Abstract
The Causal Dynamical Triangulation model of quantum gravity (CDT) has a transfer matrix, relating spatial geometries at adjacent (discrete lattice) times. The transfer matrix uniquely determines the theory. We show that the measurements of the scale factor of the (CDT) universe are well described by an effective transfer matrix where the matrix elements are labeled only by the scale factor. Using computer simulations we determine the effective transfer matrix elements and show how they relate to an effective minisuperspace action at all scales.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Ambjørn, J. Jurkiewicz and R. Loll, Quantum gravity as sum over spacetimes, Lect. Notes Phys. 807 (2010) 59 [arXiv:0906.3947] [INSPIRE].
J. Ambjørn, J. Jurkiewicz and R. Loll, The universe from scratch, Contemp. Phys. 47 (2006) 103 [hep-th/0509010] [INSPIRE].
J. Ambjørn, A. Görlich, J. Jurkiewicz and R. Loll, Nonperturbative quantum gravity, arXiv:1203.3591 [INSPIRE].
J. Ambjørn, J. Jurkiewicz and R. Loll, Dynamically triangulating Lorentzian quantum gravity, Nucl. Phys. B 610 (2001) 347 [hep-th/0105267] [INSPIRE].
J. Ambjørn, J. Jurkiewicz and R. Loll, A nonperturbative Lorentzian path integral for gravity, Phys. Rev. Lett. 85 (2000) 924 [hep-th/0002050] [INSPIRE].
P. Hořava, Quantum gravity at a Lifshitz point, Phys. Rev. D 79 (2009) 084008 [arXiv:0901.3775] [INSPIRE].
P. Hořava and C.M. Melby-Thompson, General covariance in quantum gravity at a Lifshitz point, Phys. Rev. D 82 (2010) 064027 [arXiv:1007.2410] [INSPIRE].
P. Hořava and C.M. Melby-Thompson, Anisotropic conformal infinity, Gen. Rel. Grav. 43 (2011) 1391 [arXiv:0909.3841] [INSPIRE].
P. Hořava, General covariance in gravity at a Lifshitz point, Class. Quant. Grav. 28 (2011) 114012 [arXiv:1101.1081] [INSPIRE].
S. Weinberg, Ultraviolet divergences in quantum theories of gravitation, in General relativity: Einstein centenary survey, S.W. Hawking and W. Israel eds., Cambridge University Press, Cambridge U.K. (1979), pg. 790.
H. Kawai and M. Ninomiya, Renormalization group and quantum gravity, Nucl. Phys. B 336 (1990)115 [INSPIRE].
H. Kawai, Y. Kitazawa and M. Ninomiya, Scaling exponents in quantum gravity near two-dimensions, Nucl. Phys. B 393 (1993) 280 [hep-th/9206081] [INSPIRE].
H. Kawai, Y. Kitazawa and M. Ninomiya, Ultraviolet stable fixed point and scaling relations in (2 + ϵ)-dimensional quantum gravity, Nucl. Phys. B 404 (1993) 684 [hep-th/9303123] [INSPIRE].
H. Kawai, Y. Kitazawa and M. Ninomiya, Renormalizability of quantum gravity near two-dimensions, Nucl. Phys. B 467 (1996) 313 [hep-th/9511217] [INSPIRE].
T. Aida, Y. Kitazawa, H. Kawai and M. Ninomiya, Conformal invariance and renormalization group in quantum gravity near two-dimensions, Nucl. Phys. B 427 (1994) 158 [hep-th/9404171] [INSPIRE].
M. Reuter, Nonperturbative evolution equation for quantum gravity, Phys. Rev. D 57 (1998) 971 [hep-th/9605030] [INSPIRE].
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] [INSPIRE].
M. Reuter and F. Saueressig, Functional renormalization group equations, asymptotic safety and quantum Einstein gravity, arXiv:0708.1317 [INSPIRE].
M. Niedermaier and M. Reuter, The asymptotic safety scenario in quantum gravity, Living Rev. Rel. 9 (2006) 5 [INSPIRE].
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] [INSPIRE].
D.F. Litim, Fixed points of quantum gravity, Phys. Rev. Lett. 92 (2004) 201301 [hep-th/0312114] [INSPIRE].
M. Niedermaier, The asymptotic safety scenario in quantum gravity: an introduction, Class. Quant. Grav. 24 (2007) R171 [gr-qc/0610018] [INSPIRE].
T. Regge, General relativity without coordinates, Nuovo Cim. 19 (1961) 558 [INSPIRE].
F. David, Planar diagrams, two-dimensional lattice gravity and surface models, Nucl. Phys. B 257 (1985) 45 [INSPIRE].
V. Kazakov, A.A. Migdal and I. Kostov, Critical properties of randomly triangulated planar random surfaces, Phys. Lett. B 157 (1985) 295 [INSPIRE].
J. Ambjørn, B. Durhuus and J. Fröhlich, Diseases of triangulated random surface models and possible cures, Nucl. Phys. B 257 (1985) 433 [INSPIRE].
D. Boulatov, V. Kazakov, I. Kostov and A.A. Migdal, Analytical and numerical study of the model of dynamically triangulated random surfaces, Nucl. Phys. B 275 (1986) 641 [INSPIRE].
J. Ambjørn, B. Durhuus, J. Fröhlich and P. Orland, The appearance of critical dimensions in regulated string theories, Nucl. Phys. B 270 (1986) 457 [INSPIRE].
A. Billoire and F. David, Microcanonical simulations of randomly triangulated planar random surfaces, Phys. Lett. B 168 (1986) 279 [INSPIRE].
V. Knizhnik, A.M. Polyakov and A. Zamolodchikov, Fractal structure of 2D quantum gravity, Mod. Phys. Lett. A 3 (1988) 819 [INSPIRE].
