Abstract
Recently there has been a surge of interest in studying Lorentzian quant urn cosmology using Picard-Lefschetz methods. The present paper aims to explore the Lorentzian path-integral of Gauss-Bonnet gravity in four spacetime dimensions with metric as the field variable. We employ mini-superspace approximation and study the variational problem exploring different boundary conditions. It is seen that for mixed boundary conditions non-trivial effects arise from Gauss-Bonnet sector of gravity leading to additional saddle points for lapse in some case. As an application of this we consider the No-boundary proposal of the Universe with two different settings of boundary conditions) and compute the transition amplitude using Picard-Lefschetz formalism. In first case the transition amplitude is a superposition of a Lorentzian and a Euclidean geometrical configuration leading to interference incorporating non-perturbative effects coming from Gauss-Bonnet sector of gravity. In the second case involving complex initial momentum we note that the transition amplitude is an analogue of Hartle-Hawking wave-function with non-perturbative correction coming from Gauss-Bonnet sector of gravity.
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G. ’t Hooft and M.J.G. Veltman, One loop divergencies in the theory of gravitation, Ann. Henri PoincareÁ 20 (1974) 69.
S. Deser, H.-S. Tsao and P. van Nieuwenhuizen, Nonrenormalizability of Einstein Yang-Mills Interactions at the One Loop Lev el, Phys. Lett. B 50 (1974) 491 [INSPIRE].
S. Deser and P. van Nieuwenhuizen, One Loop Divergences of Quantized Einstein-Maxwell Fields, Phys. Rev. D 10 (1974) 401 [INSPIRE].
S. Deser and P. van Nieuwenhuizen, Nonrenormalizability of the Quantized Dirac-Einstein System, Phys. Rev. D 10 (1974) 411 [INSPIRE].
M.H. Goroff and A. Sagnotti, Quantum gravity at two loops, Phys. Lett. B 160 (1985) 81 [INSPIRE].
M.H. Goroff and A. Sagnotti, The Ultraviolet Behavior of Einstein Gravity , Nucl. Phys. B 266 (1986) 709 [INSPIRE].
A.E.M. van de Ven, Two loop quantum gravity , Nucl. Phys. B 378 (1992) 309 [INSPIRE].
K.S. Stelle, Renormalization of Higher Derivative Quantum Gravity, Phys. Rev. D 16 (1977) 953 [INSPIRE].
A. Salam and J.A. Strathdee, Remarks on High- energy Stability and Renormalizability of Gravity Theory, Phys. Rev. D 18 (1978) 4480 [INSPIRE].
J. Julve and M. Tonin, Quantum Gravity with Higher Derivative Terms, Nuovo Cim. B 46 (1978) 137 [INSPIRE].
G. Narain and R. Anishetty, Short Distance Freedom of Quantum Gravity, Phys. Lett. B 711 (2012) 128 [arXiv: 1109. 3981] [INSPIRE].
G. Narain and R. Anishetty, Unitary and Renormalizable Theory of Higher Derivative Gravity, J. Phys. Conf. Ser. 405 (2012) 012024 [arXiv:1210. 0513] [INSPIRE].
G. Narain, Signs and Stability in Higher-Derivative Gravity, Int. J. Mod. Phys. A 33 (2018) 1850031 [arXiv: 1704 . 05031] [INSPIRE].
G. Narain, Exorcising Ghosts in Induced Gravity, Eur. Phys. J. C 77 (2017) 683 [arXiv: 1612 .04930] [INSPIRE].
A. Codello and R. Percacci, Fixed points of higher derivative gravity, Phys. Rev. Lett. 97 (2006) 221301 [hep-th/0607128] [INSPIRE].
M.R. Niedermaier, Gravitational Fixed Points from Perturbation Theory, Phys. Rev. Lett. 103 (2009) 101303 [INSPIRE].
A. Salvio and A. Strumia, Agravity, JHEP 06 (2014) 080 [arXiv: 1403. 4226] [INSPIRE].
D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498 [INSPIRE].
