The Union of Quantum Field Theory and Non-equilibrium Thermodynamics Thesis by Anthony Bartolotta In Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2018 Defended May 24, 2018 ii c 2018 Anthony Bartolotta ORCID: 0000-0003-4971-9545 All rights reserved iii Acknowledgments My time as a graduate student at Caltech has been a journey for me, both professionally and personally. This journey would not have been possible without the support of many individuals. First, I would like to thank my advisors, Sean Carroll and Mark Wise. Without their support, this thesis would not have been written. Despite entering Caltech with weaker technical skills than many of my fellow graduate students, Mark took me on as a student and gave me my first project. Mark also granted me the freedom to pursue my own interests, which proved instrumental in my decision to work on non-equilibrium thermodynamics. I am deeply grateful for being provided this priviledge and for his con- tinued input on my research direction. Sean has been an incredibly effective research advisor, despite being a newcomer to the field of non-equilibrium thermodynamics. Sean was the organizing force behind our first paper on this topic and connected me with other scientists in the broader community; at every step Sean has tried to smoothly transition me from the world of particle physics to that of non-equilibrium thermody- namics. My research would not have been nearly as fruitful without his support. I would also like to thank the other two members of my thesis and candidacy com- mittees, John Preskill and Keith Schwab. Both took time out of their busy schedules to attend my defenses and provide insightful comments on my research. Throughout my graduate career, I had the support of a wonderful set of collab- orators, without whom many projects would not have been possible. I would like to thank: Sean Carroll, Sebastian Deffner, Stefan Leichenauer, Dibyendu Mandal, Jason Pollack, Michael Ramsey-Musolf, and Nicole Yunger-Halpern. Coming from a particle physics background, I am an interloper in the non-equilibrium thermodynamics community. Despite this fact, I was quickly welcomed by the commu- nity. I would like to thank the many participants of the annual “Information Engines at the Frontiers of Nanoscale Thermodynamics” workshop at the Telluride Science Re- search Conference, for providing such a welcoming environment and making me feel at home in a field that was not my own. While I cannot thank everyone individually, I would like to extend my deep graditude to Sebastian Deffner. Without Sebastian, I never would have met the wider non-equilibrium thermodynamics community. Fur- iv thermore, Sebastian has been an indispensible source of career advice and support as I transition out of graduate school. I would like to thank the wonderful community in the Caltech High-Energy Theory group. Clifford Cheung for answering all of my questions about the analytic properties of field theories, and Carol Silberstein for maintaining a semblance of order in the group. I would also like to thank the many PhD students and postdocs in the group for making my time here far more enjoyable: Ning Bao, Charles Cao, Aidan Chatwin- Davies, Enrico Herrmann, Matt Heydeman, Murat Kologlu, Stefan Leichenauer, Jason Pollack, Grant Remmen, Alexander Ridgway, Chia-Hsien Shen, and Ashmeet Singh. I would especially like to thank my long-time officemate Jason Pollack for putting up with my incessant stream of physics questions, and more importantly, for not completely giving up on me after I accidentally glued myself to the printer. For keeping me grounded, and constantly foiling my plans as DM, I would like to thank the Friday night dinner and D&D group: Dustin Anderson, Kevin Barkett, Daniel Brooks, Jeremy Brouillet, and Jason Pollack. For reminding me that there is a world outside of physics and that it is not the only possible topic of conversation, I would like to thank my friends in other departments: Ania Baetica, Thomas Catanach, Niangjun Chen, Albert Chern, Ioannis Filippidis, Yorie Nakahira, and Yong Sheng Soh. For supporting my decision to pursue a career in science since before I can re- member, I would like to thank my parents, Gene and Andrea Bartolotta, and brother, Vince Bartolotta. I would also like to express my deep gratitude to my parents for their continued support in spite of the damage I caused to the house and yard with my “experiments”. Finally, I would like to thank my incredible wife, Yoke Peng Leong. Without her support (and occasional nagging to take a break), graduate school would have been a far different experience. Even in the darkest times of graduate school, Yoke Peng supported me and was a calm voice of reason to my anxiety. I am