Abstract
In this paper, we will make an attempt to clarify the relation between three-dimensional euclidean loop quantum gravity with vanishing cosmological constant and quantum field theory in the continuum. We will argue, in particular, that in three spacetime dimensions the discrete spectra for the geometric boundary observables that we find in loop quantum gravity can be understood from the quantisation of a conformal boundary field theory in the continuum without ever introducing spin networks or triangulations of space. At a technical level, the starting point is the Hamiltonian formalism for general relativity in regions with boundaries at finite distance. At these finite boundaries, we choose specific conformal boundary conditions (the boundary is a minimal surface) that are derived from a boundary field theory for an SU(2) boundary spinor, which is minimally coupled to the spin connection in the bulk. The resulting boundary equations of motion define a conformal field theory with vanishing central charge. We will quantise this boundary field theory and show that the length of a one-dimensional cross section of the boundary has a discrete spectrum. In addition, we will introduce a new class of coherent states, study the quasi-local observables that generate the quasi-local Virasoro algebra and discuss some strategies to evaluate the partition function of the theory.
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References
A. Ashtekar, Lectures on Non-Pertubative Canonical Gravity, World Scientific (1991).
C. Rovelli, Quantum Gravity, Cambridge University Press, Cambridge U.K. (2008).
T. Thiemann, Modern Canonical Quantum General Relativity, Cambridge University Press, Cambridge U.K. (2008).
A. Ashtekar and J. Lewandowski, Representation theory of analytic holonomy C * algebras, in Knots and Quantum Gravity, J.Baez ed., Oxford University Press, Oxford U.K. (1993) [gr-qc/9311010] [INSPIRE].
A. Ashtekar and J. Lewandowski, Projective techniques and functional integration for gauge theories, J. Math. Phys. 36 (1995) 2170 [gr-qc/9411046] [INSPIRE].
H. Nicolai, K. Peeters and M. Zamaklar, Loop quantum gravity: An Outside view, Class. Quant. Grav. 22 (2005) R193 [hep-th/0501114] [INSPIRE].
B. Dittrich and T. Thiemann, Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?, J. Math. Phys. 50 (2009) 012503 [arXiv:0708.1721] [INSPIRE].
W. Wieland, Fock representation of gravitational boundary modes and the discreteness of the area spectrum, Annales Henri Poincaré 18 (2017) 3695 [arXiv:1706.00479] [INSPIRE].
S. Carlip, Conformal field theory, (2 + 1)-dimensional gravity and the BTZ black hole, Class. Quant. Grav. 22 (2005) R85 [gr-qc/0503022] [INSPIRE].
S. Carlip, Quantum Gravity in 2 + 1 Dimensions, Cambridge University Press, Cambridge U.K. (2003).
J.D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].
W. Wieland, Quantum gravity in three dimensions, Witten spinors and the quantisation of length, Nucl. Phys. B 930 (2018) 219 [arXiv:1711.01276] [INSPIRE].
W. Wieland, New boundary variables for classical and quantum gravity on a null surface, Class. Quant. Grav. 34 (2017) 215008 [arXiv:1704.07391] [INSPIRE].
A.J. Speranza, Local phase space and edge modes for diffeomorphism-invariant theories, JHEP 02 (2018) 021 [arXiv:1706.05061] [INSPIRE].
M. Geiller, Edge modes and corner ambiguities in 3d Chern-Simons theory and gravity, Nucl. Phys. B 924 (2017) 312 [arXiv:1703.04748] [INSPIRE].
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
M. Ammon and J. Erdmenger, Gauge/gravity duality: Foundations and applications, Cambridge University Press, Cambridge U.K. (2015) [INSPIRE].
B. Dittrich, C. Goeller, E.R. Livine and A. Riello, Quasi-local holographic dualities in non-perturbative 3d quantum gravity I — Convergence of multiple approaches and examples of Ponzano-Regge statistical duals, arXiv:1710.04202 [INSPIRE].
B. Dittrich, C. Goeller, E.R. Livine and A. Riello, Quasi-local holographic dualities in non-perturbative 3d quantum gravity II — From coherent quantum boundaries to BMS3 characters, arXiv:1710.04237 [INSPIRE].
B. Dittrich, C. Goeller, E.R. Livine and A. Riello, Quasi-local holographic dualities in non-perturbative 3d quantum gravity, Class. Quant. Grav. 35 (2018) 13LT01 [arXiv:1803.02759] [INSPIRE].
G. Barnich, H.A. Gonzalez, A. Maloney and B. Oblak, One-loop partition function of three-dimensional flat gravity, JHEP 04 (2015) 178 [arXiv:1502.06185] [INSPIRE].
R.M. Wald and A. Zoupas, A General definition of ‘conserved quantities’ in general relativity and other theories of gravity, Phys. Rev. D 61 (2000) 084027 [gr-qc/9911095] [INSPIRE].
J.D. Brown and J.W. York Jr., Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D 47 (1993) 1407 [gr-qc/9209012] [INSPIRE].
L.B. Szabados, Quasi-Local Energy-Momentum and Angular Momentum in GR: A Review Article, Living Rev. Rel. 7 (2004) 4 [INSPIRE].
S. Carlip and C. Teitelboim, The Off-shell black hole, Class. Quant. Grav. 12 (1995) 1699 [gr-qc/9312002] [INSPIRE].
E. Bianchi and W. Wieland, Horizon energy as the boost boundary term in general relativity and loop gravity, arXiv:1205.5325 [INSPIRE].