D. Lovelock, The four-dimensionality of space and the Einstein tensor, J. Math. Phys. 13 (1972) 874 [INSPIRE].
C. Lanczos, A Remarkabl e property of the Riemann-Christoffel tensor in four dimensions, Annals M ath. 39 (1938) 842 [INSPIRE].
J. York, Boundary terms in the action principles of general relativity, Found. Phys. 16 (1986) 249 [INSPIRE].
J.D. Brown and J.W. York, Jr., The Microcanonical functional integral. 1. The Gravitational field, Phys. Rev . D 47 (1993) 1420 [gr-qc/9209014] [INSPIRE].
C. Krishnan and A. Raju, A Neumann Boundary Term for Gravit y, Mod. Phys. Lett. A 32 (2017) 1750077 [arXiv:1605 . 01603] [INSPIRE].
E. Witten, A Note On Boundary Conditions In Euclidean Gravity, arXiv: 1805 . 11559 [INSPIRE].
C. Krishnan, S. Maheshwari and P.N. Bala Subramanian, Robin Gravit y, J. Phys. Conf. Ser. 883 (2017) 012011 [arXiv:1702. 01429] [INSPIRE].
I.A. Batalin and G.A. Vilkovisky, Relativistic S Matrix of Dynamical Systems with Boson and Fermion Constraints, Phys. Lett. B 69 (1977) 309 [INSPIRE].
C. Teitelboim, Quantum Mechanics of the Gravitational Field, Phys. Rev. D 25 (1982) 3159 [INSPIRE].
C. Teitelboim, The Proper Time Gauge in Quantum Theory of Gravitation, Phys. Rev. D 28 (1983) 297 [INSPIRE].
J.J. Halliwell, Derivation of the Wheeler-De Witt Equation from a Path Integral for Minisuperspace Models, Phys. Rev . D 38 (1988) 2468 [INSPIRE].
C. Teitelboim, Causality Versus Gauge Invariance in Quantum Gravity and Supergravity, Phys. Rev. Lett. 50 (1983) 705 [INSPIRE].
G.W. Gibbons, S.W. Hawking and M.J. Perry, Path Integrals and the Indefiniteness of the Gravitational Action, Nucl. Phys. B 138 (1978) 141 [INSPIRE].
P. Candelas and D.J. Raine, Feynman Propagator in Curved Space- Time, Phys. Rev. D 15 (1977) 1494 [INSPIRE].
M. Visser, How to Wick rotate generic curved spacetime, arXiv:1702.05572 [INSPIRE].
A. Baldazzi, R. Percacci and V. Skrinjar, Quantum fields without Wick rotation, Symmetry 11 (2019) 373 [arXiv: 1901.01891] [INSPIRE].
A. Baldazzi, R. Percacci and V. Skrinjar, Wicked metrics, Class. Quant. Grav. 36 (2019) 105008 [arXiv:1811.03369] [INSPIRE].
S.W. Hawking, The Boundary Conditions of the Universe, Pontif. Acad. Sci. Scr. Varia 48 (1982) 563, PRINT-82-0179, Cambridge U.K. (1982).
J.B. Hartle and S.W. Hawking, Wave Function of the Universe, Adv. Ser. Astrophys. Cosmol. 3 (1987) 174 [Phys. Rev. D 28 (1983) 2960] [INSPIRE].
J. Feldbrugge, J.-L. Lehners and N. Turok, Lorentzian Quantum Cosmology, Phys. Rev. D 95 (2017) 103508 [arXiv:1703.02076] [INSPIRE].
J. Feldbrugge, J.-L. Lehners and N. Turok, No smooth beginning for spacetime, Phys. Rev. Lett. 119 (2017) 171301 [arXiv:1705. 00192] [INSPIRE].
J. Feldbrugge, J.-L. Lehners and N. Turok, No rescue for the no boundary proposal: Pointers to the future of quantum cosmology, Phys. Rev. D 97 (2018) 023509 [arXiv: 1708. 05104] [INSPIRE].
