Abstract
It was recently shown that the homogeneous and isotropic cosmology of a massless scalar field coupled to general relativity exhibits a new hidden conformal invariance under Mobius transformation of the proper time, additionally to the invariance under time-reparamterization. The resulting Noether charges form a \( \mathfrak{sl}\left(2,\mathbb{R}\right) \) Lie algebra, which encapsulates the whole kinematics and dynamics of the geometry. This allows to map FLRW cosmology onto conformal mechanics and formulate quantum cosmology in CFT1 terms. Here, we show that this conformal structure is embedded in a larger \( \mathfrak{so} \)(3, 2) algebra of observables, which allows to present all the Dirac observables for the whole gravity plus matter sectors in a unified picture. Not only this allows one to quantize the system and its whole algebra of observables as a single irreducible representation of \( \mathfrak{so} \)(3, 2), but this also gives access to a scalar field operator \( \hat{\phi} \) opening the door to the inclusion of non-trivial potentials for the scalar field. As such, this extended conformal structure might allow to perform a group quantization of inflationary cosmological backgrounds.
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References
J. Ben Achour and E.R. Livine, Cosmology as a CFT1, JHEP 12 (2019) 031 [arXiv:1909.13390] [INSPIRE].
J. Ben Achour and E.R. Livine, Protected SL(2, ℝ) Symmetry in Quantum Cosmology, JCAP 09 (2019) 012 [arXiv:1904.06149] [INSPIRE].
J. Ben Achour and E.R. Livine, Polymer Quantum Cosmology: Lifting quantization ambiguities using a SL(2, ℝ) conformal symmetry, Phys. Rev. D 99 (2019) 126013 [arXiv:1806.09290] [INSPIRE].
J. Ben Achour and E.R. Livine, Thiemann complexifier in classical and quantum FLRW cosmology, Phys. Rev. D 96 (2017) 066025 [arXiv:1705.03772] [INSPIRE].
B. Pioline and A. Waldron, Quantum cosmology and conformal invariance, Phys. Rev. Lett. 90 (2003) 031302 [hep-th/0209044] [INSPIRE].
M. Bojowald, Harmonic cosmology: How much can we know about a universe before the big bang?, Proc. Roy. Soc. Land. A 464 (2008) 2135 [arXiv:0710.4919] [INSPIRE].
M. Bojowald, Dynamical coherent states and physical solutions of quantum cosmological bounces, Phys. Rev. D 75 (2007) 123512 [gr-qc /0703144] [INSPIRE].
J. Erdmenger, R. Meyer and J.-H. Park, Spacetime Emergence in the Robertson- Walker Universe from a Matrix model, Phys. Rev. Lett. 98 (2007) 261301 [arXiv:0705.1586] [INSPIRE].
B. Baytas, M. Bojowald and S. Crowe, Equivalence of models in loop quantum cosmology and group field theory, Universe 5 (2019) 41 [arXiv:1811.11156] [INSPIRE].
V. de Alfaro, S. Fubini and G. Furlan, Conformal Invariance in Quantum Mechanics, Nuovo Cim. A 34 (1976) 569 [INSPIRE].
C. Chamon, R. Jackiw, S.-Y. Pi and L. Santos, Conformal quantum mechanics as the CFT1 dual to AdS2, Phys. Lett. B 701 (2011) 503 [arXiv:1106.0726] [INSPIRE].
R. Jackiw and S.Y. Pi, Conformal Blocks for the 4-Point Function in Conformal Quantum Mechanics, Phys. Rev. D 86 (2012) 045017 [Erratum ibid. D 86 (2012) 089905] [arXiv:1205.0443] [INSPIRE].
C. Rovelli, Partial observables, Phys. Rev. D 65 (2002) 124013 [gr-qc/0110035] [INSPIRE].
B. Dittrich, Partial and complete observables for canonical general relativity, Class. Quant. Grav. 23 (2006) 6155 [gr-qc/0507106] [INSPIRE].
E.R. Livine and M. Martin-Benito, Group theoretical Quantization of Isotropic Loop Cosmology, Phys. Rev. D 85 (2012) 124052 [arXiv:1204.0539] [INSPIRE].
M. Dupuis, L. Freidel, E.R. Livine and S. Speziale, Holomorphic Lorentzian Simplicity Constraints, J. Math. Phys. 53 (2012) 032502 [arXiv:1107.5274] [INSPIRE].
J. Martin, C. Ringeval, R. Trotta and V. Vennin, The Best Inflationary Models After Planck, JCAP 03 (2014) 039 [arXiv:1312.3529] [INSPIRE].
J.E. Lidsey, Cosmology and the Korteweg-de Vries Equation, Phys. Rev. D 86 (2012) 123523 [arXiv:1205.5641] [INSPIRE].
J.E. Lidsey, Inflationary Cosmology, Diffeomorphism Group of the Line and Virasoro Coadjoint Orbits, arXiv:1802.09186 [INSPIRE].
A. Ashtekar and P. Singh, Loop Quantum Cosmology: A Status Report, Class. Quant. Grav. 28 (2011) 213001 [arXiv:1108.0893] [INSPIRE].
M. Bojowald, Loop quantum cosmology, Living Rev. Rel. 11 (2008) 4 [INSPIRE].
I. Agullo and P. Singh, Loop Quantum Cosmology, in Loop Quantum Gravity: The First 30 Years, A. Ashtekar and J. Pullin eds., World Scientific, New York U.S.A. (2017), pg. 183 [arXiv:1612.01236] [INSPIRE].
M. Bojowald, D. Brizuela, H.H. Hernandez, M.J. Koop and H.A. Morales-Tecotl, High-order quantum back-reaction and quantum cosmology with a positive cosmological constant, Phys. Rev. D 84 (2011) 043514 [arXiv:1011.3022] [INSPIRE].
D. Brizuela, A formalism based on moments for classical and quantum cosmology, J. Phys. Conf. Ser. 600 (2015) 012017.
D. Brizuela, Statistical moments for classical and quantum dynamics: formalism and generalized uncertainty relations, Phys. Rev. D 90 (2014) 085027 [arXiv:1410.5776] [INSPIRE].
A. Alonso-Serrano, M. Bojowald and D. Brizuela, Quantum approach to a Bianchi I singularity, arXiv:2001.11488 [INSPIRE].
D. Brizuela and U. Muniain, A moment approach to compute quantum-gravity effects in the primordial universe, JCAP 04 (2019) 016 [arXiv:1901.08391] [INSPIRE].
G. Lindblad and B. Nagel, Continuous bases for unitary irreducible representations of SU(1,1), Ann. Henri Poincaré 13 (1970) 27.
N. Bodendorfer and D. Wuhrer, Renormalisation with SU(1, 1) coherent states on the LQC Hilbert space, arXiv:1904.13269 [INSPIRE].
N. Bodendorfer and F. Haneder, Coarse graining as a representation change, Phys. Lett. B 792 (2019) 69 [arXiv:1811.02792] [INSPIRE].
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ArXiv ePrint: 2001.11807
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Achour, J.B., Livine, E.R. Conformal structure of FLRW cosmology: spinorial representation and the \( \mathfrak{so} \) (2, 3) algebra of observables. J. High Energ. Phys. 2020, 67 (2020). https://doi.org/10.1007/JHEP03(2020)067
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DOI: https://doi.org/10.1007/JHEP03(2020)067