DEGREE PROJECT IN ENGINEERING , SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2021

On the Necessity of Quantized 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 is combining the of quantum with . 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 , 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 di￿erent conclusions on the ma￿er [2, 3]. In this work, their arguments are analyzed, compared, and combined. Assuming the 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 Sammanfa￿ning

E￿av de storsta¨ olosta¨ problemen i teoretisk fysik ar¨ a￿kombinera kvantmekaniken med den allmanna¨ relativitetsteorin. En central fraga˚ ar¨ hur ska beskrivas som e￿ kvantfalt,¨ men ocksa˚ om en kvantfaltsbeskrivning¨ av gravitation ens ar¨ nodv¨ andig.¨ Det pag˚ ar˚ forskning som fors¨ oker¨ a￿svara 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¨ a￿besvara om gravitationsfaltet¨ behover¨ kvantiseras, och kommer till olika slutsatser [2, 3]. I de￿a 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 o￿en 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 a￿ernoons 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 .

iv Contents

1 Introduction 1 1.1 ...... 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 A￿empts 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 E￿ective￿ 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

AE￿ective ￿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 [6–8], were presented already several decades ago. Only recently, however, thanks to the advancement in measurement technology, have they been experimentally veri￿ed [9–11]. Both these discoveries are strong indications that the that explain the phenomena are correct descriptions of . General relativity, predicting black holes, describes how massive objects interact with space and time, while the , predicting the , is based on 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 di￿cult to ￿nd feasible measurements where quantum gravitational e￿ects are present [16– 18]. ￿ese e￿ects have to di￿er from what can be predicted by our current theories, which are already so accurate.

￿e di￿culty in seeing these quantum gravitational phenomena is a question of scales.￿ e scales in which quantum mechanics and general relativity become relevant are quite di￿erent. 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 di￿cult 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 di￿erently 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 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 li￿it up. Going above the scale of the Earth, gravity stands as the major , keeping the planets in orbit around the sun.

It is now perhaps easier to see why the e￿ects 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 e￿ects, 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 2, Newton’s gravitational constant ⌧, and the reduced Planck’s constant \ — one could construct units of mass, length, and time. In standard SI units these can be wri￿en as

\2 8 Planck mass : < = 2.2 10 kg, (1.1) ? ⌧ ⇡ ⇥

\⌧ 35 Planck length : ; = 1.6 10 m, (1.2) ? 23 ⇡ ⇥

…\⌧ 44 Planck time : C = 5.4 10 s. (1.3) ? 25 ⇡ ⇥ … ￿e signi￿cance of these units, now called the Planck units, comes from the fact that they are constructed exclusively from fundamental… constants. ￿e in vacuum 2 3.00 108 m/s is the fundamental speed limit of any information transfer, and no massive ⇡ ⇥ object can be accelerated to this speed without in￿nite energy. Newton’s gravitational constant ⌧ 6.67 10 11 m3/kg s2 is the coupling strength of the gravitational force which ⇡ ⇥ 2 CHAPTER 1. INTRODUCTION apply to all massive objects. ￿e reduced Planck’s constant \=⌘ 2c 1.05 10 34 kg m2/s / ⇡ ⇥ (pronounced “h-bar”) is a factor 1 2c times the proportionality constant ⌘ between the energy / and frequency of a . Overly simpli￿ed, 2 is from relativity, ⌧ is from gravity, \ is from quantum mechanics, and they are all universal constants [20]. In this sense, it is reasonable to think that these units might have some connection to quantum e￿ects of gravity. It is therefore expected that quantum e￿ects on gravitational interactions become important at this scale [21].

Looking at the values of the Planck units in (1.1) – (1.3) we see that the Planck mass is quite small, about the mass of for example an iron ball with the diameter of a human hair, or a ball of Styrofoam 1 mm across1.￿is is maybe not quite massive enough for one to be able to feel the weight in one’s hand, but still large enough for the balls to be visible with the naked eye. But if this mass scale is accessible even to the naked eye, how come we are not seeing quantum e￿ects of gravity everywhere? Well, this mass scale seems to be right in the transition between the region of gravity and the region of quantum mechanics. One has 4 not been able to see gravitational e￿ects between objects less massive than about 10

While the Planck mass is quite small, it is still within the reach of our senses. ￿e Planck length in Eq. (1.2) and the Planck time in Eq. (1.3), on the other hand, are ridiculously small. In fact, the human scale of about one meter is a billion times closer to the size of the whole universe, 8.8 1026 m, than to the Planck length [24]. We humans are closer in ⇥ size to the proton, 8.4 10 16 m, than the proton is to the Planck length [25]. ￿e Planck ⇥ time is then the time light takes to travel one Planck length. ￿ese scales are far beyond what is currently measurable [26], and some argue that they could signify a fundamental lower bound on space and time resolution [27].

1￿e volume of a sphere can be wri￿en in two ways as c33 6 = < d, where 3 is the diameter, < is the mass, / / and d is the density of the sphere. ￿is gives a formula for the diameter as 3 = 3 6< cd. Plugging the densities / 7874 kg/m3 for iron and 1000 kg/m3 for Styrofoam into the formula gives approximately the stated values. p 3 CHAPTER 1. INTRODUCTION

1.2 Problem Formulation

￿e search for a uni￿ed theory of physics, a theory that combines the power of quantum mechanics and general relativity, has been ongoing for almost a century. A part of this search is to answer the question of whether the gravitational ￿eld is a quantum ￿eld, like the other of nature seem to be (see Chapter 2). One approach to this question is through Gedankenexperiments, or thought experiments, where the rules of di￿erent physical theories can be combined, tested, and taken to their limits.

In this thesis, we will thoroughly analyze two recent papers using Gedankenexperiments to answer the question of gravitational ￿eld quantization.￿ e￿rst paper by Baym and Ozawa from 2009 [2] presents a thought experiment where a paradox involving the interplay of gravity and quantum mechanics is resolved without the need of a quantized gravitational ￿eld, while a quantization was needed for a similar case considering the electromagnetic ￿eld. ￿e second paper by Belenchia et al. from 2018 [3], on the other hand, comes to the opposite conclusion. Using a similar thought experiment they ￿nd that a quantization of the gravitational ￿eld is needed to solve the paradox, analogously to the electromagnetic case. ￿e goal going into this project was to analyze and compare their arguments, and ultimately see if both papers are correct but talk about slightly di￿erent things, if one of them made a mistake, or if something else is at play. In this thesis it is found, contrary to the common assumption among physicists, that the gravitational ￿eld is not necessarily quantized.￿ is stands in contrast to the conclusion drawn in [3], which states that the quantization of the gravitational ￿eld is what solves the paradox. It is important to stress that our result does not mean that the ￿eld is necessarily not quantized, only that Belenchia et al.’s argument is not su￿cient to conclude that it is.

In the course of the thesis we will also do a deep dive into a famous paper by Bohr and Rosenfeld from 1933 [1] on the quantization of the electromagnetic ￿eld. ￿is paper uses a Gedankenexperiment and physical arguments to show that the uncertainties which come from applying quantum mechanics to the measurement of the electromagnetic ￿eld are the same as those coming from quantizing the ￿eld. ￿e reason for including an analysis of this paper in the thesis is to understand how earlier roadblocks in physics have been overcome by using thought experiments.

In summary, the goal of this project is to study two recent papers that come to di￿erent conclusions when tackling the problem of quantization of the gravitational ￿eld using Gedankenexperiments, to compare their arguments with a focus on the di￿erence between

4 CHAPTER 1. INTRODUCTION them, and to put their work in a historical context.

1.3 Outline

In Chapter 2, a brief summary of quantum mechanics, general relativity, and ￿eld quantization is presented. ￿is is not a comprehensive introduction to the subjects, but rather a non-technical presentation of the main points relevant for the thesis. ￿e scope of the background theory is such that the reader should be able to understand and appreciate the Gedankenexperiments that will be discussed. It will also give some historical perspective on the problem on quantum gravity and ￿eld quantization.

In Chapter 3, the method and procedure of the thesis is presented. ￿e scope of the thesis is also discussed. ￿e main methodological problems are ￿rstly how one can compare physical arguments that speak of slightly di￿erent things with slightly di￿erent assumptions, and secondly how Gedankenexperiments can be used to gain insights into a theory.

In Chapter 4, we present a short summary of “On the ￿estion of the Measurability of Electromagnetic Field ￿antities” by Bohr and Rosenfeld from 1933 [1]. We try to elucidate their main argument, which can be hard to grasp when reading the text. It is also compared to a paper by Bronstein that came the year a￿er [28], and some new calculations are done.

￿e next two chapters are presentations of two recent papers that use Gedankenexperiments to probe table-top quantum gravity. In Chapter 5, part of the paper “Two-slit di￿raction with highly charged particles: Niels Bohr’s consistency argument that the electromagnetic ￿eld must be quantized” by Baym and Ozawa from 2009 [2] is presented. Most of the ￿rst part dealing with the electromagnetic case are skipped and only the gravitational case is considered. ￿is is done because the quantization of the electromagnetic ￿eld is already an established fact, and their method in the electromagnetic case di￿ers largely from that used for the gravitational consideration. ￿e details of calculations and logical assumptions are discussed.

In chapter 6, we present “￿antum superposition of massive objects and the quantization of gravity” by Belenchia et al. from 2018 [3]. Here, both the electromagnetic and gravitational case is considered since the two arguments are analogous. ￿ere is also a part about the measurability of the electromagnetic ￿eld which refers to Baym and Ozawa’s explanation, which again ties back to Bohr and Rosenfeld’s paper on the subject. Here we also ￿nd what seems to be an error, which is discussed further in the last chapter.

5 CHAPTER 1. INTRODUCTION

Chapter 7 contains further calculations building upon the results of the presented Gedankenexperiments. By combining ideas from these papers, we ￿nd that the assumption of a quantized gravitational ￿eld used by Belenchia et al. to resolve the paradox is not necessary, leaving their argument inconclusive. ￿is is the main result of this thesis project and is presented in Section 7.3. At the same time, when correcting for a conceptual mistake done by Baym and Ozawa, which was commented on by Belenchia et al., the former’s argument actually becomes stronger.

Lastly, in Chapter 8, we discuss the mentioned results and what they mean for quantum gravity. We discuss the justi￿cation of using the Planck length as a minimal distance. We also try to view Bohr and Rosenfeld’s paper in perspective to the more recent papers. Finally some hopes and ideas for future work are presented.

6 Chapter 2

Theoretical Background

2.1 ￿antum Mechanics

When two waves of water overlap, one becomes superposed on the other, which means that their amplitudes interfere and add to each other. Without interacting they pass through each other unchanged, but what we see is their superposition. Similarly, quantum states are described by their , which is a superposition of eigenfunctions of the Schrodinger¨ equation [29]. ￿e eigenfunctions span a Hilbert space and are orthogonal to each other [30]. ￿is means they too can be superposed without interacting. A common example of a two-dimensional Hilbert space is the quantum property of a particle, which can be denoted spin up and spin down . Just as with ocean waves, di￿erent quantum |"i |#i states from the same Hilbert space can form a superposition, giving the total state k of the | i particle, as for instance 1 k = , (2.1) | i p2 (|"i + |#i) where the factor 1 p2 is a normalization factor. / If now another property of the particle, like the position, is made to be linked to the spin state, we would get an entangled state. When one hears about a particle being “in two places at once”, it means the particle is in such a spatial superposition (this is discussed more in the next section). One way to create a spatial superposition is to send a particle in the state k | i above, through a Stern-Gerlach experiment which involves an inhomogeneous magnetic ￿eld [31]. ￿e direction of the force on a particle inside this ￿eld will depend on its spin direction. ￿is means that the two parts of the state k will be pushed in opposite directions of each | i other, resulting in the position state being maximally entangled with the spin state. If the

7 CHAPTER 2. THEORETICAL BACKGROUND position states are right ' and le￿ ! , the particle state can become | i | i 1 k = ! ' . (2.2) | i p2 (|"i| i + |#i| i)

￿at the spin states and position states are maximally entangled means that if one were to measure either, the other would be know with certainty as well [30]. For instance if one were to measure the particle a￿er it went through the Stern-Gerlach apparatus as being to the right, one would instantaneously know that the particle would be spin down. Similarly, if one measures its spin, the position would become known without measuring it explicitly.

Until one performs such a measurement the superposition persists, but doing the measurement forces the superposition to “choose” only one of the eigenfunctions [32]. One therefore says that the measurement collapses the wave function. ￿is way of thinking is called the Copenhagen interpretation of quantum mechanics and is the most common interpretation and the basis of the more advanced quantum ￿eld theory. ￿ere are also several other interpretations that try to make sense of the unintuitive mathematics and properties of quantum mechanics [32, 33]. Going forward we will however use the language of the Copenhagen interpretation.

Another important concept in quantum mechanics that is not present in is that of uncertainty. ￿ere seems to be a fundamental limit to how accurate the information one can have of some parts of a system can be [34]. ￿is is manifested in complementary variables, or (not to be confused with the concept of complementarity in Section 2.1.2). ￿ese pairs of variables have a common limit on their minimal error and both can therefore not be perfectly known at the same time. ￿e most common example of such a pair is the position at some axis, say G, and the momentum along the same axis, ?G . Technically speaking, the operators corresponding to these observables do not commute, Gˆ,?ˆ = Gˆ?ˆ ?ˆ Gˆ < 0, which leads to the famous relation known as Heisenberg’s uncertainty [ G ] G G principle,

G? \, (2.3) G where G and ?G are the uncertainties in the position and momentum respectively [34]. In this way, the fact that the observables are non-commuting results in an uncertainty relation which limits the measurability of the system [35].

8 CHAPTER 2. THEORETICAL BACKGROUND

2.1.1 ￿antum Interference

When learning about quantum mechanics, the particle-wave duality of ma￿er is one of the “weird quirks” of quantum mechanics presented to students [36]. Light waves are also particles called . Particles can behave as waves, but are still detected as particles. Sometimes as waves, sometimes as particles, seems to be how light and ma￿er behaves. If light waves are particles, though mass-less, they should carry momentum and be able to knock electrons out of their atoms. If particles are waves they should be able to interfere with one another, and even with themselves.

