Quantum gravity
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Beyond the Standard Model
Quantum gravity is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature (electromagnetism, weak interaction, and strong interaction), with general relativity, the theory of the fourth fundamental force: gravity. One ultimate goal hoped to emerge as a result of this is a unified framework for all fundamental forces— called a "theory of everything" (TOE).
Contents [hide]
1 Overview
1.1 Effective field theories
1.2 Quantum gravity theory for the highest energy scales
2 Quantum Mechanics and General Relativity
2.1 The graviton
2.2 Nonrenormalizability of gravity
2.3 QG as an effective field theory
2.4 Spacetime background dependence
2.4.1 String theory
2.4.2 Background independent theories
2.5 Fields vs particles
2.6 Points of tension
3 Candidate theories
3.1 String theory
3.2 Loop quantum gravity
3.3 Other candidates
4 Weinberg-Witten theorem
5 In popular culture
6 See also
7 Notes
8 References
9 External links
[edit] Overview
This section does not cite any references or sources. (December 2007)
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Unsolved problems 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?
Much of the difficulty in merging these theories at all energy scales comes from the different assumptions that these theories make on how the universe works. Quantum field theory depends on particle fields embedded in the flat space-time of special relativity. General relativity models gravity as a curvature within space-time that changes as a gravitational mass moves. Historically, the most obvious way of combining the two (such as treating gravity as simply another particle field) ran quickly into what is known as the renormalization problem. In the old-fashioned understanding of renormalization, gravity particles would attract each other and adding together all of the interactions results in many infinite values which cannot easily be cancelled out mathematically to yield sensible, finite results. This is in contrast with quantum electrodynamics where, while the series still do not converge, the interactions sometimes evaluate to infinite results, but those are few enough in number to be removable via renormalization.
[edit] Effective field theories
In recent decades, however, this antiquated understanding of renormalization has given way to the modern idea of effective field theory. All quantum field theories come with some high-energy cutoff, beyond which we do not expect that the theory provides a good description of nature. The "infinities" then become large but finite quantities proportional to this finite cutoff scale, and correspond to processes that involve very high energies near the fundamental cutoff. These quantities can then be absorbed into an infinite collection of coupling constants, and at energies well below the fundamental cutoff of the theory, to any desired precision only a finite number of these coupling constants need to be measured in order to make legitimate quantum-mechanical predictions.This same logic works just as well for the highly successful theory of low-energy pions as for quantum gravity. Indeed, the first quantum-mechanical corrections to graviton-graviton scattering and Newton’s law of gravitation have been explicitly computed (although they are so astronomically small that we may never be able to measure them), and any more fundamental theory of nature would need to replicate these results in order to be taken seriously. In fact, gravity is in many ways a much better quantum field theory than the Standard Model, since it appears to be valid all the way up to its cutoff at the Planck scale. (By comparison, the Standard Model is expected to start to break down above its cutoff at the much smaller scale of around 1000 GeV.)
While confirming that quantum mechanics and gravity are indeed consistent at reasonable energies (in fact, the complete structure of gravity can be shown to arise automatically from the quantum mechanics of spin-2 massless particles), this way of thinking makes clear that near or above the fundamental cutoff of our effective quantum theory of gravity (the cutoff is generally assumed to be of order the Planck scale), a new model of nature will be needed. That is, in the modern way of thinking, the problem of combining quantum mechanics and gravity becomes an issue only at very high energies, and may well require a totally new kind of model.
[edit] Quantum gravity theory for the highest energy scales
The general approach taken in deriving a theory of quantum gravity that is valid at even the highest energy scales is to assume that the underlying theory will be simple and elegant and then to look at current theories for symmetries and hints for how to combine them elegantly into an overarching theory. One problem with this approach is that it is not known if quantum gravity will be a simple and elegant theory (that resolves the conundrum of special and general relativity with regard to the uniformity of acceleration and gravity, in the former case and spacetime curvature in the latter case).
Such a theory is required in order to understand those problems involving the combination of very large mass or energy and very small dimensions of space, such as the behavior of black holes, and the origin of the universe.
[edit] Quantum Mechanics and General Relativity
Gravity Probe B (GP-B) will measure spacetime curvature near Earth to test related models in application of Einstein’s general theory of relativity.