A. Vilenkin, Creation of Universes from Nothing, Phys. Lett. B 117 (1982) 25 [INSPIRE].
A. Vilenkin, The Birth of Inflationary Universes, Phys. Rev. D 27 (1983) 2848 [INSPIRE].
A. Vilenkin, Quantum Creation of Universes, Phys. Rev. D 30 (1984) 509 [INSPIRE].
S.W. Hawking, The Quantum State of the Universe, Adv. Ser. Astrophys. Cosmol. 3 (1987) 236 [Nucl. Phys. B 239 (1984) 257] [INSPIRE].
G.W. Gibbons, The Einstein Action of Riemannian Metrics and Its Relation to Quantum Gravity and Thermodynamics, Phys. Lett. A 61 (1977) 3 [INSPIRE].
J.J. Halliwell and J. 1ouko, Steepest Descent Contours in the Path Integral Approach to Quantum Cosmology. 1. The de Sitter Minisuperspace Model, Phys. Rev. D 39 (1989) 2206 [INSPIRE].
J.J. Halliwell and J.B. Hartle, Integration Contours for the No Boundary Wave Function of the Universe, Phys. Rev. D 41 (1990) 1815 [INSPIRE].
J.J. Halliwell and J.B. Hartle, Wave functions constructed from an invariant sum over histories satisfy constraints, Phys. Rev. D 43 (1991) 1170 [INSPIRE].
A. Di Tucci and J.-L. Lehners, No-Boundary Proposal as a Path Integral with Robin Boundary Conditions, Phys. Rev. Lett. 122 (2019) 201302 [arXiv: 1903.06757] [INSPIRE].
A. Di Tucci, J.-L. Lehners and L. Sberna, No-boundary prescriptions in Lorentzian quantum cosmology, Phys. Rev . D 100 (2019) 123543 [arXiv: 1911. 06701] [INSPIRE].
N. Deruelle and L. Farina-Busto, The Lovelock Gravitational Field Equations in Cosmology, Phys. Rev. D 41 (1990) 3696 [INSPIRE].
F.R. Tangherlini, Schwarzschild field in n dimensions and the dimensionality of space problem, Nuovo Cim. 27 (1963) 636 [INSPIRE].
F. Tangherlini, Dimensionality of Space and the Pulsating Universe, Nuovo Cim. B 91 (1986) 209.
A. Di Tucci, M.P. Heller and J.-L. Lehners, Lessons for quantum cosmology from anti-de Sitter black holes, Phys. Rev. D 102 (2020) 086011 [arXiv: 2007 .04872] [INSPIRE].
E. Witten, Analytic Continuation Of Chern-Simons Theory, AMS/IP Stud. Adv. Math. 50 (2011) 347 [arXiv:1001. 2933] [INSPIRE].
E. Witten, A New Look At The Path Integral Of Quantum Mechanics, arXiv: 1009.6032 [INSPIRE].
G. Basar, G.V. Dunne and M. Ünsal, Resurgence theory, ghost-instantons, and analytic continuation of path integrals, JHEP 10 (2013) 041 [arXiv: 1308 .1108] [INSPIRE].
Y. Tanizaki and T. Koike, Real-time Feynman path integral with Picard-Le fschetz theory and its applications to quantum tunneling, Annals Phys. 351 (2014) 250 [arXiv:1406. 2386] [INSPIRE].
J.J. Halliwell and S.W. Hawking, The Origin of Structure in the Universe, Adv. Ser. Astrophys. Cosmol. 3 (1987) 277 [Phys. Rev. D 31 (1985) 1777] [INSPIRE].
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Narain, G. On Gauss-Bonnet gravity and boundary conditions in Lorentzian path-integral quantization. J. High Energ. Phys. 2021, 273 (2021). https://doi.org/10.1007/JHEP05(2021)273
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DOI: https://doi.org/10.1007/JHEP05(2021)273