A commonly used illustration of the wave-particle duality of ma￿er is the double slit experiment, since this duality is a prominent feature. It was ￿rst introduced by￿ omas Young in 1801 [37], and was used to try to end the debate of whether light is a wave or a particle. ￿ere was evidence present for both views, with for instance Newton and his corpuscular theory being in the particle camp [38]. Young, on the other hand, found some faults in the corpuscular theory and devised his experiment to argue for light being a wave. His original experiment di￿ers slightly from the one that will be presented here, but the principle is the same. ￿e setup of the experiment is that a beam of light is hi￿ing a wall with two small slits, and a￿er the wall a screen (see Figure 2.1.1). He noticed that the screen showed an interference pa￿ern, like that seen from two waves of water interfering, and concluded that light must be a wave. ￿e distance between the fringes of the interference pa￿ern can be calculated using the parameters of the experiment as

X = _3 !, (2.4) 5 / where X5 is the fringe distance, _ is the wavelength of the incoming wave, 3 is the separation of the slits, and ! is the distance between the slits and the screen

Many years later, Einstein presented the photo-electric e￿ect, where light seemed to behave very particle-like [39]. de Broglie also introduced the de Broglie wavelength of a particle

\ _ = , (2.5) ? where ? is the momentum of the particle [40]. Feynman described what would happen if one were to send an electron through a double slit experiment as a part of his famous lecture series at Caltech [36]. At the time it was just a thought experiment, and Feynman himself thought it would be impossible to actually test it for real, but with improving technology it

9 CHAPTER 2. THEORETICAL BACKGROUND was ultimately tested and con￿rmed in [41]. ￿e results are as follows. When an electron cannon sends one electron at a time through a double slit towards a screen, one electron at a time hits the screen at a speci￿c location. ￿is shows the particle-like behaviour of the electron. ￿e same would happen were it a bullet from a gun, to follow Feynman’s analogy [36].

It is only when one lets this process continue for a while that the “weird quantum behaviour” emerges. From what starts o￿as seemingly random noise on the screen a pa￿ern soon emerges. In Figure 2.1.2, taken from [41], we see the screen of a single electron interference experiment a￿er respectively 209, 1004, and 6235 electron impacts have hit the screen.￿ ere are areas of alternately high and low density of electron collisions, similar to how it would look if the electron was a wave. ￿is is where the wave-like features emerge. So how can this be true? Is the electron a particle when observed but a wave when not? Is the electron going through both slits at the same time in a spooky superposition and interacts with itself? Is the electron just guided by a more fundamental underlying wave? Well, these are still open questions [42]. We can calculate what should happen and experimentally verify that it does, but what it means or what it says about the actual shape of elementary particles remains up for debate.

Electron cannon

δf d

Double slit L Screen

Figure 2.1.1: Schematic view of a double slit experiment using electrons. An electron cannon sends electrons one at a time through two small slits and onto a screen where the impact is detected. A￿er some time an interference pa￿ern in the shape of alternating areas of more and less frequent impacts emerge. ￿is experiment illustrates both the wave-like and the particle- like features of electrons. ￿e distance between the fringes can be calculated as X = _3 !, 5 / where _ is the de Broglie wavelength of the incoming particle/wave, 3 is the separation of the slits, and ! is the distance between the slits and the screen. Figure by E. Rydving

10 CHAPTER 2. THEORETICAL BACKGROUND

Figure 2.1.2: ￿e build-up of consecutive single-electron double slit experiments gives rise to an interference pa￿ern, as if each particle went through both slits and interfered with itself. ￿e result a￿er 209, 1004, and 6235 electron impacts are shown. Edited ￿gure from [41].

An important aspect to keep in mind when talking about measurements in quantum mechanics is that results are o￿en the average over many tests. Since quantum mechanical measurements seem to be probabilistic in nature one o￿en implicitly refers to the average when one talks about the outcome [41]. For instance in a double slit experiment one would say the outcome of the experiment is an interference pa￿ern on the screen. However, the outcome of a single-particle double slit experiment is a single point on the screen, as mentioned above, and only a very large number of such single particle experiments gives a visible interference pa￿ern, as with a beam of light (i.e. a large number of photons).

So which is it, particle or wave? Both, or neither, seems to be the correct answer [43]. Light particles and electron waves seem just as real as light waves and electron particles. ￿e problem is that the questions itself includes concepts that we are familiar to, waves and particles. However, nothing says that the quantum world has to be anything like what we are familiar with [44]. When speaking of an electron in a double slit experiment one tends to use sentences like “it goes through the slits as a wave, but it hits the screen as a particle”. It is, however, important to always keep in mind that this way of speaking is used only to make

11 CHAPTER 2. THEORETICAL BACKGROUND it simpler for us to get some kind of mental image of what is going on. Speaking of a spatial superposition as a particle “being at two places at the same time” is also just a way of ge￿ing a mental image. What actually occurs at this scale, we still do not know.

￿is may feel unful￿lling, that we can only talk of quantum mechanics in terms of not quite accurate analogies and probabilistic measurements. ￿is is probably also why quantum mechanics has gained the reputation of being extremely di￿cult, if not impossible, to understand. Luckily, where intuition and understanding lacks, mathematics gets the job done. ￿is has given rise to the “shut up and calculate”-mentality which is common in the ￿eld [45]. ￿is way of thinking works when one wants to calculate some quantity using established formulas, but when studying the basic principles of the theory it is important not to forget what assumptions are made. Going forward, we have to keep in mind the gap between physical reality and the language used to describe it.

2.1.2 Decoherence of a ￿antum State

When a superposition is measured the corresponding wave function collapses and we get our measurement result, according to the Copenhagen interpretation. ￿e details of the exact mechanics of this process, or if it is a physical process at all, has still not found a consensus [46–48]. Most a￿empts at tackling the seem to have a “for all practical purposes”-approach [49]. ￿ere are also arguments for gravity playing a vital role in the question [50], suggesting that a new theory is necessary for a detailed description. However, if one simply creates a and let it be without explicitly measuring it, it will still interact li￿le by li￿le with its surroundings and thereby lose its quantum features. ￿is process is called decoherence and is connected to the loss of information about the system to the surroundings [51]. For a superposition to be able to interfere and create an interference pa￿ern it has to remain in a superposition until the measurement. ￿is is for instance why quantum computers, which rely on superpositions being upheld for a long time, need to be so cold (mK scale) [52].

A natural question is then: how can any quantum superposition remain superposed for more than an instant? An electrically charged particle, for instance, is at all times coupled to the electromagnetic ￿eld and therefore constantly interacts with its environment. ￿e environment should then a￿ect the superposition, which would lead to immediate decoherence. However, as stated in [53], this is not the case. ￿e￿eld will be put in a superposition which becomes entangled with the particle’s superposition. Since the two parts

12 CHAPTER 2. THEORETICAL BACKGROUND of the ￿eld’s superposition are directly coupled to the corresponding position of the particle, and the ￿eld is not radiated, it does not decohere. Or rather, it undergoes a so called false decoherence, where the coupling to the ￿eld makes the state look decohered. But if the spatial superposition is adiabatically recombined, the ￿eld superposition will also recombine and we are le￿with the original state. ￿is makes it possible to perform interference experiments with a spatial superposition that is held apart for a longer period of time, even using charged particles.

What will lead to decoherence, on the other hand, is if the ￿eld radiates. To check whether a spatially separated superposition has decohered or not, one can recombine the two parts of the superposition in an interference experiment. For this, the particle has to be accelerated so that the separated parts of the wave function again overlap [54]. However, if radiation through bremsstahlung (radiation from accelerating particles) is emi￿ed in this recombination the superposition will decohere and the interference experiment will fail [55].

An explanation for why this leads to decoherence is that information about the system leaks to the surroundings. ￿is is also connected to the concept of complementarity, which Bohr pushed as being a fundamental aspect of quantum mechanics [56]. It states that there is a trade-o￿between the information one can have about the outcome of quantum measurement, and the level of entanglement in that system [57]. For the state k de￿ned in the Eq. (2.2), | i for instance, the level of entanglement is maximal, so we can have no information about the outcome of a measurement. If we then were to do a measurement, we would have full information but no longer an entangled state. ￿is can also be a gradual process. Imagine that the maximally entangled state k is only slightly disturbed, say by measuring the spin | i in a direction di￿erent from up or down.￿ en we would get some information about the state, that it is more likely to be spin up than down for instance, but it would no longer be maximally entangled.

To give an example, take the double slit experiment described in the last section. If one sends electrons through the slits, they make a pa￿ern as if they went through both slits and interacted with themselves. If one however would try to trick the experiment by checking at the exit of one slit if the electron goes through this slit or the other, one would get which- path information of the electron. By the principle of complementarity the superposition of the electron going through both slits would then decohere, making the pa￿ern on the screen the same as that of a non-quantum particle [36].

13 CHAPTER 2. THEORETICAL BACKGROUND

2.2 General Relativity

￿antum mechanics, and its subsequent development into quantum ￿eld theory and the Standard Model, have given some of the most accurate predictions in physics. It is, however, not the whole truth for one very obvious reason: it misses gravity. As will be mentioned in Section 2.3.1, Newton introduced the concept of a ￿eld to describe gravitational interactions. In his theory, it would essentially feel the same standing on the Earth and being pulled down by Earth gravity, as it would feel standing on a (rumble-free) rocket ship accelerating at the same rate as Earth’s gravitational acceleration. Einstein noticed this, but took it one step further; what if one of these situations essentially is the same as the other? Or similarly, one cannot use any physical experiments to di￿erentiate the case when one is in a closed chamber in free-fall on Earth, from the case when one is in a closed chamber ￿oating in outer space, far from any strong gravitational ￿eld [58].

From this idea, called the , and many others, as well as the math of Riemann surfaces describing curved spaces, Einstein managed to piece together his theory of general relativity [4]. Here, gravity is not described as a force ￿eld, but rather it is the consequence of itself being curved. ￿e curvature of spacetime de￿nes how massive bodies in free-fall move, while massive bodies cause the curving of spacetime. ￿is sentence points at one of the di￿culties of general relativity, apart from the mathematical formalism itself being infamously complicated; it is highly non-linear.

What follows is a very brief, non-technical introduction of some of the central quantities in general relativity which will be of use later. We start by introducing the metric tensor

6`a, which can be thought of as a generalization of the gravitational potential in Newtonian gravity. ￿e indices ` and a can take the values 0, 1, 2, or 3, with 0 denoting the one of time and 1, 2, and 3 denoting the three of space. In this way, writing 6`a implies a4 4 matrix with 10 unique values, since it is symmetric. From taking derivatives of the ⇥ metric one can de￿ne the Christo￿el symbol as

1 W = 6WX m 6 m 6 m 6 , (2.6) `a 2 ` aX + a `X X `a where m := m mG ` is the partial derivative on coordinate G `, and repeated indices are ` / implicitly summed over (Einstein’s summation convention). ￿e Christo￿el symbols say how di￿erent points in spacetime are connected. From these, one can in turn de￿ne the Riemann

14 CHAPTER 2. THEORETICAL BACKGROUND curvature tensor

W = m W m W W _ W _ , (2.7) RX`a ` aX a `X + `_ aX a_ `X which describes the curvature of spacetime. ￿e connection between this curvature and the ma￿er in it, is the main feature of general relativity. By contracting on the ￿rst and third index we get the Ricci tensor

= W , (2.8) R`a R`Wa on which we can take the trace with respect to the metric to get the Ricci scalar as

= 6`a . (2.9) R R`a

￿ese three s all describe some aspect of the curvature of spacetime, with the number of R indices telling them apart. However, in Chapter 6 we will mention “the Riemann tensor R” without indices. ￿is still refers to the Riemann tensor as de￿ned above.

￿e theory of general relativity culminates in the Einstein ￿eld equations, which simpli￿ed (se￿ing the to zero) can be wri￿en as

1 8c⌧ 6 = ) , (2.10) R`a 2R `a 24 `a where )`a is the stress-energy tensor which describe the presence of ma￿er, i.e. energy. ￿is equation describes how spacetime curves in the presence of ma￿er. Another important equation is the geodesic deviation equation

d2G ` dGW dGX = ` , (2.11) dB2 WX dB dB where B is the parameter of a spacetime curve. It describes how an object in free-fall will move through a curved spacetime. We can now see that “spacetime tells ma￿er how to move, ma￿er tells spacetime how to curve”, as famously said by John Wheeler. While there exists solutions to these sets of di￿erential equations, they are notoriously hard to ￿nd and mostly include some kind of simpli￿cation or idealization [59].

Luckily, when considering masses on the scale of everyday life, like cars, apples, or smaller particles, the non-linearities are so small that they can rightfully be neglected and a

15 CHAPTER 2. THEORETICAL BACKGROUND

Newtonian consideration is su￿cient. ￿e gravitational considerations in the following sections are of this sort. In other words, the gravitational calculations in this work are on a sort of relativistic Newtonian basis. ￿at means the equations for ￿eld strengths and forces are taken from classical Newtonian gravity, but the ￿nite speed of propagation of the force is still taken into consideration. We will also brie￿y consider a quantum version of linear gravity which involves the as a [17, 60].

Another important idea in relativity is that of . Since there is an ultimate speed limit of information transfer, equal to the speed of light in vacuum 2, an can only in￿uence other events in its causal future, and it can only be in￿uenced by events in its causal past [61]. Two ongoing events that are spatially separated cannot in￿uence each other any faster than the time it take for light to travel between them.

2.3 Field ￿antization

2.3.1 Classical Fields and Newtonian Gravity

When thinking about forces, it is easiest to imagine some object pushing on another by direct contact. Newton realized when he was creating his theory of gravity that the mechanism that makes things, like apples, fall to the ground was also a force. ￿is force was the same as the one keeping the moon in orbit around the Earth. In these cases there are however no direct contact, so the force is said to be mediated by a ￿eld.A￿eld has a value for all points in space and in￿uences, and is in￿uenced by, all object it is coupled to. For Newton’s gravitational ￿eld these objects include anything with mass.￿ at means that an apple, the Earth, and the moon are all pulling on, and being pulled by, each other. ￿e strength of this pulling force is given by the ⌧ = 6.67 10 11 m3/kg s2, called Newton’s gravitational ⇥ constant.

More than a hundred years later, Faraday noticed that something similar seemed to be true for electric and magnetic phenomena. Any charged body in￿uences and is in￿uenced by any other, and also by nearby magnets, through what was to be called the electromagnetic ￿eld. A few decades later, Maxwell summarized this idea in the partial di￿erential equations known as Maxwell’s equations [62]. ￿ey describe the interplay between the electromagnetic ￿eld and bodies coupled to it. A feature that appears in these equations is that speed of propagation of electromagnetic waves, i.e. the speed of light, has a ￿xed value of 2 3.00 108 m/s in ⇡ ⇥ vacuum.

16 CHAPTER 2. THEORETICAL BACKGROUND

2.3.2 ￿antization of the Electromagnetic Field

At the start of the 20th century, a large shi￿happened in physics as a new area was gradually being discovered. Planck had already in the year 1900 used energy quanta to solve a problem that occurred in the theory of black-body radiation, where the energy calculations gave an in￿nite radiation energy for certain temperatures [63]. If light was emi￿ed in small energy packets, or quanta, the calculations instead matched observations. ￿e energy ⇢ of each packet was proportional to the frequency a of the emi￿ed light as

⇢ = ⌘a, (2.12) where ⌘ = 6.63 10 34 Js is the proportionality constant, now called Planck’s constant. ⇥

￿is idea of light as quantized energy seemed to be fruitful in other areas than just black bodies. When it was observed that some materials would give rise to an electric current when shined with light, Einstein applied the idea of light as a series of quantized energy packets he called the light quantum and the theory matched the observations [39]. Also in this quantum idea gave good results. If one assumed that the energy levels of atoms could only take speci￿c values, such that when jumping between them energy quanta of speci￿c magnitudes would be emi￿ed, calculations again seemed to match observations perfectly [64, 65].

