On the Necessity of Quantized Gravity

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STOCKHOLM SWEDEN 2021,

On the Necessity of Quantized Gravity

A critical comparison of Baym & Ozawa (2009) and Belenchia et al. (2018)

ERIK RYDVING

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

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Erik Rydving – erydving@kth.se Engineering Physics

KTH Royal Institute of Technology

Place for Project

Stockholm, Sweden

Examiner

Gunnar Bj¨ork

�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

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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

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Abstract

One of the main unsolved problems in theoretical physics is combining the theory of quantum mechanics with general relativity. A central question is how to describe gravity as a quantum

�eld, but also whether a quantum �eld description of gravity is necessary in the �rst place. �ere is now an ongoing search for a Gedankenexperiment that would answer this question of quantization, similar to what Bohr and Rosenfeld’s classic argument did for the electromagnetic case [1]. Two recent papers use Gedankenexperiment arguments to decide whether it is necessary to quantize the gravitational �eld, and come to di�erent conclusions on the ma�er [2, 3]. In this work, their arguments are analyzed, compared, and combined.

Assuming the Planck length as a fundamental lower bound on distance measurability, we �nd that a quantum �eld theory of gravity is not a logical necessity, in contrary to the conclusion drawn in [3].

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Sammanfa�ning

E� av de största olösta problemen i teoretisk fysik är a� kombinera kvantmekaniken med den allmänna relativitetsteorin. En central fr˚aga är hur gravitation ska beskrivas som e�

kvantfält, men ocks˚a om en kvantfältsbeskrivning av gravitation ens är nödvändig. Det p˚ag˚ar forskning som försöker a� svara p˚a fr˚agan om kvantisering med hjälp av tankeexperiment, likt hur Bohr och Rosenfelds klassiska argument svarade p˚a fr˚agan om det elektromagnetiska fältets kvantisering [1]. Tv˚a moderna artiklar använder argument med tankeexperiment för a� besvara om gravitationsfältet behöver kvantiseras, och kommer till olika slutsatser [2, 3]. I de�a examensarbete analyseras, jämförs och kombineras deras argument. Det visas a�, s˚a länge Planck-längden kan antas som en nedre gräns för mätbarhet av avst˚and, är inte en kvantfältsteori för gravitation en logisk nödvändighet, i motsä�ning till konklusionen i [3].

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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 universe.

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Chapter 1 Introduction

Physics has come a long way. Some theoretical predictions, like black holes [4, 5] and the Higgs boson [6–8], were presented already several decades ago. Only recently, however, thanks to the advancement in measurement technology, have they been experimentally veri�ed [9–11]. Both these discoveries are strong indications that the theories that explain the phenomena are correct descriptions of nature. General relativity, predicting black holes, describes how massive objects interact with space and time, while the standard model, predicting the Higgs boson, is based on quantum mechanics and describes subatomic particles and their interactions. Both theories give extremely accurate descriptions of nature and far outperform their predecessors [12, 13].

�e problem is: our two best theories are in many cases not compatible with each other [14, 15]. �is fact is the basis for the almost century-long search for �antum Gravity, a theory which combines quantum mechanics with gravity. Many questions remain, partly because it is 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

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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 size of the Earth the weakness of gravity becomes apparent. Even having the mass of a whole planet pulling at a glass of water, one can easily win the tug-of-war and li� it up.

Going above the scale of the Earth, gravity stands as the major force, keeping the planets in orbit around the sun.

It is now perhaps easier to see why the 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 light 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

Planck mass : <? =

…\2

⌧ ⇡ 2.2 ⇥ 10 ⁸ kg, (1.1)

Planck length : ;? =

…\⌧

2³ ⇡ 1.6 ⇥ 10 ³⁵ m, (1.2)

Planck time : C_? =

…\⌧

2⁵ ⇡ 5.4 ⇥ 10 ⁴⁴ s. (1.3)

�e signi�cance of these units, now called the Planck units, comes from the fact that they are constructed exclusively from fundamental constants. �e speed of light in vacuum 2 ⇡ 3.00 ⇥ 10⁸m/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

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apply to all massive objects. �e reduced Planck’s constant \ = ⌘/2c ⇡ 1.05 ⇥ 10 ³⁴kg m²/s (pronounced “h-bar”) is a factor 1/2c times the proportionality constant⌘ between the energy and frequency of a photon. 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 not been able to see gravitational e�ects between objects less massive than about 10⁴ <_? [22]. Similarly one has not been able to see quantum interference e�ects (see Section 2.1.1) for object more massive than about 10 ¹⁵ <? [23]. To bridge this gap, many clever ideas for experiments have been suggested, but this scale still seems to lie outside of what is currently possible to reach [17]. In this work the Planck mass appears partly as a consequence of playing with units, and partly as a sort of transitional scale between the microscopic (quantum) and macroscopic (classical) world.

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 observable 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 =p6</cd. Plugging the densities³ 7874 kg/m³for iron and 1000 kg/m³for Styrofoam into the formula gives approximately the stated values.

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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 forces 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

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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.

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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.

