How far are we from the quantum theory of gravity

An assessment is offered of the progress that the major approaches to quantum gravity have made towards the goal of constructing a complete and satisfactory theory. The emphasis is on loop quantum gravity and string theory, although other approaches are discussed, including dynamical triangulation models (euclidean and lorentzian) regge calculus models, causal sets, twistor theory, non-commutative geometry and models based on analogies to condensed matter systems. We proceed by listing the questions the theories are expected to be able to answer. We then compile two lists: the first details the actual results so far achieved in each theory, while the second lists conjectures which remain open. By comparing them we can evaluate how far each theory has progressed, and what must still be done before each theory can be considered a satisfactory quantum theory of gravity. We find there has been impressive recent progress on several fronts. At the same time, important issues about loop quantum gravity are so far unresolved, as are key conjectures of string theory. However, there is a reasonable expectation that experimental tests of lorentz invariance at Planck scales may in the near future make it possible to rule out one or more candidate quantum theories of gravity.

[1]  J. Jurkiewicz,et al.  Lorentzian 3d gravity with wormholes via matrix models , 2001, hep-th/0106082.

[2]  The cubic matrix model and a duality between strings and loops , 2000, hep-th/0006137.

[3]  S. Adler A formula for the induced gravitational constant , 1980 .

[4]  L. Freidel,et al.  Nonperturbative summation over 3D discrete topologies , 2002, hep-th/0211026.

[5]  Paul Adrien Maurice Dirac,et al.  Lectures on Quantum Mechanics , 2001 .

[6]  Giorgio Parisi,et al.  The theory of non-renormalizable interactions: The large N expansion , 1975 .

[7]  A fixed point for quantum gravity , 1982 .

[8]  M theory as a holographic field theory , 1997, hep-th/9712130.

[9]  The Quantum Physics of Black Holes: Results from String Theory , 2000, gr-qc/0105063.

[10]  Quantum Spin Dynamics (QSD) , 1996, gr-qc/9606089.

[11]  Green-Schwarz string in AdS5 × S5: semiclassical partition function , 2000, hep-th/0001204.

[12]  R. Sorkin,et al.  Evidence for a continuum limit in causal set dynamics , 2000, gr-qc/0003117.

[13]  The Maldacena conjecture and Rehren duality , 2001, hep-th/0106073.

[14]  NAKED SINGULARITIES AND THE WILSON LOOP , 2002, hep-th/0204180.

[15]  Shamit Kachru,et al.  De Sitter vacua in string theory , 2003, hep-th/0301240.

[16]  J. Polchinski String Theory: Preface , 1998 .

[17]  Physics with nonperturbative quantum gravity: Radiation from a quantum black hole , 1996, gr-qc/9603064.

[18]  P. Veenhuis The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics , 1995 .

[19]  A. Sen On the existence of neutrino ’’zero‐modes’’ in vacuum spacetimes , 1981 .

[20]  S. Sarkar Possible astrophysical probes of quantum gravity , 2002, gr-qc/0204092.

[21]  ON SIGNATURES OF SHORT DISTANCE PHYSICS IN THE COSMIC MICROWAVE BACKGROUND , 2002, hep-th/0202142.

[22]  Rovelli,et al.  Weaving a classical metric with quantum threads. , 1992, Physical review letters.

[23]  E. Witten Quantum Gravity In De Sitter Space , 2001, hep-th/0106109.

[24]  J. McMaster,et al.  The Elegant Universe , 1999 .

[25]  K. Wilson,et al.  The Renormalization group and the epsilon expansion , 1973 .

[26]  E. D'hoker,et al.  Unitarity of closed superstring perturbation theory , 1990 .

[27]  河東 泰之,et al.  A.Connes:Noncommutative Geometry , 1997 .

[28]  A Tentative theory of large distance physics , 2002, hep-th/0204131.

[29]  Lee Smolin,et al.  The Bekenstein Bound, Topological Quantum Field Theory and Pluralistic Quantum Field Theory , 1995 .

[30]  C. Rovelli,et al.  Spacetime as a Feynman diagram: the connection formulation , 2000, gr-qc/0002095.

[31]  Graphical evolution of spin network states , 1996, gr-qc/9606013.

[32]  David H. Wolpert,et al.  No free lunch theorems for optimization , 1997, IEEE Trans. Evol. Comput..

[33]  T. Konopka,et al.  Observational limits on quantum geometry effects , 2002, hep-ph/0201184.

[34]  An Introduction to spin foam models of quantum gravity and BF theory , 1999, gr-qc/9905087.

[35]  Lee Smolin The present moment in quantum cosmology: challenges to the arguments for the elimination of time , 2000 .

[36]  R. Sorkin,et al.  “Renormalization” transformations induced by cycles of expansion and contraction in causal set cosmology , 2000, gr-qc/0009063.

