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Quantum mechanics and relativity: the enigma.

The interaction of all matter in the occurs on the basis of four fundamental forces.

These four forces are gravity, , the strong force and the weak force.

Gravity, causing the attraction of massive objects; electromagnetism, causing to repel and like charges to repel; the strong force, causing the in nuclei to be attracted to one another; and the weak force, causing the emission of radiation from unstable nuclei. Every phenomenon we witness is governed by these distinguished four forces. In the universe which is governed by these four forces, there are two widely accepted which agree with experimental evidence in their own fields, and are able to make accurate predictions.

The first of these theories is the Theory of Relativity, devised by Einstein in two parts, Special

Relativity and . His paper on was published in 1905, and was the birth of the famous equation E=mc2, which states the relationship between and . This paper also made the prediction that as an object’s velocity tended towards the , then to a stationary observer, a interval relative to the object would be significantly greater relative to the stationary observer (i.e., if one of two twins went on a rocket at near the speed of light, when they return to Earth the twin from the rocket will have aged less than the twin who stayed on Earth). This was then proven experimentally in the Around-the-World Relativistic Sagnac Experiment. In this experiment, a slowly moving portable atomic clock was moved Eastwards around the earth’s equator once, and was shown to lag an atomic clock on the earth’s surface by about 207.4x10-9 seconds (D. W.

Allan, Apr. 5th, 1985). The paper on General Relativity was released in 1916, and predicted the bending of light due to gravity, which was actually observed shortly after the paper was released, when there was an eclipse of the sun, and the positions of stars in the sky shifted to be visible when they were actually hidden directly by the sun (Thompson, 2012, p. 39).

However, although many successful predictions were devised from the Theory of Relativity, there are in fact still problems with infinities in the theory. For example, the theory predicts that in two circumstances, the density of matter will become infinite, and hence so will the force of gravity. These circumstances are those in a , and those in the universe’s infancy. For the strength of the gravitational field to be infinite is clearly not a real phenomenon, so perhaps this suggests that the Theory of Relativity is not applicable in all situations, leading to the notion that a new theory must be devised to incorporate the conditions where Relativity breaks down (Smolin, 2007, p. 5).

The second of these is , which was also discovered by Einstein in 1905, in his famous paper on the Photoelectric effect. This paper explored the relationship between firing electromagnetic radiation onto a metal surface, and detecting electrons being emitted from that surface. The revolutionary feature of this experiment was the discovery of electromagnetic radiation appearing as discrete quanta instead of as a continuous wave. This was demonstrated experimentally by the measurement of the threshold frequency for different metals, where electron emission from a metal only began after electromagnetic radiation with a certain energy was directed at the metal. This proved that light behaved as discrete quanta, however in other experiments light was proven to act as a wave, and so although quantum mechanics can make accurate predictions, it still includes elements of uncertainty. Quantum mechanics also includes serious issues with infinities, for example when using quantum mechanics to describe the electromagnetic field.

As the electromagnetic field has a value at every point in , then in a finite volume there is an infinite number of points, and so an infinite number of variables. This rapidly to predictions of infinite numbers. Infinities suggest that there is a problem with the theory itself, implying that perhaps it is somewhat incomplete. (Smolin, 2007, p. 6)

What do we already know?

A theory of should incorporate ideas of gravity on a quantum scale, for example the interaction of sub-atomic particles due to their mass. An important question to ask, then, is how do we know that gravity does have such an attractive effect on sub-atomic particles?

An extremely sensitive piece of equipment called an X-ray interferometer was developed at

Cornell university in 1964 by Ulrich Bonse and Michael Hart. It was originally believed that the interferometer would not work for , which unlike X-rays have mass, and so it was 10 years later in 1974 when the first interferometer was operated, by Ulrich

Bonse, Helmut Rauch and W. Triemer. The neutron interferometer was an apparatus which can detect the phase difference of neutron waves over very small distances. It is made out of a single crystal up to 10cm in length, which must be free of any misalignments and faults in the regular crystalline structure. (Greenberger & Overhauser, 1980)

The neutron interferometer sensitive enough to detect the change in wavelength of a neutron beam of wavelength approximate to 10-8cm of 0.5λ in relation to the other beam.