deeply grateful for her help in getting though graduate school and cannot imagine anyone else I would rather have at my side going forward. My work was supported in part by the Walter Burke Institute for Theoretical Physics at Caltech and by the United States Department of Energy under Grant DE- SC0011632. v Abstract Quantum field theory is the language used to describe nature at its most fundamen- tal scales; while thermodynamics is a framework to describe the collective behavior of macroscopic systems. Recent advances in non-equilibrium thermodynamics have enabled this framework to be applied to smaller systems operating out of thermal equi- librium. This thesis is concerned with both quantum field theory and non-equilibrium thermodynamics independently and with their intersection. First, a purely phenomenological application of quantum field theory is explored in the context of the upcoming Mu2E experiment. This experiment will look for rare decays which would indicate the presence of physics beyond the Standard Model. Using the language of effective field theories, a next-to-leading order analysis of the conversion rate is performed. The focus then shifts to an apparent paradox in the Bayesian interpretation of sta- tistical mechanics. For a Bayesian observer making measurements of an open system, the Shannon entropy decreases, in apparent violation of the Second Law of Thermody- namics. It is shown that rather than utilizing the entropy, which can decrease under Bayesian updates, the Second Law for a Bayesian observer can be rephrased in terms of a cross-entropy which is always non-negative. Finally, the intersection of quantum field theory and non-equilibrium thermody- namics is examined. Using quantum work fluctuation theorems, an investigation of how these frameworks can be applied to a driven quantum field theory is performed. For a time-dependent variant of λφ4, analytic expressions for the work distribution functions at one-loop order are derived. These expressions are shown to satisfy the quantum Jarzynski equality and Crooks fluctuation theorem. vi Published Content and Contributions With the exception of the Introduction, the body of this thesis is adapted from articles that originally appeared in other forms. All articles are available as arXiv preprints. All authors contributed equally. • A. Bartolotta and M. J. Ramsey-Musolf, Coherent µ − e Conversion at Next-to- Leading Order, arXiv:1710.0212 • A. Bartolotta, S. M. Carroll, S. Leichenauer, and J. Pollack, Bayesian second law of thermodynamics, Phys. Rev. E 94 (Aug, 2016) 022102, [arXiv:1508.0242] • A. Bartolotta and S. Deffner, Jarzynski equality for driven quantum field theories, Phys. Rev. X 8 (Feb, 2018) 011033, [arXiv:1710.0082] vii Contents Acknowledgments iii Abstractv Published Content and Contributions vi Contents vii List of Figures and Tables ix Introduction1 Coherent µ − e Conversion at Next-to-Leading Order4 2.1 Introduction5 2.2 Quark-Level CLFV Lagrangian 10 2.3 Chiral Power Counting and Chiral Lagrangians 11 2.4 Scalar-Mediated Conversion 13 2.5 Approximate One-body Interaction 17 2.6 Vector-Mediated Conversion 19 2.7 Hadronic Uncertainties 20 2.8 Wavefunctions of the Muon and Electron 21 2.9 Nuclear Density Distributions 23 2.10 Calculation of the Branching Ratio 24 2.11 Discussion and Analysis 26 2.12 Conclusions 28 2.A Chiral Lagrangian 31 2.B Low-Energy Constants of the Isoscalar Strange Operators 34 2.C Values of Low-Energy Constants and Physical Quantities 36 2.D Momentum Dependence of Approximate One-Body Interaction 37 2.E Nuclear Density Parameters 40 2.F Model Independent Overlap Integrals 42 2.G Formula for the Branching Ratio 45 viii The Bayesian Second Law of Thermodynamics 46 3.1 Introduction 47 3.2 Setup 51 3.2.1 The System and Evolution Probabilities 51 3.2.2 Measurement and Bayesian Updating 53 3.2.3 The Reverse Protocol and Time Reversal 57 3.2.4 Heat Flow 59 3.3 Second Laws from Relative Entropy 60 3.3.1 The Ordinary Second Law from Positivity of Relative Entropy 60 3.3.2 A Refined Second Law from Monotonicity of Relative Entropy 62 3.4 The Bayesian Second Law 63 3.4.1 Cross-Entropy Formulation of the BSL 64 3.4.2 Alternate Formulations of the BSL 66 3.4.3 A Refined BSL from Monotonicity of Relative Entropy 68 3.5 Bayesian Integral Fluctuation Theorems 69 3.6 Applications 73 3.6.1 Special Cases 73 3.6.2 Diffusion of a Gaussian in n Dimensions. 75 3.6.3 Randomly Driven Harmonic Oscillator 76 3.7 Discussion 79 3.A Oscillator Evolution 83 Jarzynski Equality for Driven Quantum Field Theories 86 4.1 Introduction 87 4.2 Preliminaries: Two-Time Measurement Formalism 90 4.3 Restricted Field Theories 91 4.4 Projection Operators And Finite-Time Transitions 93 4.5 Renormalization of time-dependent