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  • Online publication date: August 2012

11 - Loop quantum gravity and cosmology


“It would be permissible to look upon the Hamiltonian form as the fundamental one, and there would then be no fundamental four-dimensional symmetry in the theory. One would have a Hamiltonian built up from four weekly [sic] vanishing functions, given by [the Hamiltonian and diffeomorphism constraints]. The usual requirement of four-dimensional symmetry in physical laws would then get replaced by the requirement that the functions have weakly vanishing P.B.'s, so that they can be provided with arbitrary coefficients in the equations of motion, corresponding to an arbitrary motion of the surface on which the state is defined.” P. A. M. Dirac, in “The theory of gravitation in Hamiltonian form,” Proc. Roy. Soc. A 246 (1958) 333–43.


In its different incarnations, quantum gravity must face a diverse set of fascinating problems and difficulties, a set of issues best seen as both challenges and opportunities. One of the main problems in canonical approaches, for instance, is the issue of anomalies in the gauge algebra underlying space-time covariance. Classically, the gauge generators, given by constraints, have weakly vanishing Poisson brackets with each other: they vanish when the constraints are satisfied. After quantization, the same behavior must be realized for commutators of quantum constraints (or for Poisson brackets of effective constraints), or else the theory becomes inconsistent due to gauge anomalies. If and how canonical quantum gravity can be obtained in an anomaly-free way is an important question, not yet convincingly addressed in full generality.

[1] Artymowski, M., Lalak, Z. and Szulc, L. 2009. Loop quantum cosmology corrections to inflationary models. JCAP, 0901, 004.
[2] Ashtekar, A. 1987. New Hamiltonian formulation of general relativity. Phys. Rev. D, 36(6), 1587–602.
[3] Ashtekar, A. and Lewandowski, J. 1997. Quantum theory of geometry I: Area operators. Class. Quantum Grav., 14, A55–A82.
[4] Ashtekar, A. and Lewandowski, J. 1998. Quantum theory of geometry II: Volume operators. Adv. Theor. Math. Phys., 1, 388–429.
[5] Ashtekar, A. and Lewandowski, J. 2004. Background independent quantum gravity: A status report. Class. Quantum Grav., 21, R53–R152.
[6] Ashtekar, A., Lewandowski, J., Marolf, D., Mourão, J. and Thiemann, T. 1995. Quantization of diffeomorphism invariant theories of connections with local degrees of freedom. J. Math. Phys., 36(11), 6456–93.
[7] Ashtekar, A., Baez, J. C., Corichi, A. and Krasnov, K. 1998. Quantum geometry and black hole entropy. Phys. Rev. Lett., 80, 904–7.
[8] Ashtekar, A., Bojowald, M. and Lewandowski, J. 2003. Mathematical structure of loop quantum cosmology. Adv. Theor. Math. Phys., 7, 233–68.
[9] Ashtekar, A., Pawlowski, T. and Singh, P. 2006. Quantum nature of the Big Bang: An analytical and numerical investigation. Phys. Rev. D, 73, 124038.
[10] Bahr, B. and Dittrich, B. 2009a. Breaking and restoring of diffeomorphism symmetry in discrete gravity.
[11] Bahr, B. and Dittrich, B. 2009b. Improved and perfect actions in discrete gravity.
[12] Banerjee, K. and Date, G. 2005. Discreteness corrections to the effective Hamiltonian of isotropic loop quantum cosmology. Class. Quant. Grav., 22, 2017–33.
[13] Barbero, J. F. 1995. Real Ashtekar variables for Lorentzian signature space-times. Phys. Rev. D, 51(10), 5507–10.
[14] Bardeen, J. M. 1980. Gauge-invariant cosmological perturbations. Phys. Rev. D, 22, 1882–905.
[15] Barrau, A. and Grain, J. 2009. Cosmological footprint of loop quantum gravity. Phys. Rev. Lett., 102, 081301.
[16] Bentivegna, E. and Pawlowski, T. 2008. Anti-deSitter universe dynamics in LQC. Phys. Rev. D, 77, 124025.
[17] Bergmann, P. G. 1961. Observables in general relativity. Rev. Mod. Phys., 33, 510–14.
[18] Bojowald, M. 2001a. Absence of a singularity in loop quantum cosmology. Phys. Rev. Lett., 86, 5227–30.
[19] Bojowald, M. 2001b. Inverse scale factor in isotropic quantum geometry. Phys. Rev. D, 64, 084018.
[20] Bojowald, M. 2001c. Loop quantum cosmology IV: Discrete time evolution. Class. Quantum Grav., 18, 1071–88.