Furthermore, the microscopic effect this has on the amplitude of the neutron wave due to the change in phase of the wave can be translated into the macroscopic change in the relative counting rate of the detectors. (Greenberger & Overhauser, 1980)

The importance of this device was massive in measuring the effect of gravity on the phase of a neutron wave. Before the effect of gravity was directly measured in 1975, in the COW experiment, were already aware of the effect gravity caused neutrons to fall, no different to any other object which possessed mass. However, this was understood on a classical basis, not quantized, which is the problem I mentioned in the introduction: we are trying to explain gravity in a quantized theory, in order to extend quantum mechanics to involve the fourth force of . The importance of this experiment was that it could demonstrate the effect of gravity on the wave nature of a neutron. To do this, the interferometer was rotated about the incident beam, to give a difference in the gravitational potentials of the split beam. What the experiment demonstrated, was that as the height difference increased, the difference in phase of the neutron waves increased. (Greenberger

& Overhauser, 1980 p 71) state that this phase difference had to be separated from a classical side effect of gravity. Hence, the COW experiment proves that a weak gravitational field can shift the phase of a neutron wave, which shows that gravity appears in

Schrödinger’s equation like any other force would. (Greenberger & Overhauser, 1980)

Although this experimental evidence does prove directly the existence of gravity influencing quantum mechanical properties, we are looking for a theory which always holds. Gravity can already be perturbatively quantized, quantized meaning understood on a quantum, discrete level; and a perturbative theory is an approximation of the original equation, which may be too complicated to solve. The issue becomes apparent when calculations are made at very high , exceeding the Planck energy. The theory really has no predictive power at these high energies. Hence, the theory is unsatisfactory as it does not describe all phenomenon, and so cannot be considered a fundamental theory. (Hossenfelder, 2015)

The reason that we can’t use experiments to directly understand the effects of quantum gravity is that the conditions where general relativity and quantum mechanics both essentially crossover is at the ~10-33cm, and the Planck energy~1019GeV

(Ashtekar, 2005). In comparison to this figure, the current most powerful accelerator, the Large in , can generate collisions at 13TeV, which is approximately ≈104GeV, so the Planck Energy is roughly 1015 greater than the energy the Large can collide particles with.

However, there are future plans for new particle accelerators, for example the Future

Circular Collider, which would have a circumference of 100km (the circumference of the

Large Hadron Collider is 27km), allowing particles to reach energies of 100TeV, roughly 10 times greater than the energies reached in the . Although this would be a major improvement compared to the Large Hadron Collider, it would not bring us much closer to the Planck energy. (Anon., 2021)

So, to conclude, gravity can have an effect on quantum mechanical events, which has been demonstrated by the neutron interferometer, by shifting their phases when the two neutron beams were at different heights. However, the quantized theory of gravity which we currently have breaks down at the Planck energy. To understand gravity completely in a quantized manner would solve the conundrum of the different which quantum mechanics and general relativity both paints. This new theory must either be an extension of the first, or a new theory entirely. Already, there is an outstanding new theory which has potential to unify , but lacks any experimental evidence to show any legitimacy. This theory is called , which will be a topic of later discussion. Introduction to theories of everything

Throughout human history, we have sought reason for the occurring phenomena which transpire in the world around us, some of those explanations being Theories of

Everything. It could be argued that some of the frst Theories of Everything were fashioned in mythological culture. On reading mythological accounts, there is a prodigious sense that everything had a place; there was no uncertainty (Barrow, 2007).

The weakness of these Mythological Theories is that every line of reasoning tends to end at a Deity, instead of ending at a reasonable explanation (Barrow, 2007) . These

Mythological Theories, then, really have no predictive power. A good Theory of

Everything is a theory which has real explanatory power, and can make reliable and testable predictions, unlike the mythological accounts. Arguably, the Theory of

Everything we are seeking will be a mathematical expression, but there are some assumptions we have made which could suggest this is, in fact, not possible.

One of these signifcant assumptions that we have made is that there is a single expression which can entice every phenomenon, and hence be used to predict and explain everything (everything meaning every event in spacetime). Such an assumption is very easy to make, as we have already found so much order and predictability in the world which we live. However, could it be possible that the universe doesn’t behave in this way, and that there isn’t an equation which can explain everything?

(Barrow, 2007 p.11) states that is no more than the transformation of lists from observational data into abbreviated form by the recognition of patterns. This leads to the awareness that any list of data which can be condensed into a formula, or can be given an abbreviated representation, is named algorithmically compressible. Any string of numbers which cannot be abbreviated in such a way, then, must be random.

