ASPECTS OF SCALE INVARIANCE IN PHYSICS AND BIOLOGY Vasyl Alba A DISSERTATION PRESENTED TO THE FACULTY OF PRINCETON UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY RECOMMENDED FOR ACCEPTANCE BY THE DEPARTMENT OF PHYSICS Advisers: William Bialek and Juan Maldacena September 2017 © Copyright by Vasyl Alba, 2017. All rights reserved. Abstract We study three systems that have scale invariance. The first system is a conformal field theory in d > 3 dimensions. We prove that if there is a unique stress-energy tensor and at least one higher-spin conserved current in the theory, then the correlation functions of the stress-energy tensors and the conserved currents of higher-spin must coincide with one of the following possibilities: a) a theory of n free bosons, b) a theory of n free fermions or c) d−2 a theory of n 2 -forms. The second system is the primordial gravitational wave background in a theory with inflation. We show that the scale invariant spectrum of primordial gravitational waves is isotropic only in the zero-order approximation, and it gets a small correction due to the primordial scalar fluctuations. When anisotropy is measured experimentally, our result will allow us to distinguish between different inflationary models. The third system is a biological system. The question we are asking is whether there is some simplicity or universality underlying the complexities of natural animal behavior. We use the walking fruit fly (Drosophila melanogaster) as a model system. Based on the result that unsupervised flies’ behaviors can be categorized into one hundred twenty-two discrete states (stereotyped movements), which all individuals from a single species visit repeatedly, we demonstrated that the sequences of states are strongly non-Markovian. In particular, correlations persist for an order of magnitude longer than expected from a model of random state-to-state transitions. The correlation function has a power-law decay, which is a hint of some kind of criticality in the system. We develop a generalization of the information bottleneck method that allows us to cluster these states into a small number of clusters. This more compact description preserves a lot of temporal correlation. We found that it is enough to use a two-cluster representation of the data to capture long-range correlations, which opens a way for a more quantitative description of the system. Usage of the maximal entropy method allowed us to find a description that closely resembles a famous inverse- square Ising model in 1d in a small magnetic field. iii Acknowledgements I would like to thank the Universe for this realization of our world because I was lucky to be at Princeton and I had the privilege to work with William Bialek and Juan Maldacena. I am indebted to Juan Maldacena, my adviser, for the interesting journey to the very frontier of the High Energy Physics, as well as for a huge motivation that I got during numerous discussions with him. He exposed me to a completely new and unbelievable level of thinking about the subject. I am indebted to William Bialek, my adviser, for introducing me to a theoretical bio- physics; he taught a class in spring 2012, and as, I see it now, he planted an idea of working in the field of Biological Physics. It took a few years for the idea to grow. So, I incredibly grateful to Bill for all his help and support that lead to a smooth transition into a new field, when I decided to do it. I am very thankful to Bill for all time he spent teaching me how to work in a completely new field. Every meeting with him led to a substantial jump in my understanding of the questions we discussed. His fine taste for problems in the world of living organisms will always be the highest standard for me, that will influence my research for years. I am very grateful to Sasha Zhiboedov for uncountable hours of discussions that we had in the Institute’s ”dungeons,” and for all his help that made my first years at Princeton extremely pleasant. I thank Lyman Page, who was my adviser for an experimental project. It was a really pleasant experience. Also, I would like to thank him for serving on my pre-thesis and FPO committee. I thank Herman Verlinde for serving on my pre-thesis and FPO committee, and for numerous pieces of advice that he gave to me during these years at Princeton. I thank Igor Klebanov for being a reader of this dissertation. iv I would like to thank my collaborators for a wonderful experience that I had: Kenan Diab, Gordon Berman, Joshua Shaevitz. I am thankful to Ben Machta for help with my simulations. I would like to thank the Department of Physics for awarding me with Joseph Henry Merit Prize, and Kusaka Memorial Prize in Physics. I am extremely grateful to Andrei Mironov and Alexei Morozov for introducing