DOCSLIB.ORG

Constraints on Cosmology and Quantum Gravity from Quantum Mechanics and Quantum Field Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Constraints on Cosmology and Quantum Gravity from Quantum Mechanics and Quantum Field Theory Constraints on Cosmology and Quantum Gravity from Quantum Mechanics and Quantum Field Theory Thesis by Jason Pollack In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2017 Defended May 19, 2017 ii c 2017 Jason Pollack ORCID: 0000-0003-4754-4905 All rights reserved iii Acknowledgments Many individuals and groups of people helped shape my time at Caltech: • The Friday night dinner/drinks/D&D group: Dustin Anderson, Kevin Barkett, Tony Bartolotta, Jonathan Blackman, Daniel Brooks, and Jeremy Brouillet. • The Sunday night TV-watching group: Joe Lozier, Dan Pershey, and Paul Plucinsky. • My parents, for their continuing support of my pursuit of academia, despite the transformation over the last five years of even the titles of my papers into incomprehensibility. • My fellow Ph 20/21/22 TAs: Casey Handmer, Jenia Mozgunov, and Masha Okounkova; and • The Ph 20/21/22 instructors, Alan Weinstein and Tom Prince, who were always willing to let us do our own thing and practically run the course, making it both great fun and great experience to TA. • Senior grad students: Tristan Mckinney, Ingmar Saberi, Chia Hsien Shen, and especially Kim Boddy. • Junior grad students: Charles Cao, Aidan Chatwin-Davies, and Ashmeet Singh. • Grad students in my year: Enrico Herrmann, Nick Hunter-Jones, Murat Kologlu, Grant Remmen, and especially my officemate, Tony Bartolotta. • Postdocs: Stefan Leichenauer in my first years at Caltech and Ning Bao more recently, mentors, inspirations, and dispensers of invaluable advice on how to survive grad school and take the next steps forward into academia. • Cliff Cheung and Mark Wise, for sharing some of their vast knowledge of quan- tum field theory with me over the years, and for always having their doors open for me to wander in and ask a question when I was stuck. iv • The Cosmological Quantum Mechanics reading group, the impetus for the ideas for a number of the papers presented herein, and also responsible for the on- going development of what seems to be a unique and fruitful set of ideas on how to think about the emergence of spacetime and structure from the Hilbert space. I’m glad to have been a part of it from the beginning, and I wouldn’t be surprised if fifty years from now we’re all writing up our recollections of its its first steps. • Everyone who participated in our many wide-ranging lunch conversations, some of them even about physics, at the tables outside of Chandler and the bookstore over the last five years. We never succeeded in solving the cosmological constant problem once and for all, but it was still worth trying. • Everyone involved with the Caltech GSC, who let me progress from wandering into a meeting lured by free food in 2013 to chairing Caltech’s grad student government in 2015-6, and discover in the process that I both enjoyed and was good at a whole set of skills and subjects I’d never contemplated before: – The wonderful faculty, deans, administrators, and staff I had the oppor- tunity to work with and even to select, and who encouraged me rather than saying I was crazy when I just showed up and started doing things. In the grad office and Caltech adminstration: Felicia Hunt, Kate McAn- ulty, Doug Rees, Tom Rosenbaum, Joe Shepherd, and Cindy Weinstein. In Physics and PMA: Olga Batygin, Cliff Cheung, Fiona Harrison, Lynne Hillenbrand, Sofie Leon, and Gil Refael. I was privileged to be a part of the committees that picked Doug, Kate, and Sofie for their positions, and so I get to claim a (very) small part of the credit for all of the positive change they’ve created, and will continue to create, at Caltech! – GSC Steering buddies: Sunita Darbe, Henry Ngo, Andrew Robbins, Emily Blythe, Natalie Higgins, Gina Duggan. – Finally, especially, Alicia Lanz and Allison Strom. I’m so, so, grateful for your friendship over these last two years, and for all the great conversations and shenanigans it entailed. I wish it had started earlier, and I look forward to it continuing for a long, long time! • All of my collaborators, especially: v – Paul Steinhardt and David Spergel, who helped me bridge the gap from Princeton to Caltech and were endlessly patient and encouraging coau- thors on a paper that took much longer to come to fruition than it should have (entirely my fault!) but which I’m very glad I persisted with; – Stefan Leichenauer, for guiding me through my first paper and pushing me out of my comfort zone to explore new fields; – Kim Boddy, for permitting me to jump on board with her collaboration with Sean, enduring endless stupid arguments about grammar, serving as an inspiration that there really was life after Caltech, and sticking with the