[37]  R. Penrose,et al.  Twistor theory: An approach to the quantisation of fields and space-time , 1973 .

[38]  Notes for a brief history of quantum gravity , 2000, gr-qc/0006061.

[39]  Reinaldo J. Gleiser,et al.  Astrophysical limits on quantum gravity motivated birefringence , 2001 .

[40]  M. Duff,et al.  Quantum gravity in 2 + ε dimensions , 1978 .

[41]  M. Reuter,et al.  Flow equation of quantum Einstein gravity in a higher derivative truncation , 2002 .

[42]  A. Migdal,et al.  Simulations of Four-Dimensional Simplicial Quantum Gravity as Dynamical Triangulation , 1992 .

[43]  Rovelli,et al.  Spin networks and quantum gravity. , 1995, Physical review. D, Particles and fields.

[44]  The semiclassical limit of loop quantum cosmology , 2001, gr-qc/0105113.

[45]  Rodolfo Gambini,et al.  Nonstandard optics from quantum space-time , 1999 .

[46]  Z. Burda,et al.  Quantum gravity represented as dynamical triangulations , 1992 .

[47]  Alejandro Perez,et al.  Spin Foam Models for Quantum Gravity , 2003, gr-qc/0301113.

[48]  F. Markopoulou Coarse graining in spin foam models , 2002, gr-qc/0203036.

[49]  A. Sen Non-BPS States and Branes in String Theory , 1999 .

[50]  C. Vafa,et al.  A Perturbative Window into Non-Perturbative Physics , 2002, hep-th/0208048.

[51]  N. N. Khuri Sign of the induced gravitational constant , 1982 .

[52]  C. J. Isham,et al.  Topos Perspective on the Kochen-Specker Theorem: I. Quantum States as Generalized Valuations , 1998, quant-ph/9803055.

[53]  A Large N reduced model as superstring , 1996, hep-th/9612115.

[54]  A. Ashtekar Lectures on Non-Perturbative Canonical Gravity , 1991 .

[55]  Lee Smolin The strong and weak holographic principles , 2001 .

[56]  Statistical geometry of random weave states , 2001, gr-qc/0101080.

[57]  C. Rovelli,et al.  Relational Quantum Mechanics , 2006 .

[58]  T. Banks,et al.  M theory observables for cosmological space-times , 2001 .

[59]  A. Klemm,et al.  Landau-Ginzburg string vacua , 1992 .

[60]  Julian Barbour The End of Time , 1999 .

[61]  Polchinski Dirichlet Branes and Ramond-Ramond charges. , 1995, Physical review letters.

[62]  The holographic Weyl anomaly , 1998, hep-th/9806087.

[63]  Superfluid analogies of cosmological phenomena , 2000, gr-qc/0005091.

[64]  T. Jacobson New variables for canonical supergravity , 1988 .

[65]  An interferometric gravitational wave detector as a quantum-gravity apparatus , 1998, gr-qc/9808029.

[66]  Jorge Alfaro,et al.  Loop quantum gravity and light propagation , 2002 .

[67]  J. Ellis,et al.  Potential Sensitivity of Gamma-Ray Burster Observations to Wave Dispersion in Vacuo , 1997, astro-ph/9712103.

[68]  Loop constraints: A habitat and their algebra , 1997, gr-qc/9710016.

[69]  J. Maldacena The Large-N Limit of Superconformal Field Theories and Supergravity , 1997, hep-th/9711200.

[70]  D. Gross,et al.  String perturbation theory diverges. , 1988, Physical review letters.

[71]  Geometry eigenvalues and the scalar product from recoupling theory in loop quantum gravity. , 1996, Physical review. D, Particles and fields.

[72]  Lee Smolin,et al.  Quantum geometry with intrinsic local causality , 1998 .

[73]  L. Susskind The Anthropic Landscape of String Theory , 2003, hep-th/0302219.

[74]  J. Baez,et al.  Positivity of spin foam amplitudes , 2001, gr-qc/0110044.

[75]  M theory as a matrix model: A Conjecture , 1996, hep-th/9610043.

[76]  B R Greene,et al.  String theory. , 1994, Proceedings of the National Academy of Sciences of the United States of America.

[77]  R. Sorkin,et al.  Classical sequential growth dynamics for causal sets , 1999, gr-qc/9904062.

[78]  A. Ashtekar New perspectives in canonical gravity , 1988 .

[79]  Saurya Das,et al.  General logarithmic corrections to black-hole entropy , 2001, hep-th/0111001.

[80]  Coherent states for canonical quantum general relativity and the infinite tensor product extension , 2001, gr-qc/0102038.

[81]  James B. Hartle,et al.  Wave Function of the Universe , 1983 .

[82]  The Wall of the cave , 1998, hep-th/9809057.

[83]  Holography in a quantum spacetime , 1999, hep-th/9910146.