F. David, Conformal field theories coupled to 2D gravity in the conformal gauge, Mod. Phys. Lett. A 3 (1988) 1651 [INSPIRE].
J. Distler and H. Kawai, Conformal field theory and 2D quantum gravity or who’s afraid of Joseph Liouville?, Nucl. Phys. B 321 (1989) 509 [INSPIRE].
P.H. Ginsparg and G.W. Moore, Lectures on 2D gravity and 2D string theory, in Proceedings, Recent directions in particle theory, Boulder U.S.A. (1992), pg. 277 [YCTP-P23-92] [LA-UR-92-3479] [hep-th/9304011] [INSPIRE].
P. Di Francesco, P.H. Ginsparg and J. Zinn-Justin, 2D gravity and random matrices, Phys. Rept. 254 (1995) 1 [hep-th/9306153] [INSPIRE].
J. Ambjørn and S. Varsted, Entropy estimate in three-dimensional simplicial quantum gravity, Phys. Lett. B 266 (1991) 285 [INSPIRE].
M. Agishtein and A.A. Migdal, Three-dimensional quantum gravity as dynamical triangulation, Mod. Phys. Lett. A 6 (1991) 1863 [Erratum ibid. A 6 (1991) 2555] [INSPIRE].
D. Boulatov and A. Krzywicki, On the phase diagram of three-dimensional simplicial quantum gravity, Mod. Phys. Lett. A 6 (1991) 3005 [INSPIRE].
J. Ambjørn, D. Boulatov, A. Krzywicki and S. Varsted, The vacuum in three-dimensional simplicial quantum gravity, Phys. Lett. B 276 (1992) 432 [INSPIRE].
J. Ambjørn and S. Varsted, Three-dimensional simplicial quantum gravity, Nucl. Phys. B 373 (1992)557 [INSPIRE].
J. Ambjørn and J. Jurkiewicz, Four-dimensional simplicial quantum gravity, Phys. Lett. B 278 (1992) 42 [INSPIRE].
M. Agishtein and A.A. Migdal, Simulations of four-dimensional simplicial quantum gravity, Mod. Phys. Lett. A 7 (1992) 1039 [INSPIRE].
P. Bialas, Z. Burda, A. Krzywicki and B. Petersson, Focusing on the fixed point of 4D simplicial gravity, Nucl. Phys. B 472 (1996) 293 [hep-lat/9601024] [INSPIRE].
B.V. de Bakker, Further evidence that the transition of 4D dynamical triangulation is first order, Phys. Lett. B 389 (1996) 238 [hep-lat/9603024] [INSPIRE].
S. Catterall, R. Renken and J.B. Kogut, Singular structure in 4D simplicial gravity, Phys. Lett. B 416 (1998) 274 [hep-lat/9709007] [INSPIRE].
J. Ambjørn and R. Loll, Nonperturbative Lorentzian quantum gravity, causality and topology change, Nucl. Phys. B 536 (1998) 407 [hep-th/9805108] [INSPIRE].
J. Ambjørn, J. Correia, C. Kristjansen and R. Loll, On the relation between Euclidean and Lorentzian 2D quantum gravity, Phys. Lett. B 475 (2000) 24 [hep-th/9912267] [INSPIRE].
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] [INSPIRE].
J. Ambjørn, J. Jurkiewicz and R. Loll, Reconstructing the universe, Phys. Rev. D 72 (2005) 064014 [hep-th/0505154] [INSPIRE].
J. Ambjørn, A. Görlich, J. Jurkiewicz and R. Loll, Planckian birth of the quantum de Sitter universe, Phys. Rev. Lett. 100 (2008) 091304 [arXiv:0712.2485] [INSPIRE].
J. Ambjørn, A. Görlich, J. Jurkiewicz and R. Loll, The nonperturbative quantum de Sitter universe, Phys. Rev. D 78 (2008) 063544 [arXiv:0807.4481] [INSPIRE].
J. Ambjørn, A. Görlich, S. Jordan, J. Jurkiewicz and R. Loll, CDT meets Hořava-Lifshitz gravity, Phys. Lett. B 690 (2010) 413 [arXiv:1002.3298] [INSPIRE].
J. Ambjørn, A. Görlich, J. Jurkiewicz and R. Loll, Geometry of the quantum universe, Phys. Lett. B 690 (2010) 420 [arXiv:1001.4581] [INSPIRE].
J. Ambjørn, S. Jordan, J. Jurkiewicz and R. Loll, A second-order phase transition in CDT, Phys. Rev. Lett. 107 (2011) 211303 [arXiv:1108.3932] [INSPIRE].
J. Ambjørn, J. Jurkiewicz and R. Loll, Semiclassical universe from first principles, Phys. Lett. B 607 (2005) 205 [hep-th/0411152] [INSPIRE].
J. Ambjørn et al., The semiclassical limit of causal dynamical triangulations, Nucl. Phys. B 849 (2011)144 [arXiv:1102.3929] [INSPIRE].
B.A. Berg, Introduction to multicanonical Monte Carlo simulations, Fields Inst. Commun. 26 (2000) 1 [cond-mat/9909236] [INSPIRE].
L. Bogacz, Z. Burda and B. Waclaw, Quantum widening of CDT universe, arXiv:1204.1356 [INSPIRE].
J. Laiho and D. Coumbe, Evidence for asymptotic safety from lattice quantum gravity, Phys. Rev. Lett. 107 (2011) 161301 [arXiv:1104.5505] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1205.3791
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Ambjørn, J., Gizbert-Studnicki, J., Görlich, A. et al. The transfer matrix in four-dimensional CDT. J. High Energ. Phys. 2012, 17 (2012). https://doi.org/10.1007/JHEP09(2012)017
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2012)017