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Two Quantum Effects in the Theory of Gravitation

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Two Quantum Effects in the Theory of Gravitation Two Quantum Effects in the Theory of Gravitation by Sean Patrick Robinson S.B., Physics, Massachusetts Institute of Technology (1999) Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2005 © Sean Patrick Robinson, MMV. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. MASSCHUSETSSE OF TECHNOLOGY JUN 0 7 2005 Au th or ............................. ....-............. ...............LIBRARIES Department of Physics May 19, 2005 Certified by -/ 'I C ertifi ed by......... .............. -... ............................. Frank Wilczek Herman Feshbach Professor of Physics Thesis Supervisor ~~~~~ ~~~~/ A ccepted by ........................... Professr Thom/Greytak Associate Department Head fo ducation Two Quantum Effects in the Theory of Gravitation by Sean Patrick Robinson Submitted to the Department of Physics on May 19, 2005, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Abstract We will discuss two methods by which the formalism of quantum field theory can be included in calculating the physical effects of gravitation. In the first of these, the consequences of treating general relativity as an effective quantum field theory will be examined. The primary result will be the calculation of the first-order quantum gravity corrections to the /3 functions of arbitrary Yang-Mills theories. These correc- tions will effect the high-energy phenomenology of such theories, including the details of coupling constant unification. Following this, we will address the question of how to form effective quantum field theories in classical gravitational backgrounds. We follow the prescription that effective theories should provide a description of exper- imentally accessible degrees of freedom with all other degrees of freedom integrated out of the theory. We will show that this prescription appears to fail for a scalar field in a black hole background because of an anomaly generated in general covariance at the black hole horizon. This anomaly is repaired and the effective field theory is saved, however, by the inevitable presence of Hawking radiation in the quantum theory. Thesis Supervisor: Frank Wilczek Title: Herman Feshbach Professor of Physics 3 4 __I__ Acknowledgments The following body of work has benefited from the input of many individuals. Obvious among these is my thesis advisor, Frank Wilczek. I would like to thank the Nobel Foundation for making this an interesting year to write a thesis. I also need to recognize the other members of my thesis committee, Eddie Farhi and Roman Jackiw. Others who made significant contributions to the development of this work, but are not specifically cited within, include Brett Altschul, Ted Baltz, Serkan Cabi, Qudsia Ejaz, Ian Ellwood, Michael Forbes, Brian Fore, Vishesh Khemani, Joydip Kundu, Vivek Mohta, Brain Patt, Dru Renner, Jessie Shelton, and Ari Turner. I especially thank Michael Forbes for reading an early draft of this thesis. Finally, I would like to acknowledge the exceptional support and motivation provided by my wife, daughter, and parents, the importance of which cannot be overstated. In celebration of the completion of this thesis, I compose the following cautionary limerick 1 : When working with quantum gravity, There's not much that is easy to see. The math's so opaque that it's easy to make an occasionalerror, or three. This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative research agreement DE-FC02-94ER40818. 'Incidentally, Speak in limerick? Well, maybe I did. But if I did, it was only to kid. Speaking in limerick is sort of a gimmick behind which real intentions are hid. 5 6 _··^I_________ Contents 1 Overture 13 1.1 Quantum General Relativity and Yang-MillsTheory. 14 1.2 Black Holes and Effective Field Theory ......... 17 2 Gravitational Corrections to Yang-Mills /3 Functions 23 2.1 Introduction . .. 23 2.1.1 One-loop Divergences ....................... 26 2.1.2 Asymptotic Safety ....................... .......... 27 2.2 Technical Preliminaries ......................... .......... 28 2.2.1 Background Field Theory ................... ......... 28 2.2.2 Definition of Newton's Constant . .... 29 2.3 Setup .............................................. 31 2.4 Expanding the Action ......................... .......... 32 2.4.1 Expanding the Non-Polynomial Terms . ... 32 2.4.2 Expanding the Einstein-Hilbert Action ............. 33 2.4.2.1 Curvature with Background Derivatives . .... 33 2.4.2.2 Some Useful Definitions and Identities ....... .. 34 2.4.2.3 Expansion of Curvature . .............. 