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## Quantum Gravity (2004)

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Citations: | 571 - 11 self |

### Citations

416 | Introduction to Modern Canonical Quantum General Relativity,
- Thiemann
- 2007
(Show Context)
Citation Context ... theories, with modifications in regularization methods appropriate to theories without background metrics. More recently, all the key results have been confirmed using mathematically rigorous methods=-=[7, 8]-=-. In this review I employ a non-rigorous style, grounded on intuitions about the behavior of gauge fields that particle physicists and condensed matter physicists will find familiar. But the reader sh... |

406 |
Lectures on Quantum Mechanics, Belfer Graduate
- Dirac
- 1964
(Show Context)
Citation Context ...art of the low energy effective theory, the dynamics of the fields will be invariant under active diffeomorphisms of the manifold. The reason for this was worked out first in detail by Dirac, and his =-=[24]-=- is still a good source for understanding the connection between background independence and diffeomorphism invariance3 . We note that what is necessarily assumed from general relativity is only the p... |

259 | Background independent quantum gravity: a status report - Ashtekar, Lewandowski - 2004 |

180 | Relativistic spin networks and quantum gravity.
- Barrett, Crane
- 1998
(Show Context)
Citation Context ...work are four valent graphs. They then are each dual to a three dimensional triangulation 20 in which each node is dual to tetrahedra, whose four faces are dual to the four edges incident on that node=-=[59, 60]-=-. Each face of the dual triangulation is now labeled by a representation, while each tetrahedron is labeled by an invariant. There are then spin foams constructed from a four dimensional triangulation... |

147 |
Space-time as a causal set,
- Bombelli, Lee, et al.
- 1987
(Show Context)
Citation Context ...en in [34, 35]. There are several different approaches to a background independent quantum theory of gravity. Besides loop quantum gravity, others include dynamical triangulations[37, 38], causal sets=-=[36]-=- and the Gambini-Pullin discrete quantization approach to quantum gravity[39]. Although these are independently motivated research programs, some of their results are relevant for loop quantum gravity... |

147 |
Particle Creation by Black
- Hawking
- 1975
(Show Context)
Citation Context ...f a black hole or cosmological horizon[18]. This picture reproduces completely and explains the results on the thermodynamic and quantum properties of horizons from the work of Bekenstein[89], Hawking=-=[90]-=- and Unruh[91]. This picture is completely general and applies to all black hole and cosmological horizons. Finally, there is a first result indicating that black hole singularities are eliminated by ... |

119 |
The Life of the Cosmos,
- Smolin
- 1997
(Show Context)
Citation Context ...er should be aware that the word “background independence” has other connotations besides that given here. More about this as an answer to a FAQ. 3 This is also described in detail in [6] and in [25]-=-=[29]-=-. 5theories. This means that the configuration variable is a gauge field, and the metric information is contained in the conjugate momenta to the gauge field. This extends to coupling with all the kn... |

107 |
Real Ashtekar variables for Lorentzian signature space-times”,
- Barbero, F
- 1995
(Show Context)
Citation Context ...shtekar[2]. By now several different connections are used in loop quantum gravity. These include the self-dual part of the spacetime connection[1, 2], and a real SU(2) connection introduced by Barbero=-=[47]-=- and exploited by Thiemann[48]. There are also alternate formulations that use both the left and right handed parts of the spacetime connection, [31, 49]. (19) 12D ∧ B i = 0 (20) where D is the SU(2)... |

96 |
sum over surfaces* form of loop quantum gravity,” Phys. Rev. D56
- Reisenberger, Rovelli
- 1997
(Show Context)
Citation Context ...es such as light cones and horizons. • A spin foam as a one dimension higher analogue of a Feynman diagram in which spin networks play the role normally played by incoming and outgoing particle states=-=[57]-=-. Propagators for edges are two dimensional spacetime surfaces, they meet along edges which are propagators for nodes. Each surface is labeled by a representation, as the edge it propagates. Similarly... |

95 |
On the separation of Einsteinian substructures,”
- Plebanski
- 1977
(Show Context)
Citation Context ... 3. The action is, S topological ∫ = The field equations which follow are M (B i ∧ Fi − Λ 2 Bi ∧ Bi) (18) F i = ΛB i 13 This kind of formulation of general relativity was first discovered by Plebanksi=-=[44]-=- and later independently by [45, 46]. The corresponding simplification of the Hamiltonian theory was independently discovered by Sen[1] and formalized by Ashtekar[2]. By now several different connecti... |

88 |
Topological Investigations of Quantized Gauge Theories,”. Presented at Les Houches
- Jackiw
(Show Context)
Citation Context ...mal field theory. It is thus not known if the state is normalizable in the physical inner product of the exact 28 A related state in the context of Yang-Mills theory was previously discussed by Jackiw=-=[105]-=-. Some properties of Jackiw’s state were discussed by Witten[106]. More about this in a FAQ. 29 For a possible Λ = 0 analogue of the Kodama state, see [104]. 23theory. One can ask whether the project... |

87 | Spin foam models
- Baez
- 1998
(Show Context)
Citation Context ...over, the representation πAILRS is irreducible (that is, not only one vector is cyclic but every vector is cyclic) which excludes the existence of spurious superselection sectors. 20called spin foams=-=[13]-=--[15],[52]-[60] 24 . The histories by which spin network states evolve to other spin network states, called spin foam histories, are explicitly known. A spin foam history is a labeled combinatorial st... |

73 |
Nonstandard optics from quantum space-time,”
- Gambini, Pullin
- 1999
(Show Context)
Citation Context ...ted[20] and has already led to some predictions for effects observable in CMB spectra[20, 21]. And there are published predictions for observable Planck scale deviations from energy momentum relations=-=[22, 23]-=- that imply predictions for experiments in progress such as AUGER and GLAST. For those whose interest is more towards formal speculations concerning supersymmetry and higher dimensions than experiment... |

