Abstract
In Regge calculus, the space–time manifold is approximated by certain abstract simplicial complex, called a pseudomanifold, and the metric is approximated by an assignment of a length to each 1-simplex. In this paper for each pseudomanifold, we construct a smooth manifold which we call a manifold with defects. This manifold emerges from the purely combinatorial simplicial complex as a result of gluing geometric realizations of its n-simplices followed by removing the simplices of dimension \(n-2\). The Regge geometry is encoded in a boundary data of a BF theory on this manifold. We consider an action functional which coincides with the standard BF action for suitably regular manifolds with defects and fields. We show that the action evaluated at solutions of the field equations satisfying certain boundary conditions coincides with an evaluation of the Regge action at Regge geometries defined by the boundary data. As a result, the degrees of freedom of Regge calculus are traded for discrete degrees of freedom of topological BF theory.
Article PDF
Similar content being viewed by others
References
Rovelli, C., Vidotto, F.: Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory. Cambridge University Press, Cambridge (2014)
Rovelli, C.: Quantum Gravity. Cambridge University Press, Cambridge (2004)
Baez, J.C.: An introduction to spin foam models of quantum gravity and BF theory. Lect. Notes Phys. 543, 25–94 (2000)
Perez, A.: Spin foam models for quantum gravity. Class. Quantum Gravity 20, R43 (2003)
Perez, A.: The spin foam approach to quantum gravity. Living Rev. Relat. 16, 3 (2013)
Rovelli, C.: Zakopane lectures on loop gravity. PoS QGQGS2011, 003 (2011)
Engle, J.: Springer Handbook of Spacetime, Ch. Spin Foams. Springer, Berlin (2014)
Rovelli, C.: Loop quantum gravity: the first twenty five years. Class. Quantum Gravity 28, 153002 (2011)
Ashtekar, A., Reuter, M., Rovelli, C.: General Relativity & Gravitation: a Centennial Perspective. Pennsylvania State University (2015)
Thiemann, T.: Modern Canonical Quantum General Relativity. Cambridge University Press, Cambridge (2007)
Ponzano, G., Regge, T.: Spectroscopic and group theoretical methods in physics: Racah memorial volume. In: Bloch, F., Cohen, S., De Shalit, A., Sambursky, S., Talmi, I. (eds.) Semiclassical Limit of Racah Coefficients. North-Holland Publishing Co., Amsterdam (1968)
Engle, J., Livine, E., Pereira, R., Rovelli, C.: LQG vertex with finite Immirzi parameter. Nucl. Phys. B 799, 136–149 (2008)
Freidel, L., Krasnov, K.: A new spin foam model for 4D gravity. Class. Quantum Gravity 25, 125018 (2008)
Barrett, J.W., Crane, L.: Relativistic spin networks and quantum gravity. J. Math. Phys. 39, 3296–3302 (1998)
Barrett, J.W., Crane, L.: A Lorentzian signature model for quantum general relativity. Class. Quantum Gravity 17, 3101–3118 (2000)
Barrett, J.W., Dowdall, R.J., Fairbairn, W.J., Hellmann, F., Pereira, R.: Lorentzian spin foam amplitudes: graphical calculus and asymptotics. Class. Quantum Gravity 27(16), 165009 (2010)
Bianchi, E., Regoli, D., Rovelli, C.: Face amplitude of spinfoam quantum gravity. Class. Quantum Gravity 27, 185009 (2010)
Kaminski, W., Kisielowski, M., Lewandowski, J.: Spin-foams for all loop quantum gravity. Class. Quantum Gravity 27, 095006 (2010)
Kaminski, W., Kisielowski, M., Lewandowski, J.: The EPRL intertwiners and corrected partition function. Class. Quantum Gravity 27, 165020 (2010)
Bahr, B., Hellmann, F., Kaminski, W., Kisielowski, M., Lewandowski, J.: Operator spin foam models. Class. Quantum Gravity 28, 105003 (2011)
Engle, J.: Proposed proper Engle–Pereira–Rovelli–Livine vertex amplitude. Phys. Rev. D87(8), 084048 (2013)
Engle, J.: A spin-foam vertex amplitude with the correct semiclassical limit. Phys. Lett. B 724, 333–337 (2013)