A natural question to ask is then: if light is waves in the electromagnetic ￿eld described by Maxwell’s equations, but also spatially localized energy packets, how does this ￿t together? ￿e foundation of the mathematical quantization of the electromagnetic ￿eld was laid by in 1927 [66]. He used creation and annihilation operators to quantize the electromagnetic ￿eld, describing it as a sea of photons.

￿ere have also been e￿orts to create a quantum ￿eld theory of .￿ is was ￿rst a￿empted by Bronstein in 1936 [67], just a few years a￿er the electromagnetic ￿eld was successfully quantized, and later by Feynman in 1963 [68]. ￿is kind of quantized gravity treats the gravitational ￿eld as a sea of , the force carrier particle of the gravitational ￿eld, and only works for a weak ￿eld. It is therefore not a complete theory of quantum gravity, but rather a tool which can be used to look at some weak ￿eld cases [17, 60], as will be done in this thesis (see Section 6.3).

17 CHAPTER 2. THEORETICAL BACKGROUND

2.4 A￿empts at￿ antum Gravity

While both quantum mechanics and general relativity have been shown to give accurate predictions [12, 13, 69, 70], it is still apparent that something more is needed as a unifying theory. In their respective scale both theories describe the world extremely well, and both are shown to give Newtonian physics in the classical limit [71, 72]. However, problems arise in situations where both theories are needed to describe some physical behaviour. An easy to grasp example is the question of the spacetime curvature produced by a particle in a spatial superposition. If the particle is massive it will according to general relativity curve spacetime, but there is no consensus of what a quantum superposition of curved would look like [73]. Also on the high energy scale of black holes we know that something more is needed. In general relativity the center of a is represented by a singularity, a point of in￿nite density. ￿ese unavoidable spacetime singularities are thought to be unphysical, and a theory of quantum gravity might give some resolution to this problem [74, 75].

￿e search for a theory of quantum gravity has been a challenge in theoretical physics for almost a century. Many a￿empts at quantum gravity have been put forward, such as super- theory, , non-commutative geometry, and others [21]. Most a￿empts try to quantize gravity [15] while some instead argue in favor of “gravitizing” quantum mechanics [76]. What is common for all approaches is that it seems very di￿cult to test any quantum gravitational e￿ects [16]. A collider experiment, similar to how many e￿ects of quantum ￿eld theory have been tested, would need energies on the scale of the Planck energy ⇢ = \25 ⌧ 1028 eV for the quantum e￿ects of gravity to become comparable to ? / ⇠ other forces [18].p ￿e at CERN, currently the most energetic collider, is only on the scale of 1012 eV. ￿is suggests that collider experiments might not be a feasible approach.

Recent developments of table-top Gedankenexperiments presents a possibility of probing quantum gravitational e￿ects without going to extreme scales. Instead, these proposed experiments a￿ack the problem of quantum gravity from a lower energy approach [77, 78]. By combining techniques from precision measurements of quantum mechanics with some aspects of gravity, one could be able to get restrictions on some parameters or theories of quantum gravity. It has for instance been found that simply measuring gravitational coupling between quantum superpositions is in itself a sign of quantum gravity [79]. Some of the proposals of this kind are meant as suggestions for real life experiments [80] while others are meant to test the theoretical limits of the theories. ￿e Gedankenexperiments analyzed in

18 CHAPTER 2. THEORETICAL BACKGROUND this thesis are of the la￿er kind.

19 Chapter 3

Method

3.1 Comparing Arguments

In this project, I have read and in detail analyzed two recent papers on Gedankenexperiments concerning the quantization of the gravitational ￿eld. Firstly, “Two-slit di￿raction with highly charged particles: Niels Bohr’s consistency argument that the electromagnetic ￿eld must be quantized” by Baym and Ozawa from 2009 [2], and, secondly, “￿antum superposition of massive objects and the quantization of gravity” by Belenchia et al. from 2018 [3]. I also read and tried to understand two older papers: “On the question of the measurability of the electromagnetic ￿eld quantities” by Bohr and Rosenfeld from 1933 [1], and “On the relativistic extension of the indeterminacy principle” by Bronstein from 1934 [28]. Going forward, referring to “the two papers” will mean the former two, contemporary papers.

￿e two papers use somewhat di￿erent assumptions to a￿ack the problem from slightly di￿erent angles, one building upon the other. An important aspect of this work is then whether these two papers are comparable in the ￿rst place. Both papers use a Gedankenexperiment with two parties. One party is an interference experiment where a charged or massive particle is put into a spatial superposition, and the two parts of the superposition are then made to interfere. ￿e other party of the Gedankenexperiment is a detector used to measure the electromagnetic or gravitational ￿eld from the particle in a superposition. When the distance between the two sections is large enough for information not to be able to be sent between them in the time duration of the interference experiment, a paradox arises. Either complementary, a fundamental feature of quantum mechanics, or causality, a fundamental feature of relativity, seems to be violated.

20 CHAPTER 3. METHOD

For the case of a charged particle, both papers introduce a quantized electromagnetic ￿eld with vacuum ￿uctuations and quantized radiation to resolve the paradox. ￿is can be used as a strong argument for the necessity of a quantum ￿eld theory of electromagnetics. However, for the gravitational case the resolution to the paradox di￿ers between the papers. Baym and Ozawa come to the conclusion that because of the relative weakness of the gravitational force for a small mass, the detector cannot di￿erentiate the two superposed positions of the particle, and the paradox will therefore not be present in the ￿rst place. If instead the mass of the particle is large enough for the detector to be able to distinguish, then it is too massive for a successful interference experiment. On the other hand, Belenchia et al. proposes a solution in analogy to the electromagnetic case. If the gravitational ￿eld has vacuum ￿uctuations and quantized radiation, the paradox is resolved.

In this thesis I have compared the two arguments and tried to see where they di￿er, where they concur, why they have come to di￿erent conclusion, and ultimately to ￿nd out which conclusion is the right one. I have done this by ￿rst summarizing the papers in Chapters 5 and 6, where I did the calculations that where le￿out from the papers. I have then combined parts of both arguments to make my own calculations, and discussed several parts of the presented Gedankenexperiments in more detail.

In terms of time, the main part of the project was reading the two papers, doing all the calculations, and convincing myself of their physical and logical structure. Since the physics and mathematics used is not much more complicated than what one would be taught in an introductory course in and quantum mechanics, the di￿culty was of a more conceptual and logical nature. Seeing whether the assumptions made and the formulas used were reasonable, and looking for logical loop-holes was a large part of the e￿ort. Since the two articles had di￿erent conclusions to a similar problem it was clear from the start that something was amiss, and my struggle was in ￿nding this point of discord.

3.2 Thought Experiments

￿ought experiments, or Gedankenexperiments as they are o￿en called, can be used to test the limits of theories in physics. While the word is also used for stories or situations provoking thought and re￿ection for the reader, like the trolley problem [81] or the ship of ￿eseus, this is not the kind we will look at. Rather, we are here interested in thought experiments which are similar in form as physical experiments, even thought they might be infeasible or practically impossible to perform. While real experiments can show how well a theory stands

21 CHAPTER 3. METHOD up against reality, a thought experiment of this kind can show how well it stands up against itself. By ￿nding some ￿aw in the inner logic of a theory and expressing it as a paradox, one is forced to rethink the logic to see where it breaks down.

￿is is the logic used in the two Gedankenexperiments presented in the upcoming chapters 5 and 6. For the electromagnetic case, we are ￿rst presented a situation where it seems that either causality or complementarity has to be violated. ￿e resolution is to quantize the electromagnetic ￿eld, which makes the paradox go away. In the gravitational case the paradoxical situation is the same, with either causality or complementarity being seemingly violated, but the resolution di￿ers between the two papers. An argument like this thus shows that the assumed theory must be wrong in some way, or something must be missing. Here we also see that di￿erent resolutions can be presented for the same paradox. ￿e question then becomes if the two resolutions are in fact the same but with di￿erent words, if one resolution is stronger and leaves the other redundant, or if they both are su￿cient and mutually exclusive. ￿e answer in our case is of the second kind.

3.3 Scope of the Thesis

￿e subject of quantum gravity is a vibrant and lively ￿eld with new ideas being born with rapid frequency. In this thesis we will not try to give any sort of full overview of the ￿eld. We will also not try to answer the question of whether the gravitational ￿eld is quantized or not. ￿e scope is instead to discuss whether the arguments presented in the next chapters hold up against themselves and each other. Within this scope one can, however, answer the question of whether or not the arguments to be presented can rightfully speak in favor of or against a quantized ￿eld. In the discussion in Chapter 8, the results from the analysis will be set up against a larger picture and we consider what can be said in general terms from these results.

22 Chapter 4

Bohr & Rosenfeld

4.1 My Take on Bohr & Rosenfeld

￿is almost 80 years old paper is famous for being di￿cult to get through and even more di￿cult to understand. Not only is the language convoluted and the physics complicated, but it is also unclear what the actual conclusion is. A￿er reading the paper a few times and discussing its implications with my supervisors, this is the extent of my understanding:

￿e main question under consideration is, in their words, concerning the connection between the limitation on the measurability of the electromagnetic ￿eld quantities and the of ￿elds. In an earlier book by Heisenberg [34], he a￿empted to show that this connections is similar to that between limitations in measurements of complementary quantities, represented by Heisenberg’s @? \, and quantum ⇠ mechanics. In other words, the fact that one cannot measure both the position and momentum of a quantum particle to an unlimited accuracy is tightly connected to the quantum nature of ma￿er.

At the time, the quantum mechanics we know today was still being developed, both through theoretical and experimental work. ￿e uncertainty principle can be found with purely physical arguments, but it can also be rigorously derived by assuming quantum mechanical commutation relations like @,? = 8\.￿e limitation in measurability which one can ￿nd [ ] independently of quantum arguments, thus can be explained by quantum mechanics. ￿is is a strong argument in favor of the quantum nature of ma￿er. ￿e main question is then whether the same is true for ￿elds, which had been found to have a limitation on measurability.

Before answering this, Bohr and Rosenfeld state the importance of not using a point charge

23 CHAPTER 4. BOHR & ROSENFELD to measure the electromagnetic ￿eld. In a quantum theory of electrodynamics, the ￿eld quantities are not represented by point functions, but are rather de￿ned over spacetime regions, they say. ￿ey criticise an earlier paper by Landau and Peierls [82], where the restrictions on the measurability of the ￿eld exceed those found from assuming a quantum ￿eld theory. ￿eir critique is based on the fact that Landau and Peierls use a single electrically charged mass point. Bohr and Rosenfeld state that one can only speak meaningfully of the average of the value of an electromagnetic ￿eld component in a region of spacetime, and not of the value at a point.

￿ey then tackle the main question by using purely physical arguments, as well as the uncertainty relation for ma￿er, which is separate from the ￿eld consideration. ￿ey ￿nd that the limitation of measurability of the ￿eld stemming from uncertainty in the test body corresponds exactly to that found by assuming a quantum ￿eld theory of electromagnetics. By this the conclusion is that the electromagnetic ￿eld is quantized. In other words, they assume that the uncertainty principle holds for the test mass used in the ￿eld measurement, and from this derive the same uncertainty relations that they ￿nd when quantizing the ￿eld directly.

4.2 Bronstein and Gravity

One year a￿er Bohr and Rosenfeld published their paper, Matvej Bronstein wrote a paper that builds on their result [28]. His original paper is available only in Russian and German, so in a combined e￿ort with my supervisor Prof. Aurell, we created an English translation. ￿is version can be found in Appendix B.

To summarize, Bronstein uses the ideas and equations from Bohr and Rosenfeld’s paper to ￿nd an formula for the uncertainty in the momentum as

⌘ 42 ? CG, (4.1) G ⇠ G + + where 4 is the charge of the test particle, + is the volume of the test particle, C is the duration of the momentum measurement, and G is the uncertainty in position of the test particle. As in Bohr and Rosenfeld, Bronstein got the uncertainty of the ￿eld component from the equation

? ⇢ G , (4.2) G ⇠ d+) where d is the charge density of the test particle and ) is the duration of the ￿eld

24 CHAPTER 4. BOHR & ROSENFELD measurement. ￿is means that the highest possible accuracy of a ￿eld measurement is obtained by minimizing the uncertainty in momentum. Doing this minimization gives

⌘+ G (4.3) ⇠ 42C as the minimizing value, which in turn gives a minimal ￿eld uncertainty as

1 ⌘C ⇢ . (4.4) ( G )<8= ⇠ ) +

While this expression can seemingly be made as small as desired by choosing a su￿ciently small C, Bronstein ￿nds two mechanisms that prevent this.

Firstly, there is G < 2C.￿is is because the uncertainty in position cannot be larger than when the particle is moving at the speed of light for the duration of the momentum measurement. Combined with Eq. (4.3) and (4.4) gives

\2 3 ⇢ & / . (4.5) G <8= 1 3 2 3 1 3 ( ) 4 / )+ / d /

Secondly, we have G + 1 3 because the movement of the particle during the measurement ⌧ / is assumed to be very small. ￿is gives the limit

\ ⇢ , (4.6) G <8= 4 3 ( ) )+ / d where the important di￿erence going forward is the di￿erence in the power of d.￿e right hand side of the ￿rst equation can also be rewri￿en to ⌘2 3 )+1 342 3 while the second can / / / / be rewri￿en to ⌘ )+1 34. We here see that for a given time ) and volume + , the di￿erence in / / the two limits lies in the order of the charge factor.

To make the accuracy of the ￿eld measurement as large as possible for a given spacetime volume, we would therefore want a charge density that is as large as possible. ￿is means that the ￿rst equation, Eq. (4.5), will be what limits the accuracy of ￿eld measurement, with a larger charge giving be￿er accuracy. Bronstein’s conclusion is thus that the limiting factor will be in the atomic nature of ma￿er, namely that the amount of charge one can insert into a given spacetime volume is limited.

￿e conclusion of Bohr and Rosenfeld’s consideration of the measurability of the electromagnetic ￿eld components was that the ￿eld should be quantized for consistency.

25 CHAPTER 4. BOHR & ROSENFELD

Bronstein builds on this argument, and ￿nds that the exact limit on measurability will come from the limit on the charge density of an object. Two years later Bronstein also considered using arguments similar to Bohr and Rosenfeld’s to quantize the gravitational ￿eld, but realized it would not be doable [67].

