SISSA–INTERNATIONAL SCHOOL OF ADVANCED STUDIES DOCTORAL THESIS Exploring Spacetime Phenomenology: From Lorentz Violations to Experimental Tests of Non-locality Author: Supervisor: Alessio BELENCHIA Prof. Stefano LIBERATI A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Astroparticle Physics August 31, 2016 iii Declaration of Authorship The research presented in this thesis was mainly conducted in SISSA – International School for Advanced Studies between September 2012 and June 2016. This thesis is the result of the author own work, as well as the outcome of scientific collaborations stated below, except where explicit ref- erence is made to the results of others. The content of this thesis is based on the following research papers published in refereed Journals, conference proceedings or preprints available on arxiv.org: A. Belenchia, S. Liberati, and A. Mohd • Emergent gravitational dynamics in a relativistic Bose-Einstein condensate In: Phys. Rev. D90.10, p. 104015, arXiv: 1407.7896 [gr-qc]. A. Belenchia, D.M.T. Benincasa, and S. Liberati • Nonlocal Scalar Quantum Field Theory from Causal Sets In: JHEP 03, p. 036. arXiv: 1411.6513 [gr-qc]. A. Belenchia, D.M.T. Benincasa, A. Marcianó, and L. Modesto • Spectral Dimension from Causal Set Nonlocal Dynamics In: Phys.Rev. D93 (2016) 044017, arXiv: 1507.00330 [gr-qc] A. Belenchia, A. Gambassi and S. Liberati • Lorentz violation naturalness revisited In: JHEP (2016), arXiv: 1601.06700 [hep-th] A. Belenchia, D.M.T. Benincasa, Stefano Liberati, Francesco Marin, • Francesco Marino, and Antonello Ortolan Tests of Quantum Gravity induced non-locality via opto-mechanical quan- tum oscillators In: Phys.Rev.Lett.116 (2016), arXiv: 1512.02083 [gr-qc] A. Belenchia, • Universal behaviour of generalized Causal-set d’Alembertian in curved space- time In: Class.Quant.Grav. (2016), arXiv: 1510.04665 [gr-qc] A. Belenchia, D.M.T. Benincasa, and F. Dowker • The continuum limit of a 4-dimensional causal set scalar d’Alembertian In: arXiv: 1510.04656 [gr-qc] A. Belenchia • Causal set, non-locality and phenomenology In: arXiv:1512.08485 [gr-qc], To appear in the proceedings of the 14th Mar- cel Grossmann meeting, Rome (July, 2015) A. Belenchia, D.M.T. Benincasa, E. Martin-Martinez and M. Saravani • Low-Energy Signatures of Non-Local Field Theories Accepted in: PRD rapid communication; In: arXiv:1605.0397 v “A mathematical friend of mine said to me the other day half in jest: “The math- ematician can do a lot of things, but never what you happen to want him to do just at the moment.” Much the same often applies to the theoretical physicist when the experimental physicist calls him in. What is the reason for this peculiar lack of adaptability?” Albert Einstein vii SISSA–INTERNATIONAL SCHOOL OF ADVANCED STUDIES Abstract Doctor of Philosophy Exploring Spacetime Phenomenology: From Lorentz Violations to Experimental Tests of Non-locality by Alessio BELENCHIA This thesis deals primarily with the phenomenology associated to quan- tum aspects of spacetime. In particular, it aims at exploring the phenomeno- logical consequences of a fundamental discreteness of the spacetime fabric, as predicted by several quantum gravity models and strongly hinted by many theoretical insights. The first part of this work considers a toy-model of emergent spacetime in the context of analogue gravity. The way in which a relativistic Bose– Einstein condensate can mimic, under specific configurations, the dynamics of a scalar theory of gravity will be investigated. This constitutes proof-of- concept that a legitimate dynamical Lorentzian spacetime may emerge from non-gravitational (discrete) degrees of freedom. Remarkably, this model will emphasize the fact that in general, even when arising from a relativis- tic system, any emergent spacetime is prone to show deviations from exact Lorentz invariance. This will lead us to consider Lorentz Invariance Viola- tions as first candidate for a discrete spacetime phenomenology. Having reviewed the current constraints on Lorentz Violations and stud- ied in depth viable resolutions of their apparent naturalness problem, the second part of this thesis focusses on models based on Lorentz invariance. In the context of Casual Set theory, the coexistence of Lorentz invariance and discreteness leads to an inherently nonlocal scalar field theory over causal sets well approximating a continuum spacetime. The quantum as- pects of the theory in flat spacetime will be studied and the consequences of its non-locality will be spelled out. Noticeably, these studies will lend support to a possible dimensional reduction at small scales and, in a clas- sical setting, show that the scalar field is characterized by a universal non- minimal coupling when considered in