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Beyond the Standard Model
Standard Model

Quantum gravity (QG) is the field of theoretical physics attempting to unify quantum mechanics with general relativity in a self-consistent manner, or more precisely, to formulate a self-consistent theory which reduces to ordinary quantum mechanics in the limit of weak gravity (potentials much less than c2) and which reduces to Einsteinian general relativity in the limit of large actions (action much larger than reduced Planck's constant). The theory must be able to predict the outcome of situations where both quantum effects and strong-field gravity are important (at the Planck scale, unless extra dimensional theories are correct). Motivation for quantizing gravity comes from the remarkable success of the quantum theories of the other three fundamental interactions. Although some quantum gravity theories such as string theory and other so-called theories of everything also attempt to unify gravity with the other fundamental forces, others such as loop quantum gravity make no such attempt at unification, they simply quantize the gravitational field while keeping it separate from other force fields.

Observed physical phenomena in the early 21st century can be described well by quantum mechanics or general relativity, without needing both. This can be thought of as due to an extreme separation of scales at which they are important. Quantum effects are usually important only for the "very small", that is, for objects no larger than ordinary molecules. General relativistic effects, on the other hand, show up only for the "very large" bodies such as collapsed stars. (Planets' gravitational fields, as of 2009, are well-described by linearized gravity, so strong-field effects, any effects of gravity beyond lowest nonvanishing order in φ/c2, have not been observed even in the gravitational fields of planets and main sequence stars). Classical physics seems to be adequate over an enormous range of masses of objects from about 10−23 to 1030 kg. Thus there is a want of experimental evidence relating to quantum gravity, but the "gap" spans 53 orders of magnitude.

Contents

Overview

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 meshing 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.

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Effective field theories

Quantum gravity can be treated as an effective field theory. Effective 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-scattering and Newton's law of gravitation have been explicitly computed[1] (although they are so astronomically small that we may never be able to measure them). 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.[citation needed])

While confirming that quantum mechanics and gravity are indeed consistent at reasonable energies, it is 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. Specifically, 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.

Quantum gravity theory for the highest energy scales

The general approach to deriving a quantum gravity theory that is valid at even the highest energy scales is to assume that such a theory will be simple and elegant and, accordingly, to study symmetries and other clues offered by current theories that might suggest ways to combine them into a comprehensive, unified theory. One problem with this approach is that it is unknown whether quantum gravity will actually conform to a simple and elegant theory, as it should resolve the dual conundrums of special relativity with regard to the uniformity of acceleration and gravity, and general relativity with regard to spacetime curvature.

Such a theory is required in order to understand problems involving the combination of very high energy and very small dimensions of space, such as the behavior of black holes, and the origin of the universe.

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.

The 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 demonstrate that the structure of general relativity essentially follows inevitably from the quantum mechanics of interacting theoretical spin-2 massless particles [2][3][4][5][6] (called gravitons).

While there is no concrete proof of the existence of gravitons, 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.

The dilaton

The dilaton made its first appearance in Kaluza-Klein theory, a five-dimensional theory that combined gravitation and electromagnetism. Generally, it appears in string theory. More recently, it has appeared in the lower-dimensional many-bodied gravity problem [7] based on the field theoretic approach of Roman Jackiw. The impetus arose from the fact that complete analytical solutions for the metric of a covariant N-body system have proven elusive in General Relativity. To simplify the problem, the number of dimensions was lowered to (1+1) namely one spatial dimension and one temporal dimension. This model problem, known as R=T theory [8] (as opposed to the general G=T theory) was amenable to exact solutions in terms of a generalization of the Lambert W function. It was also found that the field equation governing the dilaton (derived from differential geometry) was none other than the Schrödinger equation and consequently amenable to quantization [9]. Thus, one had a theory which combined gravity, quantization and even the electromagnetic interaction, promising ingredients of a fundamental physical theory. It is worth noting that the outcome revealed a previously unknown and already existing natural link between general relativity and quantum mechanics. However, this theory needs to be generalized in (2+1) or (3+1) dimensions although, in principle, the field equations are amenable to such generalization. It is not yet clear what field equation will govern the dilaton in higher dimensions. This is further complicated by the fact that gravitons can propagate in (3+1) dimensions and consequently that would imply gravitons and dilatons exist in the real world. Moreover, detection of the dilaton is expected to be even more elusive than the graviton. However, since this approach allows for the combination of gravitational, electromagnetic and quantum effects, their coupling could potentially lead to a means of vindicating the theory, through cosmology and perhaps even experimentally.

Nonrenormalizability of gravity

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 [10] . 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.

QG as an effective field theory

In an effective field theory, all but the first few of the infinite set of parameters in a non-renormalizable 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.

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,[11] in which the only physically relevant information is the relationship between different events in space-time.

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

String theory

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

String theory 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.

