DEGREE PROJECT IN ENGINEERING PHYSICS, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2021 On the Necessity of Quantized Gravity A critical comparison of Baym & Ozawa (2009) and Belenchia et al. (2018) ERIK RYDVING KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES Author Erik Rydving – [email protected] Engineering Physics KTH Royal Institute of Technology Place for Project Stockholm, Sweden Examiner Gunnar Bjork¨ antum- and Biophotonics KTH Royal Institute of Technology Supervisor Erik Aurell Department of Computational Science and Technology KTH Royal Institute of Technology Co-supervisor Igor Pikovski Department of Physics Stockholm University KTH ROYAL INSTITUTE OF TECHNOLOGY M.Sc. esis On the Necessity of antized Gravity A critical comparison of Baym & Ozawa (2009) and Belenchia et al. (2018) Erik O. T. Rydving Supervisor: Erik Aurell, Co-supervisor: Igor Pikovski May 2021 i Abstract One of the main unsolved problems in theoretical physics is combining the theory of quantum mechanics with general relativity. A central question is how to describe gravity as a quantum eld, but also whether a quantum eld description of gravity is necessary in the rst place. ere is now an ongoing search for a Gedankenexperiment that would answer this question of quantization, similar to what Bohr and Rosenfeld’s classic argument did for the electromagnetic case [1]. Two recent papers use Gedankenexperiment arguments to decide whether it is necessary to quantize the gravitational eld, and come to dierent conclusions on the maer [2, 3]. In this work, their arguments are analyzed, compared, and combined. Assuming the Planck length as a fundamental lower bound on distance measurability, we nd that a quantum eld theory of gravity is not a logical necessity, in contrary to the conclusion drawn in [3]. ii Sammanfaning Eav de storsta¨ olosta¨ problemen i teoretisk fysik ar¨ akombinera kvantmekaniken med den allmanna¨ relativitetsteorin. En central fraga˚ ar¨ hur gravitation ska beskrivas som e kvantfalt,¨ men ocksa˚ om en kvantfaltsbeskrivning¨ av gravitation ens ar¨ nodv¨ andig.¨ Det pag˚ ar˚ forskning som fors¨ oker¨ asvara pa˚ fragan˚ om kvantisering med hjalp¨ av tankeexperiment, likt hur Bohr och Rosenfelds klassiska argument svarade pa˚ fragan˚ om det elektromagnetiska faltets¨ kvantisering [1]. Tva˚ moderna artiklar anvander¨ argument med tankeexperiment for¨ abesvara om gravitationsfaltet¨ behover¨ kvantiseras, och kommer till olika slutsatser [2, 3]. I dea examensarbete analyseras, jamf¨ ors¨ och kombineras deras argument. Det visas a, sal˚ ange¨ Planck-langden¨ kan antas som en nedre grans¨ for¨ matbarhet¨ av avstand,˚ ar¨ inte en kvantfaltsteori¨ for¨ gravitation en logisk nodv¨ andighet,¨ i motsa¨ning till konklusionen i [3]. iii Acknowledgements First I want to thank my supervisors for their constant support and many fruitful discussions. My weekly meetings with Erik Aurell always forced me to make my arguments clear and presentable, and his critical questions oen hit exactly where they where necessary. Igor Pikovski let me be part of his team and suggested to join an Essay contest which also helped solidify my thoughts further. I also want to thank my study friends and fellow physicists, especially Ludvig and Marcus with whom I spent countless early mornings and late aernoons in the library. ank you Yuri for a daily dose of love and happiness. Lastly I want to thank my family. My parents for always being supportive of even my smallest achievement, while keeping rm but healthy high expectations, and my brother Martin for many thought- provoking discussions about the fundamentals of the universe. iv Contents 1 Introduction 1 1.1 Planck Units ..................................... 2 1.2 Problem Formulation ................................ 4 1.3 Outline ........................................ 5 2 Theoretical Background 7 2.1 antum Mechanics ................................ 7 2.1.1 antum Interference ........................... 9 2.1.2 Decoherence of a antum State ..................... 12 2.2 General Relativity .................................. 14 2.3 Field antization .................................. 16 2.3.1 Classical Fields and Newtonian Gravity ................. 16 2.3.2 antization of the Electromagnetic Field ................ 17 2.4 Aempts at antum Gravity ........................... 18 3 Method 20 3.1 Comparing Arguments ............................... 20 3.2 Thought Experiments ................................ 21 3.3 Scope of the Thesis ................................. 22 4 Bohr & Rosenfeld 23 4.1 My Take on Bohr & Rosenfeld ........................... 23 4.2 Bronstein and Gravity ................................ 24 5 Baym & Ozawa 28 5.1 Introduction ..................................... 