Bianchi, E., Hellmann, F.: The construction of spin foam vertex amplitudes. SIGMA 9, 008 (2013)
Plebanski, J.F.: On the separation of Einsteinian substructures. J. Math. Phys. 18, 2511–2520 (1977)
Reisenberger, M.P., Rovelli, C.: ’Sum over surfaces’ form of loop quantum gravity. Phys. Rev. D 56, 3490–3508 (1997)
Rovelli, C.: The projector on physical states in loop quantum gravity. Phys. Rev. D 59, 104015 (1999)
Noui, K., Perez, A.: Three-dimensional loop quantum gravity: physical scalar product and spin foam models. Class. Quantum Gravity 22, 1739–1762 (2005)
Engle, J., Han, M., Thiemann, T.: Canonical path integral measures for Holst and Plebanski gravity. I. Reduced phase space derivation. Class. Quantum Gravity 27, 245014 (2010)
Han, M., Thiemann, T.: On the relation between Rigging inner product and master constraint direct integral decomposition. J. Math. Phys. 51, 092501 (2010)
Han, M., Thiemann, T.: On the relation between operator constraint-, master constraint-, reduced phase space-, and path integral quantisation. Class. Quantum Gravity 27, 225019 (2010)
Dittrich, B., Hohn, P.A.: From covariant to canonical formulations of discrete gravity. Class. Quantum Gravity 27, 155001 (2010)
Alesci, E., Thiemann, T., Zipfel, A.: Linking covariant and canonical LQG: new solutions to the Euclidean scalar constraint. Phys. Rev. D 86, 024017 (2012)
Thiemann, T., Zipfel, A.: Linking covariant and canonical LQG II: spin foam projector. Class. Quantum Gravity 31, 125008 (2014)
Ashtekar, A., Marolf, D., Mourao, J., Thiemann, T.: Constructing Hamiltonian quantum theories from path integrals in a diffeomorphism-invariant context. Class. Quantum Gravity 17(23), 4919 (2000)
Bianchi, E.: Loop quantum gravity a la Aharonov–Bohm. Gen. Relativ. Gravit. 46, 1668 (2014)
Haggard, H.M., Han, M., Kamiński, W., Riello, A.: SL(2, C) Chern–Simons theory, a non-planar graph operator, and 4D loop quantum gravity with a cosmological constant: semiclassical geometry. Nucl. Phys. B 900, 1–79 (2015)
Han, M.: 4D quantum geometry from 3D supersymmetric gauge theory and holomorphic block. JHEP 01, 065 (2016)
Haggard, H.M., Han, M., Kamiński, W., Riello, A.: Four-dimensional quantum gravity with a cosmological constant from three-dimensional holomorphic blocks. Phys. Lett. B 752, 258–262 (2016)
Han, M., Huang, Z.: Loop-quantum-gravity simplicity constraint as surface defect in complex Chern–Simons theory. Phys. Rev. D 95, 104031 (2017)
Penrose, R.: Angular momentum: an approach to combinatorial space-time. In: Bastin, T. (ed.) Quantum Theory and Beyond. Cambridge University Press, pp 151–180 (1971)
De Pietri, R., Petronio, C.: Feynman diagrams of generalized matrix models and the associated manifolds in dimension 4. J. Math. Phys. 41, 6671–6688 (2000)
Freidel, L.: Group field theory: an overview. Int. J. Theor. Phys. 44, 1769–1783 (2005)
Ben Geloun, J., Gurau, R., Rivasseau, V.: EPRL/FK group field theory. Europhys. Lett. 92, 60008 (2010)
Krajewski, T., Magnen, J., Rivasseau, V., Tanasa, A., Vitale, P.: Quantum corrections in the group field theory formulation of the EPRL/FK models. Phys. Rev. D 82, 124069 (2010)
Oriti, D., Ryan, J.P., Thürigen, J.: Group field theories for all loop quantum gravity. New J. Phys. 17, 023042 (2015)
Kisielowski, M., Lewandowski, J., Puchta, J.: Feynman diagrammatic approach to spin foams. Class. Quantum Gravity 29, 015009 (2012)
Regge, T.: General relativity without coordinates. Il Nuovo Cimento (1955–1965) 19(3), 558–571 (1961)
Aharonov, Y., Bohm, D.: Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115, 485–491 (1959)
Sorkin, R.D.: Development of simplectic methods for the metrical and electromagnetic fields. Ph.D. thesis, California Institute of Technology (1974)
Friedberg, R., Lee, T.D.: Derivation of Regge’s action from Einstein’s theory of general relativity. Nucl. Phys. B242, 145 (1984). [,213(1984)]