me to the beautiful world of theoretical physics, and for all their help and great support at the Institute for Theoretical and Experiment Physics (Moscow), without whom I would never reach this point. There is not enough space in this dissertation to write about each and every remarkable person who helped me or influenced me in a good way. I am very thankful to all my friends and colleagues, especially, at the Department of Physics and at the Icahn laboratories. All of you helped me a lot in my continuous journey towards the understanding of this beautiful Nature. Last but not least, I am extremely grateful to all my family for their constant love and support that helped me to get through all the difficulties of the graduate school. v Хто думає про науку, той любить її, а хто її любить, той ніколи не “ перестає вчитися, хоча б зовні він і здавався бездіяльним. The one who thinks about science loves it, and the one who loves it will never cease to learn, even though he might seem to be outwardly idle. ” Hryhorii Savych Skovoroda, (1722 – 1794), Kharkiv vi To my family. vii Contents Abstract ....................................... iii Acknowledgements ................................. iv List of Figures .................................... xii 1 Introduction 1 1.1 Symmetry of the system as a defining principle ............... 4 1.2 How can we look into the past? ........................ 7 1.3 Searching for principles ............................ 12 1.4 An Overview of the Dissertation ....................... 14 Higher-Spin Theories ............................. 14 Anisotropy of the Primordial Gravitational Waves .............. 16 Physics of Behavior .............................. 18 Publications and preprints .............................. 19 Public presentations ................................. 19 2 Constraining conformal field theories with a higher spin symmetry in d > 3 dimensions 20 2.1 Introduction .................................. 20 2.2 Definition of the lightcone limits ....................... 25 viii 2.3 Charge conservation identities ........................ 29 2.4 Quasi-bilocal fields: basic properties ..................... 33 2.5 Quasi-bilocal fields: correlation functions .................. 39 2.5.1 Symmetries of the quasi-bilocal operators .............. 40 2.5.2 Correlation functions of the bosonic quasi-bilocal .......... 42 2.5.3 Correlation functions of the fermionic and tensor quasi-bilocal ... 44 2.5.4 Normalization of the quasi-bilocal correlation functions ...... 50 2.6 Constraining all the correlation functions ................... 51 2.7 Discussion and conclusions .......................... 56 3 Anisotropy of gravitational waves 59 3.1 Introduction .................................. 59 3.2 Gravity waves from inflation at leading order ................ 60 3.3 Sachs–Wolfe effect .............................. 63 3.3.1 Geodesic Equations in Different Gauges ............... 64 3.4 Anisotropy .................................. 66 4 Exploring a strongly non-Markovian behavior 70 4.1 Introduction .................................. 70 4.2 Experimental setup and initial analysis .................... 72 4.3 Formalization of behavior description .................... 74 4.4 Is there any memory in the system or can we use HMM/MC? ........ 78 4.4.1 Markov process ............................ 79 4.4.2 Hidden Markov Model ........................ 80 4.5 Clustering ................................... 82 4.6 Maximal entropy model ............................ 90 4.7 Simulation results ............................... 94 ix A Appendices for Chapter 2 99 A.1 Form factors as Fourier transforms of correlation functions ......... 99 A.2 Uniqueness of three-point functions in the tensor lightcone limit ...... 102 A.3 Uniqueness of hs22i for s ≥ 4 ........................ 104 A.4 Transformation properties of bilocal operators under K− .......... 105 A.4.1 Fermionic case ............................ 106 A.4.2 Tensor case .............................. 107 A.5 Proof that Oq exists .............................. 108 A.6 The free Maxwell field in five dimensions .................. 110 B Appendices for Chapter 3 112 B.1 Evolution ................................... 112 B.1.1 Solution during inflation ....................... 114 B.1.2 Solution during the first non-inflationary era ............ 115 B.1.3 Solutions during consecutive eras .................. 116 B.1.4 Solutions for an arbitrary history ................... 116 B.2 Particle creation ................................ 119 B.2.1 The first method to calculate Bogolyubov coefficients ....... 119 B.2.2 More conventional method of Bogolyubov coefficient calculation . 121 B.2.3 Graviton density ........................... 122 B.2.4 Spectrum ..............................