[21] Bojowald, M. 2002a. Isotropic loop quantum cosmology. Class. Quantum Grav., 19, 2717–41.
[22] Bojowald, M. 2002b. Quantization ambiguities in isotropic quantum geometry. Class. Quantum Grav., 19, 5113–30.
[23] Bojowald, M. 2004. Spherically symmetric quantum geometry: states and basic operators. Class. Quantum Grav., 21, 3733–53.
[24] Bojowald, M. 2006. Loop quantum cosmology and inhomogeneities. Gen. Rel. Grav., 38, 1771–95.
[25] Bojowald, M. 2007a. Dynamical coherent states and physical solutions of quantum cosmological bounces. Phys. Rev. D, 75, 123512.
[26] Bojowald, M. 2007b. Large scale effective theory for cosmological bounces. Phys. Rev. D, 75, 081301(R).
[27] Bojowald, M. 2007c. What happened before the big bang?Nature Physics, 3(8), 523–5.
[28] Bojowald, M. 2008a. The dark side of a patchwork universe. Gen. Rel. Grav., 40, 639–60.
[29] Bojowald, M. 2008b. How quantum is the big bang?Phys. Rev. Lett., 100, 221301.
[30] Bojowald, M. 2008c. Loop quantum cosmology. Living Rev. Relativity, 11, 4.
[31] Bojowald, M. 2008d. Quantum nature of cosmological bounces. Gen. Rel. Grav., 40, 2659–83.
[32] Bojowald, M. and Das, R. 2008. Fermions in loop quantum cosmology and the role of parity. Class. Quantum Grav., 25, 195006.
[33] Bojowald, M. and Hossain, G. 2007. Cosmological vector modes and quantum gravity effects. Class. Quantum Grav., 24, 4801–16.
[34] Bojowald, M. and Hossain, G. 2008. Quantum gravity corrections to gravitational wave dispersion. Phys. Rev.D, 77, 023508.
[35] Bojowald, M. and Kagan, M. 2006. Singularities in isotropic non-minimal scalar field models. Class. Quantum Grav., 23, 4983–90.
[36] Bojowald, M. and Kastrup, H. A. 2000. Symmetry reduction for quantized diffeomorphism invariant theories of connections. Class. Quantum Grav., 17, 3009–43.
[37] Bojowald, M. and Reyes, J. D. 2009. Dilaton gravity, Poisson sigma models and loop quantum gravity. Class. Quantum Grav., 26, 035018.
[38] Bojowald, M. and Skirzewski, A. 2006. Effective equations of motion for quantum systems. Rev. Math. Phys., 18, 713–45.
[39] Bojowald, M. and Skirzewski, A. 2008. Effective theory for the cosmological generation of structure. Adv. Sci. Lett., 1, 92–8.
[40] Bojowald, M. and Strobl, T. 2003. Poisson geometry in constrained systems. Rev. Math. Phys., 15, 663–703.
[41] Bojowald, M. and Tavakol, R. 2008. Recollapsing quantum cosmologies and the question of entropy. Phys. Rev.D, 78, 023515.
[42] Bojowald, M. and Tsobanjan, A. 2009. Effective constraints for relativistic quantum systems. Phys. Rev. D, to appear.
[43] Bojowald, M., Hernández, H. H. and Morales-Técotl, H. A. 2006. Perturbative degrees of freedom in loop quantum gravity: Anisotropies. Class. Quantum Grav., 23, 3491–516.
[44] Bojowald, M., Hernández, H. and Skirzewski, A. 2007a. Effective equations for isotropic quantum cosmology including matter. Phys. Rev.D, 76, 063511.
[45] Bojowald, M., Hernández, H., Kagan, M., Singh, P. and Skirzewski, A. 2007b. Formation and evolution of structure in loop cosmology. Phys. Rev. Lett., 98, 031301.
[46] Bojowald, M., Cartin, D. and Khanna, G. 2007c. Lattice refining loop quantum cosmology, anisotropic models and stability. Phys. Rev.D, 76, 064018.
[47] Bojowald, M., Hossain, G., Kagan, M. and Shankaranarayanan, S. 2008. Anomaly freedom in perturbative loop quantum gravity. Phys. Rev.D, 78, 063547.
[48] Bojowald, M., Sandhöfer, B., Skirzewski, A. and Tsobanjan, A. 2009a. Effective constraints for quantum systems. Rev. Math. Phys., 21, 111–54.
[49] Bojowald, M., Hossain, G., Kagan, M. and Shankaranarayanan, S. 2009b. Gauge invariant cosmological perturbation equations with corrections from loop quantum gravity. Phys. Rev.D, 79, 043505.