[edit] The 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 applies 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 in fact demonstrate that the structure of general relativity essentially follows inevitably from the quantum mechanics of interacting theoretical spin-2 massless particles (called gravitons).
While there is no concrete proof of the existence of gravitons, all quantized theories of matter necessitate their existence.[citation needed] Supporting this theory is the observation that all other fundamental forces have one or more messenger particles, except gravity, leading researchers to believe that at least one most likely does exist; they have dubbed these hypothetical particles gravitons. Many of the accepted notions of a unified theory of physics since the 1970s, including string theory, superstring theory, M-theory, loop quantum gravity, all assume, and to some degree depend upon the existence of the graviton. Many researchers view the detection of the graviton as vital to validating their work. CERN plans to dedicate a large timeshare to search for the graviton using the Large Hadron Collider.
[edit] Nonrenormalizability of gravity
Further information: Renormalization
Historically, many believed that general relativity was in fact fundamentally inconsistent with quantum mechanics. General relativity, like electromagnetism, is a classical field theory. One might expect that, as with electromagnetism, there should be a corresponding quantum field theory.
However, gravity is nonrenormalizable. 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 infinitely many independent parameters needed to define the theory. For a given choice of those parameters, one could make sense of the theory, but since we can never do infinitely many experiments to fix the values of every parameter, we do not 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.
As explained below, there is a way around this problem by treating QG as an effective field theory.
Any meaningful theory of quantum gravity that makes sense and is predictive at all energy scales must have some deep principle that reduces the infinitely many unknown parameters 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 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.
[edit] QG 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 indeed a predictive quantum field theory[1]. (A very similar situation occurs for the very similar effective field theory of low-energy pions.) Furthermore, many theorists agree that even the Standard Model should really be regarded as an effective field theory as well, with "nonrenormalizable" interactions suppressed by large energy scales and whose effects have consequently not been observed experimentally.
Recent work[1] has shown that 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. Such predictions would need to be replicated by any candidate theory of high-energy quantum gravity.
[edit] 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,[2] 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-dynamical) 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.
[edit] 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 theoryString theory started out as a generalization of quantum field theory where instead of point particles, string-like objects propagate in a fixed spacetime background. 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.
[edit] 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.
[edit] Fields vs particles
Quantum field theory on curved (non-Minkowskian) backgrounds, while not a quantum theory of gravity, has shown that some of the assumptions of quantum field theory cannot be carried over to curved spacetime[citation needed], let alone to full-blown quantum gravity. In particular, the vacuum, when it exists, is shown to depend on the path of the observer through space-time (see Unruh effect).
Also, some argue that in curved spacetime, the field concept is seen to be fundamental over the particle concept (which arises as a convenient way to describe localized interactions). However, since it appears possible to regard curved spacetime as consisting of a condensate of gravitons, there is still some debate over which concept is truly the more fundamental.
[edit] Points of tension
There are two other points of tension between quantum mechanics and general relativity.
First, classical general relativity breaks down at singularities, and quantum mechanics becomes inconsistent with general relativity in a neighborhood of singularities (however, no one is certain that classical general relativity applies near singularities in the first place).
Second, it is not clear how to determine the gravitational field of a particle, since under the Heisenberg uncertainty principle of quantum mechanics its location and velocity cannot be known with certainty. The resolution of these points may come from a better understanding of general relativity[3].