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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 wave function, which is a superposition of eigenfunctions of the Schr¨odinger 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 spin property of a particle, which can be denoted spin up |"i and spin down |#i. Just as with ocean waves, di�erent quantum states from the same Hilbert space can form a superposition, giving the total state |ki of the particle, as for instance

|ki = 1

p2 (|"i + |#i), (2.1)

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 |ki 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 |ki will be pushed in opposite directions of each other, resulting in the position state being maximally entangled with the spin state. If the

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position states are right |'i and le� |!i, the particle state can become

|ki = 1

p2 (|"i |!i + |#i |'i) . (2.2)

�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 classical mechanics 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 observables (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, [ ˆG, ˆ?_G] = ˆG ˆ?G ˆ?_GˆG < 0, which leads to the famous relation known as Heisenberg’s uncertainty principle,

G ?G \, (2.3)

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].

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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 photons. 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

X₅ = _3/!, (2.4)

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

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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.

L d

δ_f Electron cannon

Double slit 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 X₅ = _3/!, 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

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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

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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 measurement problem 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 quantum superposition 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

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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 charge 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 |ki de�ned in the Eq. (2.2), 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 |ki is only slightly disturbed, say by measuring the spin 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].

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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 equivalence principle, 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 spacetime 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 dimension of time and 1, 2, and 3 denoting the three dimensions of space. In this way, writing 6`aimplies a 4⇥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

W

`a = 1

26^WX m`6_aX + m^a6_`X m_X6`a , (2.6) 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

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curvature tensor

R^W_X`a = m` W

aX m_a _`X^W + _{`_}^W aX^_ W a_

_

`X, (2.7)

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

R^`a = R^W`Wa, (2.8)

on which we can take the trace with respect to the metric to get the Ricci scalar as

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

�ese three Rs all describe some aspect of the curvature of spacetime, with the number of 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 cosmological constant to zero) can be wri�en as

R`a 1

2R6_`a = 8c⌧

2⁴ )_`a, (2.10)

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^`

dB² = _WX^` dG^W dB

dG^X

dB , (2.11)

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

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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 graviton as a force carrier [17, 60].

Another important idea in relativity is that of causality. Since there is an ultimate speed limit of information transfer, equal to the speed of light in vacuum 2, an event 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 coupling constant ⌧ = 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.

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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 ³⁴Js 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 atomic physics 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 Paul Dirac 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 linearized gravity. �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 gravitons, 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).

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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 spacetimes 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 black hole 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- string theory, loop quantum gravity, 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 ⇢? =p

\2⁵/⌧ ⇠ 10²⁸ eV for the quantum e�ects of gravity to become comparable to other forces [18]. �e Large Hadron Collider at CERN, currently the most energetic collider, is only on the scale of 10¹²eV. �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

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this thesis are of the la�er kind.

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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.

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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 classical physics 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

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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.

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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 quantum theory 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 uncertainty principle @ ? ⇠ \, 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

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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

?_G ⇠ ⌘ G +4²

+ C G, (4.1)

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 ⇠ ?G

d+), (4.2)

where d is the charge density of the test particle and ) is the duration of the �eld

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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 ⇠ ⌘+

4² C (4.3)

as the minimizing value, which in turn gives a minimal �eld uncertainty as

( ⇢G)<8= ⇠ 1 )

⌘ C

+ . (4.4)

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 < 2 C. �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

( ⇢G)<8= & \^2/3

4^1/3)+^2/3d^1/3. (4.5)

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

( ⇢G)<8=

\

)+^4/3d, (4.6)

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/)+^1/34^2/3 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.

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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 ⇠ ?G

d+), (4.7)

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

( 6^G)^<8= & ⌘^2/3

)+^1/3<^2/3 (4.8)

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

" = AB2²

2⌧ , (4.9)

it will collapse into a black hole with an event horizon at AB. Inserting this into Eq. (4.8) we get the minimal physically possible uncertainty in a gravitational �eld measurement as

( 6^G)^<8= & (2⌧⌘)^2/3

)AB2^4/3 . (4.10)

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

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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].

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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

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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 electromagnetism is necessary for consistency in physics.

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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 G₀ = '. Going forward, a dot over a variable represent a time derivative §5 = m5 /mC, while an apostrophe represents a space derivative 5⁰ = m5 /mG. �e following calculations follow those done in Baym and Ozawa’s paper [2], with some intermediate steps added.

Considering the equations of motion of the two mirrors, we have

"•G+ = "q⁰(G+) 1

2"l²[G+ G (],

"•G = "q⁰(G ) + 1

2"l²[G+ G (],

(5.1)

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

2 •[ = 1

2l²[ q⁰(G0+ (( + [)/2). (5.2)

�e last term is linearized in [ and q⁰⁰ as

q⁰(G0+ (( + [)/2) ⇡ q⁰(G0) + q⁰⁰(G0)(( + [)/2 ⇡ q⁰(G0) + q⁰⁰(G0)(/2, (5.3) where we assume ( is small, and [ and q⁰⁰are even smaller. �is gives the equation of motion

On the Necessity of Quantized Gravity