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Prospects for Infrared Quantum Gravity: from Cosmology to Black Holes

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Prospects for Infrared Quantum Gravity: from Cosmology to Black Holes University of Massachusetts Amherst ScholarWorks@UMass Amherst Doctoral Dissertations Dissertations and Theses November 2016 Prospects for Infrared Quantum Gravity: From Cosmology to Black Holes Basem K. Mahmoud El-Menoufi University of Massachusetts Amherst Follow this and additional works at: https://scholarworks.umass.edu/dissertations_2 Part of the Elementary Particles and Fields and String Theory Commons, and the Quantum Physics Commons Recommended Citation Mahmoud El-Menoufi, Basem K., "Prospects for Infrared Quantum Gravity: From Cosmology to Black Holes" (2016). Doctoral Dissertations. 779. https://doi.org/10.7275/8609577.0 https://scholarworks.umass.edu/dissertations_2/779 This Open Access Dissertation is brought to you for free and open access by the Dissertations and Theses at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. PROSPECTS FOR INFRARED QUANTUM GRAVITY: FROM COSMOLOGY TO BLACK HOLES A Dissertation Presented by BASEM MAHMOUD EL-MENOUFI Submitted to the Graduate School of the University of Massachusetts Amherst in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY September 2016 Physics Department c Copyright by Basem Mahmoud El-Menoufi 2016 All Rights Reserved PROSPECTS FOR INFRARED QUANTUM GRAVITY: FROM COSMOLOGY TO BLACK HOLES A Dissertation Presented by BASEM MAHMOUD EL-MENOUFI Approved as to style and content by: John F. Donoghue, Chair Lorenzo Sorbo, Member Jennie Traschen, Member Houjun Mo, Member Rory Miskimen, Department Chair Physics Department I have not failed. I've just found 10,000 ways that won't work. ACKNOWLEDGMENTS First and for most, I thank my Lord Allah whose grace and mercy were the only reason I was able to complete this dissertation. I would also like to express my gratitude towards everyone who helped me in writing this dissertation. On the faculty side, I would like to thank Professor John F. Donoghue for his supervision during my Ph.D. Also, I am grateful to Professors Jennie Traschen, David Kastor, Lorenzo Sorbo, Barry Holstein, Eugene Golowich, Michael Ramsey-Musolf for our collaboration and their classroom instruction during various stages in my Ph.D. Last but not least, many thanks to Professor Houjun Mo for being a member of my committee. I am quite indebted to Gregory Ovanesyan, Ben Ett and Graham White for their collaboration and emotional support during a tough stage in my Ph.D. I would also like to thank my friends Dominique Cambou, Mohamed Anber, Selcuk Yassar, Sheema Rahmanseresht, Alphan Aksoyoglu, Peker Milas, Amir Azadi. Many thanks to the physics department staff Jane Knapp, Mary Ann Ryan, Kris- tine Reopell and Ingrid Pollard. v ABSTRACT PROSPECTS FOR INFRARED QUANTUM GRAVITY: FROM COSMOLOGY TO BLACK HOLES SEPTEMBER 2016 BASEM MAHMOUD EL-MENOUFI B.Sc., THE AMERICAN UNIVERSITY IN CAIRO M.Sc., UNIVERSITY OF MASSACHUSETTS AMHERST Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor John F. Donoghue Although perturbatively non-renormalizable, general relativity is a perfectly valid quantum theory at low energies. Treated as an effective field theory one is able to make genuine quantum predictions by applying the conventional rules of quantum field theory. The low energy degrees of freedom and couplings of quantum gravity are fully dictated by the symmetries of general relativity. To realize the full EFT treatment one has to supplement the theory with experimental input necessary to fix the Wilson coefficients of the most general Lagrangian. In spite of the fact that this is not feasible, one can still extract the leading quantum corrections which are precisely induced by the low-energy fluctuations of the massless graviton. The long- distance portion of loops is non-analytic in momentum space or equivalently non-local in position space. In this thesis we are going to study the construction, properties and phenomenology of these non-local effects. vi TABLE OF CONTENTS Page ACKNOWLEDGMENTS ............................................. v ABSTRACT .......................................................... vi LIST OF TABLES .................................................... xi LIST OF FIGURES.................................................. xii CHAPTER 1. INTRODUCTION ................................................. 1 2. QED TRACE ANOMALY, NON-LOCAL LAGRANGIANS AND QUANTUM EQUIVALENCE PRINCIPLE VIOLATIONS .................................................. 9 2.1 Introduction . .9 2.2 The background field method and the non-local effective action . 