[edit] Candidate theories
There are a number of proposed quantum gravity theories.[4] 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.[5]
[edit] String theory
Main article: String theory
Projection of a Calabi-Yau manifold, one of the ways of compactifying the extra dimensions posited by string theoryOne suggestive starting point are ordinary quantum field theories which, after all, 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,[6] gravity turns out to be much more problematic at higher energies. Where, 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,[7] 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.[8]
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.[9] 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.[10] The theory is successful in that one mode will always correspond to a graviton, the messenger particle of gravity; however, the price to pay are unusual features such as six extra dimensions of space in addition to the usual three.[11] 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[12] form part of a hypothesized eleven-dimensional model known as M-theory, which would constitute a uniquely defined and consistent theory of quantum gravity.[13]
[edit] Loop quantum gravity
Main article: loop quantum gravity
Simple spin network of the type used in loop quantum gravityAnother approach to quantum gravity starts with the canonical quantization procedures of quantum theory. Starting with the initial-value-formulation of general relativity (cf. the section on evolution equations, above), the result is an analogue of the Schrödinger equation: the Wheeler-deWitt equation which, regrettably, turns out to be ill-defined.[14] A major break-through came with the introduction of what are now known as Ashtekar variables, which represent geometric gravity using mathematical analogues of electric and magnetic fields.[15] The resulting candidate for a theory of quantum gravity is Loop quantum gravity, in which space is represented by a network structure called a spin network, evolving over time in discrete steps.[16]
[edit] Other candidates
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[17]. Examples include:
Acoustic metric and other analog models of gravity
An Exceptionally Simple Theory of Everything
Causal Dynamical Triangulation
Dynamical triangulations,[18]
Causal sets,[19]
The Omega Point and the quantum gravity Theory of Everything
path-integral based models of quantum cosmology.[20]
Process physics
Regge calculus
Supergravity
Twistor models[21]
[edit] Weinberg-Witten theorem
There is a theorem in quantum field theory called the Weinberg-Witten theorem which places some constraints on theories of composite gravity/emergent gravity.
[edit] In popular culture
The famous spoof of postmodernism by Alan Sokal (see Sokal Affair) was entitled Transgressing the Boundaries: Toward a Transformative Hermeneutics of Quantum Gravity.
Quantum gravity is said to be the reason behind the dual realities in the anime film The Place Promised in Our Early Days.
Hard science fiction author Greg Egan proposed in his short story The Planck Dive a concept of quantum gravity in which spacetime itself is a function of networks of all possible world lines and gravity is merely the alteration of probabilities associated with individual worldlines according the increased or decreased incidences of quantum phase shift due to time dilation which is a result of interactions with virtual particles.
[edit] See also
Centauro event
Hawking Radiation
M-theory
Quantum field theory in curved spacetime
Semiclassical gravity
List of quantum gravity researchers
Abraham-Lorentz force
Invariance mechanics
Black hole electron
[edit] Notes
^ a b [gr-qc/9512024] Introduction to the Effective Field Theory Description of Gravity
^ Smolin, Lee (2001). Three roads to quantum gravity. Basic Books, 20-25. ISBN 0-465-08735-4. Pages 220-226 are annotated references and guide for further reading.
^ [astro-ph/0506506] Singularity-Free Collapse through Local Inflation
^ A timeline and overview can be found in Rovelli 2000.
^ E.g. Ashtekar 2007, Schwarz 2007.
^ See Donoghue 1995.
^ Cf. chapters 17 and 18 of Weinberg 1996.
^ Cf. Goroff & Sagnotti 1985.
^ An accessible introduction at the undergraduate level can be found in Zwiebach 2004; more complete overviews can be found in Polchinski 1998a and Polchinski 1998b.
^ E.g. Ibanez 2000.
^ 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.
^ E.g. Weinberg 2000, ch. 31.
^ E.g. Townsend 1996, Duff 1996.
^ Cf. section 3 in Kuchař 1973.
^ See Ashtekar 1986, Ashtekar 1987.
^ For a review, see Thiemann 2006; more extensive accounts can be found in Rovelli 1998, Ashtekar & Lewandowski 2004 as well as in the lecture notes Thiemann 2003.
^ See e.g. the systematic expositions in Isham 1994 and Sorkin 1997.
^ See Loll 1998.
^ See Sorkin 2005.
^ Cf. Hawking 1987.
^ See ch. 33 in Penrose 2004 and references therein.