curved spacetimes. Finally, the phenomenological possibilities for detecting this non-locality will be investigated. First, by considering the related spontaneous emission of particle detectors, then by developing a phenomenological model to test nonlocal effects using opto-mechanical, non-relativistic systems. In both cases, one could be able to cast in the near future stringent bounds on the non-locality scale. ix Acknowledgements There are several people that I want to thank. First of all, my supervisor Stefano Liberati for his guide in these four years, his patience, advices and helpful suggestions. I am much indebted to my other wonderful collabo- rator, and friend, Dionigi Benincasa who has shared with me most of these four years and guided me. I have to thank also Arif Mohd, Fay Dowker and Eduardo Martin-Martinez for fruitful collaborations and discussions. I want to thank Marco, Mauro, Bruno, Federico, Giovanni, Cristiano, Eolo, Guillaume, Elena, Serena, Kate, Claudia, Fiamma, Giovanna and Cristina and all the other friends I have met in SISSA and Trieste in these years. It has been wonderful to share with them the days of the PhD and for sure I will miss them as well as this wonderful city. A special thanks goes to Clara, with which I share beautiful memories more than with everyone else, even if our paths have unfortunately grown apart. Of course, all this would not have been possible without the constant support of my family during the years of the University and Ph.D: my mother Nadia, my father Sirio, my sister Tania and her soul-mate Paolo, my aunts, uncles and cousins (also my dog Charlie which is with us since 17 years). They have been a constant presence in these years. I owe them everything, and for this I dedicate this thesis to them. xi Contents Declaration of Authorship iii Abstract vii Acknowledgements ix Preface1 1 Introduction5 1.1 Quantum Gravity and Phenomenology............5 1.1.1 Discreteness of spacetime................7 1.2 Analogue Gravity.........................9 1.2.1 The ancestor of all analogue models..........9 1.2.2 Bose–Einstein condensate analogue.......... 10 1.2.3 Emergent dynamics?................... 14 1.3 Lorentz Invariance Violations.................. 15 1.3.1 An experimental window on Quantum Gravity... 16 1.3.2 The naturalness problem of LIV............ 17 1.4 Causal Set Theory and Non-Locality.............. 19 1.4.1 Kinematics......................... 20 1.4.2 Lorentz Invariance and Non-locality.......... 20 1.4.3 CS Phenomenology (so far)............... 23 1.4.4 Dynamics......................... 24 2 Analogue Gravity as a toy model of Emergent spacetime 25 2.1 Complex scalar field theory: relativistic BEC......... 26 2.2 Relativistic BEC as an analogue gravity model........ 28 2.2.1 Dynamics of perturbations: acoustic metric..... 29 2.3 Relation to previous results................... 29 2.4 Emergent Nordström gravity.................. 31 2.4.1 Stress Energy Tensor and Newton constant...... 33 2.5 Summary.............................. 35 2.6 Discussion............................. 35 3 Lorentz Invariance Violations naturalness 39 3.1 LIV percolation: previous results................ 40 3.1.1 LI case........................... 42 3.1.2 LIV case.......................... 43 3.2 Fermion self-energy........................ 46 3.2.1 Particles with equal masses (R = 1) and same viola- tion (f = f~)........................ 49 3.2.2 Particles with different masses (R = 1) and same vio- 6 lation (f = f~)....................... 50 3.2.3 Violation only on the scalar field (f~ = 1)........ 50 xii 3.3 Separation of scales........................ 51 3.3.1 Sharp LI cutoff...................... 52 3.3.2 Smooth LI cutoff..................... 54 3.4 Dissipation and LIV naturalness................ 56 3.4.1 The general setting.................... 58 3.4.2 Sharp LI cutoff...................... 59 3.4.3 Smooth LI cutoff..................... 61 3.5 Summary and Discussion.................... 61 4 Causal Set theory and Non-locality 65 4.1 Nonlocal d’Alembertians..................... 66 4.2 Massive Extension: Nonlocal Klein-Gordon Equation.... 72 4.3 Huygens’ Principle and the Nonlocal d’Alembertians.... 73 4.4 Free Scalar Nonlocal QFT.................... 75 4.4.1 2 Dimensions....................... 78 4.4.2 4 Dimensions....................... 81 4.4.3 Renormalisation..................... 86 4.5 Summary and Outlook...................... 86 5 Curved Spacetime and Dimensional reduction in CS theory 89 5.1 Curved Spacetime d’Alembertians............... 89 5.1.1 General set-up...................... 90 5.1.2 Universality of R=2 factor:............... 94 − 5.1.3 Summary......................... 100 5.2 Dimensional Reduction...................... 101 5.2.1 Momentum Space Nonlocal