72 | Quantum geometry of isolated horizons and black hole entropy,”
- Ashtekar, Baez, et al.
- 2000
(Show Context)
Citation Context ...t a number of applications to problems of physical interest are under development. There is by now a well understood detailed description of the quantum physics of black hole and cosmological horizons=-=[16, 17, 18]-=- that reproduces the Bekenstein-Hawking results on the relationship between area and entropy[19]. There is under development an approach to quantum cosmology that shows that cosmological singularities... |

72 |
Barrett–Crane model from a Boulatov–Ooguri field theory over a homogeneous space, Nuclear Phys.
- Pietri, Freidel, et al.
- 2000
(Show Context)
Citation Context ...e by showing that there for every spin foam model there is a matrix model (more precisely a dynamical system on a group manifold) such that the spin foams are the Feynman diagrams of that matrix model=-=[58]-=-. Each foam is then a Feynman diagram, and like in ordinary quantum field theory, there are sums over intermediate states on internal propagators. In the spin foam case these are sums over representat... |

72 |
Exactly Soluble Diffeomorphism Invariant Theories
- Horowitz
- 1989
(Show Context)
Citation Context ...methods can also be used to solve quantum gravity in 2+1 dimensions[50] and in some 1 + 1 dimensional reductions of the theory[113]. They also work to solve a large class of topological field theories=-=[114, 115]-=-, giving results equivalent to those achieved by other methods. Further, loop methods applied to lattice gauge theories yields results equivalent to those achieved by other methods[116]. 34. Spin foam... |

70 | Quantum Spin Dynamics (QSD),
- Thiemann
- 1998
(Show Context)
Citation Context ...ntum theory of gravity. 5. Among the operators that have been constructed and found to be finite on H diffeo is the Hamiltonian constraint (or, as it is often called, the Wheeler de Witt equation[74]-=-=[76]-=-.) Not only can the Wheeler deWitt equation be precisely defined, it can be solved exactly. Several infinite sets of solutions have been constructed, as certain superpositions of the spin network basi... |

69 |
Emergence of a 4D world from causal quantum gravity
- Ambjørn, Jurkiewicz, et al.
(Show Context)
Citation Context ...triangulations. This work shows that Lorentzian and Euclidean path integrals for quantum gravity fall into different universality classes, with the Lorentzian path integrals being more convergent[38]-=-=[42]-=-. For the first time there are results that suggest that a dynamical 3 + 1 spacetime may emerge from critical behavior of a path integral over combinatorially defined spacetimes[42]. 214.3 Results on... |

67 | Linking topological quantum field theory and nonperturbative quantum gravity
- Smolin
- 1995
(Show Context)
Citation Context ...s may be incorporated in the theory including timelike boundaries, in the presence of both positive and negative cosmological constant, and null boundaries such as black hole and cosmological horizons=-=[16]-=--[18, 31, 49]. In all these cases the boundary states and observables are understood in terms of structures derived from Chern-Simons theory. 17. The boundary theory provides a detailed microscopic de... |

57 | Quantum gravity with a positive cosmological constant. 2002, arXiv:hep-th/0209079v1. arXiv.org, e-Print Arch. High Energy Phys. Theor. http://arxiv.org/abs/hep-th/0209079 (accessed 3
- Smolin
- 2010
(Show Context)
Citation Context ...ntum constraint equations, whose semiclassical limit exists[100]. 28 That limit describes deSitter or anti-deSitter spacetime. Solutions obtained by perturbing around this state, in both gravitational=-=[11]-=- and matter fields[101], reproduce, at long wavelength, quantum field theory in curved spacetime and the quantum theory of long wave length, free gravitational waves on deSitter or anti-deSitter space... |

54 |
Geometry eigenvalues and the scalar product from recoupling theory in loop quantum gravity” Phys
- DePietri, Rovelli
- 1996
(Show Context)
Citation Context ...iffeomorphism invariant observables have been constructed. After a suitable regularization procedure these turn out to be represented by finite operators on H diffeo , the space of spin network states=-=[10, 63, 64]-=-. In the case of general relativity and supergravity, these include the volume of the universe, the area of the boundary of the universe, or of any surface defined by the values of matter fields. Thes... |

50 |
The Chern-Simons Invariant as the Natural Time Variable for Classical and Quantum Cosmology
- Smolin, Soo
- 1995
(Show Context)
Citation Context ...ons, whose semiclassical limit exists[100]. 28 That limit describes deSitter or anti-deSitter spacetime. Solutions obtained by perturbing around this state, in both gravitational[11] and matter fields=-=[101]-=-, reproduce, at long wavelength, quantum field theory in curved spacetime and the quantum theory of long wave length, free gravitational waves on deSitter or anti-deSitter spacetime 29 . It is not yet... |

49 |
Louapre: Ponzano–Regge model revisited. I: Gauge fixing, observables and interacting spinning particles
- Freidel, David
(Show Context)
Citation Context ...foam amplitudes[57]. 14. The spin foam models have been extended to include gauge and spinor degrees of freedom. Recently spin foam models with matter have been extensively studied in 2 + 1 dimensions=-=[81]-=-. 15. Spin foam models appropriate for Lorentzian quantum gravity, called causal spin foams, have quantum analogues of all the basic features of general relativistic spacetimes 25 . These include dyna... |

49 |
q-deformation of Poincar’e algebra
- Lukierski, Ruegg, et al.
- 1991
(Show Context)
Citation Context .... Such theories are called modified special relativity or doubly special relativity[123, 126, 127, 125]. Examples are given by some forms of noncommutative geometry, for example, κ−Minkowski spacetime=-=[124]-=-. In all such theories energy and momentum conservation become non-linear which, of course, effects the analysis of the experiments. Among the experiments which either already give sufficient sensitiv... |