collaboration even once she’d moved to Hawai’i; – Ning Bao, for coming in to the Cosmo QM group and catalyzing an incred- ible amount of new research, for always pushing to take the half-formed idea we’d come up with at lunch or in group meeting and turn it into a paper, for endless irrelevant and irreverent comments to keep me on my toes, and for that whole year you’d sneak into my office and stand up the Superman flash drive on my desk; – Grant Remmen, for being just as obsessed with grammar and punctuation as I am (much to Ning’s chagrin), for being the resident GR expert while still knowing how to translate crazy differential geometry and incredibly confusing terminology into something I could get physical intuition for, and for inadvertently getting me involved with the GSC; – Aidan Chatwin-Davies, for putting up with me barging into his office at random times with bizarre questions, for reporting on Sean’s whereabouts, for suggestions for paper titles I’m sad we never used, for giving me inside info on Vancouver and UBC, not to mention Japan, for all the gaming hilarity, and for having way too many ways for us to get in touch with each other (I count 9); – Tony Bartolotta, for similarly letting me snipe his collaboration, being willing to put in the time to buckle down and work through the calcu- lations when the rest of us made excuses, constantly letting me bounce ideas and questions off of him, from notation to formatting to rants about QFT in curved space—and, no less importantly, for letting me dragoon him into DMing, for the endless chats about the craziness of American politics, and for putting up with me teasing him endlessly about that time vi he glued himself to the printer and that time he baked his laptop in the oven; • And, of course, my advisor, Sean Carroll. From the time I wandered into his office before the beginning of my first year and was immediately ushered into a collaboration with him and Stefan, through what must have been dozens of hours of meetings with him and Kim where we hashed out just what the late- time structure of de Sitter space and its derivatives looked like, to his unabating support of my career as I applied for postdocs, I couldn’t have asked for a more supportive and collaborative academic partner and mentor. That’s not even to mention the many speaker dinners, or Cosmo QM, or his patient toleration of my decision to become GSC chair. As I continue on with my career to UBC and wherever comes after that, Sean will be the model I’ll always try to emulate for how to collaborate, how to advise, and how to do physics. vii Abstract Typical cosmological states have structure, obey to very good approximation the laws of classical physics on large scales, and are far from equilibrium. Typical quantum- mechanical states have none of these properties. If the universe is described by a state in a Hilbert space, the state and its Hilbert space must therefore obey a number of constraints to describe realistic cosmological spacetimes. In particular, they must admit a quantum-to-classical transition via decoherence that allows for the emergence of classical spacetimes, and such spacetimes must obey gravitational constraints, in particular on the entanglement entropy of subsystems within them. The papers collected in this thesis are concerned with these constraints. We investigate two holographic correspondences inspired by AdS/CFT, the AdS-MERA correspondence, which suggests that anti-de Sitter space may be given a discretized description as a tensor network, and the ER=EPR duality, which identified entangled qubits with wormholes connecting them. In the former case, we use holographic entropy bounds to severely constrain the properties of any such tensor network; in the latter case we prove a new general-relativistic area theorem which states that an area corresponding to the entanglement entropy in wormhole geometries is exactly conserved. We use information-theoretic constraints to show that under mild assumptions about the black hole interior an observer falling beyond the horizon is unable to verify the claimed cloning of information in the firewall paradox before reaching the singularity. Finally, we analyze the decoherence structures of late-time de Sitter space and early- time slow-roll eternal inflation. We show that in the former case a universe with an infinite-dimensional Hilbert space and a positive cosmological constant inevitably reaches a maximum-entropy state from which no further branching or decoherence is possible, forbidding the existence of dynamical quantum fluctuations at late time. In the latter case, gravitational-strength interaction among inflaton modes leads to decoherence of sufficiently super-Hubble modes, which we argue backreacts to cause different histories of cosmological evolution on different branches and hence creates the conditions necessary for eternal inflation.