For example, (Wheeler, 2019, p. 463) uses binary digits to describe algorithmic compressibility. Consider a list of binary data points with a regular pattern for example

A) 101101101101101101, and B) 100101100000111001. The storage of each of these data sets will require the same amount of storage, i.e. 18 bits. However, the information coded in

A can be represented as C) repeat “101” 6 times. If the length of C is smaller than the length of A, then it can be said that C is an algorithmic compression of A. (Wheeler,

2019, p. 463) goes on to state that, according to the algorithmic theory, this is what laws of nature are: algorithms in algorithmic compressions of nature. When viewed from this perspective, it seems somewhat more comprehendible why some philosophers are able argue that the laws of nature are simply not compressible, and therefore why some could draw the conclusion that the laws of nature are simply incompressible on an empirical level.

Hence, one of the assumptions that we make when we say we are looking for a , in mathematical terms, is that the universe does have, on a fundamental level, an algorithm of some form to determine specifc outcomes. Obviously, in physics today it is assumed that this is true; that the universe must be inherently predictable.

Perhaps this is a naïve assumption, but it is an assumption we must make in order to develop a Theory of Everything. (Barrow, 2007)

The goal of science, arguably being a Theory of Everything, can be split into two diferent fragments. (Hawking, 2016, p.12) states that frst there are laws that tell us how the universe changes with time, and secondly there is the question of the initial state of the universe. The mythological Theories of Everything were inadequate in the sense that they explained neither of the two enigmas. A scientifcally accurate Theory of Everything should be able to answer both of these questions, and hence have real predictive power.

Another important factor to register is that a good theory must agree with experiment.

If it can’t be proven experimentally at all, then it could just not exist. A good theory is one which makes many predictions that can be disproven through experiment or observation, and so falsify the theory. A theoretical mathematical framework I will write about later on is , which is intertwined with a current candidate for a Theory of Everything: M-Theory. The signifcance of supersymmetry is that as

(Gubser, 2010, p.5) states, supersymmetry might simply not be there. This is a good theory as it can be denied through experiment, and therefore rejected as a real concept.

Why do we need a Theory of Everything?

It could be argued that a Theory of Everything will echo the completion of physics, as it would signify an entire understanding of the universe. It would conclude man’s age-old desire for knowledge, and settle the curiosity which is undoubtedly the reason why the importance of scientifc research is driven into so many students. However, there is reason to believe that such a theory may not be that useful today. The justifcation for why quantum mechanics has survived as a successful theory without including any mention of gravity is that the conditions in which we exist do not compare to the conditions where gravity would have noticeable efect on a quantum scale. As I have previously mentioned, the only situations where the efects of gravity would become signifcant are in a black hole singularity, or the initial conditions of the universe. Therefore, it is a rational deduction that currently it is not essential to have a complete theory of everything, as quantum mechanics still holds as a good approximation for the afairs that we would fnd practical uses of understanding.

Congruently, ideas surrounding reductionism can be used to resist the need for an ultimate theory, as if quantum mechanics can do a good enough job of explaining the most basic phenomenon which we can replicate in the conditions on earth, and we can explain each occurrence in a level system, i.e., empirical biology is complicated chemistry, and empirical chemistry is complicated physics (quantum mechanics), then it is possible to ignore the incomplete description. In this sense, the only reason to continue seeking a quantum theory of gravity is for the aesthetic of a complete picture.

This line of reasoning illustrates how a theory of Quantum Gravity could be perceived as somewhat irrelevant currently, since the two widely accepted theories we have today are still extremely accurate, and there is no reason why we can’t use them to deal with problems in regular conditions.

Scientifc Development through Unifcation and Reductionism

An important subject to analyse is the scientifc development of the past. In 1897 it was declared that all of physics was complete, and patent ofces were closing. It was thought that there were no more discoveries to be made, which in hindsight was an obvious mistake, as over 100 years later there is much still left unknown. In 1905, Einstein released four famous papers, one which gave another outlook on the quantum world: “the photoelectric efect”. It then became apparent that physics was not complete at all, consequently leading to a new era of scientifc development.

In fact, it was only 1905 when the two main candidates for explaining the world around us began developing.

About 30 years earlier, Maxwell released a paper on electromagnetism, which connected two phenomenon which were previously regarded as separate. The process of connecting foreign or disparate concepts, and explaining them in the same framework is called unifcation. In this specifc case, Maxwell unifed the electric force and the magnetic force by describing them as manifestations of principally the same thing: light. Essentially, light is just electromagnetic radiation, and this unifcation explained a broad range of physical phenomena. There is a longer history of unifcation in physics, all of which have been somewhat rewarding in physics.