[edit] References
Ashtekar, Abhay (2007), Loop Quantum Gravity: Four Recent Advances and a Dozen Frequently Asked Questions, arΧiv:0705.2222
Ashtekar, Abhay & Lewandowski, Jerzy (2004), “Background Independent Quantum Gravity: A Status Report”, Class. Quant. Grav. 21: R53-R152, arΧiv:gr-qc/0404018
Ashtekar, Abhay (1986), “New variables for classical and quantum gravity”, Phys. Rev. Lett. 57: 2244–2247, DOI 10.1103/PhysRevLett.57.2244
Ashtekar, Abhay (1987), “New Hamiltonian formulation of general relativity”, Phys. Rev. D36: 1587–1602, DOI 10.1103/PhysRevD.36.1587
Carlip, Steven (2001), “Quantum Gravity: a Progress Report”, Rept.Prog.Phys. 64: 885, arΧiv:gr-qc/0108040
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, arΧiv:gr-qc/9512024, ISBN 9810229089
Duff, Michael (1996), “M-Theory (the Theory Formerly Known as Strings)”, Int. J. Mod. Phys. A11: 5623–5642, arΧiv:hep-th/9608117
Goroff, Marc H. & Sagnotti, Augusto (1985), “Quantum gravity at two loops”, Phys. Lett. 160B: 81–86
Hawking, Stephen W. (1987), “Quantum cosmology”, in Hawking, Stephen W. & Israel, Werner, 300 Years of Gravitation, Cambridge University Press, pp. 631–651, ISBN 0-521-37976-8
Ibanez, L. E. (2000), “The second string (phenomenology) revolution”, Class. Quant. Grav. 17: 1117–1128, arΧiv:hep-th/9911499
Isham, Christopher J. (1994), “Prima facie questions in quantum gravity”, in Ehlers, Jürgen & Friedrich, Helmut, Canonical Gravity: From Classical to Quantum, Springer, ISBN 3-540-58339-4
Kuchař, Karel (1973), “Canonical Quantization of Gravity”, in Israel, Werner, Relativity, Astrophysics and Cosmology, D. Reidel, pp. 237–288, ISBN 90-277-0369-8
Loll, Renate (1998), “Discrete Approaches to Quantum Gravity in Four Dimensions”, Living Rev. Relativity 1, <
http://www.livingreviews.org/lrr-1998-13>. Retrieved on 9 March 2008
Polchinski, Joseph (1998a), String Theory Vol. I: An Introduction to the Bosonic String, Cambridge University Press, ISBN 0-521-63303-6
Polchinski, Joseph (1998b), String Theory Vol. II: Superstring Theory and Beyond, Cambridge University Press, ISBN 0-521-63304-4
Rovelli, Carlo (2000), Notes for a brief history of quantum gravity, arΧiv:gr-qc/0006061
Rovelli, Carlo (1998), “Loop Quantum Gravity”, Living Rev. Relativity 1, <
http://www.livingreviews.org/lrr-1998-1>. Retrieved on 13 March 2008
Schwarz, John H. (2007), String Theory: Progress and Problems, arΧiv:hep-th/0702219
Sorkin, Rafael D. (2005), “Causal Sets: Discrete Gravity”, in Gomberoff, Andres & Marolf, Donald, Lectures on Quantum Gravity, Springer, arΧiv:gr-qc/0309009, ISBN 0-387-23995-2
Sorkin, Rafael D. (1997), “Forks in the Road, on the Way to Quantum Gravity”, Int. J. Theor. Phys. 36: 2759–2781, arΧiv:gr-qc/9706002
Thiemann, Thomas (2006), Loop Quantum Gravity: An Inside View, arΧiv:hep-th/0608210
Thiemann, Thomas (2003), “Lectures on Loop Quantum Gravity”, Lect. Notes Phys. 631: 41–135
Townsend, Paul K. (1996), Four Lectures on M-Theory, arΧiv:hep-th/9612121
Trifonov, Vladimir (2008), “GR-friendly description of quantum systems”, Int. Journal of Theor. Phys. 47 (2): 492–510, DOI: 10.1007/s10773-007-9474-3
Weinberg, Steven (1996), The Quantum Theory of Fields II: Modern Applications, Cambridge University Press, ISBN 0-521-55002-5
Zwiebach, Barton (2004), A First Course in String Theory, Cambridge University Press, ISBN 0-521-83143-1
[edit] External links
Quantum Gravity Perimeter Institute for Theoretical Physics
Prima facie questions in quantum gravity Chris J. Isham
Quantum Gravity in Stanford Encyclopedia of Philosophy
The shape of things to come New Scientist, July 30 2005
Quantum gravity articles
Quantum Gravity pages by Lee Smolin (no longer active, archived at
http://web.archive.org/web/20060428001322/http://www.qgravity.org/)
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