28 5.2 Electromagnetic Case ................................ 29 5.3 Gravitational Case .................................. 30 6 Belenchia et al. 34 6.1 Introduction ..................................... 34 6.1.1 The Paradox ................................. 35 v CONTENTS 6.2 Electromagnetic Case ................................ 35 6.2.1 antum Fluctuations ........................... 36 6.2.2 antum Radiation ............................. 39 6.2.3 Solving the Paradox ............................. 41 6.3 Gravitational Case .................................. 42 6.3.1 antum Fluctuations ........................... 42 6.3.2 antum Radiation ............................. 45 6.3.3 Solving the Paradox ............................. 46 7 Critical Comparative Analysis and New Results 47 7.1 Details of the Interference Experiments ...................... 47 7.2 Calculation of Eective adrupole Moment ................... 49 7.3 Distinguishability of Bob and Interference .................... 50 7.4 Radiation of Alice’s Particle and Interference ................... 52 8 Discussion and Conclusion 54 8.1 Planck Length as a Lower Bound .......................... 54 8.2 Bohr & Rosenfeld in Perspective .......................... 55 8.3 Conclusion ...................................... 56 8.4 Future Work ..................................... 58 Bibliography 59 Appendices 66 AEective adrupole Moment of Two Masses in a Spatial Superposition 66 B Translation of Bronstein (1934) 69 C Essay for Gravity Research Foundation 73 vi Chapter 1 Introduction Physics has come a long way. Some theoretical predictions, like black holes [4, 5] and the Higgs boson [6–8], were presented already several decades ago. Only recently, however, thanks to the advancement in measurement technology, have they been experimentally veried [9–11]. Both these discoveries are strong indications that the theories that explain the phenomena are correct descriptions of nature. General relativity, predicting black holes, describes how massive objects interact with space and time, while the standard model, predicting the Higgs boson, is based on quantum mechanics and describes subatomic particles and their interactions. Both theories give extremely accurate descriptions of nature and far outperform their predecessors [12, 13]. e problem is: our two best theories are in many cases not compatible with each other [14, 15]. is fact is the basis for the almost century-long search for antum Gravity, a theory which combines quantum mechanics with gravity. Many questions remain, partly because it is dicult to nd feasible measurements where quantum gravitational eects are present [16– 18]. ese eects have to dier from what can be predicted by our current theories, which are already so accurate. e diculty in seeing these quantum gravitational phenomena is a question of scales. e scales in which quantum mechanics and general relativity become relevant are quite dierent. On the scale of a human, things of interest are mostly in the orders of magnitude around a meter. Anything smaller than a millimeter becomes dicult to see clearly and control accurately. Similarly, anything larger than about a hundred meters becomes too large to be seen as a single entity. Within these scales our lives take place, and most physical phenomena we encounter can be explained using the laws of physics discovered before the 1 CHAPTER 1. INTRODUCTION 20th century. If we try looking at smaller scales, our intuitions start breaking down. Friction and surface tension work dierently from our everyday experience, and mass becomes less relevant. Smaller still, and we enter the realm of antum Mechanics. Here, even the concept of “here” breaks down. ings are cloudy, uncertain, and unlike anything we are used to. Going in the other direction and looking at larger scales, we see that things change more slowly. Now surface tension becomes less relevant and mass plays a larger role. On the scale of the size of the Earth the weakness of gravity becomes apparent. Even having the mass of a whole planet pulling at a glass of water, one can easily win the tug-of-war and liit up. Going above the scale of the Earth, gravity stands as the major force, keeping the planets in orbit around the sun. It is now perhaps easier to see why the eects of quantum gravity are hard to come by; they involve combining the two opposite ends of the scale! In this project, we will nevertheless try to probe these eects, using clever thought experiments [2, 3]. We will see that it is not as easy as some might think. 1.1 Planck Units A scale which will be of great importance in this work is the Planck scale, rst introduced by Planck in 1899 [19]. He realized that using three fundamental constants of nature — the speed of light 2, Newton’s gravitational constant ⌧, and the