Load more

Recommended publications
  • Estimating Remaining Lifetime of Humanity Abstract 1. Introduction
    Estimating remaining lifetime of humanity Yigal Gurevich [email protected] Abstract In this paper, we estimate the remaining time for human existence, applying the Doomsday argument and the Strong Self-Sampling Assumption to the reference class consisting of all members of the Homo sapiens, formulating calculations in traditional demographic terms of population and time, using the theory of parameter estimation and available paleodemographic data. The remaining lifetime estimate is found to be 170 years, and the probability of extinction in the coming year is estimated as 0.43%. 1. Introduction Modern humans, Homo sapiens, exist according to available data for at least 130,000 years [4], [5]. It is interesting, however, to estimate the time remaining for the survival of humanity. To determine this value there was proposed the so-called doomsday argument [1] - probabilistic reasoning that predicts the future of the human species, given only an estimate of the total number of humans born so far. This method was first proposed by Brandon Carter in 1983 [2]. Nick Bostrom modified the method by formulating the Strong Self-Sampling Assumption (SSSA): each observer-moment should reason as if it were randomly selected from the class of all observer-moments in its reference class. [3]. In this paper, we apply the SSSA method to the reference class consisting of all members of our species, formulating calculations in traditional demographic terms of population and time, using the parameter estimation theory and the available paleodemographic data. 1 To estimate the remaining time t we will fulfill the assumption that the observer has an equal chance to be anyone at any time.
    [Show full text]
  • Anthropic Measure of Hominid (And Other) Terrestrials. Brandon Carter Luth, Observatoire De Paris-Meudon
    Anthropic measure of hominid (and other) terrestrials. Brandon Carter LuTh, Observatoire de Paris-Meudon. Provisional draft, April 2011. Abstract. According to the (weak) anthropic principle, the a priori proba- bility per unit time of finding oneself to be a member of a particular popu- lation is proportional to the number of individuals in that population, mul- tiplied by an anthropic quotient that is normalised to unity in the ordinary (average adult) human case. This quotient might exceed unity for conceiv- able superhuman extraterrestrials, but it should presumably be smaller for our terrestrial anthropoid relations, such as chimpanzees now and our pre- Neanderthal ancestors in the past. The (ethically relevant) question of how much smaller can be addressed by invoking the anthropic finitude argument, using Bayesian reasonning, whereby it is implausible a posteriori that the total anthropic measure should greatly exceed the measure of the privileged subset to which we happen to belong, as members of a global civilisation that has (recently) entered a climactic phase with a timescale of demographic expansion and technical development short compared with a breeding gen- eration. As well as “economist’s dream” scenarios with continual growth, this finitude argument also excludes “ecologist’s dream” scenarios with long term stabilisation at some permanently sustainable level, but it it does not imply the inevitability of a sudden “doomsday” cut-off. A less catastrophic likelihood is for the population to decline gradually, after passing smoothly through a peak value that is accounted for here as roughly the information content ≈ 1010 of our genome. The finitude requirement limits not just the future but also the past, of which the most recent phase – characterised by memetic rather than genetic evolution – obeyed the Foerster law of hyperbolic population growth.