Let us now consider an argument based on the Bronstein paper we just summarized, but for the case of the gravitational ￿eld. What prevents us from doing in￿nitely accurate gravitational ￿eld measurements? ￿is is assuming Bohr and Rosenfeld’s arguments, which Bronstein used to get his equations, can be used analogously in the gravitational case. If we use the exact same equations, only changing 4 to < for the mass of the test particle, and changing d to mean the mass density of the test particle, we would get a similar equation to Eq. (4.2), ? 6 G , (4.7) G ⇠ d+) for the accuracy of a gravitational ￿eld measurement. When it comes to the uncertainty of the momentum, the ￿rst term would be the same as in Eq. (4.1) since the Heisenberg uncertainty principle G?G & ⌘ is always true. ￿e second term is more elusive and hard to grasp, but it is said to come from the uncertainty in the radiation, i.e. the ￿eld created by the test particle as it is accelerated during the measurement. Assuming something similar can be said in the gravitational case, the rest of the argument should be able to be made in a similar fashion and we arrive at the inequality ⌘2 3 6 & / (4.8) G <8= 1 3 2 3 ( ) )+ / < / as the limit of accuracy in measurement of the ￿eld component.

As we know, there is a limit to how much mass can be concentrated into a given volume of space without it collapsing into a black hole, given by the Swartzchild radius. For a given sphere of radius AB, if its mass is more than

A 22 " = B , (4.9) 2⌧ it will collapse into a black hole with an at AB. Inserting this into Eq. (4.8) we get the minimal physically possible uncertainty in a gravitational ￿eld measurement as

2⌧⌘ 2 3 6 & ( ) / . (4.10) G <8= 4 3 ( ) )AB2 /

For a set density of the test particle, that of an object on the brink of collapsing into a black

26 CHAPTER 4. BOHR & ROSENFELD hole, we see that it is the size (or equivalently mass) of the particle and the time of the measurement that decide the limits of the measurement accuracy. Assuming we want to use a gravitational ￿eld measurement in the same way as is done in the Gedankenexperiments presented in the next two chapters, we would need to probe quantum mechanical e￿ects at the same time. ￿is would, however, greatly limit the size of the test particle and the time duration of the experiment since these e￿ects are more di￿cult to both obtain and sustain for a more massive particle [23].

27 Chapter 5

Baym & Ozawa

5.1 Introduction

￿e result of the gravitational analysis in the paper by Baym and Ozawa [2] is that even without quantization of the ￿eld, if using a mass large enough to be able to distinguish the upper and lower path of a double slit experiment, the interference fringes on the screen will be smaller than a Planck length and therefore not measurable. ￿e conclusion is that, unlike in the electromagnetic case, the consistency of the presented thought experiment is not dependent on the quantization of the gravitational ￿eld.

￿e setup of the thought experiment in question is shown in Figure 5.1.1. A particle of mass < is travelling a distance ! through a double slit of separation 3 towards a screen where an interference pa￿ern will be seen. A distance ' from the upper slit there is a detector to measure the gravitational ￿eld to get which-path information of the particle.￿ e detection will take place in the causal future of the interference experiment, which means that time it takes for the particle to hit the screen ! E, where E is the characteristic speed of the particle, / must be smaller that the time it takes for information about the experiment to reach the detector ' 2. Assuming causality holds, if the interference experiment is successful and / which-path information is gained one has a violation of complementarity (introduced in Section 2.1.2). In other words, if the detector for instance would measure that the particle went through the upper slit, this information should have decohered the superposition and it should not have been able to create the interference pa￿ern which is already there. On the other hand, if having the detector present would somehow make the interference pa￿ern disappear, this would be a violation of causality (introduced in Section 2.2) since the interference experiment

28 CHAPTER 5. BAYM & OZAWA

Figure 5.1.1: ￿e setup of the thought experiment. A particle travels a distance ! from the point 0, throught a wall with two slits of separation 3, onto a point 1 on a screen. A ￿eld detector is positioned in the plane of the slits a distance ' away, such that the detection happens in the causal future of the interference experiment. A paradox arises where either complementarity or causality seems to be violated. Figure from [2].

happens before the detection. ￿is is the paradox which the paper sets out to resolve.

5.2 Electromagnetic Case

Without going into details, the case of an electrically charged particle going through the double slit, with the detector being an electromagnetic ￿eld detector, the paradox is solved by the quantization of the ￿eld. In a quantized electromagnetic ￿eld there are vacuum ￿uctuations which set a limit on the sensitivity on the detector. If the charge of the test particle is too small, the detector will not be able to get which-path information since the ￿eld signal from the particle will be buried in noise from the ￿uctuations. ￿is means the paradox is evaded. On the other hand, if the charge is too large the particle will emit quantized radiation when “rounding the corner” of the slit and thereby decohere such that there will be no interference pa￿ern. Again, the paradox is evaded. ￿us, with vacuum ￿uctuations and quantized radiation, which come with quantizing the electromagnetic ￿eld, the paradox is resolved. ￿is is a strong argument for why a quantum ￿eld theory of the is necessary for consistency in physics.

29 CHAPTER 5. BAYM & OZAWA

5.3 Gravitational Case

In the gravitational case we have a charge-less particle of positive mass and a detector to measure the di￿erence of the gravitational ￿eld from a particle going through the upper and the lower slit. To do this measurement we use a highly sensitive laser interferometer. Two mirrors of mass " are positioned a distance ( apart such that their midpoint is a distance ' away from the upper slit, in line with the same plane as the slits (G-direction, upwards in ￿gure 5.1.1).￿ e deviation of the mirrors from the equilibrium distance ( is given by [ C . ( ) As the massive particle takes the upper or lower path it will create a Newtonian gravitational ￿eld q G,C which will alter the positions of the mirrors slightly. We assume the mirrors ( ) are tied together with a spring such that they work as a harmonic oscillator with frequency l. We de￿ne the undisturbed position of the mirrors in comparison with the upper slit as G = ' ( 2 and G = ' ( 2 for the lower and upper mirror respectively, and their midpoint / + + / as G = '. Going forward, a dot over a variable represent a time derivative 5 = m5 mC, while 0 § / an apostrophe represents a space derivative 5 = m5 mG.￿e following calculations follow 0 / those done in Baym and Ozawa’s paper [2], with some intermediate steps added.

Considering the equations of of the two mirrors, we have

1 2 "G = "q0 G "l G G ( , •+ ( +)2 [ + ] (5.1) 1 2 "G = "q0 G "l G G ( , • ( )+2 [ + ] where the ￿rst term of the right hand side is the Newtonian force from the massive particle and the second term is the harmonic oscillator trying to go back to its equilibrium position. ￿e di￿erence in signs between the two equations comes from the fact that as the mirrors move away from each other the lower mirror will be accelerated upwards while the upper mirror will be accelerated downwards, to get to the equilibrium position. We now write

G = G0 ( [ 2 to get for G ± ±( + )/ +

1 1 2 [ = l [ q0 G ( [ 2 . (5.2) 2 • 2 ( 0 +( + )/ )

￿e last term is linearized in [ and q00 as

q0 G ( [ 2 q0 G q00 G ( [ 2 q0 G q00 G ( 2, (5.3) ( 0 +( + )/ )⇡ ( 0)+ ( 0)( + )/ ⇡ ( 0)+ ( 0) / where we assume ( is small, and [ and q00 are even smaller. ￿is gives the equation of motion

30 CHAPTER 5. BAYM & OZAWA as 1 1 2 1 [ = l [ q00 G ( q0 G . (5.4) 2 • 2 2 ( 0) ( 0) We then do the same procedure for the equation we get for G in (5.1) to get

1 1 2 [ = l [ q0 G ( [ 2 . (5.5) 2 • 2 ( 0 ( + )/ )

￿e linearized￿ eld becomes

q0 G ( [ 2 q0 G q00 G ( 2, (5.6) ( 0 ( + )/ )⇡ ( 0) ( 0) / which gives the equation

1 1 2 1 [ = l [ q00 G ( q0 G . (5.7) 2 • 2 + 2 ( 0) ( 0)

By now taking the di￿erence between (5.4) and (5.7) we get the equation

2 [ = l [ q00 G (. (5.8) • ( 0)

(In the original paper the factor l2 was not squared, but this must have been a typo.)

It is then assumed [ 0 = [ 0 = 0 and that the potential is constant during the measurement, ( ) 0( ) which would mean the particle comes in from in￿nity, or at least from so far away that this assumption is reasonable. It also means the particle is virtually on the upper or lower track during the whole experiment, and does not move from a centered position up (or down) and then back again. ￿is approximation leads to simpler calculations and a higher lower bound, which in this case is alright. We also assume q 0 = 0 and solve the di￿erential 0( ) equation.

As a homogeneous solution we get

[ = sin lC ⌫ cos lC, (5.9) ⌘ + and a particular solution is 2 [ = q00 G ( l . (5.10) ? ( 0) / Using the initial conditions we also get

q G ( = 0,⌫= 00 ( 0) , (5.11) l2

31 CHAPTER 5. BAYM & OZAWA which gives the solution in the paper

1 cos lC [ = q00 G ( . (5.12) ( 0) l2

￿e case C = 0 will here correspond to the instance the ￿eld from the double slit experiment can be felt by the measuring apparatus. Since q00 is the second derivative of the potential it will go as 1 '3.￿e needed accuracy for q is the di￿erence of the￿ eld from the upper and / 00 lower path. ￿is becomes

1 1 q00 G = ⌧2< ( 0) '3 ' 3 3 ✓ ( + ) ◆ (5.13) 3'23 3'32 33 ⌧<3 = 2⌧< + , '6 3'53 3'432 '333 ⇠ '4 + + + where < is the mass of the particle, ' is the distance from the upper slit to the measuring apparatus, and we do not consider the extent of the measuring apparatus. From the Taylor expansion of cos lC we can see that

lC 2 lC 4 lC 2 1 cos lC = 1 1 ( ) ( ) ... ( ) . (5.14) + 2 24 +  2

￿us the accuracy needed for [ becomes

⌧<3 lC 2 ⌧<3() 2 ⌧<3 [ ( ( ) < < , (5.15)  '4 l2 '4 '22 where the last inequality holds if () 2 '3 < 1 22.￿is can be split into ) ' 1 2 and ( '. / / /  /  ￿e￿rst inequality comes from the fact that we do not want any back-reaction from the measurement so we assumed ' 2) , with) ! E where ! is the distance the particle travels ⇡ / and E its speed. ￿e second inequality ( ' means the distance between the mirrors of the  measuring apparatus is smaller than that to the slits. ￿is is reasonable since we considered ( to be very small when doing the Taylor expansion. We can write

⌧< 2 3 2 ⌧<2 3 2 [ 2 = ;2 , (5.16) ( )  22 ' \2 ? ' ✓ ◆ ✓ ◆ ✓ ◆ since ; = \⌧ 23.￿is gives ? / p [ 2 ' 2 < 2 , (5.17) ; 3  < ✓ ? ◆ ✓ ◆ ✓ ? ◆

32 CHAPTER 5. BAYM & OZAWA with < = ⌧ \2. For distinguishability we need [ ; > 1, which gives that the mass of ? / / ? the particlep has to be larger than the Planck mass by a factor of ' 3.￿e Planck length limit / here enters through a consideration of the standard quantum limit, which is a limit on the accuracy of a measurement of the displacement of a massive body [83]. For the mirrors this can be wri￿en as X[ > \) ", where) is the time between measuring the relative positions / of the mirrors, and " isp their mass. For the mirror not to turn into a black hole we need the mass to satisfy " < (22 4⌧, which gives the quantum limit as /

X[ < ; 2) (. (5.18) ? / p Considering the fringe pa￿ern on the screen of the double slit experiment, we have that the distance between fringes is ! X = _ . (5.19) 5 3 ￿e wavelength is in this case the de Broglie wavelength of the particle,

⌘ \) _ = , (5.20)

33 Chapter 6

Belenchia et al.

6.1 Introduction

Belenchia et al. [3] present an argument with a thought experiment that suggests that the gravitational ￿eld should be quantized. In summary, if the gravitational ￿eld lacks quantized radiation or quantum ￿uctuations, the thought experiment in Figure 6.1.1 will lead to a violation of causality or complementarity for di￿erent cases.

Figure 6.1.1: ￿e arrangement of the though experiment taken from [3]. Alice has a spatial superposition with distance 3 and performs a recombination in time ). A distance ⇡ away Bob has a particle in a trap that he can either keep closed, or open to measure its change in position XG a￿er a time )⌫. Figure from [3].

We have two parties, Alice and Bob, each with a particle, and separated by the distance ⇡.

34 CHAPTER 6. BELENCHIA ET AL.

Alice has in the past performed a Stern-Gerlach experiment to create a spatially separated superposition with the distance of separation 3. Bob has his particle in a trap which, if released, will be a￿ected by the ￿eld from Alice’s particle and thereby have its position changed. At a time C = 0, Bob decides to keep his trap closed, or to open it and measure the change in position of his particle a￿er a time )⌫.￿e di￿erence in position of Bob’s particle given which path Alice’s particle takes is indicated by XG. At the same time, Alice starts to recombine her particle and checks for quantum interference a￿er a time ).

6.1.1 The Paradox

To see how this thought experiment seems to cause a paradox, we assume 2) < ⇡ and

2)⌫ < ⇡ which means Alice and Bob will ￿nish their respective experiments before the information of this has the time to reach the other party (shown by the striped diagonal line in Figure 6.1.1). If now Bob would open his trap, the particle would be a￿ected by the ￿eld from the state of Alice’s particle a time ⇡ 2 ago. Assuming Alice prepared her superposition long / before this, the ￿eld that Bob’s particle feels is that of the superposition. Bob’s particle will then become entangled with Alice’s because the ￿eld from the right part of the superposition will be stronger than that of the le￿, leading to di￿erent ￿nal positions of Bob. If Bob can di￿erentiate these ￿nal positions he can get which-path information of Alice’s particle. ￿at is, by measuring his particle’s position he will instantaneously “collapse” Alice’s superposition by making her particle choose one path or the other. Alice will then not be able to observe quantum interference in her experiment. Since Bob keeping his trap closed would not cause this to happen, Bob can e￿ectively send one bit of information to Alice superluminally, close or open. ￿is would violate relative causality which states that no information can be sent faster than 2.

On the other hand, if Alice’s superposition would not collapse and she could successfully perform her quantum interference experiment, this would be a violation of complementarity. One cannot have complete information of the position while still having the particle in a spatial superposition (see Section 2.1.2).

6.2 Electromagnetic Case

We ￿rst assume that the particles of Alice and Bob are charged with @ and @⌫ respectively. When Alice separates her particle and creates the superposition, the state of the electromagnetic ￿eld will also turn into a superposition of the ￿eld from a particle to the

35 CHAPTER 6. BELENCHIA ET AL. le￿ U and the ￿eld from a particle to the right U . Since these ￿elds are di￿erent and | !i | 'i directly coupled to the position of the particle, the total superposition can avoid decoherence. ￿is is discussed in [53] in more detail. When the superposition is made and Alice’s particle is entangled with its environment through the ￿eld, a so called false decoherence takes place. As long as the superposition is brought together adiabatically, the di￿erent states of the environment will also recombine and quantum interference between the di￿erent parts of the superposition can still be observed. ￿e initial state of Alice’s particle is then given by

1 = ! U! ' U' k ⌫ , (6.1) | i p2 | i |#i | i + | i |"i | i ⌦ | i ⇥ ⇤ where ! ' is the wavefunction of the le￿(right) part of Alice’s superposition, | i (| i) shows that the spatial position is maximally entangled to the spin of the particle, |#i (|"i) and U U is the static ￿eld produced by the charge of the particle as described above. | !i (| 'i) k is the initial state of Bob’s particle, which is separable from Alice’s. We assume that Bob’s | i⌫ trap is such that Alice’s particle is not a￿ected by Bob’s, and vise versa, if the trap remains closed.