Gibbons, G.W., Hawking, S.W.: Action integrals and partition functions in quantum gravity. Phys. Rev. D15(10), 2752 (1977)
York Jr., J.W.: Role of conformal three-geometry in the dynamics of gravitation. Phys. Rev. Lett. 28(16), 1082 (1972)
Aref’eva, I.Y.: Non-Abelian Stokes formula. Theor. Math. Phys. 43(1), 353–356 (1980)
Montvay, I., Münster, G.: Quantum Fields on a Lattice. Cambridge University Press, Cambridge (1997)
Thurston, W.P., Levy, S.: Three-Dimensional Geometry and Topology, vol. 1. Princeton university press, Princeton (1997)
Khatsymovsky, V.: Tetrad and self-dual formulations of Regge calculus. Class. Quantum Gravity 6(12), L249–L255 (1989)
Bander, M.: Functional measure for lattice gravity. Phys. Rev. Lett. 57, 1825 (1986)
Pontryagin, L.S.: Foundations of Combinatorial Topology. Courier Corporation, Chelmsford (1999)
Lee, J.: Introduction to Topological Manifolds, vol. 940. Springer, Berlin (2010)
Pseudo-manifold. Encyclopedia of mathematics: http://www.encyclopediaofmath.org/index.php?title=Pseudo-manifold&oldid=24541. Accessed 13 Jan 2017
Spanier, E.H.: Algebraic Topology, vol. 55. Springer, Berlin (1994)
Lazebnik, F.: On a regular simplex in \(\mathbb{R}^{n}\). http://www.math.udel.edu/~lazebnik/papers/simplex.pdf. Accessed 12 Feb 2017
Freudenthal, H.: Simplizialzerlegungen von beschrankter flachheit. Ann. Math. Second Ser. 43(3), 580–582 (1942)
Edelsbrunner, H., Grayson, D.R.: Edgewise subdivision of a simplex. Discrete Comput. Geom. 24(4), 707–719 (2000)
Wieland, W.M.: A new action for simplicial gravity in four dimensions. Class. Quantum Gravity 32(1), 015016 (2015)
Minkowski, H.: Allgemeine lehrsätze über die convexen polyeder. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse 1897, 198–220 (1897)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. 1. Wiley, New York (1963)
Barrett, J.W., Foxon, T.J.: Semiclassical limits of simplicial quantum gravity. Class. Quantum Gravity 11(3), 543 (1994)
Cheeger, J., et al.: Spectral geometry of singular Riemannian spaces. J. Differ. Geom. 18(4), 575–657 (1983)
Wintgen, P.: Normal cycle and integral curvature for polyhedra in Riemannian manifolds. In: Soos, Gy., Szenthe, J. (eds.) Differential Geometry. North-Holland Publishing Co., Amsterdam (1982)
Cheeger, J., Muller, W., Schrader, R.: On the curvature of piecewise flat spaces. Commun. Math. Phys. 92, 405 (1984)
Cattaneo, A.S., Mnev, P., Reshetikhin, N.: A cellular topological field theory (2017) arXiv:1701.05874
Dittrich, B., Geiller, M.: A new vacuum for loop quantum gravity. Class. Quantum Gravity 32(11), 112001 (2015)
Dittrich, B., Geiller, M.: Flux formulation of loop quantum gravity: classical framework. Class. Quantum Gravity 32(13), 135016 (2015)
Bahr, B., Dittrich, B., Geiller, M.: A new realization of quantum geometry (2015) arXiv:1506.08571
Dittrich, B., Geiller, M.: Quantum gravity kinematics from extended TQFTs. New J. Phys. 19(1), 013003 (2017)
Delcamp, C., Dittrich, B.: From 3D TQFTs to 4D models with defects. J. Math. Phys. 58(6), 062302 (2017)
Reisenberger, M.P.: Classical Euclidean general relativity from ’left-handed area = right-handed area’. Class. Quantum Gravity 16, 1357 (1999)
Ding, Y., Han, M., Rovelli, C.: Generalized spinfoams. Phys. Rev. D 83, 124020 (2011)
Wieland, W.: Discrete gravity as a topological gauge theory with light-like curvature defects. JHEP 5, 142 (2017)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Carlo Rovelli.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Kisielowski, M. Relation Between Regge Calculus and BF Theory on Manifolds with Defects. Ann. Henri Poincaré 20, 1403–1437 (2019). https://doi.org/10.1007/s00023-018-0747-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-018-0747-6