    [Show full text]
  • UC Santa Barbara Other Recent Work
    UC Santa Barbara Other Recent Work Title Geopolitics, History, and International Relations Permalink https://escholarship.org/uc/item/29z457nf Author Robinson, William I. Publication Date 2009 Peer reviewed eScholarship.org Powered by the California Digital Library University of California OFFICIAL JOURNAL OF THE CONTEMPORARY SCIENCE ASSOCIATION • NEW YORK Geopolitics, History, and International Relations VOLUME 1(2) • 2009 ADDLETON ACADEMIC PUBLISHERS • NEW YORK Geopolitics, History, and International Relations 1(2) 2009 An international peer-reviewed academic journal Copyright © 2009 by the Contemporary Science Association, New York Geopolitics, History, and International Relations seeks to explore the theoretical implications of contemporary geopolitics with particular reference to territorial problems and issues of state sovereignty, and publishes papers on contemporary world politics and the global political economy from a variety of methodologies and approaches. Interdisciplinary and wide-ranging in scope, Geopolitics, History, and International Relations also provides a forum for discussion on the latest developments in the theory of international relations and aims to promote an understanding of the breadth, depth and policy relevance of international history. Its purpose is to stimulate and disseminate theory-aware research and scholarship in international relations throughout the international academic community. Geopolitics, History, and International Relations offers important original contributions by outstanding scholars and has the potential to become one of the leading journals in the field, embracing all aspects of the history of relations between states and societies. Journal ranking: A on a seven-point scale (A+, A, B+, B, C+, C, D). Geopolitics, History, and International Relations is published twice a year by Addleton Academic Publishers, 30-18 50th Street, Woodside, New York, 11377.
    [Show full text]
  • It from Qubit Simons Collaboration on Quantum Fields, Gravity, and Information
    It from Qubit Simons Collaboration on Quantum Fields, Gravity, and Information GOALS Developments over the past ten years have shown that major advances in our understanding of quantum gravity, quantum field theory, and other aspects of fundamental physics can be achieved by bringing to bear insights and techniques from quantum information theory. Nonetheless, fundamental physics and quantum information theory remain distinct disciplines and communities, separated by significant barriers to communication and collaboration. Funded by a grant from the Simons Foundation, It from Qubit is a large-scale effort by some of ​ ​ ​ ​ the leading researchers in both communities to foster communication, education, and collaboration between them, thereby advancing both fields and ultimately solving some of the deepest problems in physics. The overarching scientific questions motivating the Collaboration include: ● Does spacetime emerge from entanglement? ● Do black holes have interiors? Does the universe exist outside our horizon? ● What is the information-theoretic structure of quantum field theories? ● Can quantum computers simulate all physical phenomena? ● How does quantum information flow in time? MEMBERSHIP It from Qubit is led by 16 Principal Investigators from 15 institutions in 6 countries: ​ ● Patrick Hayden, Director (Stanford University) ​ ● Matthew Headrick, Deputy Director (Brandeis University) ​ ● Scott Aaronson (MIT) ​ ● Dorit Aharonov (Hebrew University) ​ ● Vijay Balasubramanian (University of Pennsylvania) ​ ● Horacio Casini (Bariloche
    [Show full text]
  • The End of the World: the Science and Ethics of Human Extinction John Leslie
    THE END OF THE WORLD ‘If you want to know what the philosophy professors have got to say, John Leslie’s The End of the World is the book to buy. His willingness to grasp the nettles makes me admire philosophers for tackling the big questions.’ THE OBSERVER ‘Leslie’s message is bleak but his touch is light. Wit, after all, is preferable to the desperation which, in the circumstances, seems the only other response.’ THE TIMES ‘John Leslie is one of a very small group of philosophers thoroughly conversant with the latest ideas in physics and cosmology. Moreover, Leslie is able to write about these ideas with wit, clarity and penetrating insight. He has established himself as a thinker who is unafraid to tackle the great issues of existence, and able to sift discerningly through the competing— and frequently bizarre—concepts emanating from fundamental research in the physical sciences. Leslie is undoubtedly the world’s expert on Brandon Carter’s so-called Doomsday Argument—a philosophical poser that is startling yet informative, seemingly outrageous yet intriguing, and ultimately both disturbing and illuminating. With his distinctive and highly readable style, combined with a bold and punchy treatment, Leslie offers a fascinating glimpse of the power of human reasoning to deduce our place in the universe.’ PAUL DAVIES, PROFESSOR OF NATURAL PHILOSOPHY, UNIVERSITY OF ADELAIDE, AUTHOR OF THE LAST THREE MINUTES ‘This book is vintage John Leslie: it presents a bold and provocative thesis supported by a battery of arguments and references to the most recent advances in science. Leslie is one of the most original and interesting thinkers today.