48 |
Loops, Knots, Gauge Theories and Quantum Gravity” (Cambridge Monographis
- Gambini, Pullin
- 1996
(Show Context)
Citation Context ...th a list of open problems. For those interested in a more detailed introduction to loop quantum gravity, there are several textbooks, monographs and review papers now available. Early monographs are =-=[4, 5]-=-. A textbook by Rovelli[6] is about to appear from Cambridge University Press, with an early draft available on line at http://www.cpt.univ-mrs.fr∼rovelli/rovelli.html. Comprehensive introductions to ... |

47 |
Topological quantum field theories, Publ
- Atiyah
- 1989
(Show Context)
Citation Context ...methods can also be used to solve quantum gravity in 2+1 dimensions[50] and in some 1 + 1 dimensional reductions of the theory[113]. They also work to solve a large class of topological field theories=-=[114, 115]-=-, giving results equivalent to those achieved by other methods. Further, loop methods applied to lattice gauge theories yields results equivalent to those achieved by other methods[116]. 34. Spin foam... |

45 | Quantum geometry with intrinsic local causality, Phys.Rev. D58
- Markopoulou, Smolin
- 1998
(Show Context)
Citation Context ... by abstract spin networks31 . The graphs on which the spin networks are based are defined combinatorially, so that the need to specify the topology and dimension of the spatial manifold is eliminated=-=[52, 55]-=-. In such a theory the dimension and topology are dynamical, and different states may exist whose coarse grained descriptions reveal manifolds of different dimensions and topology. The histories of th... |

45 | World Sheet Formulations of Gauge Theories and Gravitation,
- Reisenberger
- 1994
(Show Context)
Citation Context ...work are four valent graphs. They then are each dual to a three dimensional triangulation 20 in which each node is dual to tetrahedra, whose four faces are dual to the four edges incident on that node=-=[59, 60]-=-. Each face of the dual triangulation is now labeled by a representation, while each tetrahedron is labeled by an invariant. There are then spin foams constructed from a four dimensional triangulation... |

45 | Coherent states for canonical quantum general relativity and the infinite tensor product extension
- Sahlmann, Thiemann, et al.
- 2001
(Show Context)
Citation Context ...rgy behavior 20. A large class of putative ground states can be constructed in Hkin which have course grained descriptions which reproduce the geometry of flat space, or any slowly varying metric[93]-=-=[97]-=-. Linearizing the quantum theory around these states yields linearized quantum gravity, for gravitons with wavelength long compared to the Planck length[96]. 21. Excitations of matter fields on these ... |

44 | Black hole entropy in loop quantum gravity,
- Meissner
- 2004
(Show Context)
Citation Context ...t then the horizon quantum geometry has maximal symmetry, as does the classical geometry. If one includes states where the quantum geometry is arbitrarily asymmetric, the match does not appear to work=-=[84]-=-, but in that case the quasi normal mode spectrum should be altered as well. 22 (37)spectrum fills in and becomes continuous in the limit of infinite black hole mass. This fine structure stands as an... |

43 |
The physical Hamiltonian in nonperturbative quantum gravity,
- Rovelli, Smolin
- 1994
(Show Context)
Citation Context ...e quantum theory of gravity. 5. Among the operators that have been constructed and found to be finite on H diffeo is the Hamiltonian constraint (or, as it is often called, the Wheeler de Witt equation=-=[74]-=--[76].) Not only can the Wheeler deWitt equation be precisely defined, it can be solved exactly. Several infinite sets of solutions have been constructed, as certain superpositions of the spin network... |

42 | An Analytical computation of asymptotic Schwarzschild quasinormal frequencies
- Motl
- 2003
(Show Context)
Citation Context ...he loop quantum gravity description of horizons. The value of the asymptotic quasi normal mode frequency was at first known only numerically, but it has been very recently derived analytically by Motl=-=[83]-=-. Once Dreyer’s argument fixes the Imirzi parameter, the Bekenstein-Hawking relation (37) is predicted exactly for black hole and cosmological horizons 27 . 18. Corrections to the Bekenstein entropy h... |

40 |
Nature 418
- Amelino-Camelia
- 2002
(Show Context)
Citation Context ...he lorentz transformations are realized non-linearly when acting on the energy and momentum eigenstates of the theory. Such theories are called modified special relativity or doubly special relativity=-=[123, 126, 127, 125]-=-. Examples are given by some forms of noncommutative geometry, for example, κ−Minkowski spacetime[124]. In all such theories energy and momentum conservation become non-linear which, of course, effect... |

36 |
Einstein’s Search for General Covariance, 1912–1915”, in Einstein and the History of General Relativity, edited by D
- Stachel
- 1989
(Show Context)
Citation Context ... reader should be aware that the word “background independence” has other connotations besides that given here. More about this as an answer to a FAQ. 3 This is also described in detail in [6] and in =-=[25]-=--[29]. 5theories. This means that the configuration variable is a gauge field, and the metric information is contained in the conjugate momenta to the gauge field. This extends to coupling with all t... |

33 | Partial observables, - Rovelli - 2002 |

31 | Notes for a brief history of quantum gravity - Rovelli - 2000 |

30 |
in Quantum Gravity and Cosmology, eds J Perez-Mercader et al, World Scienti
- Smolin
- 1992
(Show Context)
Citation Context ...iffeomorphism invariant observables have been constructed. After a suitable regularization procedure these turn out to be represented by finite operators on H diffeo , the space of spin network states=-=[10, 63, 64]-=-. In the case of general relativity and supergravity, these include the volume of the universe, the area of the boundary of the universe, or of any surface defined by the values of matter fields. Thes... |

29 |
New Perspectives in Canonical Gravity,” Bibliopo ersity dge
- Ashtekar
- 2004
(Show Context)
Citation Context ... are background independent, in that no fixed spacetime metric is needed to describe their quantum dynamics. Instead, the geometry of space and time is coded in the degrees of freedom of a gauge field=-=[4]-=--[15]. While a number of technical issues remain open, enough is known about the physics of loop quantum gravity that a number of applications to problems of physical interest are under development. T... |