6.2.1 ￿antum Fluctuations

Belenchia et al. states that when averaged over a spacetime region of scale ', the quantum ￿uctuations of the electric ￿eld are of order

1 ⇢ , (6.2) ⇠ '2 where we use 2 =\=n0 = 1, and ' is the linear dimension of the spacetime region we are looking at. To see why this is the case we have to look at the explanation made in Baym and Ozawa [2], which in turn is based on Bohr and Rosenfeld’s argument [1]. Landau and Peierls had earlier found that trying to measure the e￿ect of a￿ eld on a point charge in the quantum range would lead to diverging uncertainty [82]. So instead of a point charge, Bohr and Rosenfeld imagined taking the average over a spacetime region. ￿eir measurement setup is that one has two cubical boxes of the same volume + = '3 and mass M, but with opposite charge &. Box ⌫ is ￿xed while Box can move freely, but because of the Coulomb a￿raction the two boxes stick together in their . Assuming there are no other forces, if A is moved a distance G from its original position, the restoring Coulomb force to bring it back would be 4c&2G = , (6.3) ' +

36 CHAPTER 6. BELENCHIA ET AL. when the still overlap. ￿is distance dependence come from the fact that the boxes and ⌫ overlap such that there is no restoring force when G = 0, and the distance between the two opposite “charge centers” will stay the same as long as the boxes overlap. We then add an external electric ￿eld ⇢ which gets averaged over the boxes, and look at the change in momentum of box a￿er a time ) 0. From Newton’s second law of motion we ￿rst have

4c&2G ? = = &⇢ , (6.4) § + ’ which gives the momentum of a￿er a time ) 0 as

2 4c& G) 0 ? = &⇢)0 . (6.5) +

Here we assume the change in position G to be small enough to be negligible. Rearranged we get the ￿eld as 4c&G ? ⇢ = . (6.6) + + &) 0 ￿is is where we go into the quantum realm. Given the uncertainty relation G? & 1 \ we ( ) also get a minimal certainty in the electric ￿eld. By writing

4c&G 1 ⇢ & , (6.7) + + &) 0G and minimizing on G we get

m ⇢ 4c& 1 + ( ) = = 0 G = . (6.8) m G + &) G2 ) 4c&2) ( ) 0 0 Inserted into the expression in Eq. 6.7 we get a lower limit on the precision of a measurement of the electric ￿eld as 1 c\ ⇢ & = 4 . (6.9) +)0 n0+)0 ! Assuming we are measuring the ￿eld from an experiment that takes place during time ) ,we should at most have) 0 ) since a longer measurement would not give any extra information. ⇠ … In the same way the measuring box should at most be of size ' ) (in natural units) since ⇠ a larger box would again not give any more information. We now think of ' and ) 0 as the sides of a four-dimensional measurement box in spacetime and see that since ) ' we have 0 ⇠

1 1 \2 1 ⇢ 4 = 2 = 2 , (6.10) ⇠ ' ' n0 ' !

37 … CHAPTER 6. BELENCHIA ET AL. as was assumed in Belenchia et al.

To see how this translates to an uncertainty in position we use Newton’s equation of motion for charged particle in an electric ￿eld

<G = @⇢, (6.11) • and integrate over a spacetime scale ' to get

@⇢ @⇢ 2 @ \ @ G = dCdC0 ' = , (6.12) < ⇠ < ⇠ < n 23 < π' π' 0 ! because of Eq. (6.2). We see that this is independent of the spacetime-scale '. We have thus found that we cannot use an electromagnetic ￿eld to localize a classical particle be￿er than to its charge radius @ <. Below this scale the ￿uctuations of the ￿eld are to large to de￿ne / the position exactly. Going back to the thought experiment, this means the ￿eld from Alice’s particle must produce a relative displacement of Bob’s particle

\ @ > ⌫ XG @⌫ <⌫ = 3 (6.13) / n02 <⌫ ! for him to be able to get which-path information.

Now looking at Alice’s particle, the spatial superposition creates an e￿ective dipole with e￿ective dipole moment = @ 3.￿e word “e￿ective” is being used since we are not D dealing with an actual dipole, but rather something that mathematically looks the same as a dipole. As we are looking not at the strength of the ￿eld but rather the di￿erence in ￿eld strength from the two parts of the superposition, one can still think of this as an e￿ective dipole. ￿e di￿erence of the two￿ elds is the ￿eld from a positive charge minus the ￿eld from another positive charge ⇢ = ⇢ ⇢ , which mathematically is the same as the ￿eld from 1 2 a positive charge plus the ￿eld from a negative charge ⇢ = ⇢ ⇢ , as is the case for a 1 +( 2) dipole. Following the calculations in Belenchia et al, with some intermediate steps added, we

￿rst have the electric ￿eld felt at r from a point charge in r0 as

@ r r0 ⇢ r r0 = . (6.14) ( ) 4cn r r 3 0 | 0| We put Alice’s particle in the origin and Bob’s particle on the G-axis in position r = ⇡, 0, 0 . ( ) Furthermore, we assume a distance ⇡ from the midpoint of Alice’s spacial superposition

38 CHAPTER 6. BELENCHIA ET AL. to Bob’s box, and therefore a distance 3 2 between the midpoint and each part of the / superposition. ￿is gives

@ @ X⇢ = ⇢! ⇢' = ⇢ ⇡ 3 2 ⇢ ⇡ 3 2 ( + / ) ( / )⇠ ⇡ 3 2 ⇡ 3 2 | + 2 | | 2 | 2 2 3 3 (6.15) @ ⇡ 2 @ ⇡ 2 2@ ⇡3 @ 3 + = = D . ⇣ ⌘ ⇣ 2 ⌘ 3 4 3 3 2 3 2 4 2 2 ⇠ ⇡ ⇡ ⇡ ⇡ ⇡ 3 2 2 + ⇣ ⇥ ⇤ ⌘ ⇥ ⇤ ￿us we have that the di￿erence of the two parts of the electric ￿eld that Bob’s particle feels is of order ⇡3 = n ⇡3 . To see how this e￿ects Bob’s particle we use Newton’s equation D/ ( D/ 0 ) of motion for a particle in an electric ￿eld to get the di￿erence in change of position that these two ￿elds would create.

@⌫ @⌫ 2 @⌫ 2

D) 2 > 1. (6.17) ⇡3 ⌫

If this inequality is broken, Bob’s particle will not be able to get which-path information from the ￿eld emi￿ed by Alice’s particle because the vacuum ￿uctuations will be larger than the needed precision. ￿e expression in Eq. (6.17) is given in natural units, that is with

2 =\=n0 = 1. Reintroducing SI units and doing dimensional analysis gives the inequality

n \ D) 2 > 0 . (6.18) ⇡3 ⌫ 23 We see that for a smaller e￿ective dipole moment or large distance between Alice and Bob, the ￿eld will not be strong enough for Bob’s particle to feel the di￿erence of the two paths. … Similarly, given the other parameters, a certain time is needed for the di￿erence of the two ￿elds from Alice’s particle to a￿ect Bob’s particle enough for its positional di￿erence to be measurable.

6.2.2 ￿antum Radiation

When charges are accelerated, they radiate energy. When Alice recombines her particle she has to accelerate the two parts of the superposition from a relatively constant distance to start moving together. ￿erefore, her particles will radiate when being recombined, which would

39 CHAPTER 6. BELENCHIA ET AL. entangle her state with the emi￿ed photons, making her superposition decohere. ￿is is due to the complementarity relation between decoherence and entanglement: an increase in one necessarily leads to a decrease in the other. However, if the recombination is done slowly enough, that is adiabatically, maybe she can evade the radiation. ￿e energy ￿ux from a classical dipole is of the order 2.￿is can for instance be seen from Larmor’s equation ⇠(D• ) [84] for the power emi￿ed by an accelerated charge, which states

2 @202 % = 3 , (6.19) 3 4cn02 where 0 is the acceleration. As we have set 2 = n = 1, and we have @202 = @232 = 2 we 0 • (D• ) see that the relation % 2 holds. To get the total energy we can rewrite this as ⇠(D• ) 2 2 2 = dCdC0 = ) = D , (6.20) D D• D• )(D• ) ) 2 π) π) ! since the recombination will take time ), and we assume a constant accelerations of the particle. Now, integrating the ￿ux over the recombination time we get the total radiated energy during the recombination is of order

2 2 1 D2 ) = 3 D2 ) . (6.21) E⇠ ) ! n02 ) ! © ™ ≠ Æ ￿e calculations up to this point have been classical, but we now consider that the radiation ´ ¨ will come in the shape of photons. For a process of time ), the lowest possible frequency of a radiated photon is l 1 ) .￿is is because the time needed to create one period of such 0 ⇠ / a photon is at least ). Assuming the largest number of emi￿ed photons, the total radiated energy would be proportional to this frequency and the total number of emi￿ed photons as

l0# .￿is means E⇠ 2 # E D . (6.22) ⇠ l ⇠ ) 0 ✓ ◆ If we want no emi￿ed photons we need # < 1 which in turn gives the limit

D < 1. (6.23) )

Doing dimensional analysis on this expression gives

3 D < n0\2 . (6.24) ) p 40 CHAPTER 6. BELENCHIA ET AL.

￿inking intuitively on this expression we see that if we have a too strong dipole that needs to be recombined in a too small time frame, the acceleration has to be large, which would lead to radiation of photons.

6.2.3 Solving the Paradox

With the two inequalities we found in Eq.s (6.17) and (6.23) the apparent paradox can be solved. Still using natural units, we consider the case ) < ⇡,)⌫ < ⇡ for which we got a violation of either causality or complementarity, and ￿rst look at > ) . For this case D we have from (6.23) that Alice’s particle will emit radiation as it is recombined, which will make her superposition entangle with the environment and therefore decohere. ￿is means Alice’s particle will decohere independently of whether Bob chooses to release his particle or not.

Assuming the electric ￿eld was not quantized, and would not emit quantized radiation that entangles Alice’s particle with the environment, Alice would be able to perform her recombination if Bob’s trap were closed. Had Bob instead opened his trap, he would have been able to get which-path information. If Alice still would be able to perform the recombination without decoherence, this would be a violation of the complementarity principle which in this case states that Alice’s particle cannot remain in a superposition and at the same time fully entangled with Bob’s. If, instead, Alice’s superposition would decohere because of Bob opening the trap, this would violate relativistic causality since the information would have had to reach Alice superluminally. ￿us, quantized radiation is in this case necessary for the solution of the paradox.

If we instead assume < ) , we get that ) 2 < ⇡3, so from Eq. (6.17) we have that Bob D D ⌫ will be unable to measure his particle accurately enough to get which-path information from Alice’s particle’s electric ￿eld. For this case Alice will be able to perform her recombination adiabatically, again independently of what Bob chooses to do. If we here assume there were no quantum ￿uctuations in the ￿eld, we would not get the inequality in Eq. (6.17). ￿is means Bob would be able to get which-path information, and we get the same problems of causality and complementarity as described in the last paragraph. ￿us, quantum ￿uctuations are also necessary for a consistent solution.

41 CHAPTER 6. BELENCHIA ET AL.

6.3 Gravitational Case

Still following the paper by Belenchia et al., we now assume Alice and Bob have charge-less particles of mass < and <⌫ respectively, and that the experimental setup is otherwise the same. What we will show is that the same arguments for why both quantum ￿uctuations and quantum radiation are needed for consistency, hold for the gravitational case as well. If this is true, it would be supportive evidence for a quantized gravitational ￿eld. One di￿erence from the electromagnetic case is that one cannot block a gravitational ￿eld, so Bob’s particle would create an e￿ect on Alice’s particle and change the phase factor of the di￿erent parts of Alice’s superposition di￿erently, even from inside the trap. However, since Bob’s particle would remain stationary it would not become entangled and ruin the superposition.

6.3.1 ￿antum Fluctuations

For this part we assume gravity as a linearized quantum ￿eld, as discussed in Section 2.3.2. It is a widely accepted result that our understanding of the physical world breaks down around the Planck scale [85]. Furthermore, since general relativity lacks a background , one can only talk about the relative distance between two particles, which thus cannot be speci￿ed more precisely than a Planck length. Belenchia et al. give an argument for this which follows a similar pa￿ern as the one used for the electromagnetic case.

We ￿rst look at how the position of a body will ￿uctuate. In general relativity one cannot de￿ne a position without relating it to something else, so we look at the distance between two bodies. ￿e geodesic equation captures how free-falling world-lines move towards each other. In other words it compares the distances between neighboring world-lines. Considering two bodies separated by the distance ', the deviation of this distance is proportional to the bodies’ relative acceleration '.￿rough the geodesic deviation equation this is proportional to the ⇠ • Riemann tensor times the distance, so ' '.￿is give, analogous to Eq. (6.11), R • ⇠R

' ' (6.25) • ⇠ R

Integrating over a time ' = ' 2 we get the deviations as ( / )

' '3. (6.26) ⇠ R

Belenchia et al. then states that when averaged over a spacetime region of dimension ' the Riemann tensor should ￿uctuate as ; '3.￿us the geodesic deviation ￿uctuates as ; . R⇠ ?/ ? 42 CHAPTER 6. BELENCHIA ET AL.

Since Bob would need an accuracy above this level we have

XG > ;% (6.27) as a lower bound for the di￿erence in position of Bob’s particle given Alice’s particle being to the right or to the le￿. Since we have ; = ⌧\ 23 we also set ⌧ = 1 to get the equation % / p XG > 1. (6.28)

In this case we cannot think of Alice’s particle as an e￿ective dipole. ￿is is because we have to consider the conservation of energy/mass of Alice’s system, which means the center of mass must stay constant. ￿us, as her particle moves to the right (le￿), her lab moves to the le￿ (right) just the right amount for the center of mass to be conserved. Since the mass of the lab is much larger than that of the particle, " < , the position change of the lab will be very small and therefore not introduce any other e￿ects.￿ is involvement of the lab/environment in the superposition of Alice will not decohere her particle for the same reason as coupling with the static ￿eld does not cause decoherence. In [53] the concept of false decoherence in discussed. ￿is is when coupling with the environment leads to decoherence of the reduced , but the state still does not decohere in the sense that quantum interference e￿ects can be seen if the di￿erent parts of the superposition are brought together slowly enough (see Section 2.1.2). ￿e lack of a consideration of the center of mass conservation is something Belenchia et al. criticize in the similar earlier works by Baym and Ozawa [2] and Mari et al. [54].