    [Show full text]
  • A Paradox Regarding Monogamy of Entanglement
    A paradox regarding monogamy of entanglement Anna Karlsson1;2 1Institute for Advanced Study, School of Natural Sciences 1 Einstein Drive, Princeton, NJ 08540, USA 2Division of Theoretical Physics, Department of Physics, Chalmers University of Technology, 412 96 Gothenburg, Sweden Abstract In density matrix theory, entanglement is monogamous. However, we show that qubits can be arbitrarily entangled in a different, recently constructed model of qubit entanglement [1]. We illustrate the differences between these two models, analyse how the density matrix property of monogamy of entanglement originates in assumptions of classical correlations in the construc- tion of that model, and explain the counterexample to monogamy in the alternative model. We conclude that monogamy of entanglement is a theoretical assumption, not necessarily a phys- ical property, and discuss how contemporary theory relies on that assumption. The properties of entanglement entropy are very different in the two models — a priori, the entropy in the alternative model is classical. arXiv:1911.09226v2 [hep-th] 7 Feb 2020 Contents 1 Introduction 1 1.1 Indications of a presence of general entanglement . .2 1.2 Non-signalling and detection of general entanglement . .3 1.3 Summary and overview . .4 2 Analysis of the partial trace 5 3 A counterexample to monogamy of entanglement 5 4 Implications for entangled systems 7 A More details on the different correlation models 9 B Entropy in the orthogonal information model 11 C Correlations vs entanglement: a tolerance for deviations 14 1 Introduction The topic of this article is how to accurately model quantum correlations. In quantum theory, quan- tum systems are currently modelled by density matrices (ρ) and entanglement is recognized to be monogamous [2].
    [Show full text]
  • Quantum-Corrected Rotating Black Holes and Naked Singularities in (2 + 1) Dimensions
    PHYSICAL REVIEW D 99, 104023 (2019) Quantum-corrected rotating black holes and naked singularities in (2 + 1) dimensions † ‡ Marc Casals,1,2,* Alessandro Fabbri,3, Cristián Martínez,4, and Jorge Zanelli4,§ 1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, CEP 22290-180, Brazil 2School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland 3Departamento de Física Teórica and IFIC, Universidad de Valencia-CSIC, C. Dr. Moliner 50, 46100 Burjassot, Spain 4Centro de Estudios Científicos (CECs), Arturo Prat 514, Valdivia 5110466, Chile (Received 15 February 2019; published 13 May 2019) We analytically investigate the perturbative effects of a quantum conformally coupled scalar field on rotating (2 þ 1)-dimensional black holes and naked singularities. In both cases we obtain the quantum- backreacted metric analytically. In the black hole case, we explore the quantum corrections on different regions of relevance for a rotating black hole geometry. We find that the quantum effects lead to a growth of both the event horizon and the ergosphere, as well as to a reduction of the angular velocity compared to their corresponding unperturbed values. Quantum corrections also give rise to the formation of a curvature singularity at the Cauchy horizon and show no evidence of the appearance of a superradiant instability. In the naked singularity case, quantum effects lead to the formation of a horizon that hides the conical defect, thus turning it into a black hole. The fact that these effects occur not only for static but also for spinning geometries makes a strong case for the role of quantum mechanics as a cosmic censor in Nature.