Background independent theories

Loop quantum gravity is the fruit of an effort to formulate a background-independent quantum theory.

Topological quantum field theory provided an example of background-independent quantum theory, but with no local degrees of freedom, and only finitely many degrees of freedom globally. This is inadequate to describe gravity in 3+1 dimensions which has local degrees of freedom according to general relativity. In 2+1 dimensions, however, gravity is a topological field theory, and it has been successfully quantized in several different ways, including spin networks.

Semi-classical quantum gravity

Quantum field theory on curved (non-Minkowskian) backgrounds, while not a full quantum theory of gravity, has shown many promising early results. In an analagous 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 least energy (and may or may not contain particles). See Quantum field theory in curved spacetime for a more complete discussion.

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 the 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[12].

Candidate theories

There are a number of proposed quantum gravity theories.[13] 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.[14]

String theory

Projection of a Calabi-Yau manifold, one of the ways of compactifying the extra dimensions posited by string theory

One suggested starting point is ordinary quantum field theories which, 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,[15] 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,[16] 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.[17]

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.[18] 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.[19] 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 for space and one for time.[20] 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[21] form part of a hypothesized eleven-dimensional model known as M-theory, which would constitute a uniquely defined and consistent theory of quantum gravity.[22]

Loop quantum gravity

Simple spin network of the type used in loop quantum gravity

Another 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 some argue is ill-defined.[23] 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.[24] 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.[25]

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[26]. Examples include:

Weinberg-Witten theorem

In quantum field theory, the Weinberg-Witten theorem places some constraints on theories of composite gravity/emergent gravity.

See also

Notes

  1. ^ a b c [gr-qc/9512024] Introduction to the Effective Field Theory Description of Gravity
  2. ^ Kraichnan, R. H. (1955), "Special-Relativistic Derivation of Generally Covariant Gravitation Theory", Phys. Rev. 98: 1118–1122, doi:10.1103/PhysRev.98.1118 
  3. ^ Gupta, S. N. (1954), "Gravitation and Electromagnetism", Phys. Rev. 96: 1683–1685, doi:10.1103/PhysRev.96.1683 
  4. ^ Gupta, S. N. (1957), "Einstein's and Other Theories of Gravitation", Rev. Mod. Phys. 29: 334–336, doi:10.1103/RevModPhys.29.334 
  5. ^ Gupta, S. N. (1962), "Quantum Theory of Gravitation", Recent Developments in General Relativity, Pergamon Press, pp. 251–258 
  6. ^ Deser, S. (1970), "Self-Interaction and Gauge Invariance", Gen. Relativ. Gravit. 1: 9–18, doi:10.1007/BF00759198 
  7. ^ T. Ohta and R.B. Mann (1996). "Canonical reduction of two-dimensional gravity for particle dynamics", Class. Quantum Grav. 13: 2585-2603.[1] Arxiv article [2].
  8. ^ A. Sikkemma and R.B. Mann (1991). "Gravitation and cosmology in two-dimensions", Class. Quantum Grav. 8: 219-236.
  9. ^ P.S. Farrugia, R.B. Mann, and T.C. Scott (2007). "N-body Gravity and the Schrödinger Equation", Class. Quantum Grav. 24: 4647-4659,[3]; Arxiv article [4].
  10. ^ Feynman, R. P.; Morinigo, F. B., Wagner, W. G., & Hatfield, B. (1995). Feynman lectures on gravitation. Addison-Wesley. ISBN 0201627345. 
  11. ^ Smolin, Lee (2001), Three roads to quantum gravity, Basic Books, pp. 20–25, ISBN 0-465-08735-4  Pages 220-226 are annotated references and guide for further reading.
  12. ^ [astro-ph/0506506] Singularity-Free Collapse through Local Inflation
  13. ^ A timeline and overview can be found in Rovelli 2000.
  14. ^ E.g. Ashtekar 2007, Schwarz 2007.
  15. ^ See Donoghue 1995.
  16. ^ Cf. chapters 17 and 18 of Weinberg 1996.
  17. ^ Cf. Goroff & Sagnotti 1985.
  18. ^ 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.
  19. ^ E.g. Ibanez 2000.
  20. ^ 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.
  21. ^ E.g. Weinberg 2000, ch. 31.
  22. ^ E.g. Townsend 1996, Duff 1996.
  23. ^ Cf. section 3 in Kuchař 1973.
  24. ^ See Ashtekar 1986, Ashtekar 1987.
  25. ^ 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.
  26. ^ See e.g. the systematic expositions in Isham 1994 and Sorkin 1997.
  27. ^ See Loll 1998.
  28. ^ See Sorkin 2005.
  29. ^ Cf. Hawking 1987.
  30. ^ See ch. 33 in Penrose 2004 and references therein.
  31. ^ See Daniele Oriti and references therein.

References

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