28 | Dual formulation of spin network evolution
- Markopoulou
(Show Context)
Citation Context ...cterizations of a history, F. ∂F = Γout − Γin (33) • A spin foam as a causal history. A causal spin foam is a succession of local moves mi that transform the initial state Γint to the final state Γout=-=[52]-=--[56]. The moves mi are considered the discrete analogues of the events of a continuous spacetime. They have a partial order, which give the history a discrete analogue of the causal structure of a Lo... |

27 |
On quantum statistical mechanics of a Schwarzschild black hole,”
- Krasnov
- 1998
(Show Context)
Citation Context ...t a number of applications to problems of physical interest are under development. There is by now a well understood detailed description of the quantum physics of black hole and cosmological horizons=-=[16, 17, 18]-=- that reproduces the Bekenstein-Hawking results on the relationship between area and entropy[19]. There is under development an approach to quantum cosmology that shows that cosmological singularities... |

26 |
Canonical quantization of General Relativity in discrete space-times
- Gambini, Pullin
(Show Context)
Citation Context ...ndent quantum theory of gravity. Besides loop quantum gravity, others include dynamical triangulations[37, 38], causal sets[36] and the Gambini-Pullin discrete quantization approach to quantum gravity=-=[39]-=-. Although these are independently motivated research programs, some of their results are relevant for loop quantum gravity, because they concern models which can be understood as arising from spin fo... |

25 | A holographic formulation of quantum general relativity,” Phys
- Smolin
(Show Context)
Citation Context ..., and a real SU(2) connection introduced by Barbero[47] and exploited by Thiemann[48]. There are also alternate formulations that use both the left and right handed parts of the spacetime connection, =-=[31, 49]-=-. (19) 12D ∧ B i = 0 (20) where D is the SU(2) gauge covariant derivative. The second term contains a quadratic function of Bi , which can be expressed as S constraint ∫ = φijB M i ∧ B j where φij is... |

23 | Spectrum of the volume operator in quantum gravity”, Nucl. Phys
- Loll
- 1996
(Show Context)
Citation Context ...erator that measures angles in the quantum geometry[71] and the lengths of curves in Σ[72]. 3. The area, volume and length operators have discrete, finite spectra, valued in terms of the Planck length=-=[63, 64, 65, 72]-=-. There is hence a smallest possible volume, a smallest possible area, and a smallest possible length, each of Planck scale. The spectra have been computed in closed form. 4. The area and volume opera... |

22 |
Spin Networks and Quantum Gravity", gr-qc/9505006 L. Smolin, "The Future of Spin Networks
- Rovelli, Smolin
(Show Context)
Citation Context ... a gauge field on a manifold Σ, has an orthonormal basis, |{Γ} > whose elements are in one to one correspondence with the diffeomorphism equivalence classes of embeddings of spin networks[61], into Σ =-=[62]-=-. The inner product is given by < {Γ}|{Γ ′ } >= δ{Γ}{Γ ′ }. (36) Here {Γ} refers to the equivalence class of graphs under piecewise smooth diffeomorphisms of Σ. In the case of pure general relativity ... |

21 |
Finiteness of a spin foam model for euclidean quantum general relativity
- Perez
- 2001
(Show Context)
Citation Context ... has two parts, a sum over graphs representing histories of spin networks, and, on each, a sum over the labels. The sums over labels are known from both analytic and numerical results to be convergent=-=[78, 79]-=- for some spin foam models, including some corresponding to the quantization of the Einstein equations in 2 + 1 and 3 + 1 dimensions. 12. For some spin foam model in 2 + 1 dimensions, it has been show... |

21 |
General logarithmic corrections to black-hole entropy,”
- Das, Majumdar, et al.
- 2002
(Show Context)
Citation Context ...eter, the Bekenstein-Hawking relation (37) is predicted exactly for black hole and cosmological horizons 27 . 18. Corrections to the Bekenstein entropy have been calculated and found to be logarithmic=-=[86]-=-. 19. Suitable approximate calculations reproduce the Hawking radiation. They further predict a discrete fine structure in the Hawking spectrum[87, 88]. At the same time, the 27 Two caveats should be ... |

21 |
Physics with non-perturbative quantum gravity: radiation from a quantum black
- Barreira, Carfora, et al.
- 1996
(Show Context)
Citation Context ...ropy have been calculated and found to be logarithmic[86]. 19. Suitable approximate calculations reproduce the Hawking radiation. They further predict a discrete fine structure in the Hawking spectrum=-=[87, 88]-=-. At the same time, the 27 Two caveats should be mentioned. 1) Dreyer’s calculation leads also to the conclusion that transitions where punctures, i.e. ends of spin networks, are added or subtracted t... |

21 |
et al., Limits to Quantum Gravity Effects from Observations of TeV
- Biller
- 1999
(Show Context)
Citation Context ...le modification of the energy-momentum relations, but there are proposals for forms of such theories that do achieve this. 2. A similar anomaly is possibly indicated in Tev photons coming from blazers=-=[130]-=-. Similar remarks apply as to the explanatory power of Scenarios A) and B) in the event that the anomaly exists. 273. A consequence of (39) can be an energy dependent speed of light. This effect can ... |

20 |
Non-perturbative summation over 3D discrete topologies”, Phys
- Freidel, Louapre
- 2003
(Show Context)
Citation Context ...to the quantization of the Einstein equations in 2 + 1 and 3 + 1 dimensions. 12. For some spin foam model in 2 + 1 dimensions, it has been shown that the sum over spin foam histories is Borel summable=-=[80]-=-. 13. The physical inner product, which is the inner product on solutions to all the constraints, has an exact expression, given in terms of a summation over spin foam amplitudes[57]. 14. The spin foa... |