[84]  Strings in flat space and pp waves from ${\cal N}=4$ Super Yang Mills , 2002, hep-th/0202021.

[85]  C. Soo,et al.  Ashtekar's variables and the topological phase of quantum gravity , 1991 .

[86]  Worldsheet formulations of gauge theories and gravity , 1994, gr-qc/9412035.

[87]  A. Ashtekar,et al.  2+1 quantum gravity as a toy model for the 3+1 theory , 1989 .

[88]  C. Rovelli,et al.  Gravitons from loops: non-perturbative loop-space quantum gravity contains the graviton-physics approximation , 1994 .

[89]  J. Maldacena,et al.  Large N Field Theories, String Theory and Gravity , 1999, hep-th/9905111.

[90]  A. Polyakov,et al.  Gauge Theory Correlators from Non-Critical String Theory , 1998, hep-th/9802109.

[91]  Planck-scale models of the Universe , 2002, gr-qc/0210086.

[92]  L. Smolin,et al.  Causal evolution of spin networks , 1997 .

[93]  C. Rovelli What is observable in classical and quantum gravity , 1991 .

[94]  S. Kachru,et al.  Vacuum Energy Cancellation in a Non-supersymmetric String , 1998, hep-th/9807076.

[95]  T. Thiemann Introduction to Modern Canonical Quantum General Relativity , 2001, gr-qc/0110034.

[96]  The physical Hamiltonian in nonperturbative quantum gravity. , 1993, Physical review letters.

[97]  Stuart Kauffman,et al.  Combinatorial dynamics in quantum gravity , 1998 .

[98]  Edward Witten Anti-de Sitter Space, Thermal Phase Transition, And Confinement in Gauge Theories , 1998 .

[99]  D. Gray,et al.  Cim , 1987 .

[100]  Eleven-dimensional supergravity as a constrained topological field theory , 2000, hep-th/0003285.

[101]  Fotini Markopoulou The Internal Description of a Causal Set:¶What the Universe Looks Like from the Inside , 2000 .

[102]  T. Jacobson,et al.  A strong astrophysical constraint on the violation of special relativity by quantum gravity , 2003, Nature.

[103]  A. Ashtekar,et al.  New variables for classical and quantum gravity. , 1986, Physical review letters.

[104]  John C. Baez,et al.  Spin foam models , 1997, gr-qc/9709052.

[105]  A Strong coupling test of S duality , 1994, hep-th/9408074.

[106]  Linking topological quantum field theory and nonperturbative quantum gravity , 1995, gr-qc/9505028.

[107]  An Insider's guide to quantum causal histories , 1999, hep-th/9912137.

[108]  J. Jurkiewicz,et al.  A Lorentzian Cure for Euclidean Troubles , 2002, hep-th/0201104.

[109]  T.Banks Cosmological Breaking of Supersymmetry? , 2000, hep-th/0007146.

[110]  P. Townsend,et al.  Unity of superstring dualities , 1994, hep-th/9410167.

[111]  L. Smolin The fate of black hole singularities and the parameters of the standard models of particle physics and cosmology , 1994, gr-qc/9404011.

[112]  C. Rovelli,et al.  GRAVITONS AS EMBROIDERY ON THE WEAVE , 1992 .

[113]  D. Friedan On two-dimensional conformal invariance and the field theory of strings , 1985 .

[114]  Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity , 1996, gr-qc/9606088.

[115]  J. Stachel Einstein's search for general covariance, 1912 - 1915. , 1989 .

[116]  Bombelli,et al.  Space-time as a causal set. , 1987, Physical review letters.

[117]  Quantum gravity vacuum and invariants of embedded spin networks , 2003, gr-qc/0301047.

[118]  Fermion doubling and Berenstein-Maldacena-Nastase correspondence , 2003, hep-th/0302104.

[119]  L. Smolin A Holographic formulation of quantum general relativity , 1998, hep-th/9808191.

[120]  On the continuum limit of the discrete Regge model in 4d , 2002, hep-lat/0209028.

[121]  An algebraic approach to coarse graining , 2000, hep-th/0006199.

[122]  G. Horowitz Exactly soluble diffeomorphism invariant theories , 1989 .

[123]  C. Vafa,et al.  Microscopic origin of the Bekenstein-Hawking entropy , 1996, hep-th/9601029.

[124]  Isotropic loop quantum cosmology , 2002, gr-qc/0202077.

[125]  J. Maldacena The Large N limit of superconformal field theories and supergravity , 1998 .

[126]  Quantum geometry of isolated horizons and black hole entropy , 2000, gr-qc/0005126.

[127]  Wavefunction of the Universe and Chern-Simons perturbation theory , 2001, gr-qc/0109046.

[128]  Stephen W. Hawking,et al.  Particle Creation by Black Holes , 1993, Resonance.