Because the lab is brought into the picture, we think of the ￿eld from Alice’s particle as an e￿ective quadrupole where the four e￿ective masses (particle and lab in two positions) are placed on the same line. ￿e following calculations of the ￿eld felt by Bob’s particle was le￿out of the Belenchia et al. paper, so we do the calculations based on their setup and assumptions, in analogy to the electromagnetic case. To calculate the di￿erence in ￿eld from the two parts of Alice’s superposition we ￿rst consider the change G in position of the lab in comparison with that of the particle satisfying conservation of mass centre, that is

3 < 3 < "G = 0 G = . (6.29) 2 ) " 2

We have the Newtonian ￿eld 6 felt by Bob’s particle from when Alice’s particle is on the le￿

43 CHAPTER 6. BELENCHIA ET AL. as < " 6! , (6.30) ⇠ ⇡ 3 2 + ⇡ < 3 2 | + 2 | | " 2 | and on the right as < " 6' . (6.31) ⇠ ⇡ 3 2 + ⇡ < 3 2 | 2 | | + " 2 | From this we get the di￿erence in ￿eld at Bob’s particle’s position as

< " < " 6 = 6! 6' . (6.32) ⇠ ⇡ 3 2 + ⇡ < 3 2 ⇡ 3 2 ⇡ < 3 2 | + 2 | | " 2 | | 2 | | + " 2 | With the assumptions ⇡ 3 and " < we can simplify as < " < " 6 2 2 < 2 2 < ⇠ 3 + ⇡ ⇡3 3 ⇡ ⇡3 ⇡2 ⇡3 " ⇡2 ⇡3 + " + + 2 + 2 ⇣ ⌘ 4 2 ⇣ ⌘ 3 3 < < (6.33) <⇡ 3 " " < 33 3 ✓ ◆ = Q . ⇠ ⇣ 2⌘ ⇣ ⌘ ⇠ ⇡5 ⇡ ⇡4 < ⇡8 " ⇣ ⌘ ￿e result of this calculation di￿ers from Belenchia et al. by a factor 3 ⇡ and also includes / the actual expression for and e￿ective quadrupole moment as = < 32. A more detailed Q version of the calculation above is done in Appendix A. We continue calculations in this section with the answer found in the paper, namely 6 ⇡4, so that the equation will ⇠Q/ look the same. If the ￿eld strength actually should have this extra factor 3 ⇡, all arguments / in the end are stronger. For a longer discussion see Section 7.2.

From this di￿erence in ￿eld strength we can calculate the di￿erence in position of Bob’s particle being subjected to these ￿elds for a time )⌫ with Newtons second equation as

2 < XG = < 6 XG = Q dCdC0 = Q ) . (6.34) ⌫ • ⌫ ) ⇡4 ⇡4 ⌫ π)⌫ π)⌫ With the restriction from Eq. (6.28) we get

Q) 2 > 1, (6.35) ⇡4 ⌫ in a similar fashion as Eq. (6.23) for the electromagnetic case. Doing dimensional analysis we can also write this as ;? Q) 2 > . (6.36) ⇡4 ⌫ ⌧

44 CHAPTER 6. BELENCHIA ET AL.

￿is sets limit on the mass of Alice’s particle and the time span of Bob’s experiment in comparison to the distance between Alice and Bob. If the two former are to small in comparison to the la￿er, Bob will not be able to distinguish from his particle’s position whether Alice’s particle is to the le￿or right.

6.3.2 ￿antum Radiation

Since we have assumed linearized quantum gravity, Alice’s particle should radiate gravitons as it is recombined. Belenchia et al. states that the energy radiated will be given by

2 Q3 ), (6.37) E⇠ ) ! which comes from the fact that the energy emi￿ed by gravitational radiation is proportional to the square of the third derivative of the quadrupole moment. While for dipoles the wave amplitude is , for a quadrupole we instead have the amplitude as .￿e power emi￿ed ⇠ D§ ⇠ Q• is proportional to the squared derivative of the wave amplitude, which gives us % 2. ⇠(Q®) With this we have

3 = dCdC0dC00 ) = Q , (6.38) Q Q® ⇠ Q® ) Q® ) 3 π) π) π) which squared would give us an expression for the power. Multiplied by the time once again we get the total energy as in Eq. (6.37). Assuming this is true we again move to the quantum realm and say that this radiation must come in the form of gravitons of minimal frequency l = 1 ) . If we want no radiated gravitons # < 1 we get the lower bound 0 / 2 # E Q2 < 1, (6.39) ⇠ l0 ⇠ ) ! which gives < ) 2. (6.40) Q Reintroducing units we get < < 22) 2. (6.41) Q ?

￿is means that for a given recombination time ) the mass < of Alice’s particle and the distance 3 of the superposition cannot take too large values if Alice wants to be able to recombine her particle without decoherence, with the limiting factor being the emission

45 CHAPTER 6. BELENCHIA ET AL. quantum radiation.

6.3.3 Solving the Paradox

￿e rest of the argument largely follows the same structure as for the electromagnetic case (see Section 6.2.3), but with Eq.s (6.35) and (6.40) as the limiting equations for distinguishability of Bob and radiation of Alice respectively. For 2) < ⇡ and 2) < ⇡, if we have < < 22) 2 we ⌫ Q ? get that Eq. (6.35) is violated and Bob will not be also to get which-path information of Alice by measuring his particle’s position. If, on the other hand, > < 22) 2 then Eq. (6.40) is Q ? violated and Alice’s particle will radiate gravitons during her recombination. ￿e interference experiment testing the superposition of her particle will then fail independently of whether or not Bob opens his trap. We thus get a resolution to the paradox by introducing quantum ￿uctuations and quantized radiation to the gravitational ￿eld. ￿e conclusion Belenchia et al. draws from this is that the gravitational ￿eld must be quantized for consistency.

46 Chapter 7

Critical Comparative Analysis and New Results

I have now presented the ideas in the two articles which the goal of this thesis was to compare. I have also ￿lled in the details of their calculation where needed. A￿er looking at them both in detail we can start comparing their results, which seem to be di￿erent despite a very similar starting point. What we will ￿nd is that, while the critique Belenchia et al. voiced against their predecessors might be correct, they seem to have failed to bypass the main point. ￿e critique that a quadrupole should be used instead of a dipole only seems to make Baym and Ozawa’s argument stronger. To make the following text less clu￿ered, we call the experimental setup in Baym and Ozawa’s paper [2] Setup I (Figure 5.1.1), and the setup in Belenchia et al.’s paper [3] Setup II (Figure 6.1.1).

7.1 Details of the Interference Experiments

￿e setups of the two Gedankenexperiments are presented as di￿erent. Even though Belenchia et al. are commenting on the Baym and Ozawa paper, they write that their setup is inspired by a later Gedankenexperiment by Mari et al. [54]. We will here see, however, that when considering the assumptions made in the three cases, they can be considered as one and the same setup.

￿e interference experiment in Setup I is a double slit experiment, as described in Section 2.1.1. In this case, the particle is ￿rstly in a central position, then splits into a spacial superposition where each part moves through one slit, then move together to be recombined at the position

47 CHAPTER 7. CRITICAL COMPARATIVE ANALYSIS AND NEW RESULTS on the screen where the particle is measured. ￿is outwards and inwards movement of the particle is depicted in Figure 5.1.1.

In the calculations, however, it is assumed that the potential from the particle is constant throughout the experiment. ￿is means that the results are the same as for a case where the particle was put into a spatial superposition already in the past. ￿e calculations assume no potential before the experiment, and a constant potential during the experiment. ￿ese assumptions are made in order to simplify the math. ￿ey are also on the “correct side of the inequality”, such that the argument for their results would be even stronger had this assumption not been made.

Speci￿cally, if a more detailed calculation had been made, the strength of the ￿eld felt by the detector in Setup I would be slightly smaller. ￿is means an even larger mass would be needed to get the same movement in the detector, making the visibility of the fringe pa￿ern even less visible. In other words, if the results of the article holds for the simpli￿ed case, it should hold for the real case as well.

If one instead of using this assumption would change the setup to account for the same simpli￿cation, we see how Setup I is similar to Setup II. To account for the assumption that the ￿eld is zero before the experiment, one could imagine ￿xing the mirrors of the detector until a time C0 when the ￿eld from the particle being at point 0 reaches the detector. ￿is is similar to how in Setup II, Bob has a trap which he opens at a previously agreed upon time

C0. To account for the assumption that the ￿eld felt by the detector is constant throughout the experiment, one can imagine the superposition having been prepared in the past, and the separation distance 3 is constant during the experiment. We again see how this is similar to Setup II where Alice has prepared her superposition in the causal past. Bob’s particle then feels the ￿eld from the superposition of separation 3 from the moment he opens his trap until he measures his particle’s position, which exactly accounts for this assumption in Setup I.

Another apparent di￿erence in the two setups is the mechanics of the respective interference experiment itself. While Setup I explicitly uses a double slit experiment, Setup II has no explanation of the exact mechanics. ￿e two parts of the wave function of Alice’s particle are moved together and will interfere with each other. To determine if the of the particle has decohered or not, it is necessary to do this a large number of times to get an interference pa￿ern on a screen. We can therefore assume the result of Alice’s interference experiment in Setup II to be similar to that of a normal double slit experiment.

48 CHAPTER 7. CRITICAL COMPARATIVE ANALYSIS AND NEW RESULTS

￿e distance between the maxima in the fringe pa￿ern on the screen is then proportional to _U, where U can be thought of as the angle between the direction of the incoming particle- wave from either slit and _ is the wavelength [36, 86]. In the case of a double slit-like experiment, the small angle can be approximated by U 3 ! where 3 is the distance between ⇠ / the slits and ! is the distance from the slits to the screen. ￿is means the formula X _ 3 5 ⇠ ! used in (5.19) for Setup I works for Setup II as well even though it is not explicitly a double slit experiment. ￿e distance ! = E) is in this case the spatial extent of the experiment, or equivalently the duration of the experiment multiplied by a characteristic velocity of the particle.

We have thus shown that the two Gedankenexperiments can be thought of as the same, if the assumptions used are taken into account.

7.2 Calculation of E￿ective ￿adrupole Moment

￿e e￿ective dipole in Setup I and e￿ective quadrupole in Setup II comes from the fact that we are taking the di￿erence between the ￿elds from two parts of a spatial superposition. For the dipole case, the resulting di￿erence is then mathematically the same as if we had had two charges/masses of opposite sign. It was, however, correctly pointed out by Belenchia et al. [3] that the dipole moment vanishes for massive object because of the necessary conservation of the center of mass. When Alice’s particle is split into the spatial superposition, the environment around the particle has to also split ever so slightly to preserve the center of mass. ￿e calculations do however assume that the environment of Alice is also far away from Bob. ￿e two parts of Alice’s particle’s superposition and the corresponding superposition of her lab will have to be distanced from Bob or else he will be a part of her environment and will move with it.

To get an intuition of the size of the movement of Alice’s lab, let us assume the mass of her particle is equal to the Planck mass < =

6 < \2 1 10 m 17 G = 3 = ⇥ 2 10 m. (7.1) < ⌧ 1 103 kg ⇠ ⇥ ! ⇥ ￿is is an extremely small distance, but still within the sensitivity of advanced detectors, so in principle detectable… [87]. ￿e question of what constitutes “Alice’s lab” is

49 CHAPTER 7. CRITICAL COMPARATIVE ANALYSIS AND NEW RESULTS however more di￿cult to answer. Here, the important part is that it is very massive compared to the particle, and therefore moves a very small distance, which is a reasonable assumption to make.

In Section 6.3.1 Eq. (6.33) we found that our calculation of the di￿erence in gravitational￿ eld from the two parts of Alice’s superposition di￿ers from the expression found in [3] by a factor 3 ⇡. Whether this is a miscalculation is hard to say since Belenchia et al. do not show the / explicit calculation in the original paper. ￿e assumptions for the calculation in this text are simply that if Alice’s particle where to go to the right, the environment will have to move slightly to the le￿so that the center of mass is preserved. What this seems to lead to is that not only does the e￿ective dipole moment vanish, but the quadrupole is a factor 3 ⇡ weaker / than expected. A detailed calculation of this is done in Appendix A.

￿e following results will hold even if the calculations were done using the quadrupole dependence found in the original paper. Fixing this apparent error and adding the factor 3 ⇡, however, makes our argument even stronger. Going forward, we introduce the reduced / e￿ective quadrupole moment = 3 = < 33 ⇡. Any limit on this will also be a limit on Q⇤ ⇡ Q / the e￿ective quadrupole moment, but with the addition of the extra 3 ⇡ factor. We can in / this way continue our calculations as if we are using the normal quadrupole moment, so the equations will look the same as if using the calculations in [3] as a starting point.

7.3 Distinguishability of Bob and Interference

￿e logic of the argument in Setup II is that the thought experiment is divided into two cases based on whether there is radiation from Alice’s particle or not. For the radiation case her interference experiment fails regardless of Bob, but for the case of no radiation we again have a division on whether Bob will be able to distinguish the position of Alice’s particle based on a position measurement of his own. It is shown that for the case of no radiation Bob will never be able to have distinguishability. ￿is analysis does, however, lack a consideration of the actual result of Alice’s interference experiment.

If one instead ￿rst divide the cases into Bob distinguishing or not, and then look at the interference experiment of Alice, the results are di￿erent. Starting from Eq. (6.35), which comes from the limit on Bob’s distinguishability, we have with units

2 2 2 2 ⌧ ⇤) ⇤2 ) ⇤ (7.2) 4 4 2 2 2 ;?⇡

50 CHAPTER 7. CRITICAL COMPARATIVE ANALYSIS AND NEW RESULTS

Since we necessarily have ⇡ > 2)⌫ for there to be a paradox this gives

2 ⇤ > < ⇡ , (7.3) Q ? so the quadrupole moment has to be at least as strong as that of two Planck mass particles a distance ⇡ apart (with one of them having an e￿ective negative mass) for Bob to have distinguishability. In other words, when this inequality is satis￿ed, the paradox remains.

Another condition for the experiment is that ⇡ > 3.￿is inequality comes from the fact that we have ⇡ > 2)⌫ for there to be a paradox in the ￿rst place, and 2)⌫ > 3 since Alice’s particle cannot be recombined with a velocity faster than the speed of light. Looking at Eq. (7.3), we see that the mass of Alice’s particle must be much larger than the Planck mass for Bob to be able to di￿erentiate his particle’s position to more than a Planck length. Writing it out we have < ⇡3 3 > 3 > > <3

Next, we look at the fringe spacing X5 in Alice’s interference experiment. ￿e validity of using the following formula was justi￿ed in Section 7.1. Looking at the argument about fringe distance from [2] presented in Eq. (5.19), but for the corrected quadrupole we have

2 2 2 _! \ E) \)

￿is result is independent of whether Alice’s particle will radiate or not. It is the same result as what was found in [2], but with a di￿erent starting point and a much stronger argument since the upper bound is now 32 ⇡2 times smaller. ￿e consequence of this is that the paradox / is solved without the need of radiation. Even more, if one assumes the Planck length as a

51 CHAPTER 7. CRITICAL COMPARATIVE ANALYSIS AND NEW RESULTS universal lower bound on what we can say anything about, the paradox is solved without the need of a quantized gravitational ￿eld. ￿is is discussed further in Section 8.1. ￿e result that has been presented in this section shows that Belenchia et al.’s argument for the quantization of the gravitational ￿eld does not hold. ￿eir failure in considering the details of Alice’s interference experiment means that the proposed quantization of the ￿eld is in fact not necessary to solve the paradox.