20 | L.: Quantum theory from quantum gravity
- Markopoulou, Smolin
- 2004
(Show Context)
Citation Context ...rom obstructions to recovering trivial topology at low energies. It has been proposed that these may account for the existence of matter degrees of freedom[118], and perhaps even quantum theory itself=-=[119]-=-. 4.8 Comments on the results On the basis of these results, it can be claimed that loop quantum gravity is both a consistent quantization of general relativity and a physically plausible candidate fo... |

19 |
Causal evolution of spin networks,” grqc/9702025
- Markopoulou, Smolin
(Show Context)
Citation Context ...zations of a history, F. ∂F = Γout − Γin (33) • A spin foam as a causal history. A causal spin foam is a succession of local moves mi that transform the initial state Γint to the final state Γout[52]-=-=[56]-=-. The moves mi are considered the discrete analogues of the events of a continuous spacetime. They have a partial order, which give the history a discrete analogue of the causal structure of a Lorentz... |

18 | Supersymmetric spin networks and quantum supergravity
- Ling, Smolin
(Show Context)
Citation Context ... than experiment, there are also results that show how the methods of loop quantum gravity may be extended to give background independent descriptions of quantum gravity in the higher and super realms=-=[31]-=--[35]. It thus seems like a good time for an introduction to the whole approach that may help to make the basic ideas, results and methods accessible to a wider range of physicists. The main purpose o... |

18 | New variables for canonical supergravity - Jacobson - 1988 |

18 | Operators for quantized directions
- Major
- 1999
(Show Context)
Citation Context ...r fields. These operators all preserve the diffeomorphism invariance of the states[66]. Other operators also have been constructed, for example an operator that measures angles in the quantum geometry=-=[71]-=- and the lengths of curves in Σ[72]. 3. The area, volume and length operators have discrete, finite spectra, valued in terms of the Planck length[63, 64, 65, 72]. There is hence a smallest possible vo... |

18 |
communication
- Thiemann
- 1990
(Show Context)
Citation Context ... expressed in a path integral formalism, 22To my knowledge whether loop quantum gravity suffers from the fermion doubling problem is an open question. 23A more precise statement of the theorem follows=-=[77]-=-. Let A(e) be the holonomy of a classical connection along a piecewise analytic path e, let Ef(S) = ∫ Tr(f ∗ S E) be the electric flux of the classical two-form ∗E through the piecewise analytic surfa... |

18 |
Weaving a classical metric with quantum threads.
- Ashtekar, Rovelli, et al.
- 1992
(Show Context)
Citation Context ...w energy behavior 20. A large class of putative ground states can be constructed in Hkin which have course grained descriptions which reproduce the geometry of flat space, or any slowly varying metric=-=[93]-=--[97]. Linearizing the quantum theory around these states yields linearized quantum gravity, for gravitons with wavelength long compared to the Planck length[96]. 21. Excitations of matter fields on t... |

17 |
Quantum gravity represented as dynamical triangulations
- Ambjørn, Burda, et al.
- 1992
(Show Context)
Citation Context ...his direction are given in [34, 35]. There are several different approaches to a background independent quantum theory of gravity. Besides loop quantum gravity, others include dynamical triangulations=-=[37, 38]-=-, causal sets[36] and the Gambini-Pullin discrete quantization approach to quantum gravity[39]. Although these are independently motivated research programs, some of their results are relevant for loo... |

17 |
Theory of quantized directions, unpublished manuscript; in “Quantum theory and beyond
- Penrose
- 1971
(Show Context)
Citation Context ...ant states of a gauge field on a manifold Σ, has an orthonormal basis, |{Γ} > whose elements are in one to one correspondence with the diffeomorphism equivalence classes of embeddings of spin networks=-=[61]-=-, into Σ [62]. The inner product is given by < {Γ}|{Γ ′ } >= δ{Γ}{Γ ′ }. (36) Here {Γ} refers to the equivalence class of graphs under piecewise smooth diffeomorphisms of Σ. In the case of pure genera... |

15 |
Finite, diffeomorphism invariant observables in quantum gravity, Phys
- Smolin
- 1994
(Show Context)
Citation Context ...the volume of the universe, the area of the boundary of the universe, or of any surface defined by the values of matter fields. These operators all preserve the diffeomorphism invariance of the states=-=[66]-=-. Other operators also have been constructed, for example an operator that measures angles in the quantum geometry[71] and the lengths of curves in Σ[72]. 3. The area, volume and length operators have... |

15 | Observational limits on quantum geometry effects
- Konopka, Major
(Show Context)
Citation Context ...t bounds on α for photons, electrons and hadrons, again in Scenario A), only[134]. 6. Present data from the absence of vacuum Cherenkov effects puts interesting bounds on α in the case of Scenario A) =-=[128]-=-. 7. Observations of bifringence effects in polarized light from distant galaxies puts tight bounds on a possible helicity dependent α [135]. 8. Observations of phase coherence in stellar and galactic... |

13 | Quantum geometry and thermal radiation from black holes”, preprint CGPG-97/9-4
- Krasnov
(Show Context)
Citation Context ...ropy have been calculated and found to be logarithmic[86]. 19. Suitable approximate calculations reproduce the Hawking radiation. They further predict a discrete fine structure in the Hawking spectrum=-=[87, 88]-=-. At the same time, the 27 Two caveats should be mentioned. 1) Dreyer’s calculation leads also to the conclusion that transitions where punctures, i.e. ends of spin networks, are added or subtracted t... |

11 | Eleven-dimensional supergravity as a constrained topological field theory
- Ling
- 2001
(Show Context)
Citation Context ...the diffeomorphism invariance of the theory. This is true of general relativity in all dimensions and is also known to be true of supergravity, in d = 4, at least up to N = 2[31]-[33], and for d = 11 =-=[34]-=-. As a result, the techniques which give us consistent diffeomorphism invariant gauge theories can be applied to give consistent quantizations of general relativity and supergravity. Many results foll... |

