaystyle{ \ln\left(\frac{\Box}{\mu^2}\right)=\int_0^{+\infty}ds\, \left(\frac{1}{\mu^2+s}-\frac{1}{\Box+s}\right). }[/math]

By treating general relativity as an effective field theory, legitimate predictions can be made 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.[32] Moreover, one can compute the quantum gravitational corrections to classical thermodynamic properties of black holes, most importantly the entropy. A rigorous derivation of the quantum gravitational corrections to the entropy of Schwarzschild black holes was provided by Calmet and Kuipers.[38] A generalisation for charged (Reissner–Nordström) black holes was subsequently carried out by Campos Delgado.[39]

Spacetime background dependence

Main page: Physics: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 simple to grasp in principle, this is a complex 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,[40] in which the only physically relevant information is the relationship between different events in spacetime.

On the other hand, quantum mechanics has depended since its inception on a fixed background (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 dynamic 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).

Problem of time

Main page: Physics: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.[41] In contrast, general relativity treats time as a dynamical variable which relates directly with matter and moreover requires the Hamiltonian constraint to vanish.[42] 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.[43] 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.[44][45]

String theory

Main page: Physics:String theory
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.[46] 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.[47] 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.[48]

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[49] form part of a hypothesized eleven-dimensional model known as M-theory, which would constitute a uniquely defined and consistent theory of quantum gravity.[50][51] 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 page: Physics: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 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 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.[52][53] In the quantum theory, space is represented by a network structure called a spin network, evolving over time in discrete steps.[54][55][56][57]

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.[58] 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.[59][60] 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.[66]

The most widely pursued possibilities for quantum gravity phenomenology include gravitationally mediated entanglement,[67][68] violations of Lorentz invariance, imprints of quantum gravitational effects in the cosmic microwave background (in particular its polarization), and decoherence induced by fluctuations[69][70][71] in the space-time foam.[72] The latter scenario has been searched for in light from gamma-ray bursts and both astrophysical and atmospheric neutrinos, placing limits on phenomenological quantum gravity parameters.[73][74][75]

ESA's INTEGRAL satellite measured polarization of photons of different wavelengths and was able to place a limit in the granularity of space that is less than 10−48 m, or 13 orders of magnitude below the Planck scale.[76][77]

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.[78]