34 2.4.3 Expanding the Yang-Mills Action . ...... 36 2.5 Gauge-Fixing ............................... 36 2.6 Combining the Pieces ........................... 38 2.7 Compiling the Superfield ......................... 41 2.8 Renormalization ........................................... 43 2.8.1 Computation of Functional Determinants ........... ..... 46 2.8.2 Extracting the Function . ...... 49 2.9 Enlarging the Matter Sector and the Gauge Group ............. 50 2.10 Coupling Constant Unification ............................... 53 2.11 Phenomenology . ............ 54 2.12 Commentary . ............ 57 3 Black Hole Effective Field Theory 59 3.1 Introduction ...................... 59 . 3.1.1 Hawking Radiation .............. 59 3.1.2 Anomalies and Anomaly Driven Currents . 60 3.1.3 Hawking Radiation and the Conformal Anomaly 61 7 3.1.4 Effective Field Theory Framework . 62 3.2 Spacetime Prel iminaries ......................... 63 3.2.1 Sphericaal Static Metrics ..................... 63 3.2.1.1 Einstein's Equation ................... 65 3.2.1.2 Horizon Structure . ................. 67 3.2.2 Kruskal Extension ................................ 69 3.2.2.1 The Quantum Vacua ................. ....... 74 3.2.2.2 Euclidean Section ............................ 75 3.2.3 Wave Equation ................................. 75 3.2.3.1 Spherical Harmonics ................. ....... 77 3.2.3.2 Radial Wave Equation . ............... 79 3.2.3.3 Near-Horizon Action . ................ 80 3.3 Thermal Radiation .................................. 82 3.3.1 Hyperci ubic Blackbody Cavity ................. ....... 83 3.3.2 Flux Versus Energy Density ................... 86 3.3.3 Sphericaal Blackbody Cavity ................... 88 3.3.3.1 Radial Mode Density ................. ....... 90 3.3.3.2 Spectral Densities ................... 92 3.4 Calculation . 95 3.5 Commentary . 98 3.6 Blackbody Spe ctrum from an Enhanced Symmetry? ......... ... 99 4 Finale 101 4.1 Summary ....................... 101 . 4.2 Open Possibilities ................... 101 4.3 Conclusion. ...................... 102 8 List of Figures 2-1 Feynman diagrams for two typical processes contributing to the renor- malization of a Yang-Mills coupling at one-loop ........... 25 2-2 The schematic Feynman diagram represented by the functional trace -TI[Mh]. A momentum p circulates in a virtual graviton loop coupled to external gluons of momentum k .................... 46 2-3 The schematic Feynman diagram represented by the functional trace -Tr[N]. A momentum p circulates in a virtual gluon loop coupled to external gluons of momentum k ...................... 47 2-4 The schematic Feynman diagram represented by the functional trace 1Tr[O+O_]. A momentum p circulates in a virtual gluon-graviton loop coupled to external gluons of momentum k .............. ...... 48 2-5 In Figure 2-5(a), the three Yang-Mills couplings of an MSSM-like the- - 2 ory evolve as straight lines in a plot of r ' =_ 47r/g versus log10 (E) when gravitation is ignored. The initial values at Mzo - 100 GeV are set so that the lines approximately intersect at 1016 GeV. When gravity is included at one-loop, the three lines curve towards weaker coupling at high energy, but remain unified near 1016 GeV. In Figure 2-5(b), g is plotted for the same theory. All three couplings rapidly go to zero near Mp, rendering the theory approximately free above this scale ................................................ 55 3-1 Part of the causal diagram of a black hole spacetime, with inset detail of a region near the horizon ................................ 63 3-2 Sketches of three integrated mass functions and their associated h(r). In 3-2(a) the matter distribution is relatively smooth and vanishes at the origin, as in a normal star. In 3-2(b) the matter has a density singularity at the origin, but is otherwise well behaved. In 3-2(c) a potentially difficult-to-analyze situation is sketched ........... 70 3-3 In 3-3(a) typical profiles for the functions h(r) and f(r) are sketched for an asymptotically flat black hole spacetime. The horizon occurs where the functions vanish at r = rh. In 3-3(b) the corresponding profile of r is sketched along with the line r = r. Note that r diverges logarithmically at rh and approaches r at large r ............ 71 9 3-4 A sketch of a typical effective radial scattering potential. The potential for any metric qualitatively similar to the one sketched in Figure 3-3 will be qualitatively similar to the one sketched here for > 0 and d > 3. The potential falls off exponentially for negative r and as is typically dominated by the centrifugal term at large r,. r, which falls off as 1/r 2 . ......... 81 3-5 The thermal integral I(a) defined in Equation
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