Therefore, it is feasible that the way to pursue a Theory of Quantum Gravity is from a reductionist viewpoint. Allowing this stance, we can approach the current model we have -including Quantum Mechanics and Relativity- to create this map of how a reductionist perspective could perhaps face this issue. This map shows the synthesis of ideas which leads to the importance of a theory of Quantum Gravity. The arrows in this scheme represent the direction towards deeper, or more general, layers in our description of nature (Zinkernagel, 2006, p. 297). The key notion here is that the further down the diagram we look, the smaller and more general the description of the phenomenon is. Therefore, a logical conclusion would be that the deepest level, a Theory of Quantum Gravity, would be able to explain everything. Therefore, it can be argued that the higher levels are just special cases of the fundamental level, but can still be used to describe the world in essentially the same way. (Zinkernagel, 2006)

(Zinkernagel, 2006, p. 298) argues that the job of a is not just to fnd the most fundamental theory by following these arrows, but to reverse the arrows and use said theory to explain the universe from the ground up, hence reconstructing the universe from scratch.

However, let’s not accept a reductionistic approach to be fawless too soon, as there is reason to believe that in this situation, it may not be as fruitful as it has been in the past. Perhaps some of the levels of explanation are simply not reducible to the deeper levels, and can’t be explained in a smaller, more general sense. This would mean that a purely reductionist approach would certainly not to a singular Theory of

Everything.

Bohr argued that we can’t understand quantum events, for example the double slit experiment, without considering the measurement devices in a certain experimental context. For example, in the double slit experiment, if we describe the measurement devices as quantum objects, they inherit identifable uncertainties -in position and momentum- which are described in Heisenberg’s Uncertainty Principle. This would mean that the measurement devices would adapt quantum behaviour, which leads to the notorious measurement problem (Zinkernagel, 2006). If the measurement devices did act as quantum objects, then surely, we would see superpositions in the measurement outcomes, for example dials being in two positions at once, which is evidently not true. However, if we treat the measurement device as classical, where it has a defnite momentum and position, then this issue is solved. The idea of having to treat the measurement device as classical, when it is used to describe a quantum system, implies that the problem we are trying to solve (having a single Quantum

Theory which can describe everything at once) may not have an answer in the form we expect.

String Theory and Extra

String Theory is one of the leading contenders in a Theory of Quantum Gravity, as it naturally incorporates gravity in its structure. It was invented in the late 1960’s, and its purpose was to describe the strong , however it experienced problems with a massless -two particle which consistently appeared. Later, it was proposed that the unidentifed particle was the force carrier particle for the gravitational feld: the . Hence, gravity is included in the universe which String Theory describes, which is what makes it one of the leading candidates for a theory of Quantum Gravity.

(Becker, et al., 2007)

In more detail, String Theory is essentially the idea that every fundamental particle is a one-dimensional string, but each particle has diferent inherent properties due to the string vibrating at diferent frequencies, like the diferent frequencies which are present on a standing wave with two closed ends. These distinct modes could, in theory, be responsible for the diferent properties of particles. String Theory originally had problems which lead to inferences that for the Theory to be consistent it must have twenty-six dimensions, however when a new specifc type of was introduced -called supersymmetry- the number of dimensions required was reduced to eleven. This is questionable, as we obviously only see three dimensions of space and on of time. However, Kaluza-Klein theory explained how these could exist and be almost impossible to detect to humans or any other macro-scale object. Kaluza-Klein theory essentially suggests that the extra dimensions are small and curled up, which makes them hidden and inaccessible to us, however quantum objects would be able to access such dimensions. Brian Greene illustrates these extra dimensions in a very understandable way. Say you were looking at a pole from a distance, and the pole was stuck upright in the ground. To you, that pole would be essentially one-dimensional, i.e., there is negligible thickness and width, only height. Such an object would look like the y-axis of a graph, or in other words, the dimension of height. On the other hand, for an ant on the pole, it also has access to the circumference of the pole, as well as the height. Therefore, the ant has access to what is essentially an extra dimension compared to what we can access (obviously the pole is three dimensional in )(Greene, 2013). Kaluza-Klein theory was published in 1921 by Kaluza, and extended in 1926 by Klein, which was almost forty years before

String Theory was discovered. The theory did manage to unify gravity and electromagnetism (which were the only known forces at the time), but the theory had some major faws, for example the physical values predicted from theory did not correspond to the physical values which are measured accurately through experiment

(Založnik, 2012). Many unifed feld theories include Kaluza-Klein theory, as embracing the concept of extra dimensions can be fruitful. (Založnik, 2012) This is comparable to how the small, hidden dimensions are described in String

Theory, however the one extra dimension considered in Kaluza-Klein Theory is substituted for a total of 11 dimensions in String Theory.