See also


Notes

  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

References

  1. ↑ 1.0 1.1 Rovelli, Carlo (2008). "Quantum gravity". Scholarpedia 3 (5): 7117. doi:10.4249/scholarpedia.7117. Bibcode: 2008SchpJ...3.7117R. 
  2. ↑ Overbye, Dennis (10 October 2022). "Black Holes May Hide a Mind-Bending Secret About Our Universe - Take gravity, add quantum mechanics, stir. What do you get? Just maybe, a holographic cosmos.". The New York Times. https://www.nytimes.com/2022/10/10/science/black-holes-cosmology-hologram.html. Retrieved 16 October 2022. 
  3. ↑ Starr, Michelle (16 November 2022). "Scientists Created a Black Hole in The Lab, And Then It Started to Glow". ScienceAlert. https://www.sciencealert.com/scientists-created-a-black-hole-in-the-lab-and-then-it-started-to-glow. Retrieved 16 November 2022. 
  4. ↑ Kiefer, Claus (2012) (in en). Quantum gravity. International series of monographs on physics (3rd ed.). Oxford: Oxford University Press. pp. 1–4. ISBN 978-0-19-958520-5. 
  5. ↑ Mannheim, Philip (2006). "Alternatives to dark matter and dark energy" (in en). Progress in Particle and Nuclear Physics 56 (2): 340–445. doi:10.1016/j.ppnp.2005.08.001. Bibcode: 2006PrPNP..56..340M. 
  6. ↑ Nadis, Steve (2 December 2019). "Black Hole Singularities Are as Inescapable as Expected". Quanta Magazine. https://www.quantamagazine.org/black-hole-singularities-are-as-inescapable-as-expected-20191202/. 
  7. ↑ Koksma, Jurjen; Prokopec, Tomislav (2011). "The Cosmological Constant and Lorentz Invariance of the Vacuum State". arXiv:1105.6296 [gr-qc].
  8. ↑ Penrose, Roger (2007). The road to reality : a complete guide to the laws of the universe. Vintage. p. 1017. ISBN 9780679776314. OCLC 716437154. https://archive.org/details/roadtoreality00penr_319. 
  9. ↑ Rovelli, Carlo (2001). "Notes for a brief history of quantum gravity". arXiv:gr-qc/0006061.
  10. ↑ Bose, S. (2017). "Spin Entanglement Witness for Quantum Gravity". Physical Review Letters 119 (4): 240401. doi:10.1103/PhysRevLett.119.240401. PMID 29286711. Bibcode: 2017PhRvL.119x0401B. 
  11. ↑ 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. doi:10.1103/PhysRevLett.119.240402. PMID 29286752. Bibcode: 2017PhRvL.119x0402M. 
  12. ↑ Nemirovsky, J.; Cohen, E.; Kaminer, I. (5 November 2021). "Spin Spacetime Censorship". Annalen der Physik 534 (1). doi:10.1002/andp.202100348. 
  13. ↑ Hossenfelder, Sabine (2 February 2017). "What Quantum Gravity Needs Is More Experiments". http://nautil.us/issue/45/power/what-quantum-gravity-needs-is-more-experiments. Retrieved 21 September 2020. 
  14. ↑ Experimental search for quantum gravity. Cham: Springer. 2017. ISBN 9783319645360. 
  15. ↑ Carney, Daniel; Stamp, Philip C. E.; Taylor, Jacob M. (7 February 2019). "Tabletop experiments for quantum gravity: a user's manual". Classical and Quantum Gravity 36 (3): 034001. doi:10.1088/1361-6382/aaf9ca. Bibcode: 2019CQGra..36c4001C. 
  16. ↑ Danielson, Daine L.; Satishchandran, Gautam; Wald, Robert M. (2022-04-05). "Gravitationally mediated entanglement: Newtonian field versus gravitons". Physical Review D 105 (8): 086001. doi:10.1103/PhysRevD.105.086001. Bibcode: 2022PhRvD.105h6001D. https://link.aps.org/doi/10.1103/PhysRevD.105.086001. Retrieved 2022-12-11. 
  17. ↑ Wheeler, John Archibald (2010). Geons, Black Holes, and Quantum Foam: A Life in Physics. W. W. Norton & Company. pp. 235. ISBN 9780393079487. 
  18. ↑ 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. 
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  20. ↑ Kraichnan, R. H. (1955). "Special-Relativistic Derivation of Generally Covariant Gravitation Theory". Physical Review 98 (4): 1118–1122. doi:10.1103/PhysRev.98.1118. Bibcode: 1955PhRv...98.1118K. 
  21. ↑ Gupta, S. N. (1954). "Gravitation and Electromagnetism". Physical Review 96 (6): 1683–1685. doi:10.1103/PhysRev.96.1683. Bibcode: 1954PhRv...96.1683G. 
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  24. ↑ Deser, S. (1970). "Self-Interaction and Gauge Invariance". General Relativity and Gravitation 1 (1): 9–18. doi:10.1007/BF00759198. Bibcode: 1970GReGr...1....9D. 
  25. ↑ Weinberg, Steven; Witten, Edward (1980). "Limits on massless particles". Physics Letters B 96 (1–2): 59–62. doi:10.1016/0370-2693(80)90212-9. Bibcode: 1980PhLB...96...59W. 
  26. ↑ Horowitz, Gary T.; Polchinski, Joseph (2006). "Gauge/gravity duality". in Oriti, Daniele. Approaches to Quantum Gravity. Cambridge University Press. ISBN 9780511575549. OCLC 873715753. Bibcode: 2006gr.qc.....2037H. 
