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Partition Function for Quantum Gravity in 4 Dimensions As a 1/N Expansion Godwill Mbiti Kanyolo, Titus Masese

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Partition Function for Quantum Gravity in 4 Dimensions As a 1/N Expansion Godwill Mbiti Kanyolo, Titus Masese Partition function for quantum gravity in 4 dimensions as a 1/N expansion Godwill Mbiti Kanyolo, Titus Masese To cite this version: Godwill Mbiti Kanyolo, Titus Masese. Partition function for quantum gravity in 4 dimensions as a 1/N expansion. 2021. hal-03335930v2 HAL Id: hal-03335930 https://hal.archives-ouvertes.fr/hal-03335930v2 Preprint submitted on 7 Sep 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. 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Distributed under a Creative Commons Attribution - NonCommercial - NoDerivatives| 4.0 International License Partition function for quantum gravity in 4 dimensions as a 1/ expansion N Godwill Mbiti Kanyolo1, ∗ and Titus Masese2,3, † 1The University of Electro-Communications, Department of Engineering Science, 1-5-1 Chofugaoka, Chofu, Tokyo 182-8585 2Research Institute of Electrochemical Energy (RIECEN), National Institute of Advanced Industrial Science and Technology (AIST), 1-8-31 Midorigaoka, Ikeda, Osaka 563-8577, Japan 3AIST-Kyoto University Chemical Energy Materials Open Innovation Laboratory (ChEM-OIL), Yoshidahonmachi, Sakyo-ku, Kyoto-shi 606-8501, Japan Quantum gravity is the solution ascribed to rendering the geometric description of classi- cal gravity, in any dimensions, completely consistent with principles of quantum theory. The serendipitous theoretical discovery that black holes are thermodynamical objects that must participate in the second law has led to these purely gravitational objects to be dubbed, ‘the hydrogen atom for quantum gravity’, analogous to the atomic spectrum of hydrogen, effectively used by Neils Bohr and his contemporaries to successfully for- mulate quantum mechanics in the early 20th century. Here, we employ the temperature and entropy formulae describing Schwarzschild black holes to consider the emergence of Einstein Field Equations from a complex-Hermitian structure, [Ricci tensor √ 1 Yang-Mills field strength], where the gravitational degrees of freedom are the SU(± N−) colors with N Z 0 and a condensate comprising color pairs, k = N/2, appropri- ately coupled to∈ the≥ Yang-Mills gauge field. The foundations of our approach reveal the complex-Hermitian structure is analogous to Cayley-Dickson algebras, which aids in formulating the appropriate action principle for our formalism. The SU(N) gauge group is broken into an effective SU(4) SO(4) SO(1, 3) field theory on the tangent space with two terms: the Einstein-Hilbert→ action↔ and a Gauss-Bonnet topological term. Moreover, since topologically classifying all n = 4 (dimensional) Riemannian manifolds is not a clear-cut endeavor, we only consider the Euclidean path integral as the sum over manifolds with distinct topologies, h Z 0 homeomorphic to connected sums of an arbitrary number of n = 4-spheres and∈ h number≥ of n = 4-tori. Consequently, the par- tition function adopts a reminiscent form of the sum of the vacuum Feynman diagrams for a large = exp(βM/2) theory, provided = βM/2 = πN is the Schwarzschild black hole entropy,N β = 8πGM is the inverse temperature,S G is gravitational constant, M is the black hole mass and horizon area, = 2GβM = 4πGN is pixelated in units of 4πG. This leads us to conclude that theA partition function for quantum gravity is equivalent to the vacuum Feynman diagrams of a yet unidentified large theory in n = 4 dimensions. Our approach also sheds new light on the asymptoticN behavior of dark matter-dominated galaxy rotation curves (the empirical baryonic Tully-Fisher re- lation) and emergent gravity in condensed matter systems with defects such as layered materials with cationic vacancies as topological defects. CONTENTS 2. Defining the central mass 13 3. Constraint 2 14 I. Introduction 2 C. Lagrangian density 16 II. Notation and Nomenclature 4 IV. Results 18 A. Group scalars and vectors (tensors) 4 A. Generating all mass terms 18 B. Introduction to p forms 7 B. Quantum gravity 19 C. Group scalars and vectors (p forms) 10 C. 1/ expansion 19 N D. Topology 20 III. Theory 11 1. Old quantum condition 20 A. Motivation 11 2. Entropy as an adiabatic invariant 20 B. Equations of motion 12 3. Random acceleration 21 1. Constraint 1 12 4. Gauss-Bonnet theorem 22 5. Euler characteristic, χ4 = χ2 22 E. Average energy of manifolds 23 ∗ [email protected] V. Ramifications 23 † [email protected] A. Fermion/boson picture 23 2 B. Asymptotic behavior in galaxy rotation curves 25 are not entirely guaranteed to be more than an anal- C. Layered materials with cationic vacancies 27 ogy.