[129]  Carlo Rovelli,et al.  Discreteness of area and volume in quantum gravity [Nucl. Phys. B 442 (1995) 593] , 1994, gr-qc/9411005.

[130]  Donoghue,et al.  General relativity as an effective field theory: The leading quantum corrections. , 1994, Physical review. D, Particles and fields.

[131]  Quantization of diffeomorphism invariant theories of connections with local degrees of freedom , 1995, gr-qc/9504018.

[132]  Towards a loop representation for quantum canonical supergravity , 1995, hep-th/9508036.

[133]  Strings, black holes, and Lorentz contraction. , 1993, Physical review. D, Particles and fields.

[134]  Rovelli,et al.  Knot theory and quantum gravity. , 1988, Physical review letters.

[135]  J. Bekenstein Black Holes and Entropy , 1973, Jacob Bekenstein.

[136]  Superstrings in AdS in light cone gauge , 2000, hep-th/0009226.

[137]  Takehiro Azuma,et al.  Curved-space classical solutions of a massive supermatrix model , 2002 .

[138]  V. Frolov,et al.  Quantum Physics of Black Holes , 1998 .

[139]  Volume operator in discretized quantum gravity. , 1995, Physical review letters.

[140]  L. Crane A New Approach to the Geometrization of Matter , 2001, gr-qc/0110060.

[141]  Finite diffeomorphism-invariant observables in quantum gravity. , 1993, Physical review. D, Particles and fields.

[142]  L. Crane,et al.  Relativistic spin networks and quantum gravity , 1997, gr-qc/9709028.

[143]  TASI lectures on black holes in string theory , 2000, hep-th/0008241.

[144]  BF Description of Higher-Dimensional Gravity Theories , 1999, hep-th/9901069.

[145]  The statistics of string/M theory vacua , 2003, hep-th/0303194.

[146]  The complete spectrum of the area from recoupling theory in loop quantum gravity , 1996, gr-qc/9608043.

[147]  Quantum gravity with a positive cosmological constant , 2002, hep-th/0209079.

[148]  J. Lukierski,et al.  Q deformation of Poincare algebra , 1991 .

[149]  T. Yukawa,et al.  Clear evidence of a continuum theory of 4D Euclidean simplicial quantum gravity , 2001 .

[150]  J. Ellis,et al.  Quantum-gravity analysis of gamma-ray bursts using wavelets , 2002, astro-ph/0210124.

[151]  Type IIB superstring action in AdS5 × S5 background , 1998, hep-th/9805028.

[152]  G. Volovik,et al.  The Universe in a Helium Droplet , 2003 .

[153]  How Does God Play Dice? (Pre-)Determinism at the Planck Scale , 2001, hep-th/0104219.

[154]  H. Shimizu,et al.  Energy determination in the Akeno Giant Air Shower Array experiment , 2002, astro-ph/0209422.

[155]  Fotini Markopoulou Dual formulation of spin network evolution , 1997 .

[156]  David Rideout Dynamics of causal sets , 2001 .

[157]  M. Pospelov,et al.  EXPERIMENTAL CHALLENGES FOR QUANTUM GRAVITY , 2004, gr-qc/0402028.

[158]  J. Plebański On the separation of Einsteinian substructures , 1977 .

[159]  Grand unification in RS1 , 2002, hep-ph/0212028.

[160]  String theory dynamics in various dimensions , 1995, hep-th/9503124.

[161]  E. Witten Anti-de Sitter space and holography , 1998, hep-th/9802150.

[162]  O. Dreyer Quasinormal modes, the area spectrum, and black hole entropy. , 2002, Physical review letters.

[163]  Does the fine structure constant vary? A third quasar absorption sample consistent with varying α , 2002, astro-ph/0210531.

[164]  Susskind,et al.  The stretched horizon and black hole complementarity. , 1993, Physical review. D, Particles and fields.

[165]  M. Reuter,et al.  Is quantum Einstein gravity nonperturbatively renormalizable , 2002 .

[166]  V. Frolov,et al.  Statistical origin of black hole entropy in induced gravity , 1996, hep-th/9607104.

[167]  Supersymmetric spin networks , 2000, hep-th/0009020.

[168]  A. Ashtekar,et al.  Quantum geometry and black hole entropy , 1998 .

[169]  Evolution in quantum causal histories , 2003, hep-th/0302111.

[170]  Black hole condensation and the unification of string vacua , 1995, hep-th/9504145.

[171]  Time, measurement and information loss in quantum cosmology , 1993, gr-qc/9301016.

[172]  R. Pietri,et al.  Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space , 1999, hep-th/9907154.

[173]  Perturbative Quantum Gravity and its Relation to Gauge Theory , 2002, Living reviews in relativity.

[174]  T. Regge,et al.  SEMICLASSICAL LIMIT OF RACAH COEFFICIENTS. , 1969 .