Towards quantum gravity

Document Type : Review article

Author

University of Toronto

Abstract

We analyze different approaches to quantum gravity. It is stressed that nonperturbative methods to quantise gravity and the usage of diffeomorphism-invariant variables are very important. We pay attention on the Wheeler--DeWitt equation in the framework of canonical quantum gravity. The Wheeler--DeWitt equation is presented in the first order formalism with the hope that this form can solve some problems such as singularities and the ordering. Also, there is a problem of defining the time.

Keywords

Main Subjects

[1] S. Garlip, Spacetime foam: a review, arXiv:2209.14282.
[2] S. Weinberg, Ultraviolet divergences in quantum theories of gravitation, in General relativity: An Einstein centenary survey, eds. S.W. Hawking and W. Israel, Cambridge University Press (1979) 790-831.
[3] A. Ashtekar, New variables for classical and quantum gravity, Phys. Rev. Lett. 57 (1986) 2244-2247.
[4] C. Rovelli and L. Smolin, Loop space representation of quantum general relativity, Nucl. Phys. B 331 (1990) 80-152.
[5] T. Regge, General relativity without coordinates, Nuovo Cim. 19 (1961) 558-571.
[6] T. Thiemann, Modern canonical quantum general relativity, Cambridge University Press (2007).
[7] C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, W.H. Freeman, San Francisco (1973).
[8] F. David, Simplicial quantum gravity and random lattices, in Gravitation and quantizations (Les Houches Summer School, Session LVII, 1992), eds. J. Zinn-Justin and B. Julia, Elsevier, Amsterdam (1995) 679-750 [arXiv:hep-th/9303127].
[9] J. Ambjorn and R. Loll, Nonperturbative Lorentzian quantum gravity, causality and topology
change, Nucl. Phys. B 536 (1998) 407-434 [arXiv:hep-th/9805108].
[10] S. Surya, The causal set approach to quantum gravity, Living Rev. Rel. 22 (2019), 5 [arXiv:1903.11544].
[11] L. Smolin, Quantum gravity on a lattice, Nucl. Phys. B 148 (1979) 333-372.
[12] P. Menotti and A. Pelissetto, Reflection positivity and graviton doubling in Euclidean lattice gravity, Ann. Phys. 170 (1986) 287-309,
[13] I. Montvay and G. Mnster, Quantum fields on a lattice, Cambridge University Press (1994).
[14] J. Smit, Introduction to quantum fields on a lattice, Cambridge University Press (2002).
[15] D. Diakonov, Towards lattice-regularized quantum gravity, arXiv:1109.0091.
[16] M. Reuter and F. Saueressig, Quantum gravity and the functional renormalization group, Cambridge University Press (2019).
[17] M. Niedermaier and M. Reuter, The asymptotic safety scenario in quantum gravity, Living Rev. Rel. 9 (2006) 5-173.
[18] R. Percacci, An introduction to covariant quantum gravity and asymptotic safety, World Sci- entific, Singapore (2017).
[19] C. Kiefer, Quantum gravity, 3rd edition, Oxford University Press (2012).
[20] R. Loll, G. Fabiano and F. Wagner, Quantum Gravity in 30 Questions, arXiv:2206.06762.
[21] C. J. Isham, Canonical quantum gravity and the problem of time, in Integrable systems, quantum groups and quantum field theories, eds. L. A. Ibort and M. A. Rodriguez, NATO Sci.
[22] S. Carlip, Quantum gravity: A progress report, Rept. Prog. Phys. 64 (2001) 885 [arXiv:gr-qc/0108040].
[23] B. S. DeWitt, Quantum theory of gravity. I. The canonical theory, Phys. Rev. 162 (1967) 1113-1148.
[24] B. S. DeWitt, Quantum theory of gravity. II. The manifestly covariant theory, Phys. Rev. 162 (1967) 1195-1239.
[25] B. S. DeWitt, The Quantum and gravity: The Wheeler-DeWitt equation, in The eighth Marcel Grossmann meeting, Proceedings, eds. T. Piran and R. Ruffini, World Scientific, Singapore (1999) 6-25.
[26] K. V. Kucha╦śr, Canonical quantum gravity, arXiv:gr-qc/9304012.
[27] N. C. Tsamis and R. P. Woodard, The factor ordering problem must be regulated, Phys. Rev. D 36 (1987) 3641-3650.
[28] J. L. Friedman and I. Jack, Formal commutators of the gravitational constraints are not well defined: A translation of Ashtekars ordering to the Schrdinger representation, Phys. Rev. D 37 (1988) 3495-3504.
[29] Sergey I. Kruglov and Mir Faizal, Wave Function of the Universe from a Matrix Valued First-Order Formalism, Int. J. Geom. Meth. Mod. Phys. 12 (2015), 1550050 [arXiv:1408.3794].
Volume 3, Issue 2
June 2023
Pages 53-58
  • Receive Date: 25 May 2023
  • Revise Date: 20 June 2023
  • Accept Date: 30 June 2023