  27. ↑ Rothman, Tony; Boughn, Stephen (2006). "Can Gravitons be Detected?". Foundations of Physics 36 (12): 1801–1825. doi:10.1007/s10701-006-9081-9. Bibcode: 2006FoPh...36.1801R. https://link.springer.com/article/10.1007/s10701-006-9081-9. Retrieved 2020-05-15. 
  28. ↑ Feynman, Richard P. (1995) (in en-us). Feynman Lectures on Gravitation. Reading, Massachusetts: Addison-Wesley. pp. xxxvi–xxxviii, 211–212. ISBN 978-0201627343. 
  29. ↑ Hamber, H. W. (2009). Quantum Gravitation – The Feynman Path Integral Approach. Springer Nature. ISBN 978-3-540-85292-6. 
  30. ↑ Goroff, Marc H.; Sagnotti, Augusto; Sagnotti, Augusto (1985). "Quantum gravity at two loops". Physics Letters B 160 (1–3): 81–86. doi:10.1016/0370-2693(85)91470-4. Bibcode: 1985PhLB..160...81G. 
  31. ↑ Distler, Jacques (2005-09-01). "Motivation" (in en). https://golem.ph.utexas.edu/~distler/blog/archives/000639.html. 
  32. ↑ 32.0 32.1 Donoghue, John F. (1995). "Introduction to the Effective Field Theory Description of Gravity". in Cornet, Fernando. Effective Theories: Proceedings of the Advanced School, Almunecar, Spain, 26 June–1 July 1995. Singapore: World Scientific. ISBN 978-981-02-2908-5. Bibcode: 1995gr.qc....12024D. 
  33. ↑ Zinn-Justin, Jean (2007). Phase transitions and renormalization group. Oxford: Oxford University Press. ISBN 9780199665167. OCLC 255563633. 
  34. ↑ Barvinsky, A. O.; Vilkovisky, G. A. (1983). "The generalized Schwinger–DeWitt technique and the unique effective action in quantum gravity". Phys. Lett. B 131 (4–6): 313–318. doi:10.1016/0370-2693(83)90506-3. Bibcode: 1983PhLB..131..313B. 
  35. ↑ Barvinsky, A. O.; Vilkovisky, G. A. (1985). "The Generalized Schwinger–DeWitt Technique in Gauge Theories and Quantum Gravity". Phys. Rep. 119 (1): 1–74. doi:10.1016/0370-1573(85)90148-6. Bibcode: 1985PhR...119....1B. 
  36. ↑ Barvinsky, Vilkovisky, A. O.; Vilkovisky, G. A. (1987). "Beyond the Schwinger–Dewitt Technique: Converting Loops Into Trees and In-In Currents". Nuclear Physics B 282: 163–188. doi:10.1016/0550-3213(87)90681-X. Bibcode: 1987NuPhB.282..163B. 
  37. ↑ Barvinsky, A. O.; Vilkovisky, G. A. (1990). "Covariant perturbation theory. 2: Second order in the curvature. General algorithms". Nuclear Physics B 333: 471–511. doi:10.1016/0550-3213(90)90047-H. 
  38. ↑ Calmet, Kuipers, Xavier, Folkert (2021). "Quantum gravitational corrections to the entropy of a Schwarzschild black hole". Phys. Rev. D 104 (6): 6. doi:10.1103/PhysRevD.104.066012. Bibcode: 2021PhRvD.104f6012C. 
  39. ↑ Campos Delgado, Ruben (2022). "Quantum gravitational corrections to the entropy of a Reissner-Nordström black hole". Eur. Phys. J. C 82 (3): 272. doi:10.1140/epjc/s10052-022-10232-0. Bibcode: 2022EPJC...82..272C. 
  40. ↑ 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.
  41. ↑ Sakurai, J. J.; Napolitano, Jim J. (2010-07-14) (in en). Modern Quantum Mechanics (2 ed.). Pearson. p. 68. ISBN 978-0-8053-8291-4. 
  42. ↑ Novello, Mario; Bergliaffa, Santiago E. (2003-06-11) (in en). 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. https://books.google.com/books?id=vDWvUBiNgNkC. 
  43. ↑ 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)
  44. ↑ 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. 126. doi:10.1142/9789812834300_0008. ISBN 978-981-283-426-3. Bibcode: 2008mgm..conf..126A. 
  45. ↑ Schwarz, John H. (2007). "String Theory: Progress and Problems". Progress of Theoretical Physics Supplement 170: 214–226. doi:10.1143/PTPS.170.214. Bibcode: 2007PThPS.170..214S. 
  46. ↑ 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. 
  47. ↑ Ibanez, L. E. (2000). "The second string (phenomenology) revolution". Classical and Quantum Gravity 17 (5): 1117–1128. doi:10.1088/0264-9381/17/5/321. Bibcode: 2000CQGra..17.1117I. 
  48. ↑ 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.
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  74. ↑ The IceCube Collaboration; Abbasi, R.; Ackermann, M.; Adams, J.; Aguilar, J. A.; Ahlers, M.; Ahrens, M.; Alameddine, J. M. et al. (2022-11-01). "Search for quantum gravity using astrophysical neutrino flavour with IceCube" (in en). Nature Physics 18 (11): 1287–1292. doi:10.1038/s41567-022-01762-1. ISSN 1745-2473. Bibcode: 2022NatPh..18.1287I. https://www.nature.com/articles/s41567-022-01762-1. 
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  78. ↑ Cowen, Ron (30 January 2015). "Gravitational waves discovery now officially dead". Nature. doi:10.1038/nature.2015.16830. 

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