[50] Bojowald, M., Reyes, J. D. and Tibrewala, R. 2009c. Non-marginal LTB-like models with inverse triad corrections from loop quantum gravity. Phys. Rev.D, 80, 084002.
[51] Bojowald, M. 2009. Consistent loop quantum cosmology. Class. Quantum Grav., 26, 075020.
[52] Bruni, M., Dunsby, P. K. S. and Ellis, G. F. R. 1992. Cosmological perturbations and the physical meaning of gauge invariant variables. Astrophys. J., 395, 34–53.
[53] Brunnemann, J. and Fleischhack, C. 2007. On the configuration spaces of homogeneous loop quantum cosmology and loop quantum gravity.
[54] Cametti, F., Jona-Lasinio, G., Presilla, C. and Toninelli, F. 2000. Comparison between quantum and classical dynamics in the effective action formalism. Pages 431–48 of: Proceedings of the International School of Physics “Enrico Fermi”, Course CXLIII. Amsterdam: IOS Press.
[55] Campiglia, M., Di Bartolo, C., Gambini, R. and Pullin, J. 2006. Uniform discretizations: a new approach for the quantization of totally constrained systems. Phys. Rev.D, 74, 124012.
[56] Campiglia, M., Gambini, R. and Pullin, J. 2007. Loop quantization of spherically symmetric midi-superspaces. Class. Quantum Grav., 24, 3649.
[57] Copeland, E. J., Mulryne, D. J., Nunes, N. J. and Shaeri, M. 2009. The gravitational wave background from super-inflation in Loop Quantum Cosmology. Phys. Rev.D, 79, 023508.
[58] Deruelle, N., Sasaki, M., Sendouda, Y. and Yamauchi, D. 2009. Hamiltonian formulation of f(Riemann) theories of gravity.
[59] Dirac, P. A. M. 1958. The theory of gravitation in Hamiltonian form. Proc. Roy. Soc.A, 246, 333–43.
[60] Dittrich, B. 2006. Partial and complete observables for canonical general relativity. Class. Quant. Grav., 23, 6155–84.
[61] Dittrich, B. 2007. Partial and complete observables for Hamiltonian constrained systems. Gen. Rel. Grav., 39, 1891–927.
[62] Domagala, M. and Lewandowski, J. 2004. Black hole entropy from quantum geometry. Class. Quantum Grav., 21, 5233–43.
[63] Ellis, G. F. R. and Bruni, M. 1989. Covariant and gauge invariant approach to cosmological density fluctuations. Phys. Rev.D, 40, 1804–18.
[64] Ellis, G. F. R. and Maartens, R. 2004. The emergent universe: inflationary cosmology with no singularity. Class. Quant. Grav., 21, 223–32.
[65] Ellis, G. F. R., Murugan, J. and Tsagas, C. G. 2004. The emergent universe: An explicit construction. Class. Quant. Grav., 21, 233–50.
[66] Fewster, C. and Sahlmann, H. 2008. Phase space quantization and loop quantum cosmology: A Wigner function for the Bohr-compactified real line. Class. Quantum Grav., 25, 225015.
[67] Fleischhack, C. 2009. Representations of the Weyl algebra in quantum geometry. Commun. Math. Phys., 285, 67–140.
[68] Giesel, K., Hofmann, S., Thiemann, T. and Winkler, O. 2007a. Manifestly gaugeinvariant general relativistic perturbation theory: I. Foundations.
[69] Giesel, K., Hofmann, S., Thiemann, T. and Winkler, O. 2007b. Manifestly gaugeinvariant general relativistic perturbation theory: II. FRW Background and first order.
[70] Giesel, K., Tambornino, J. and Thiemann, T. 2009. LTB spacetimes in terms of Dirac observables.
[71] Grain, J., Cailleteau, T., Barrau, A. and Gorecki, A. 2009a. Fully LQC-corrected propagation of gravitational waves during slow-roll inflation.
[72] Grain, J., Barrau, A. and Gorecki, A. 2009b. Inverse volume corrections from loop quantum gravity and the primordial tensor power spectrum in slow-roll inflation. Phys. Rev.D, 79, 084015.
[73] Husain, V. and Winkler, O. 2004. On singularity resolution in quantum gravity. Phys. Rev.D, 69, 084016.
[74] Immirzi, G. 1997. Real and complex connections for canonical gravity. Class. Quantum Grav., 14, L177–L181.
[75] Jacobson, T. 2000. Trans-Planckian redshifts and the substance of the space-time river.
[76] Kaul, R. K. and Majumdar, P. 1998. Quantum black hole entropy. Phys. Lett. B, 439, 267–70.
