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  • Print publication year: 2012
  • Online publication date: August 2012

14 - Proper time is stochastic time in 2D quantum gravity

Summary

We show that proper time, when defined in the quantum theory of 2D gravity, becomes identical to the stochastic time associated with the stochastic quantization of space. This observation was first made by Kawai and collaborators in the context of 2D Euclidean quantum gravity, but the relation is even simpler and more transparent in the context of 2D gravity formulated in the framework of CDT (causal dynamical triangulations).

Introduction

Since time plays such a prominent role in ordinary flat space quantum field theory defined by a Hamiltonian, it is of interest to study the role of time even in toy models of quantum gravity where the role of time is much more enigmatic. The model we will describe in this chapter is the so-called causal dynamical triangulation (CDT) model of quantum gravity. It starts by providing an ultraviolet regularization in the form of a lattice theory, the lattice link length being the (diffeomorphism-invariant) UV cut-off. In addition, the lattice respects causality. It is formulated in the spirit of asymptotic safety, where it is assumed that quantum gravity is described entirely by “conventional” quantum field theory, in this case by approaching a non-trivial fixed point [1, 2]. It is formulated in space-times with Lorentzian signature, but the regularized space-times which are used in the path integral defining the theory allow a rotation to Euclidean space-time. The action used is the Regge action for the piecewise linear geometry.