11 | Time, measurement and information loss in quantum cosmology
- Smolin
(Show Context)
Citation Context ... computed in closed form. 4. The area and volume operators can be promoted to genuine physical observables, by gauge fixing the time gauge so that at least locally time is measured by a physical field=-=[73, 66]-=-. The discrete spectra remain for such physical observables, hence the spectra of area and volume constitute genuine physical predictions of the quantum theory of gravity. 5. Among the operators that ... |

11 | Statistical geometry of random weave states
- Bombelli
- 2000
(Show Context)
Citation Context ...state from other candidates. 8. Can you calculate graviton-graviton scattering? In principle yes. One could easily study interactions among the excitations of candidate ground states such as those in =-=[93, 94, 97, 11]-=-. The result must be finite as the discrete of area and volume impose an ultraviolet cutoff on the spectra of excitations of any candidate ground state. Presumably the results will depend on the prope... |

11 |
An algebraic approach to coarse graining
- Markopoulou
(Show Context)
Citation Context ...rstood rigorously how to construct coherent states in H kin which are peaked around classical configurations[97]. 23. Formulations of the renormalization group for spin foam models have been given in =-=[98, 99]-=-. As a byproduct of this work it is shown that while the Wilsonian renormalization group is not a group, it does have a natural algebraic setting, as a Hopf algebra. 24. For the case of non-vanishing ... |

11 | Nonperturbative dynamics for abstract (p,q) string networks,” hep-th/9712148
- Markopoulou, Smolin
(Show Context)
Citation Context ...r equivalently to local moves by which the spin networks evolve). Such generalized loop quantum gravity models have been proposed to serve as background independent formulations of string and M theory=-=[117, 55]-=-. They also offer new possibilities for the unification of physics, because the topology of spacetime is not assumed a priori, and hence must be emerge dynamically in the low energy limit. There is th... |

10 |
Non-minimal couplings, quantum geometry and black hole entropy
- Ashtekar, Corichi
- 2003
(Show Context)
Citation Context ...grees of freedom on the horizon. This has been found to work for a large class of black holes, including Schwarzschild black holes[17, 18]. LQG also gives the correct results for dilatonic black holes=-=[85]-=-. The boundary Hilbert spaces decompose into eigenspaces, one for each eigenvalue of the operator that measures the area of the boundary[16]. For each area eigenvalue, the Hilbert space is finite dime... |

10 |
R.L.(1986) "Expert Resolution
- Winkler
(Show Context)
Citation Context ...y bounces where the universe re-expands (or pre-contracts). The replacement of cosmological singularities by bounces has been confirmed in other non-perturbative studies of quantum cosmological models=-=[39, 110]-=- When couplings to a scalar field are included, there is a natural mechanism which generates Planck scale inflation as well as a graceful exit from it[20]. Using this formulation of quantum cosmology ... |

10 | Possible astrophysical probes of quantum gravity - Sarkar - 2002 |

9 | Quantum gravity vacuum and invariants of embedded spin networks
- Mikovic
- 2003
(Show Context)
Citation Context ...ls theory was previously discussed by Jackiw[105]. Some properties of Jackiw’s state were discussed by Witten[106]. More about this in a FAQ. 29 For a possible Λ = 0 analogue of the Kodama state, see =-=[104]-=-. 23theory. One can ask whether the projection into the Hilbert space of linearized gravitons on deSitter spacetime gotten by truncating the Kodama state to quadratic order is normalizable or not. Th... |

9 |
The linearization of the Kodama state
- Freidel, Smolin
(Show Context)
Citation Context ...spacetime gotten by truncating the Kodama state to quadratic order is normalizable or not. The answer is it is not in the Lorentzian case, and it is delta functional normalizable in the Euclidean case=-=[107]-=-. 25. The inverse cosmological constant turns out to be quantized in integral units, so that k = 6π/GΛ (38) is an integer[16]. This is related to a basic result, which is that in 3 + 1 dimensions, a n... |

8 |
Starodubtsev A., Quantum symmetry, the cosmological constant and Planck-scale phenomenology, Class. Quantum Grav
- Amelino-Camelia, Smolin
(Show Context)
Citation Context ...ional theory, modifications of the energy momentum relations (39) are present. They are understood as indicating, not a breaking of Poincare invariance, but a deformation of it[108]. It is conjectured=-=[11, 109]-=-, but not shown, that the same will be true of quantum gravity in 3 + 1 dimensions. This will be discussed further in the next section. To summarize, the situation with regard to the low energy limit ... |

7 |
On the existence of neutrino zero modes in vacuum spacetime
- Sen
- 1981
(Show Context)
Citation Context ...ded or assumed that this could not be done. However, in the mid 1980’s the situation changed because it was understood that general relativity could be most simply formulated as a kind of gauge theory=-=[1, 2]-=-. Given the tremendous advances in our understanding of gauge theories that had occurred in the decade previous, this made it possible to take a fresh approach to quantum gravity based on asking a sim... |

7 | How far are we from the quantum theory of gravity
- Smolin
- 2003
(Show Context)
Citation Context ...mpressive results, but those concerning the consistent unification of general relativity with quantum theory are limited so far to second order (genus two) perturbation theory around fixed backgrounds=-=[12]-=-. 38Foundation for their very generous support which made my own work possible. I am thankful especially to my colleagues at Perimeter Institute for many critical discussions. Finally, I am grateful ... |

7 |
Bojowald,”Isotropic Loop Quantum Cosmology”, Class.Quant.Grav. 19 (2002) 2717-2742, gr-qc/0202077; “Inflation from Quantum Geometry”, gr-qc/0206054; “The Semiclassical Limit of Loop Quantum
- Martin
(Show Context)
Citation Context ...s the Bekenstein-Hawking results on the relationship between area and entropy[19]. There is under development an approach to quantum cosmology that shows that cosmological singularities are eliminated=-=[20]-=- and has already led to some predictions for effects observable in CMB spectra[20, 21]. And there are published predictions for observable Planck scale deviations from energy momentum relations[22, 23... |