E. Frodden, A. Ghosh and A. Perez, Quasilocal first law for black hole thermodynamics, Phys. Rev. D 87 (2013) 121503 [arXiv:1110.4055] [INSPIRE].
T. De Lorenzo and A. Perez, Light Cone Thermodynamics, Phys. Rev. D 97 (2018) 044052 [arXiv:1707.00479] [INSPIRE].
S.A. Fulling, Nonuniqueness of canonical field quantization in Riemannian space-time, Phys. Rev. D 7 (1973) 2850 [INSPIRE].
E.R. Livine and S. Speziale, A New spinfoam vertex for quantum gravity, Phys. Rev. D 76 (2007) 084028 [arXiv:0705.0674] [INSPIRE].
W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, JHEP 09 (2016) 102 [arXiv:1601.04744] [INSPIRE].
W. Donnelly, Entanglement entropy in loop quantum gravity, Phys. Rev. D 77 (2008) 104006 [arXiv:0802.0880] [INSPIRE].
L. Freidel and S. Speziale, From twistors to twisted geometries, Phys. Rev. D 82 (2010) 084041 [arXiv:1006.0199] [INSPIRE].
W. Wieland, Twistorial phase space for complex Ashtekar variables, Class. Quant. Grav. 29 (2012) 045007 [arXiv:1107.5002] [INSPIRE].
E.R. Livine and J. Tambornino, Spinor Representation for Loop Quantum Gravity, J. Math. Phys. 53 (2012) 012503 [arXiv:1105.3385] [INSPIRE].
E. Bianchi, J. Guglielmon, L. Hackl and N. Yokomizo, Squeezed vacua in loop quantum gravity, arXiv:1605.05356 [INSPIRE].
E. Bianchi, J. Guglielmon, L. Hackl and N. Yokomizo, Loop expansion and the bosonic representation of loop quantum gravity, Phys. Rev. D 94 (2016) 086009 [arXiv:1609.02219] [INSPIRE].
A. Ashtekar, J.C. Baez and K. Krasnov, Quantum geometry of isolated horizons and black hole entropy, Adv. Theor. Math. Phys. 4 (2000) 1 [gr-qc/0005126] [INSPIRE].
N. Bodendorfer, T. Thiemann and A. Thurn, New Variables for Classical and Quantum Gravity in all Dimensions V. Isolated Horizon Boundary Degrees of Freedom, Class. Quant. Grav. 31 (2014) 055002 [arXiv:1304.2679] [INSPIRE].
A. Ghosh and D. Pranzetti, CFT/Gravity Correspondence on the Isolated Horizon, Nucl. Phys. B 889 (2014) 1 [arXiv:1405.7056] [INSPIRE].
K. Noui and A. Perez, Three-dimensional loop quantum gravity: Physical scalar product and spin foam models, Class. Quant. Grav. 22 (2005) 1739 [gr-qc/0402110] [INSPIRE].
A. Perez, The Spin Foam Approach to Quantum Gravity, Living Rev. Rel. 16 (2013) 3 [arXiv:1205.2019] [INSPIRE].
B. Dittrich and M. Geiller, Flux formulation of loop quantum gravity: Classical framework, Class. Quant. Grav. 32 (2015) 135016 [arXiv:1412.3752] [INSPIRE].
T. Thiemann, Kinematical Hilbert spaces for Fermionic and Higgs quantum field theories, Class. Quant. Grav. 15 (1998) 1487 [gr-qc/9705021] [INSPIRE].
T. Thiemann, QSD 5: Quantum gravity as the natural regulator of matter quantum field theories, Class. Quant. Grav. 15 (1998) 1281 [gr-qc/9705019] [INSPIRE].
D. Oriti, Boundary terms in the Barrett-Crane spin foam model and consistent gluing, Phys. Lett. B 532 (2002) 363 [gr-qc/0201077] [INSPIRE].
M. O’Loughlin, Boundary actions in Ponzano-Regge discretization, quantum groups and AdS 3, Adv. Theor. Math. Phys. 6 (2003) 795 [gr-qc/0002092] [INSPIRE].
G. Arcioni, M. Carfora, A. Marzuoli and M. O’Loughlin, Implementing holographic projections in Ponzano-Regge gravity, Nucl. Phys. B 619 (2001) 690 [hep-th/0107112] [INSPIRE].
D. Oriti, Group Field Theory and Loop Quantum Gravity, in Loop Quantum Gravity, The First Thirty Years, volume 4, A. Abhay and J. Pullin eds., World Scientific (2017) [arXiv:1408.7112] [INSPIRE].
E. Adjei, S. Gielen and W. Wieland, Cosmological evolution as squeezing: a toy model for group field cosmology, Class. Quant. Grav. 35 (2018) 105016 [arXiv:1712.07266] [INSPIRE].
A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].
B. Oblak, Characters of the BMS Group in Three Dimensions, Commun. Math. Phys. 340 (2015) 413 [arXiv:1502.03108] [INSPIRE].
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Wieland, W. Conformal boundary conditions, loop gravity and the continuum. J. High Energ. Phys. 2018, 89 (2018). https://doi.org/10.1007/JHEP10(2018)089
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DOI: https://doi.org/10.1007/JHEP10(2018)089