Loop quantum gravity

Beyond the Standard Model
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Standard Model
Quantum gravity
String theory
Loop quantum gravity
Causal dynamical triangulation
Canonical general relativity

Loop quantum gravity (LQG), also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity. Loop quantum gravity suggests that space can be viewed as an extremely fine fabric or network "woven" of finite quantised loops of excited gravitational fields called spin networks. When viewed over time, these spin networks are called spin foam, which should not be confused with quantum foam. A major quantum gravity contender with string theory, loop quantum gravity incorporates general relativity without requiring string theory's higher dimensions.

LQG preserves many of the important features of general relativity, while simultaneously employing quantization of both space and time at the Planck scale in the tradition of quantum mechanics. The technique of loop quantization was developed for the nonperturbative quantization of diffeomorphism-invariant gauge theory. Roughly, LQG tries to establish a quantum theory of gravity in which the very space itself, where all other physical phenomena occur, becomes quantized.

LQG is one of a family of theories called canonical quantum gravity. The LQG theory also includes matter and forces, but does not address the problem of the unification of all physical forces the way some other quantum gravity theories such as string theory do.

Contents

History of LQG

In 1986, Abhay Ashtekar reformulated Einstein's field equations of general relativity, using what have come to be known as Ashtekar variables, a particular flavor of Einstein-Cartan theory with a complex connection. In 1988, Carlo Rovelli and Lee Smolin used this formalism to introduce the loop representation of quantum general relativity, which was soon developed by Ashtekar, Rovelli, Smolin and many others. In the Ashtekar formulation, the fundamental objects are a rule for parallel transport (technically, a connection) and a coordinate frame (called a vierbein) at each point. Because the Ashtekar formulation was background-independent, it was possible to use Wilson loops as the basis for a nonperturbative quantization of gravity. Explicit (spatial) diffeomorphism invariance of the vacuum state plays an essential role in the regularization of the Wilson loop states.

Around 1990, Rovelli and Smolin obtained an explicit basis of states of quantum geometry, which turned out to be labelled by Roger Penrose's spin networks, and showed that the geometry is quantized, that is, the (non-gauge-invariant) quantum operators representing area and volume have a discrete spectrum. In this context, spin networks arose as a generalization of Wilson loops necessary to deal with mutually intersecting loops. Mathematically, spin networks are related to group representation theory and can be used to construct knot invariants such as the Jones polynomial.

Key concepts of loop quantum gravity

Main article: Quantum gravity

In the framework of quantum field theory, and using the standard techniques of perturbative calculations, one finds that gravitation is non-renormalizable in contrast to the electroweak and strong interactions of the Standard Model of particle physics. This implies that there are infinitely many free parameters in the theory and thus that it cannot be predictive.

In general relativity, the Einstein field equations assign a geometry (via a metric) to space-time. Before this, there is no physical notion of distance or time measurements. In this sense, general relativity is said to be background independent. An immediate conceptual issue that arises is that the usual framework of quantum mechanics, including quantum field theory, relies on a reference (background) space-time. Therefore, one approach to finding a quantum theory of gravity is to understand how to do quantum mechanics without relying on such a background; this is the approach of the canonical quantization/loop quantum gravity/spin foam approaches.

Simple spin network of the type used in loop quantum gravity

Starting with the initial-value-formulation of general relativity (cf. the section on General relativity#Evolution equations), the result is an analogue of the Schrödinger equation called the Wheeler-deWitt equation, which some argue is ill-defined.[1] 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.[2] 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.[3]

Though not proven, it may be impossible to quantize gravity in 3+1 dimensions without creating matter and energy artifacts. Should LQG succeed as a quantum theory of gravity, the known matter fields will have to be incorporated into the theory a posteriori. Many of the approaches now being actively pursued (by Renate Loll, Jan Ambjørn, Lee Smolin, Sundance Bilson-Thompson, Laurent Freidel, Mark B. Wise and others[4]) combine matter with geometry.

The main successes of loop quantum gravity are:

  1. It is a nonperturbative quantization of 3-space geometry, with quantized area and volume operators.
  2. It includes a calculation of the entropy of black holes.
  3. It replaces the Big Bang spacetime singularity with a Big Bounce.

These claims are not universally accepted among the physics community, which is presently divided between different approaches to the problem of quantum gravity. LQG may possibly be viable as a refinement of either gravity or geometry. Many of the core results are rigorous mathematical physics; their physical interpretations remain speculative. Three speculative physical interpretations of LQG's core mathematical results are loop quantization, Lorentz invariance, General covariance and background independence, discussed below. Another physical test for LQG is to reproduce the physics of general relativity coupled with quantum field theory, discussed under problems.

Loop quantization

At the core of loop quantum gravity is a framework for nonperturbative quantization of diffeomorphism-invariant gauge theories, which one might call loop quantization. While originally developed in order to quantize vacuum general relativity in 3+1 dimensions, the formalism can accommodate arbitrary spacetime dimensionalities, fermions,[5] an arbitrary gauge group (or even quantum group), and supersymmetry,[6] and results in a quantization of the kinematics of the corresponding diffeomorphism-invariant gauge theory. Much work remains to be done on the dynamics, the classical limit and the correspondence principle, all of which are necessary in one way or another to make contact with experiment.

In a nutshell, loop quantization is the result of applying C*-algebraic quantization to a non-canonical algebra of gauge-invariant classical observables. Non-canonical means that the basic observables quantized are not generalized coordinates and their conjugate momenta. Instead, the algebra generated by spin network observables (built from holonomies) and field strength fluxes is used.