7 |
A Lorentzian cure for Euclidean troubles”, hep-th/0201104
- Ambjorn, Dasgupta, et al.
- 1998
(Show Context)
Citation Context ...his direction are given in [34, 35]. There are several different approaches to a background independent quantum theory of gravity. Besides loop quantum gravity, others include dynamical triangulations=-=[37, 38]-=-, causal sets[36] and the Gambini-Pullin discrete quantization approach to quantum gravity[39]. Although these are independently motivated research programs, some of their results are relevant for loo... |

7 |
553, Lett. al Nuovo Cimento 38
- D23
- 1981
(Show Context)
Citation Context ...ixed there. The dynamics is formulated in terms of a diffeomorphism invariant action which is a functional only of that gauge field and its derivatives. 1. The states of the theory are known precisely=-=[67, 68, 69, 70, 7, 6]-=-. The kinematical Hilbert space H kin has been rigorously constructed and has all the properties described in the last section. The Hilbert space H diffeo of spatially diffeomorphism invariant and 18... |

7 |
A new approach to the geometrization of matter. gr-qc/0110060
- Crane
- 2001
(Show Context)
Citation Context ... then the possibility of new physics coming from obstructions to recovering trivial topology at low energies. It has been proposed that these may account for the existence of matter degrees of freedom=-=[118]-=-, and perhaps even quantum theory itself[119]. 4.8 Comments on the results On the basis of these results, it can be claimed that loop quantum gravity is both a consistent quantization of general relat... |

6 |
Ashtekar’s variables and the topological phase of quantum gravity
- Chang, Soo
- 1991
(Show Context)
Citation Context ...? • Loop quantum gravity has its simplest formulation in terms of the chiral AshtekarSen variables. The classical theory is nevertheless chirally symmetric, but it was suggested a long time ago by Soo=-=[102]-=- that there might be violations of CP and chirally asymmetric effects quantum mechanically, and evidence for such effects was found in studies of one ansatz for the ground state[22]. It would be good ... |

6 | Wave function of the Universe and Chern-Simons Perturbation Theory - Soo |

4 | Introduction to supersymmetric spin networks
- Ling
- 2002
(Show Context)
Citation Context ...m, while maintaining the diffeomorphism invariance of the theory. This is true of general relativity in all dimensions and is also known to be true of supergravity, in d = 4, at least up to N = 2[31]-=-=[33]-=-, and for d = 11 [34]. As a result, the techniques which give us consistent diffeomorphism invariant gauge theories can be applied to give consistent quantizations of general relativity and supergravi... |

4 |
Smolin,Strings as perturbations of evolving spin-networks
- Lee
(Show Context)
Citation Context ... experiment, there are also results that show how the methods of loop quantum gravity may be extended to give background independent descriptions of quantum gravity in the higher and super realms[31]-=-=[35]-=-. It thus seems like a good time for an introduction to the whole approach that may help to make the basic ideas, results and methods accessible to a wider range of physicists. The main purpose of thi... |

4 |
Quantum spin dynamics,” gr-qc/9606089, gr-qc/9606090; “Anomaly free formulation of non-perturbative 4-dimensional Lorentzian quantum gravity,” Phys
- Thiemann
- 1996
(Show Context)
Citation Context ...ferent connections are used in loop quantum gravity. These include the self-dual part of the spacetime connection[1, 2], and a real SU(2) connection introduced by Barbero[47] and exploited by Thiemann=-=[48]-=-. There are also alternate formulations that use both the left and right handed parts of the spacetime connection, [31, 49]. (19) 12D ∧ B i = 0 (20) where D is the SU(2) gauge covariant derivative. T... |

4 |
Knot theory and quantum theory, Phys. Rev. Lett 61(1988)1155; Loop representation of quantum general relativity, Nucl. Phys
- Rovelli, Smolin
(Show Context)
Citation Context ...ixed there. The dynamics is formulated in terms of a diffeomorphism invariant action which is a functional only of that gauge field and its derivatives. 1. The states of the theory are known precisely=-=[67, 68, 69, 70, 7, 6]-=-. The kinematical Hilbert space H kin has been rigorously constructed and has all the properties described in the last section. The Hilbert space H diffeo of spatially diffeomorphism invariant and 18... |

4 |
Disappearance of the Black Hole Singularity
- Modesto
(Show Context)
Citation Context ...s to all black hole and cosmological horizons. Finally, there is a first result indicating that black hole singularities are eliminated by the same mechanism that eliminates cosmological singularities=-=[92]-=-. 4.4 Results on the low energy behavior 20. A large class of putative ground states can be constructed in Hkin which have course grained descriptions which reproduce the geometry of flat space, or an... |

4 |
Gravitons as embroidery on the weave,” Int
- Iwasaki, Rovelli
- 1993
(Show Context)
Citation Context ...lat space, or any slowly varying metric[93]-[97]. Linearizing the quantum theory around these states yields linearized quantum gravity, for gravitons with wavelength long compared to the Planck length=-=[96]-=-. 21. Excitations of matter fields on these states reproduce a cut off version of the matter quantum field theories, but with a physical, Planck scale cutoff. As a result of the discreteness of area a... |

4 |
Observables for spacetimes with two Killing field symmetries”, grqc/9402019,Phys.Rev. D50
- Husain
- 1994
(Show Context)
Citation Context ...1 and several have been studied also for N = 2 [32, 33, 31]. 33. The same methods can also be used to solve quantum gravity in 2+1 dimensions[50] and in some 1 + 1 dimensional reductions of the theory=-=[113]-=-. They also work to solve a large class of topological field theories[114, 115], giving results equivalent to those achieved by other methods. Further, loop methods applied to lattice gauge theories y... |