References
[1] S., Weinberg: Ultraviolet divergences in quantum theories of gravitation, in General Relativity: Einstein Centenary Survey, eds. S. W., Hawking and W., Israel, Cambridge University Press, Cambridge, UK (1979) 790–831.
[2] A., Codello, R., Percacci, and C., Rahmede, Investigating the ultraviolet properties of gravity with a Wilsonian renormalization group equation, Annals Phys. 324 (2009) 414 [arXiv:0805.2909 [hep-th]].
M., Reuter and F., Saueressig: Functional renormalization group equations, asymptotic safety, and quantum Einstein gravity [0708.1317, hep-th].
M., Niedermaier and M., Reuter: The asymptotic safety scenario in quantum gravity, Living Rev. Rel. 9 (2006) 5.
H. W., Hamber and R. M., Williams: Nonlocal effective gravitational field equations and the running of Newton's G, Phys. Rev.D 72 (2005) 044026 [hep-th/0507017].
D. F., Litim: Fixed points of quantum gravity, Phys. Rev. Lett. 92 (2004) 201301 [hep-th/0312114].
H., Kawai, Y., Kitazawa, and M., Ninomiya: Renormalizability of quantum gravity near two dimensions, Nucl. Phys.B 467 (1996) 313–31 [hep-th/9511217].
[3] J., Ambjørn, J., Jurkiewicz, and R., Loll: Dynamically triangulating Lorentzian quantum gravity, Nucl. Phys.B 610 (2001) 347–82 [hep-th/0105267].
[4] J., Ambjørn, J., Jurkiewicz, and R., Loll: Reconstructing the universe, Phys. Rev.D 72 (2005) 064014 [hep-th/0505154].
[5] J., Ambjørn, A., Görlich, J., Jurkiewicz, and R., Loll, The nonperturbative quantum de Sitter universe, Phys. Rev.D 78 (2008) 063544 [arXiv:0807.4481 [hep-th]].
J., Ambjorn, A., Görlich, J., Jurkiewicz and R., Loll: Planckian birth of the quantum de Sitter universe, Phys. Rev. Lett. 100 (2008) 091304 [0712.2485, hep-th].
[6] J., Ambjørn, J., Jurkiewicz, and R., Loll: Emergence of a 4D world from causal quantum gravity, Phys. Rev. Lett. 93 (2004) 131301 [hep-th/0404156].
[7] J., Ambjørn, J., Jurkiewicz, and R., Loll: The universe from scratch, Contemp. Phys. 47 (2006) 103–17 [hep-th/0509010].
R., Loll: The emergence of spacetime, or, quantum gravity on your desktop, Class. Quant. Grav. 25 (2008) 114006 [0711.0273, gr-qc].
[8] J., Ambjørn, J., Jurkiewicz, and R., Loll: Semiclassical universe from first principles, Phys. Lett.B 607 (2005) 205–13 [hep-th/0411152].
Spectral dimension of the universe, Phys. Rev. Lett. 95 (2005) 171301 [hep-th/0505113].
[9] O., Lauscher and M., Reuter: Fractal spacetime structure in asymptotically safe gravity, JHEP 0510 (2005) 050 [arXiv:hep-th/0508202].
[10] P., Horava: Spectral dimension of the universe in quantum gravity at a Lifshitz point, Phys. Rev. Lett. 102 (2009) 161301 [arXiv:0902.3657 [hep-th]].
[11] T., Regge: General relativity without coordinates, Nuovo Cim. 19 (1961) 558.
[12] J., Ambjørn and R., Loll: Non-perturbative Lorentzian quantum gravity, causality and topology change, Nucl. Phys.B 536 (1998) 407–34 [hep-th/9805108].
[13] C., Teitelboim: Causality versus gauge invariance in quantum gravity and supergravity, Phys. Rev. Lett. 50 (1983) 705–8.
The proper time gauge in quantum theory of gravitation, Phys. Rev.D 28 (1983) 297–309.
[14] J., Ambjørn, R., Loll, W., Westra, and S., Zohren: Putting a cap on causality violations in CDT, JHEP 0712 (2007) 017 [0709.2784, gr-qc].
[15] J., Ambjørn, R., Loll, Y., Watabiki, W., Westra, and S., Zohren, A matrix model for 2D quantum gravity defined by causal dynamical triangulations, Phys. Lett.B 665 (2008) 252–56 [0804.0252, hep-th].
A new continuum limit of matrix models, Phys. Lett.B 670 (2008) 224 [arXiv:0810.2408 [hep-th]].
A causal alternative for c = 0 strings, Acta Phys. Polon.B 39 (2008) 3355 [arXiv:0810.2503 [hep-th]].
[16] J., Ambjørn, R., Loll, Y., Watabiki, W., Westra, and S., Zohren: A string field theory based on causal dynamical triangulations, JHEP 0805 (2008) 032 [0802.0719, hep-th].
[17] H., Kawai, N., Kawamoto, T., Mogami, and Y., Watabiki: Transfer matrix formalism for two-dimensional quantum gravity and fractal structures of space-time, Phys. Lett.B 306 (1993) 19–26 [hep-th/9302133].
N., Ishibashi and H., Kawai: String field theory of noncritical strings, Phys. Lett.B 314 (1993) 190 [arXiv:hep-th/9307045].
String field theory of c ≤ 1 noncritical strings, Phys. Lett.B 322 (1994) 67 [arXiv:hep-th/9312047].
