History of loop quantum gravity

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The history of loop quantum gravity spans more than three decades of intense research.

History

Classical theories of gravitation

gravitation published by Albert Einstein in 1915. According to it, the force of gravity is a manifestation of the local geometry of spacetime. Mathematically, the theory is modelled after Bernhard Riemann's metric geometry, but the Lorentz group of spacetime symmetries (an essential ingredient of Einstein's own theory of special relativity
) replaces the group of rotational symmetries of space. (Later, loop quantum gravity inherited this geometric interpretation of gravity, and posits that a quantum theory of gravity is fundamentally a quantum theory of spacetime.)

In the 1920s, the French mathematician Élie Cartan formulated Einstein's theory in the language of bundles and connections,[1] a generalization of Riemannian geometry to which Cartan made important contributions. The so-called Einstein–Cartan theory of gravity not only reformulated but also generalized general relativity, and allowed spacetimes with torsion as well as curvature. In Cartan's geometry of bundles, the concept of parallel transport is more fundamental than that of distance, the centerpiece of Riemannian geometry. A similar conceptual shift occurs between the invariant interval of Einstein's general relativity and the parallel transport of Einstein–Cartan theory.

Spin networks

In 1971, physicist Roger Penrose explored the idea of space arising from a quantum combinatorial structure.[2][3] His investigations resulted in the development of spin networks. Because this was a quantum theory of the rotational group and not the Lorentz group, Penrose went on to develop twistors.[4]

Loop quantum gravity

In 1982,

Yang–Mills field
theories. But Sen's work fell short of giving a full clear systematic theory and particularly failed to clearly discuss the conjugate momenta to the spinorial variables, its physical interpretation, and its relation to the metric (in his work he indicated this as some lambda variable).

In 1986–87, physicist

vierbein) at each point.[6][7] So these variable became what we know as Ashtekar variables, a particular flavor of Einstein–Cartan theory with a complex connection. General relativity theory expressed in this way, made possible to pursue quantization of it using well-known techniques from quantum gauge field theory
.

The quantization of gravity in the Ashtekar formulation was based on

background-independent, it was possible to use Wilson loops as the basis for nonperturbative quantization of gravity
.

Due to efforts by Sen and Ashtekar, a setting in which the

Hamiltonian operator on a well-defined Hilbert space was obtained. This led to the construction of the first known exact solution, the so-called Chern–Simons form or Kodama state
. The physical interpretation of this state remains obscure.

In 1988–90, Carlo Rovelli and Lee Smolin obtained an explicit basis of states of quantum geometry, which turned out to be labeled by Penrose's spin networks.[9][10] 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. Loop quantum gravity (LQG) thus became related to topological quantum field theory and group representation theory.

In 1994, Rovelli and Smolin showed that the quantum operators of the theory associated to area and volume have a discrete spectrum.[11] Work on the semi-classical limit, the continuum limit, and dynamics was intense after this, but progress was slower.

On the semi-classical limit front, the goal is to obtain and study analogues of the harmonic oscillator coherent states (candidates are known as weave states).

Hamiltonian dynamics

LQG was initially formulated as a quantization of the Hamiltonian ADM formalism, according to which the Einstein equations are a collection of constraints (Gauss, Diffeomorphism and Hamiltonian). The kinematics are encoded in the Gauss and Diffeomorphism constraints, whose solution is the space spanned by the spin network basis. The problem is to define the Hamiltonian constraint as a self-adjoint operator on the kinematical state space. The most promising work[according to whom?] in this direction is Thomas Thiemann's Phoenix Project.[12]

Covariant dynamics

Much of the recent[as of?] work in LQG has been done in the covariant formulation of the theory, called "spin foam theory." The present version of the covariant dynamics is due to the convergent work of different groups, but it is commonly named after a paper by Jonathan Engle, Roberto Pereira and Carlo Rovelli in 2007–08.[13] Heuristically, it would be expected that evolution between spin network states might be described by discrete combinatorial operations on the spin networks, which would then trace a two-dimensional skeleton of spacetime. This approach is related to state-sum models of statistical mechanics and topological quantum field theory such as the Turaeev–Viro model of 3D quantum gravity, and also to the Regge calculus approach to calculate the Feynman path integral of general relativity by discretizing spacetime.

See also

References

  1. ^ Élie Cartan. "Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion." C. R. Acad. Sci. (Paris) 174, 593–595 (1922); Élie Cartan. "Sur les variétés à connexion affine et la théorie de la relativité généralisée." Part I: Ann. Éc. Norm. 40, 325–412 (1923) and ibid. 41, 1–25 (1924); Part II: ibid. 42, 17–88 (1925).
  2. ISBN 0-12-743350-3
    .
  3. ISBN 0-521-07956-X
    .
  4. ISBN 88-7088-142-3
    .
  5. ^ Amitabha Sen, "Gravity as a spin system," Phys. Lett. B119:89–91, December 1982.
  6. ^ Abhay Ashtekar, "New variables for classical and quantum gravity," Phys. Rev. Lett., 57, 2244-2247, 1986.
  7. ^ Abhay Ashtekar, "New Hamiltonian formulation of general relativity," Phys. Rev. D36, 1587-1602, 1987.
  8. doi:10.1103/PhysRevD.10.2445
    .
  9. ^ Carlo Rovelli and Lee Smolin, "Knot theory and quantum gravity," Phys. Rev. Lett., 61 (1988) 1155.
  10. ^ Carlo Rovelli and Lee Smolin, "Loop space representation of quantum general relativity," Nuclear Physics B331 (1990) 80-152.
  11. ^ Carlo Rovelli, Lee Smolin, "Discreteness of area and volume in quantum gravity" (1994): arXiv:gr-qc/9411005.
  12. S2CID 16304158
    .
  13. ^ Jonathan Engle, Roberto Pereira, Carlo Rovelli, "Flipped spinfoam vertex and loop gravity". Nucl. Phys. B798 (2008). 251–290. arXiv:0708.1236.

Further reading

Topical reviews
Popular books
  • The End of Time: The Next Revolution in Our Understanding of the Universe
    (1999).
  • Lee Smolin, Three Roads to Quantum Gravity (2001).
  • Carlo Rovelli, Che cos'è il tempo? Che cos'è lo spazio?, Di Renzo Editore, Roma, 2004. French translation: Qu'est ce que le temps? Qu'est ce que l'espace?, Bernard Gilson ed, Brussel, 2006. English translation: What is Time? What is space?, Di Renzo Editore, Roma, 2006.
Magazine articles
Easier introductory, expository or critical works
  • Abhay Ashtekar, "Gravity and the Quantum," e-print available as gr-qc/0410054.
  • John C. Baez and Javier P. Muniain, Gauge Fields, Knots and Quantum Gravity, World Scientific (1994).
  • Carlo Rovelli, "A Dialog on Quantum Gravity," e-print available as hep-th/0310077.
More advanced introductory/expository works
Conference proceedings
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