    [Show full text]
  • Traversable Wormholes and Regenesis
    Traversable Wormholes and Regenesis The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Gao, Ping. 2019. Traversable Wormholes and Regenesis. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:42029626 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Traversable Wormholes and Regenesis A dissertation presented by Ping Gao to The Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Physics Harvard University Cambridge, Massachusetts April 2019 c 2019 | Ping Gao All rights reserved. Dissertation Advisor: Daniel Louis Jafferis Ping Gao Traversable Wormholes and Regenesis Abstract In this dissertation we study a novel solution of traversable wormholes in the context of AdS/CFT. This type of traversable wormhole is the first such solution that has been shown to be embeddable in a UV complete theory of gravity. We discuss its property from points of view of both semiclassical gravity and general chaotic system. On gravity side, after turning on an interaction that couples the two boundaries of an eternal BTZ black hole, in chapter 2 we find a quantum matter stress tensor with negative average null energy, whose gravitational backreaction renders the Einstein-Rosen bridge traversable. Such a traversable wormhole has an interesting interpretation in the context of ER=EPR, which we suggest might be related to quantum teleportation.
    [Show full text]
  • Do Black Holes Create Polyamory?
    Do black holes create polyamory? Andrzej Grudka1;4, Michael J. W. Hall2, Michał Horodecki3;4, Ryszard Horodecki3;4, Jonathan Oppenheim5, John A. Smolin6 1Faculty of Physics, Adam Mickiewicz University, 61-614 Pozna´n,Poland 2Centre for Quantum Computation and Communication Technology (Australian Research Council), Centre for Quantum Dynamics, Griffith University, Brisbane, QLD 4111, Australia 3Institute of Theoretical Physics and Astrophysics, University of Gda´nsk,Gda´nsk,Poland 4National Quantum Information Center of Gda´nsk,81–824 Sopot, Poland 5University College of London, Department of Physics & Astronomy, London, WC1E 6BT and London Interdisciplinary Network for Quantum Science and 6IBM T. J. Watson Research Center, 1101 Kitchawan Road, Yorktown Heights, NY 10598 Of course not, but if one believes that information cannot be destroyed in a theory of quantum gravity, then we run into apparent contradictions with quantum theory when we consider evaporating black holes. Namely that the no-cloning theorem or the principle of entanglement monogamy is violated. Here, we show that neither violation need hold, since, in arguing that black holes lead to cloning or non-monogamy, one needs to assume a tensor product structure between two points in space-time that could instead be viewed as causally connected. In the latter case, one is violating the semi-classical causal structure of space, which is a strictly weaker implication than cloning or non-monogamy. This is because both cloning and non-monogamy also lead to a breakdown of the semi-classical causal structure. We show that the lack of monogamy that can emerge in evaporating space times is one that is allowed in quantum mechanics, and is very naturally related to a lack of monogamy of correlations of outputs of measurements performed at subsequent instances of time of a single system.
    [Show full text]
  • Finite-Cutoff JT Gravity and Self-Avoiding Loops Arxiv
    Finite-cutoff JT gravity and self-avoiding loops Douglas Stanford and Zhenbin Yang Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305 Abstract We study quantum JT gravity at finite cutoff using a mapping to the statistical mechanics of a self-avoiding loop in hyperbolic space, with positive pressure and fixed length. The semiclassical limit (small GN ) corresponds to large pressure, and we solve the problem in that limit in three overlapping regimes that apply for different loop sizes. For intermediate loop sizes, a semiclassical effective description is valid, but for very large or very small loops, fluctuations dominate. For large loops, this quantum regime is controlled by the Schwarzian theory. For small loops, the effective description fails altogether, but the problem is controlled using a conjecture from the theory of self-avoiding walks. arXiv:2004.08005v1 [hep-th] 17 Apr 2020 Contents 1 Introduction 2 2 Brief review of JT gravity 4 3 JT gravity and the self-avoiding loop measure 5 3.1 Flat space JT gravity . .5 3.2 Ordinary (hyperbolic) JT gravity . .8 3.3 The small β limit . .8 4 The flat space regime 9 5 The intermediate regime 11 5.1 Classical computations . 11 5.2 One-loop computations . 13 6 The Schwarzian regime 18 7 Discussion 19 7.1 A free particle analogy . 20 7.2 JT gravity as an effective description of JT gravity . 21 A Deriving the density of states from the RGJ formula 23 B Large argument asymptotics of the RGJ formula 24 C Monte Carlo estimation of c2 25 D CGHS model and flat space JT 26 E Large unpressurized loops 27 1 푝 2/3 -2 푝 푝 = ℓ β large pressure effective theory Schwarzian theory flat-space (RGJ) theory 4/3 β = ℓ -3/2 = ℓ4β3/2 푝 = β 푝 푝 = ℓ-2 0 0 0 β 0 β Figure 1: The three approximations used in this paper are valid well inside the respective shaded regions.