7.4 Radiation of Alice’s Particle and Interference

We have now looked at Bob’s side and used his ability to distinguish Alice’s particle’s position from his own as a condition, to then look at Alice’s interference experiment. ￿is resulted in the realisation that quantum radiation from Alice’s particle is not a necessary factor in solving the paradox. Let us now take radiation as the ￿rst condition and look at how this a￿ects Alice’s interference experiment.

Looking at the criterion for radiation in Eq. (6.41) we can get the inequality

< ⇡2 < 2 2 <

_! \ E) X5 = = = ;? ;? 2 ;? 3 ; . ) 5 ? ⇡

So the fringe distance is larger than the Planck length multiplied by a very small factor 3 ⇡. / For the fringe distance to then be larger than the Planck length, we would need this inequality to be a “ ” rather than a “>”. To achieve that, and still not make 3 ⇡ smaller, we would / either need that 2) ⇡ for the ￿rst step in (7.6), or < < for the last inequality ⌧ ⌧ ? 52 CHAPTER 7. CRITICAL COMPARATIVE ANALYSIS AND NEW RESULTS

in (7.6). In the ￿rst case, making ⇡ larger is not preferable so we would have to make ) smaller. ￿is would however limit the mass and distance 3 of Alice’s superposition. If we now consider Bob for a moment, any one of these adjustments would make him have a harder time ge￿ing which-path information (and even then we are below the Planck length limit to begin with). What all this points at is again that the radiation argument is redundant, and not necessary for the resolution of the paradox. Even if we ￿rst consider the radiation argument, ge￿ing distinguishable fringes in Alice’s reinterference experiment would require such a weak quadrupole and such a large distance between Alice and Bob that Bob will have no chance of ge￿ing which-path information.

53 Chapter 8

Discussion and Conclusion

8.1 Planck Length as a Lower Bound

From the results found in the last chapter, we see that the paradox presented in Belenchia et al. can be resolved without introducing a quantized gravitational ￿eld. Instead, the solution relies only on the use of the Planck length as a lower bound on distance, below which our current understanding of physics is inadequate to draw any conclusions [88]. In Baym and Ozawa the Planck length limit is said to come from the standard quantum limit applied to the case of a detector mirror so massive that it just barely does not become a black hole (see text above Eq. (5.18)). In Belechia et al. it instead comes from the size of ￿uctuations in the being on the order of a Planck length, and when the geodesic deviation equation then is integrated it yields that two bodies should deviate in their relative position by ;? (see Section 6.3.1).

￿ere are in fact many ways to arrive at the Planck scale [20], and it seems to appear quite independently of the speci￿c theory of quantum gravity used [21].￿ e question relevant to this discussion is whether the Planck scale in itself is related to the need of a quantized gravitational ￿eld. While some argue that the Planck length cut-o￿is in fact an argument for a quantum ￿eld theory of gravity [27], contrarily it also appears as a cut-o￿in other contexts [89, 90]. In this work we have shown that one such argument in favor of the quantization of gravity is invalid, using only the assumption of a Planck length lower bound. As the true nature of Planck scale physics is so far from our current experimental limits, the Planck length being too small and the Planck energy being to large, it is reasonable to assume this scale as a bound in the way that has been done here.

54 CHAPTER 8. DISCUSSION AND CONCLUSION

Considering di￿erent logical possibilities, ￿rst we have the assumption made here, that the Planck length can unconditionally be thought of as a minimum distance. ￿is could be a result of spacetime having a minimum meaningful length, beyond which one can make no meaningful statements, independently of whether or not the gravitational ￿eld is quantized. ￿e Planck length is then present either as a limit to our current knowledge, but with a more complete theory of quantum gravity one can make meaningful statements even beyond, or an actual limit in a similar sense as the Heisenberg relations, a line which the Universe by some mechanics prevents us from .

Another view is that the Planck length is a limit stemming from the quantized gravitational ￿eld, or quantized spacetime [91]. Either as the size of ￿uctuations of spacetime, making anything smaller too fuzzy to be de￿ned properly. Or as the size of the “building blocks” of a discretized spacetime, the Planck length being the physically smallest distance. A quantization of this sort is o￿en done by assuming non-commutative relations between spacetime coordinates, similar to how the non-commutativity of the electric ￿eld components leads to a quantized electromagnetic ￿eld [1]. In the quantum ￿eld theory of the , introducing a discrete la￿ice space can help get rid of some in￿nities that otherwise emerge. ￿is could just be a mathematical trick, but might also be a hint towards the workings of physical reality [92].

￿e former view of the Planck length is the one taken in this thesis, but it is not necessarily meant as an argument for this being the correct view. Rather, with the e￿orts taken in the thought experiments presented here, one cannot say with certainty one way or the other.

8.2 Bohr & Rosenfeld in Perspective

While the Bohr and Rosenfeld paper has been accepted as an argument in favor of , and as a reason for looking further into this topic, it was quickly shown that the same argument could not be used for the case of the gravitational ￿eld [67]. One simple way to see this is that the lack of a body with negative mass makes the measurement process they present impossible to perform in the real world for a gravitational measurement. ￿ere has, however, been made e￿orts towards constructing commutation relations between the di￿erent components of the gravitational ￿eld, similar to those found for the electromagnetic ￿eld [93].

55 CHAPTER 8. DISCUSSION AND CONCLUSION

￿is acceptance of Bohr and Rosenfeld’s arguments can even be seen in the two main papers discussed in this thesis. When Belenchia et al. justify that the ￿uctuations of the electromagnetic ￿eld is of order 1 '2, where ' is the linear dimension of the spacetime region / in which we measure the ￿eld (see Eq. (6.2)), they refer to the explanation found in the Baym and Ozawa’s paper (and which I have put in Chapter 6 since this is where the electromagnetic case is discussed). ￿is argument again ties back to Bohr and Rosenfeld’s argument and use equations found in the classic paper. We can here see how the paper from the ’30s is still, even 90 years later, used as a reference for measurability of the electromagnetic ￿eld.

￿e setup of Bohr and Rosenfeld’s ￿eld measurement apparatus is quite complicated but at the same time also quite “low-tech”. It uses springs, beams and mirrors, but with the current knowledge of measurement technology maybe one could come up with a setup which cancels the unwanted terms of the ￿eld uncertainty without using a negative charge.￿ is could make it possible to construct a translation of the argument to the gravitational case.

￿e question then becomes if some new problem arises that is not present in the electromagnetic case. Similar to how the complementarity principle in quantum mechanics makes it impossible to “trick” a quantum system (see Section 2.1.2) something might theoretically prevent us from probing quantum gravitational phenomena until a more complete theory of quantum gravity is found. In the case of the two articles discussed in this thesis, this “something” was the Planck length limit. Perhaps, by following this road to the end, coming up with new thought experiments in the overlap of gravity and quantum mechanics, some new ground will be laid that could help in the endeavour towards a full theory of quantum gravity, and ultimately to a full understanding of the Universe.

8.3 Conclusion

In this thesis we have analyzed Gedankenexperiments that try to probe quantum e￿ects of gravity by looking at how the gravitational ￿eld from a spatially separated quantum superposition behaves. We studied a paradoxical situation involving a distant measurement of the ￿eld from an interference experiment, where either causality or complementarity seemed to be violated. Baym and Ozawa [2] resolved the paradox by noting that for the detector to be able to get which-path information from the superposition, the mass of the test particle would have to be so large that the fringes of the interference experiment become smaller than a Planck length in width, and therefore undetectable. Belenchia et al. [3] instead resolved the paradox by introducing quantum ￿uctuations and quantized radiation to the gravitational

56 CHAPTER 8. DISCUSSION AND CONCLUSION

￿eld, leading to the conclusion that the￿ eld must be quantized for consistency.

By analyzing and comparing their arguments, we have ￿rstly found that both arguments are su￿cient to solve the paradox. However, by the one assumption that one cannot do meaningful distance measurements below the Planck length, we show that Belenchia et al.’s argument of a quantized gravitational ￿eld is not necessary to resolve the paradox. Similar to the logic used in Baym and Ozawa, if which-path information from the superposition can be obtained, the test particle will have to be too massive for the interference experiment to be successful.

It is important to note, once again, that both arguments are completely su￿cient. ￿e di￿erence is that the assumptions of one is included in the other. Both papers assume distance measurements below the Planck length to be meaningless, but Belenchia has the additional assumption of quantized radiation. As discussed in Section 8.1 above, there are several lines of reasoning that can lead to a Planck length limit, not only that of a quantized ￿eld. Since the paradox can be resolved using only the assumption of the Planck length limit, the radiation assumption is seemingly redundant, making the argument with fewer assumptions the stronger one.

￿e reasoning for the Planck length limit in Belenchia et al. is that there are inevitable quantum ￿uctuations of spacetime on the scale of the Planck length, meaning distances below this scale will not be well de￿ned. ￿ey use this lower bound to say that the test particle doing the ￿eld measurement must move a physical distance longer than the Planck length for any conclusion to be drawn, but they could just as well have used the same reasoning to state that the fringes of the interference experiment must be more than a Planck length apart for any conclusion to be drawn. If they had made this connection and analyzed the details of their interference experiment, they would have reached the same conclusion as has been done here.

￿e century-old question then remains: is the gravitational ￿eld quantized? What we have found here is that the speci￿c argument used by Belenchia et al. to arrive at the answer “yes”, is in fact not conclusive. ￿is does not mean that we have shown the answer to be “no”, rather the question remains open. A decisive argument showing that there is a logical necessity to quantize gravity is thus still missing. However, with the recent advancements in the technologies needed to realize table-top quantum gravity e￿ects, the answer might come in the near future (see next section).

Physics has come a long way, but there is still a long way to go.

57 CHAPTER 8. DISCUSSION AND CONCLUSION

8.4 Future Work

While the thought experiments considered here are just that — thought experiments — other similar works looking to probe quantum gravitational e￿ects using table-top experiments are close to being experimentally realizable. In [80] and [79] a setup similar to that in Belenchia et al. [3], but with Bob’s particle replaced by an interference experiment like Alice’s, is considered. Approximate calculations show that this this type of experiment should be physically feasible [80]. It is also showed that if any gravitational e￿ect between the two superposed particles can be measured, it would be an indication for gravitational ￿eld quantization [79].

￿is type of argument could also be combined with the type of argument considered in this work to perhaps improve Bob’s accuracy. If instead of looking at how much Bob’s particle will physically move and thereby get which-path information of Alice, he could look at the relative phase shi￿of his own spatial superposition. Relative phase shi￿here indicates the di￿erence in phase shi￿between the two parts of the superposition. ￿is di￿erence can perhaps be measured to such an accuracy that Bob would be able to distinguish the two paths of Alice’s particle. If that is the case, Bob might be able to get which-path information even if his own particle has a spatial movement smaller than a Planck length. ￿at could possibly change the outcome of the analysis done in this thesis and would be a very interesting topic to study in more detail.

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65 Appendix A

E￿ective ￿adrupole Moment of Two Masses in a Spatial Superposition

We here do the full calculation of the quadrupole moment from two masses < and " in a spatial superposition on the G-axis where

< ", (A.1) ⌧

Comparing to the calculations shown in Section 6.3.1 we have < = < and " = ". For the two parts of the superposition the smaller mass < has the position G = 3 2 while the ± / larger mass has the position G = < 3 . We call the con￿guration where the small mass has ⌥" 2 the position G = 3 2 as the le￿con￿guration (L) and similarly when G = 3 2 as the right / / con￿guration (R) ￿e Newtonian gravitational acceleration felt by an observer at position G = ⇡ is given for the two con￿gurations as

⌧< ⌧" 6! , (A.2) ⇠ ⇡ 3 2 + ⇡ < 3 2 | + 2 | | " 2 | and ⌧< ⌧" 6' = , (A.3) ⇡ 3 2 + ⇡ < 3 2 | 2 | | + " 2 | where ⌧ is the Newton’s gravitational constant. To get the e￿ective quadrupole we take the di￿erence between these two ￿elds. ￿is is mathematically the same as having four masses on

66 APPENDIX A. EFFECTIVE QUADRUPOLE MOMENT OF TWO MASSES IN A SPATIAL SUPERPOSITION a row where two are positive and two and ”negative” masses. ￿e di￿erence becomes

⌧< ⌧" ⌧< ⌧" 6 = 6! 6' = . (A.4) ⇡ 3 2 + ⇡ < 3 2 ⇡ 3 2 ⇡ < 3 2 | + 2 | | " 2 | | 2 | | + " 2 | Expanding the denominators and rejecting term with a <2 "2-factor we get / 6 < " < " = 2 2 2 2 . ⌧ + 2 2 ⇡2 ⇡3 3 ⇡2 < ⇡3 < 3 ⇡2 ⇡3 3 ⇡2 < ⇡3 < 3 + + 2 " + "2 2 + 2 + " + "2 2 ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ ⇣ (A.5)⌘ We write this as a single fraction by multiplying the numerator and denominator of each term with the denominator of the three other terms, and the terms with an "-factor by < ".To / reduce clu￿er we de￿ne ` = < ".￿e total denominator then becomes / D

= ⇡2 ⇡3 32 4 ` ⇡2 `⇡3 `232 4 ` ⇡2 ⇡3 32 4 ` ⇡2 `⇡3 `232 4 . D + + / + / + / + + / (A.6) With the assumption ⇡ > 3 which is needed in our case, we get the dominant term as

= `2⇡8. (A.7) D

We now turn to the numerator , which becomes N

= <`2 ⇡2 `⇡3 `232 4 ⇡2 ⇡3 32 4 ⇡2 `⇡3 `232 4 N + / + / + + / <` ⇡2 ⇡3 32 4 ⇡2 ⇡3 32 4 ⇡2 `⇡3 `232 4 + + + / + / + + / (A.8) <`2 ⇡ 2 ⇡3 32 4 ⇡2 `⇡3 `2324 ⇡2 `⇡3 `232 4 + + / + / + + / <` ⇡2 ⇡3 32 4⇡2 `⇡3 `232 4⇡2 ⇡3 32 4 . + + / + / + / We simplify, ignoring the factor 1 4, and sort by order of ⇡ until we get a non-zero coe￿cient: /

<` = ⇡6 ` 1 ` 1 N/ ( + ) ⇡53 `2 ` `2 ` 1 1 `2 `2 ` 1 ` 1 + ( + + + + + ) (A.9) ⇡432 2`3 ` `3 2 `2 1 2`3 ` `3 2 `2 1 + ( + + + + + ) ⇡333 `3 ` `3 ` ... + ( + )+

67 APPENDIX A. EFFECTIVE QUADRUPOLE MOMENT OF TWO MASSES IN A SPATIAL SUPERPOSITION

We found a non-zero coe￿cient and can thus reintroduce ` = < " to divide the numerator / in Eq. (A.9) on the denominator in Eq. (A.7) to get

3 3 < 4 < 2 6 <⇡ 3 " " <33 3 = Q . (A.10) ⌧ ⇠ ⇣< 2 8 ⌘ ⇠ ⇡5 ⇡ ⇡4 " ⇡ We see that we get a factor 3 ⇡ more than the equation that was presented in Belenchia et / al. (Eq. (14) in the original paper [3]).