[77] Kibble, T. W. B. 1979. Geometrization of quantum mechanics. Commun. Math. Phys., 65, 189–201.
[78] Laddha, A. 2007. Polymer quantization of CGHS model – I. Class. Quant. Grav., 24, 4969–88.
[79] Laddha, A. and Varadarajan, M. 2008. Polymer parametrised field theory. Phys. Rev.D, 78, 044008.
[80] Lewandowski, J., Okolów, A., Sahlmann, H. and Thiemann, T. 2006. Uniqueness of diffeomorphism invariant states on holonomy-flux algebras. Commun. Math. Phys., 267, 703–33.
[81] Martin-Benito, M., Garay, L. J. and Mena Marugán, G. A. 2008. Hybrid quantum Gowdy cosmology: Combining loop and Fock quantizations. Phys. Rev.D, 78, 083516.
[82] Meissner, K. A. 2004. Black hole entropy in loop quantum gravity. Class. Quantum Grav., 21, 5245–51.
[83] Mielczarek, J. 2008. Gravitational waves from the Big Bounce. JCAP, 0811, 011.
[84] Mielczarek, J. 2009. The observational implications of loop quantum cosmology.
[85] Nelson, W. and Sakellariadou, M. 2007a. Lattice refining loop quantum cosmology and inflation. Phys. Rev.D, 76, 044015.
[86] Nelson, W. and Sakellariadou, M. 2007b. Lattice refining LQC and the matter Hamiltonian. Phys. Rev.D, 76, 104003.
[87] Nelson, W. and Sakellariadou, M. 2008. Numerical techniques for solving the quantum constraint equation of generic lattice-refined models in loop quantum cosmology. Phys. Rev.D, 78, 024030.
[88] Puchta, J. 2009. Ph.D. thesis, University of Warsaw.
[89] Reyes, J. D. 2009. Spherically Symmetric Loop Quantum Gravity: Connections to 2-Dimensional Models and Applications to Gravitational Collapse. Ph.D. thesis, The Pennsylvania State University.
[90] Rovelli, C. 1991a. Quantum reference systems. Class. Quantum Grav., 8, 317–32.
[91] Rovelli, C. 1991b. What is observable in classical and quantum gravity?Class. Quantum Grav., 8, 297–316.
[92] Rovelli, C. 2004. Quantum Gravity. Cambridge, UK: Cambridge University Press.
[93] Rovelli, C. and Smolin, L. 1990. Loop space representation of quantum general relativity. Nucl. Phys. B, 331, 80–152.
[94] Rovelli, C. and Smolin, L. 1994. The physical Hamiltonian in nonperturbative quantum gravity. Phys. Rev. Lett., 72, 446–9.
[95] Rovelli, C. and Smolin, L. 1995. Discreteness of area and volume in quantum gravity. Nucl. Phys.B, 442, 593–619.
Erratum: Nucl. Phys.B 456 (1995) 753.
[96] Rovelli, C. and Vidotto, F. 2008. Stepping out of homogeneity in loop quantum cosmology. Class. Quantum Grav., 25, 225024.
[97] Sabharwal, S. and Khanna, G. 2008. Numerical solutions to lattice-refined models in loop quantum cosmology. Class. Quantum Grav., 25, 085009.
[98] Sahlmann, H. 2009. This volume.
[99] Shimano, M. and Harada, T. 2009. Observational constraints of a power spectrum from super-inflation in loop quantum cosmology.
[100] Singh, P. 2006. Loop cosmological dynamics and dualities with Randall–Sundrum braneworlds. Phys. Rev.D, 73, 063508.
[101] Singh, P. and Vandersloot, K. 2005. Semi-classical states, effective dynamics and classical emergence in loop quantum cosmology. Phys. Rev.D, 72, 084004.
[102] Taveras, V. 2008. Corrections to the Friedmann equations from LQG for a universe with a free scalar field. Phys. Rev.D, 78, 064072.
[103] Thiemann, T. 1998a. Quantum spin dynamics (QSD). Class. Quantum Grav., 15, 839–73.
[104] Thiemann, T. 1998b. QSD V: Quantum gravity as the natural regulator of matter quantum field theories. Class. Quantum Grav., 15, 1281–314.
[105] Thiemann, T. 2007. Introduction to Modern Canonical Quantum General Relativity. Cambridge, UK: Cambridge University Press.
[106] Unruh, W. 1997. Time, Gravity, and Quantum Mechanics. Cambridge, UK: Cambridge University Press, pp. 23–94.
[107] Weiss, N. 1985. Constraints on Hamiltonian lattice formulations of field theories in an expanding universe. Phys. Rev.D, 32, 3228–32.