    [Show full text]
  • Diagnosing Chaos Using Four-Point Functions in Two-Dimensional Conformal Field Theory
    Diagnosing Chaos Using Four-Point Functions in Two-Dimensional Conformal Field Theory The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Roberts, Daniel A., and Douglas Stanford. "Diagnosing Chaos Using Four-Point Functions in Two-Dimensional Conformal Field Theory." Phys. Rev. Lett. 115, 131603 (September 2015). © 2015 American Physical Society As Published http://dx.doi.org/10.1103/PhysRevLett.115.131603 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/98897 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. week ending PRL 115, 131603 (2015) PHYSICAL REVIEW LETTERS 25 SEPTEMBER 2015 Diagnosing Chaos Using Four-Point Functions in Two-Dimensional Conformal Field Theory Daniel A. Roberts* Center for Theoretical Physics and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA † Douglas Stanford School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540, USA (Received 10 March 2015; revised manuscript received 14 May 2015; published 22 September 2015) We study chaotic dynamics in two-dimensional conformal field theory through out-of-time-order thermal correlators of the form hWðtÞVWðtÞVi. We reproduce holographic calculations similar to those of Shenker and Stanford, by studying the large c Virasoro identity conformal block. The contribution of this block to the above correlation function begins to decrease exponentially after a delay of ∼t − β=2π β2E E t β=2π c E ;E à ð Þ log w v, where à is the fast scrambling time ð Þ log and w v are the energy scales of the W;V operators.
    [Show full text]
  • Advanced Lectures on General Relativity
    Lecture notes prepared for the Solvay Doctoral School on Quantum Field Theory, Strings and Gravity. Lectures given in Brussels, October 2017. Advanced Lectures on General Relativity Lecturing & Proofreading: Geoffrey Compère Typesetting, layout & figures: Adrien Fiorucci Fonds National de la Recherche Scientifique (Belgium) Physique Théorique et Mathématique Université Libre de Bruxelles and International Solvay Institutes Campus Plaine C.P. 231, B-1050 Bruxelles, Belgium Please email any question or correction to: [email protected] Abstract — These lecture notes are intended for starting PhD students in theoretical physics who have a working knowledge of General Relativity. The 4 topics covered are (1) Surface charges as con- served quantities in theories of gravity; (2) Classical and holographic features of three-dimensional Einstein gravity; (3) Asymptotically flat spacetimes in 4 dimensions: BMS group and memory effects; (4) The Kerr black hole: properties at extremality and quasi-normal mode ringing. Each topic starts with historical foundations and points to a few modern research directions. Table of contents 1 Surface charges in Gravitation ................................... 7 1.1 Introduction : general covariance and conserved stress tensor..............7 1.2 Generalized Noether theorem................................. 10 1.2.1 Gauge transformations and trivial currents..................... 10 1.2.2 Lower degree conservation laws........................... 11 1.2.3 Surface charges in generally covariant theories................... 13 1.3 Covariant phase space formalism............................... 14 1.3.1 Field fibration and symplectic structure....................... 14 1.3.2 Noether’s second theorem : an important lemma................. 17 Einstein’s gravity.................................... 18 Einstein-Maxwell electrodynamics.......................... 18 1.3.3 Fundamental theorem of the covariant phase space formalism.......... 20 Cartan’s magic formula...............................
    [Show full text]