(Dougherty and Callender, 2016) Nonetheless, consistency with well-tested ther- VI. Discussion 30 modynamical behavior(Isi et al., 2021), coupled Acknowledgements 32 with experiments using analogue gravitational systems(Jacquet et al., 2020; Steinhauer, 2014) offer a References 32 robust test for the correctness of the results, albeit with major unresolved problems such as the black hole infor- mation paradox(Almheiri et al., 2021; Hawking, 1976b) I. INTRODUCTION related to the apparent violation of information conser- vation in such systems(Hawking, 2005), and the nature The gravitational field in general relativity has of the black hole microstates.(Bekenstein, 2008) In par- a geometric description that, on one hand avails a ticular, according to the no-hair theorem(Thorne et al., myriad of visual cues which aid to build physicists’ 2000), a charged-rotating (Kerr-Newman) black hole intuitions on the nature of space-time(Thorne et al., is completely characterized by its mass (M), charge 2000), but on the other hand, notoriously does not lend (Q) and angular momentum (L), implying that black itself to renormalization.(Doboszewski and Linnemann, holes are the ‘simplest’ objects in the universe, with no 2018; Feynman et al., 2018; Hamber, 2008; Shomer, other internal degrees of freedom. Moreover, Birkhoff’s 2007; Thorne et al., 2000; Zee, 2010) This has led theorem(Israel, 1967; Jebsen, 2005) guarantees that to theoretic descriptions of gravity as an emer- the simplest such black holes are static, uncharged and gent phenomenon garnering some traction(Jacobson, non-rotating (Schwarszchild) black holes. While these 1995; Kleinert, 1987; Konopka et al., 2008; Lee et al., theorems are consistent with classical general relativity, 2013; Markopoulou and Smolin, 2004; Oh et al., 2018; semi-classical approaches to black hole thermodynamics Padmanabhan, 2010; Swingle and Van Raamsdonk, pioneered by Hawking(Hawking, 1974, 1976a, 1975) 2014; Van Raamsdonk, 2010; Verlinde, 2011) over quan- introduce a quantum field theory in a fixed space-time tum field theoretic approaches since emergence appears background, leading to the serendipitous conclusion that to by-pass the need for renormalization.(Burgess, 2004; black holes not only are black bodies with a temperature, Linnemann and Visser, 2017) Arguments for a successful T = κ/2πkB proportional to their surface gravity, κ, quantum theory of gravity range from the need to under- and entropy, = /4G proportional to their surface S A stand the nature of cosmic inflation(Hartle and Hawking, area, , where G is the gravitational constant and kB 1983) and black holes(Hawking, 1976a,b, 2005), to aes- is Boltzmann’sA constant, but also pack the maximum thetic considerations in favor of the unification of gravity amount of information possible within a given space, with gauge symmetries in the Standard Model of particle often referred to as the Bekenstein bound.(Bekenstein, physics.(Ross, 2003) 2005; Bousso, 1999) The Bekenstein bound, coupled Employing a semi-classical approach, Bekenstein and with the no-hair theorem, already implies an apparent his contemporaries showed that black holes are thermo- contradiction, namely that the ‘simplest’ objects in the dynamical objects that must participate in the second universe simultaneously harbor the most amount of law.(Bardeen et al., 1973; Bekenstein, 1973; Hawking, information. 1974, 1975) As a consequence, some proponents of emer- However, the black hole information paradox is a gent gravity have argued that gravity in general relativ- far more insidious problem than the aforementioned ity may not be as fundamental as the other gauge the- apparent contradiction, since black hole entropy and ories after all, since Boltzmann’s investigations demon- temperature require the black hole to eventually evap- strated that the second law is statistical, as it emerges orate by Unruh-Hawking radiation.(Hawking, 1974; from underlying microscopic physics.(Jacobson, 1995; Unruh, 1977) In particular, the radiation is essentially Padmanabhan, 2010; Verlinde, 2011) Nonetheless, these a black body spectrum in a mixed quantum state. considerations have led to black holes being dubbed, ‘the Whence, according to the semi-classical approach, hydrogen atom for quantum gravity’(Corda and Feleppa, quantum objects in a pure quantum state which fall 2019; ’t Hooft, 2016), setting up black hole thermody- into a black hole will eventually escape as Unruh- namics as an indispensable tool, analogous to the atomic Hawking radiation in a mixed state via particle-pair spectrum of hydrogen, effectively used by Neils Bohr
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