14 2.3 Including the energy momentum tensor in the effective action . 17 2.3.1 Renormalization . 22 2.3.2 Fermions and non-universality . 23 2.3.3 Position space effective action . 24 2.4 Conformal and scaling properties of the effective action . 26 2.5 The on-shell energy-momentum matrix element at one loop . 29 2.6 Gravity and a non-linear completion of the action . 31 2.7 Quantum equivalence principle violation . 33 2.8 Conclusion . 37 3. COVARIANT NON-LOCAL ACTION FOR MASSLESS QED AND THE CURVATURE EXPANSION ....................... 39 3.1 Introduction . 39 3.2 The problem of ln 2 ............................................. 41 vii 3.3 Covariant non-local actions: General remarks . 46 3.4 Non-linear completion: Expansion in the curvature . 50 3.5 The R2 action: The elusive logarithm . 52 3.5.1 Toy example: A scalar field . 53 3.5.2 2-forms . 55 3.6 The R3 action . 57 3.6.1 Terms including 1=r2 ...................................... 57 3.6.2 Terms including (log r2)=r2 ................................ 59 3.6.3 Counter-terms for the logarithm . 60 3.7 Remarks on the trace anomaly . 61 3.7.1 Weyl transformations . 62 3.7.2 Scale transformations . 66 3.8 Summary . 67 4. NON-LOCAL QUANTUM EFFECTS IN COSMOLOGY: QUANTUM MEMORY, NON-LOCAL FLRW EQUATIONS AND SINGULARITY AVOIDANCE .......................... 69 4.1 Introduction . 69 4.2 Perturbative analysis . 71 4.3 Causal behavior . 76 4.4 Non-linear completion and quasi-local form . 81 4.5 Non-local FLRW equations . 86 4.6 Emergence of classical behavior . 88 4.7 Contracting universe and the possibility of a bounce . 93 4.8 Summary . 99 5. INFLATIONARY MAGNETOGENESIS AND NON-LOCAL ACTIONS: THE CONFORMAL ANOMALY ................. 101 5.1 Introduction . 101 5.2 The non-local action . 104 5.3 The set-up . 107 5.4 Canonical Quantization . 111 5.4.1 Solving for the mode functions . 112 5.5 The magnetic field at the end of inflation . 115 5.6 The current magnetic field . 118 5.7 The lower bound on the IGM field . 121 viii 5.8 Summary and conclusions . 122 6. QUANTUM GRAVITY OF KERR-SCHILD SPACETIMES AND THE LOGARITHMIC CORRECTION TO SCHWARZSCHILD BLACK HOLE ENTROPY .............. 124 6.1 Introduction . 124 6.2 The heat kernel for the covariant d' Alembertian . 131 6.2.1 Lowest order . 134 6.2.2 Next-to-leading order . 135 6.2.3 Next-to-next-to-leading order . 136 6.2.4 A brief comment on the result . 137 6.3 The curvature expansion . 137 6.3.1 The heat kernel at zeroth order . 138 6.3.2 The heat kernel at linear order . 139 6.3.3 The heat kernel at quadratic order. 140 6.3.3.1 Fixing the form factors . 142 6.3.4 Comments on the form factors . 144 6.4 The effective action . 144 6.4.1 The action at zeroth order . 145 6.4.2 The action at linear order . 146 6.4.3 The action at quadratic order . 148 6.4.4 The total action and renormalization . 149 6.5 The partition function and entropy . 151 6.5.1 Schwarzschild black hole . 153 6.5.2 Dimensional transmutation and final remarks . 156 6.6 Future outlook . 157 APPENDICES A. SCALE CURRENTS ............................................ 161 B. REDUCTION OF THE TRIANGLE AND BUBBLE INTEGRALS ................................................. 164 C. MOMENTUM SPACE REPRESENTATION .................... 166 D. ASPECTS OF THE IN-IN FORMALISM ........................ 168 ix E. KERR-SHILD SPACETIMES .................................... 171 F. HEAT KERNEL ................................................. 176 G. USEFUL IDENTITIES .......................................... 181 BIBLIOGRAPHY .................................................. 183 x LIST OF TABLES Table Page 4.1 Coefficients of different fields. All numbers should be divided by 11520π2..................................................... 85 6.1 The coefficients appearing in the effective action due to massless fields of various spins. All numbers are divided by 11520π2........ 150 xi LIST OF FIGURES Figure Page 2.1 Triangle diagram. 19 2.2 Bubble diagrams. 19 2.3 External leg corrections relevant for the matrix element. 30 2.4 Gravitational scattering of a photon off a static massive target. The diagram on the top is the tree level process, while the square in the bottom diagram represents the non-local effects. 35 4.1 The evolution of the scale factor and its time derivative in an expanding dust-filled universe for N=10. 90 4.2 The evolution of the scale factor and its time derivative in an expanding dust-filled universe for N=100. 91 4.3 The evolution of the scale factor and its time derivative in an expanding dust-filled universe for N=1000. 91 4.4 The evolution of the scale factor and its time derivative in an expanding.
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