Loop quantization techniques are particularly successful in dealing with topological quantum field theories, where they give rise to state-sum/spin-foam models such as the Turaev-Viro model of 2+1 dimensional general relativity. A much studied topological quantum field theory is the so-called BF theory in 3+1 dimensions. Since classical general relativity can be formulated as a BF theory with constraints, scientists hope that a consistent quantization of gravity may arise from the perturbation theory of BF spin-foam models.

Lorentz invariance

LQG is a quantization of a classical Lagrangian field theory which is equivalent to the usual Einstein-Cartan theory in that it leads to the same equations of motion describing general relativity with torsion. As such, it can be argued that LQG respects local Lorentz invariance. Global Lorentz invariance is broken in LQG just as in general relativity. A positive cosmological constant can be realized in LQG by replacing the Lorentz group with the corresponding quantum group.

General covariance and background independence

General covariance, also known as "diffeomorphism invariance", is the invariance of physical laws under arbitrary coordinate transformations. An example of this are the equations of general relativity, where this symmetry is one of the defining features of the theory. LQG preserves this symmetry by requiring that the physical states remain invariant under the generators of diffeomorphisms. The interpretation of this condition is well understood for purely spatial diffemorphisms. However, the understanding of diffeomorphisms involving time (the Hamiltonian constraint) is more subtle because it is related to dynamics and the so-called problem of time in general relativity.[7] A generally accepted calculational framework to account for this constraint is yet to be found.[8][9]

Whether or not Lorentz invariance is broken in the low-energy limit of LQG, the theory is formally background independent. The equations of LQG are not embedded in, or presuppose, space and time, except for its invariant topology. Instead, they are expected to give rise to space and time at distances which are large compared to the Planck length.

Problems

While there has been a recent proposal relating to observation of naked singularities,[10] and doubly special relativity, as a part of a program called loop quantum cosmology, as of now there is no experimental observation for which loop quantum gravity makes a prediction not made by the Standard Model or general relativity (a problem that plagues all current theories of quantum gravity).

Making predictions from the theory of LQG has been extremely difficult computationally, also a recurring problem with modern theories in physics.

Another problem is that a crucial free parameter in the theory known as the Immirzi parameter can only be computed by demanding agreement with Bekenstein and Hawking's calculation of the black hole entropy. Loop quantum gravity predicts that the entropy of a black hole is proportional to the area of the event horizon, but does not obtain the Bekenstein-Hawking formula S = A/4 unless the Immirzi parameter is chosen to give this value. A prediction directly from theory would be preferable.

Presently, no semiclassical limit recovering general relativity has been shown to exist. This means it remains unproven that LQG's description of spacetime at the Planck scale has the right continuum limit, described by general relativity with possible quantum corrections. Specifically, the dynamics of the theory is encoded in the Hamiltonian constraint, but there is no candidate Hamiltonian (quantum mechanics). [11] Other technical problems includes finding off-shell closure of the constraint algebra and physical inner product vector space, coupling to matter fields Quantum field theory, fate of the Renormalization of the graviton in Perturbation theory that lead to Ultraviolet divergence beyond 2-loops One-loop Feynman diagram in Feynman diagram. [12]. The fate of Lorentz invariance in loop quantum gravity remains an open problem. [13]

Current LQG research directions attempt to address these known problems, and includes spinfoam models [14] and Gravity as an entropic force [15].

See also

  • Loop quantum cosmology
  • Heyting algebra
  • Category theory
  • Noncommutative geometry
  • Topos theory
  • C*-algebra
  • Regge calculus
  • Double special relativity
  • Lorentz invariance in loop quantum gravity
  • Immirzi parameter
  • Invariance mechanics
  • spin foam
  • group field theory
  • string-net
  • Kodama state
  • supersymmetry
  • Loop quantum black holes

Notes

  1. ↑ Cf. section 3 in Kuchař 1973.
  2. ↑ See Ashtekar 1986, Ashtekar 1987.
  3. ↑ 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.
  4. ↑ See List of loop quantum gravity researchers
  5. ↑ Baez, John, Krasnov,Kirill : Quantization of Diffeomorphism-Invariant Theories with Fermions arXiv:hep-th/9703112
  6. ↑ Ling,Yi; Smolin, Lee : Supersymmetric Spin Networks and Quantum Supergravity http://www.arxiv.org/abs/hep-th/9904016
  7. ↑ See, e.g., Stuart Kauffman and Lee Smolin "A Possible Solution For The Problem Of Time In Quantum Cosmology" (1997). [1]
  8. ↑ See, e.g., Lee Smolin, "The Case for Background Independence", in Dean Rickles, et al. (eds.) The Structural Foundations of Quantum Gravity (2006), p 196 ff.
  9. ↑ For a highly technical explanation, see, e.g., Carlo Rovelli (2004). Quantum Gravity, p 13 ff.
  10. ↑ Goswami et al.. "Quantum evaporation of a naked singularity". Physical Review Letters. http://arxiv.org/abs/gr-qc/0506129. Retrieved 2010-01-02. 
  11. ↑ http://arxiv.org/abs/hep-th/0501114
  12. ↑ http://arxiv.org/abs/hep-th/0501114
  13. ↑ "Testing Einstein's special relativity with Fermi's short hard gamma-ray burst GRB090510". Fermi GBM/LAT Collaborations. http://arxiv.org/abs/0908.1832. Retrieved August 13, 2009. 
  14. ↑ http://math.ucr.edu/home/baez/week280.html
  15. ↑ Newtonian gravity in loop quantum gravity, Lee Smolin, 2010

References

External links