68 Appendix B

Translation of Bronstein (1934)

On the Relativistic Extension of the Indeterminacy Principle

M. Bronstein 1934

Translated from Russian to Swedish by Erik Aurell, transcribed and translated to English by Erik Rydving, March 2021.

(Submi￿ed by S. Wawilow, member of the academy, 22 January 1934)

In the recently published work by Bohr and Rosenfeld it was shown that the uncertainty in the value of the electric ￿eld one would get from a charged test particle constituting the experimental setup, can always be made unconditionally small in comparison to the uncertainty coming from the inequality G? & ⌘. However, to be able to measure the ￿eld with maximal accuracy, one should perform the experiment such that the e￿ect of the radiation on the test particle is as small as possible. For measuring the average value of EG in a volume + during a time ) by using a test particle of volume + with constant density of charge d, we have approximately

dEG+) = ?G C ) ?G C, (B.1) ( ) + ( ) 69 APPENDIX B. TRANSLATION OF BRONSTEIN (1934) from which we have ? E G . (B.2) G ⇠ d+) If the time that inevitably is needed to measure the momentum satis￿es C ) and we let G ⌧ represent the uncertainty in the coordinate corresponding to the momentum measurement, Bohr and Rosenfeld found that the e￿ect of radiation gives an uncertainty in the momentum as d2+ GC (because the electric ￿eld caused by the measurement of the momentum at the position of the test particle is of order d G C, and similarly the momentum the ￿eld gives ⇠ C to the test particle equals the multiplication of the ￿eld with d+ C).￿us we have

⌘ 42 ? CG, (B.3) G ⇠ G + + where 4 is the charge of the test particle (not the charge 4 of the electron!). Bohr and Rosenfeld shows that the second term can be made unconditionally small in comparison to the ￿rst. ￿e minimal value of ?G is found when both term have about the same order of magnitude, that is when ⌘+ G . (B.4) ⇠ 42C

￿e minimal value of ?G then becomes

⌘C ? 4 , (B.5) ( G )<8= ⇠ + and the uncertainty of the electric ￿eld is

1 ⌘C E , (B.6) ( G )<8= ⇠ ) + which for su￿ciently small C can be made as small as desired. Two circumstanced, however, prevent the uncertainty from becoming too inde ￿nitely small. Firstly, from the follows that G is smaller than 2C, that is the speed which one measures the momentum cannot exceed the speed of light 2.￿ereby we have

1 3 ⌘+ ⌘2 / 1 3 . 2C>A2C & + / . (B.7) 42C 42 ✓ ◆

70 APPENDIX B. TRANSLATION OF BRONSTEIN (1934)

Secondly, it follows from the notion of measuring the ￿eld itself that G is much smaller than the linear dimension of the test particle, that is

⌘+ 1 3 ⌘2 1 3 + / >A 2C + / . (B.8) 42C ⌧ 42

For the minimal value of the uncertainty of the ￿eld we get the following inequalities:

⌘ 1 6 2 3 2 ⌘ / ⌘ / EG <8= & = , (B.9) ( ) )+1 3 42 41 3)+2 3d1 3 / ✓ ◆ / / / and ⌘ 1 2 » 2 ⌘ / ⌘ EG <8= = . (B.10) ( ) )+1 3 42 )+4 3d / ✓ ◆ / ￿e￿rst inequality is greater or smaller than the second depending on the whether 42 is greater than ⌘2. If for instance the test particle» is an electron, then only the second condition would ma￿er (this makes the smallest value of EG be similar in form to the result by Fock and Jordan). However, since measuring the ￿eld in a given spacetime volume as accurately as possible would be achieved by using as large a charge as possible, it is the ￿rst condition that will apply. In a future relativistic quantum theory, the fundamental impossibility of measuring the ￿eld to unlimited accuracy will be connected with the atomic nature of ma￿er, i.e. the fundamental impossibility of making the charge density unconditionally large.

Concerning the measurement of the magnetic ￿eld component BG we consider in this case a charged test particle moving in the ~-direction with velocity V2. In the frame of reference that coincides with the test particle it will give rise to an electric ￿eld in the I-direction VBG and p1 V2 V ?G we get BG , where ) 0 is the time di￿erence of two momentum measurements p1 V2 ⇠ 2) 0 performed with a clock positioned in the test particle. Since it is known that ) = ) 1 V2 0 we get p 2?I BG . (B.11) ⇠ 4)E~

￿e quantity ~ = E~) , that is the disturbance of the test particle in the ~-direction during the time one measures the ￿eld, must be very small in comparison. Since ?I must be greater ⌘ than I , we get the inequality ⌘2 B & (B.12) G 4~I (￿e above equation makes a parallel to the earlier derived expression for the electric ￿eld ⌘2 which can be wri￿en on the form EG & 4G2) , where 2) is, so to speak, the disturbance of the

71 APPENDIX B. TRANSLATION OF BRONSTEIN (1934) measuring instrument during the time of the ￿eld measurement in the direction of the2C-axis.)

￿e lower limit of the uncertainty of BG is connected to two circumstances. Firstly, with the condition I & 2C and ~ . 2) , and secondly with the condition ~, I + 1 3. Since ⌧ / I must have the order of magnitude ⌘+ and since under the condition 2) + 1 3 (also ⇠ 42C ⌧ / used in Bohr and Rosenfeld) the inequality ~ . 2) makes the other inequality ~ + 1 3 ⌧ / super￿uous, we get for the minimal value of BG the same expression as for EG (i.e. 2 3 ⌘ / 1 3 2 3 1 3 for the case most favorable for» the measurement of the magnetic ￿eld for which 4 / )+ / d / we have the largest d and the largest velocity of the test particle).

72 Appendix C

Essay for Gravity Research Foundation

￿e following is an essay paper that I wrote with my supervisors Igor Pikovski and Erik Aurell for an essay competition lead by the Gravity Research Foundation.

73 On the missing necessity of a quantum field theory of gravity

1, 2, 3, 4, Erik Rydving, ⇤ Erik Aurell, † and Igor Pikovski ‡

1School of Engineering Sciences, KTH-Royal Institute of Technology, SE-114 28 Stockholm, Sweden 2Department of Computational Science and Technology, KTH-Royal Institute of Technology, AlbaNova University Center, SE-106 91 Stockholm, Sweden 3Department of Physics, Stevens Institute of Technology, Castle Point on the Hudson, Hoboken, NJ 07030, USA 4Department of Physics, Stockholm University, AlbaNova University Center, SE-106 91 Stockholm, Sweden

(Dated: March 31, 2021)

Whether gravity is quantized remains an open question. Gedankenexperiments can help shed light on this problem. One scenario is an interference experiment with a massive system that interacts gravitationally with another system. An apparent para- dox arises where for space-like separation the outcome of the interference experiment depends on actions on the distant system. A recent resolution shows that the paradox is avoided when quantizing gravitational radiation and including quantum fluctuations of the gravitational field. Here we show that this paradox can also be resolved without considering gravitational radiation, relying only on the Planck length as a limit on spatial resolution. Therefore, in contrast to conclusions previously drawn, we find that the ne- cessity for a quantum field theory of gravity does not follow from this Gedankenexperiment.

Essay written for the Gravity Research Foundation 2021 Awards for Essays on Gravitation.

⇤ [email protected]

† [email protected]

‡ Corresponding author: [email protected] 2

A full quantum theory of gravity is a major outstanding challenge in modern physics. While much important work is focused on solving this challenge, one may ask whether quantizing gravity is logically necessary for consistency with known physics. The analogous electromagnetic problem was considered prior to the development of quantum electrodynamics, by Landau and Peierls [1], and then famously by Bohr and Rosenfeld [2]. The latter authors showed that electromagnetism requires field quantization to be consistent with the uncertainty principle applied to charged par- ticles. This insight, based on a series of Gedankenexperiments, preceded the development of the full theory. A natural question is whether these arguments, or similar Gedankenexperiments, can be found for a quantum field theory of gravity. It was realized early on by Bronstein that the Bohr-Rosenfeld argument does not apply to gravity [3]. Since then, arguments in favor of field quantization [4, 5] and against [6, 7] have been put forward.

In a recent paper, Belenchia et al. [8] showed that a quantum treatment of the gravitational field solves a paradoxical situation in a Gedankenexperiment first introduced by Mari et al. [9]. The setup is a protocol between Alice and Bob, whose systems interact gravitationally (see Figure 1). Alice performs an interference experiment with a massive particle which, in the distant past, she has prepared in a superposition of spatial separation d. Bob can choose to measure, or not to measure, Alice’s gravitational field. Bob does that with a trapped massive particle that he can release or keep trapped. If Bob releases his particle, it will entangle to Alice’s particle. Alice uses atimeTA to recombine her interfering particle. If Bob decides to open his box, he measures the position of the released particle after a time TB, giving which-path information on Alice’s system, and thereby causing Alice’s interference experiment to fail. Since opening the box will a↵ect Alice’s outcome, but keeping it closed will not, Bob can e↵ectively send one bit of information to

Alice. If the distance D between Alice and Bob satisfies cTB

Belenchia et al. [8] resolve the issue by including two physical mechanisms, which together imply the need to quantize the gravitational field. One mechanism is quantum fluctuations of the gravitational field, which limits Bob’s ability to measure his particle’s position to precision better than the Planck length lp. The other mechanism is quantized gravitational radiation, which will be emitted from Alice’s interferometer and thus decohere the experiment if it is performed too fast. 3

t Alice Bob Δx

δf

TA TB Release Trap

Interfere

D

d

Light cone x xL xR

FIG. 1. The Gedankenexperiment of interest (see main text for details). Alice performs an interference experiment with a massive system in a spatial superposition of separation d,whileBobatadistanceD chooses whether to release his particle from a trap or not. If released, the particle will experience a superposition of two gravitational accelerations towards Alice (since Alice prepared her superposition in the causal past), and will reach a final separation x after time TB. When Alice’s system is recombined in a time TA to interfere, it will emit gravitational radiation. Independently, the interference fringes have spacing f which Alice must resolve.

In both cases, the relevant parameter is the e↵ective quadrupole moment md2 of Alice’s QA ⇠ system. It quantifies emission of radiation on Alice’s side, and also Bob’s ability to distinguish Alice’s two paths, since the di↵erence in gravitational acceleration g = g g will not be larger R L G A than DQ4 ,whereG is the gravitational constant. Bob will get which-path information if his freely-falling particle will di↵er in position by more than a Planck length for the two superposed paths of Alice,

x = x x >l . (1) | R L| p This gives a lower threshold for the strength of the quadrupole of Alice’s particle:

2 A D Q 2 > 2 2 , (2) mpD c TB

~c where mp = G . The right-hand side is larger than 1 since we need cTB

Since the setup of the experiment implies d

>m D2. (3) QA p For a quadrupole below this threshold, the di↵erence in field strength from Alice’s particle being on either side of the interferometer is not large enough for Bob to distinguish them. This prevents Bob from getting which-path information.

If, however, the quadrupole moment is large enough for eq. (3) to be satisfied, the paradox remains unresolved. For this case, the authors of [8] propose a solution which involves emission of radiation. As Alice’s particle is accelerated for the two parts of the superposition to recombine, it will emit radiation which, if quantized, results in a number of gravitons 2 N QA . (4) ⇠ m c2T 2 ✓ p A ◆ By emitting radiation, Alice’s system decoheres and her interference experiment fails, regardless of whether Bob opens his box or not. To avoid decoherence we need N<1, meaning there is also an upper limit on the quadrupole moment:

This solution required the introduction of quantum fluctuations and quantum radiation of the gravitational field, strongly suggesting the need for a quantum field description of gravity [8, 10]. The Planck length enters this resolution as the size of quantum fluctuations in the gravitational field, giving rise to an uncertainty in the geodesic deviation equation. The Planck length can, however, also just be assumed as a lower cuto↵beyond which no conclusions can be drawn [11]. We now show that the latter assumption is entirely sucient to resolve the paradox, without the need to consider radiation. Our resolution hence shows that the argument involving a quantum field theory of gravity is, on the contrary, not necessary to resolve this Gedankenexperiment. 5

Building on Baym and Ozawa [11] (correcting for the vanishing mass dipole moment, as pointed out in [8]), we consider the phase shift measurements in Alice’s interference experiment. Inde- pendent of the details of Alice’s experiment, the fringe spacing in the interference pattern will generically depend on the wavelength of the system, as well as the angle ↵ at which the two beams are recombined, given by /↵. Thus the fringe spacing is f ⇠

L ~ vTA mp cTA f = = lp , (6) ⇠ d mAv d mA d

where L = vTA is the characteristic length over which Alice’s particle recoheres, at a speed v over atimeTA. For the fringe pattern to be measurable, we need f >lp, which gives

This yields another upper bound on the quadrupole moment. Note that this bound is lower than that found from the radiation argument, eq. (5), by a factor d/cTA < 1. Thus for masses for which radiation would decohere the interference, Alice could anyway not distinguish her interference pattern as it would be finer than the Planck length. In other words, whether or not Alice’s particle radiates is irrelevant since she anyway cannot resolve her intereference pattern. Consequently, the introduction of quantized radiation is redundant for solving the paradox.

Our conclusion is thus that the question remains inconclusive. The only assumption we use is the inability to resolve distances smaller than the Planck length. One may argue that this alone hints at a quantum theory with fluctuations on the Planck scale [12], but without the need for quantized radiation the argument for a quantum field theory is weakened. The cut-o↵also arises in other contexts [13–15], or could just be taken as a limit on what we can predict with our current understanding of physics [11] (in the words of Sakharov: “It is natural to suppose also that [lp] determines the limit of applicability of present-day notions of space and causality.” [16]). Thus, the single assumption of a Planck length limit on measurability is not by itself a conclusive argument in favor of a quantum field theory of gravity. A Gedankenexperiment in clear favor of the necessity for a quantized gravitational field, in analogy to Bohr and Rosenfeld, remains lacking. The diculty in formulating such an argument might suggest that our current understanding of physics is insucient to address this question. 6

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