A background independent formulation of noncritical string theory, Phys. Lett.B 352 (1995) 75 [arXiv:hep-th/9503134].
M., Ikehara, N., Ishibashi, H., Kawai, T., Mogami, R., Nakayama, and N., Sasakura: String field theory in the temporal gauge, Phys. Rev.D 50 (1994) 7467 [arXiv:hep-th/9406207].
Y., Watabiki: Construction of noncritical string field theory by transfer matrix formalism in dynamical triangulation, Nucl. Phys.B 441 (1995) 119–66 [hep-th/9401096].
H., Aoki, H., Kawai, J., Nishimura, and A., Tsuchiya: Operator product expansion in two-dimensional quantum gravity, Nucl. Phys.B 474 (1996) 512–28 [hep-th/9511117].
J., Ambjørn and Y., Watabiki: Non-critical string field theory for 2D quantum gravity coupled to (p, q)–conformal fields, Int. J. Mod. Phys.A 12 (1997) 4257 [arXiv:hep-th/9604067].
[18] F., David: Loop equations and nonperturbative effects in two-dimensional quantum gravity, Mod. Phys. Lett.A 5 (1990) 1019.
[19] J., Ambjørn, J., Jurkiewicz, and Yu. M., Makeenko: Multiloop correlators for two-dimensional quantum gravity, Phys. Lett.B 251 (1990) 517–24.
[20] J., Ambjørn and Yu. M., Makeenko: Properties of loop equations for the Hermitean matrix model and for two-dimensional quantum gravity,'Mod. Phys. Lett.A 5 (1990) 1753.
[21] J., Ambjørn, L., Chekhov, C. F., Kristjansen, and Yu., Makeenko: Matrix model calculations beyond the spherical limit, Nucl. Phys.B 404 (1993) 127 [Erratum-Matrix model calculations beyond the spherical limit, Nucl. Phys. B. 449 (1995) 681] [arXiv:hep-th/9302014].
[22] B., Eynard: Topological expansion for the 1-hermitian matrix model correlation functions, JHEP 0411 (2004) 031 [arXiv:hep-th/0407261].
[23] L., Chekhov and B., Eynard: Hermitean matrix model free energy: Feynman graph technique for all genera, JHEP 0603 (2006) 014 [arXiv:hep-th/0504116].
[24] B., Eynard and N., Orantin: Invariants of algebraic curves and topological expansion, arXiv:math-ph/0702045. Topological expansion and boundary conditions, JHEP 0806 (2008) 037 [arXiv:0710.0223 [hep-th]].
[25] M., Abramowitz and I., Stegun (eds): Pocketbook of Mathematical Functions (Harri Deutsch, Frankfurt, 1984).
[26] F., David: Nonperturbative effects in matrix models and vacua of two-dimensional gravity, Phys. Lett.B 302 (1993) 403 [hep-th/9212106].
[27] M., Mariño: Nonperturbative effects and nonperturbative definitions in matrix models and topological strings, JHEP 0812 (2008) 114 [0805.3033 [hep-th]].
[28] J., Jurkiewicz and A., Krzywicki: Branched polymers with loops, Phys. Lett.B 392 (1997) 291 [hep-th/9610052].
[29] J., Ambjørn and B., Durhuus: Regularized bosonic strings need extrinsic curvature, Phys. Lett.B 188 (1987) 253–57.
[30] J., Ambjørn, S., Jain, and G., Thorleifsson: Baby universes in 2-d quantum gravity, Phys. Lett.B 307 (1993) 34–9 [hep-th/9303149].
[31] J., Ambjørn, S., Jain, J., Jurkiewicz, and C. F., Kristjansen: Observing 4-d baby universes in quantum gravity, Phys. Lett.B 305 (1993) 208 [hep-th/9303041].
[32] J., Ambjørn, R., Loll, W., Westra, and S., Zohren: Stochastic quantization and the role of time in quantum gravity, Phys. Lett.B 680 (2009) 359 [arXiv:0908.4224 [hep-th]].
[33] J., Zinn-Justin: Quantum field theory and critical phenomena, Int. Ser. Monogr. Phys. 113 (2002) 1.
[34] M., Chaichian and A., Demichev: Path Integrals in Physics, Volume II, Institute of Physics Publishing, Bristol, UK (2001).
[35] M., Ikehara, N., Ishibashi, H., Kawai, T., Mogami, R., Nakayama, and N., Sasakura: A note on string field theory in the temporal gauge, Prog. Theor. Phys. Suppl. 118 (1995) 241 [arXiv:hep-th/9409101].
[36] J., Ambjørn and C. F., Kristjansen: Nonperturbative 2-d quantum gravity and Hamiltonians unbounded from below, Int. J. Mod. Phys.A 8 (1993) 1259 [arXiv: hep-th/9205073].
[37] J., Greensite and M. B., Halpern: Stabilizing bottomless action theories, Nucl. Phys.B 242 (1984) 167.
[38] J., Ambjørn and J., Greensite: Nonperturbative calculation of correlators in 2-D quantum gravity, Phys. Lett.B 254 (1991) 66.
[39] J., Ambjørn, J., Greensite, and S., Varsted: A nonperturbative definition of 2-D quantum gravity by the fifth time action, Phys. Lett.B 249 (1990) 411.