3 |
Spin Foam Models for Quantum Gravity, Topical Review
- Perez
(Show Context)
Citation Context ... background independent, in that no fixed spacetime metric is needed to describe their quantum dynamics. Instead, the geometry of space and time is coded in the degrees of freedom of a gauge field[4]-=-=[15]-=-. While a number of technical issues remain open, enough is known about the physics of loop quantum gravity that a number of applications to problems of physical interest are under development. There ... |

3 | The End of Time, Oxford University Press, 2000; “Leibnizian Time - Barbour |

3 | Planck-scale models of the Universe
- Markopoulou
(Show Context)
Citation Context ...ial slices are spin networks, which are quantum analogues of spatial geometries 26 . 24 For the most recent review see [15]. 25 For more details on these models and the resulting physical picture, see=-=[53]-=-. 26 It should be mentioned also that there are very impressive results relevant to the spin foam program from a research program called causal dynamical triangulations. This work shows that Lorentzia... |

3 |
A length operator for canonical quantum gravity, grqc/9606092. 29 C Rovelli L Smolin, The physical hamiltonian for non-perturbative quantum gravity, Phys Rev Lett 72
- Thiemann
- 1994
(Show Context)
Citation Context ...rve the diffeomorphism invariance of the states[66]. Other operators also have been constructed, for example an operator that measures angles in the quantum geometry[71] and the lengths of curves in Σ=-=[72]-=-. 3. The area, volume and length operators have discrete, finite spectra, valued in terms of the Planck length[63, 64, 65, 72]. There is hence a smallest possible volume, a smallest possible area, and... |

3 |
Positivity of spin foam amplitudes. Class.Quant.Grav
- Baez
- 2002
(Show Context)
Citation Context ... has two parts, a sum over graphs representing histories of spin networks, and, on each, a sum over the labels. The sums over labels are known from both analytic and numerical results to be convergent=-=[78, 79]-=- for some spin foam models, including some corresponding to the quantization of the Einstein equations in 2 + 1 and 3 + 1 dimensions. 12. For some spin foam model in 2 + 1 dimensions, it has been show... |

3 |
Loop representation for lattice gauge theory,1990 Pittsburgh and Syracuse preprint
- D38
- 1988
(Show Context)
Citation Context ...d theories[114, 115], giving results equivalent to those achieved by other methods. Further, loop methods applied to lattice gauge theories yields results equivalent to those achieved by other methods=-=[116]-=-. 34. Spin foam models are known in closed form for quantum general relativity in an 25arbitrary dimension d > 4 [51]. 4.7 Other extensions of the theory 35. The mathematical language of spin network... |

2 |
P and Maartens R Loop quantum gravity effects
- Tsujikawa, Singh
(Show Context)
Citation Context ... There is under development an approach to quantum cosmology that shows that cosmological singularities are eliminated[20] and has already led to some predictions for effects observable in CMB spectra=-=[20, 21]-=-. And there are published predictions for observable Planck scale deviations from energy momentum relations[22, 23] that imply predictions for experiments in progress such as AUGER and GLAST. For thos... |

2 |
quantum gravity and light propagation”, Phys.Rev. D65
- Alfaro, Morales-Tcotl, et al.
- 2002
(Show Context)
Citation Context ...ted[20] and has already led to some predictions for effects observable in CMB spectra[20, 21]. And there are published predictions for observable Planck scale deviations from energy momentum relations=-=[22, 23]-=- that imply predictions for experiments in progress such as AUGER and GLAST. For those whose interest is more towards formal speculations concerning supersymmetry and higher dimensions than experiment... |

2 |
Kimetical quantum algebra for theories of connections and uniqueness of its representations in the diffeomorphism invariant context
- Sahlmann
(Show Context)
Citation Context ...ust, and has been achieved by several regularization procedures. Significantly, there is a completely rigorous formulation of these diffeomorphism invariant gauge theories. A recent uniqueness theorem=-=[30]-=- greatly limits the possibilities for quantum descriptions of diffeomorphism invariant gauge theories apart from the one studied in LQG. III. General relativity and all related theories, such as super... |

2 | operators for spin net gravity: Definitions and consequences,” arXiv:gr-qc/0101032 - “New |

2 |
A Non-Chaotic Quantum Bianchi IX Universe and the Quantum Structure of Classical Singularities Phys
- Bojowald, Date
- 2004
(Show Context)
Citation Context ...onding to inflating universes, in the homogeneous approximation. 31. It has recently been shown that loop quantum gravity effects eliminate the chaotic behavior of Bianchi IX models near singularities=-=[112]-=-. 4.6 Results concerning supergravity and other dimensions 32. Many of these results extend to quantum supergravity for N = 1 and several have been studied also for N = 2 [32, 33, 31]. 33. The same me... |

2 |
The AGASA collaboration), Updated AGASA event list above4x10
- Hayashida, Honda, et al.
- 1999
(Show Context)
Citation Context ...nt sensitivity to measure α and β, or are expected to by 2010, are, 1. There are apparent violations of the GZK bound observed in ultra high energy cosmic rays (UHECR) detected by the AGASA experiment=-=[121]-=-. The experimental situation is inconsistent, but the new AUGER cosmic ray detector, which is now operational, is expected to resolve the situation over the next several years. If there is a violation... |

1 |
Separable Hilbert space
- Fairbairn, Rovelli
(Show Context)
Citation Context ...rom observations III and IV above. In 3 + 1 dimensions the correspondence gives G = SU(2) or SO(3) 10 which we 9 For technical reasons, this requires that the diffeomorphisms be piecewise smooth, see =-=[43]-=-. 10 For N = 1 supergravity, G = Osp(1, 2). i 9will assume for the following. This gives us correspondences between flux of the electric field ˜ Ea i and geometric quantities. Given a surface S, it t... |