This is also an experimentally testable idea, which is always a desired property when dealing with . It was thought that in the Large Hadron Collider, in high energy collisions, some of the energy could escape into these hidden dimensions.

Originally, it was believed that these hidden dimensions would have lengths associated with the Planck length~10-35m, however after reconsidering, many physicists believed that these hidden dimensions may actually be accessible at the Large Hadron Collider, which can probe to lengths of ~10-19m. (Antoniadis, 2001)

Unfortunately, there has not yet been any evidence that this happens in the real world, and so there is no experimental evidence that Kaluza-Klein dimensions could exist.

M Theory and other Candidates for a Quantum Theory of Gravity

The eleven-dimensional Theory of Superstrings was nicknamed M-theory.

Supersymmetry has not yet been experimentally proven to exist, but it does make testable predictions (Duf, 2001). One of these predictions is that every particle in the would have a super-partner, i.e. every would have a super-partner, and every fermion would have a boson super-partner. However, this is obviously not true as we do not detect such particles in the world around us, hence supersymmetry -if it exists- is a broken symmetry (Duf, 2001). It is argued, though, that at higher energies supersymmetry may be restored, and this would be direct evidence of supersymmetry. In September 2008, when the Large Hadron Collider was initially activated, it was believed that it would be able to detect supersymmetry, or the super-partner particles that appear in unbroken supersymmetry. (Becker, et al., 2007) state that a variety of arguments, not specifc to string theory, suggest that the characteristic energy scale associated with should be related to the electroweak scale, in other words 100GeV to a few TeV. Hence, it would be feasible that the Large Hadron

Collider could detect such super-particles.

During the Large Hadron Collider’s second run, in 2015 to 2018, the ATLAS collaboration has looked for signs of the staus and ; staus are the super-partners of the third generation of electron, the tau electron. Such particles are thought to be very unstable and rare. Unfortunately, there was no data to suggest the existence of the staus (Lopes, 2019).

Consequently, there is currently no evidence for supersymmetry, and hence M-theory.

It could be argued that this is due to the energies of particle accelerators still being too low to detect such symmetry, but considering there has been no evidence of any super-particles over the last 13 years in the Large Hadron Collider, it may be time to abandon the concepts of string theory, as it may simply not be real.

This would lead to the question “if String Theory is no longer the best candidate for a theory of Quantum Gravity, then what is?”. Unfortunately, there are currently not many other promising avenues which can facilitate a Quantum Theory of Gravity as well as

String Theory, however there is a theory called which assumes that the universe is pixelated on a quantum scale, not continuous, i.e. there is a smallest length, volume and time interval. Loop Quantum Gravity promises to, instead of creating a new theory altogether, describe a possible universe where both and General Relativity could exist together (Rovelli, 2008). Nonetheless,

Loop Quantum Gravity also has certain issues associated with it, and I will not go into detail here as Loop Quantum Gravity is also not backed up by any experimental evidence, like String Theory. However, I will briefy mention that evidence has been found which denies Loop Quantum Gravity: all electromagnetic radiation has the same speed, which would -in theory- not be true in a pixelized universe.

Therefore, we can conclude this chapter by evaluating the legitimacy of any current theories which supposedly ofer the solution to the question of Quantum Gravity. I have presented, in detail, the main candidate, and another candidate which is perhaps the second most popular contender for a Quantum Theory of Gravity. The most important matter which appears whilst discussing these two theories is that they are both completely theoretical, and so no matter how convincing the case, it is never as valid as a theory which has been experimentally proven, like Quantum Mechanics.

This questions the validity of research into these two areas of physics, when perhaps other areas of physics which are not as theoretical could be much more fruitful if worked on by each physicist still enquiring into these completely theoretical frameworks. On the other hand, it is also possible that to continue exploring these complicated felds could result in other discoveries which could indirectly lead to new discoveries in theoretical physics. It is difcult to say which of these two viewpoints is more legitimate, but considering no signifcant breakthroughs in String Theory have been made since the second superstring revolution, it would be fair to induce that

String Theory is quite possibly a dead end.

Conclusion It seems apparent that there are many concepts which all tie in to create a theory of

Quantum Gravity, and so to properly assess the situation, all of them must be studied in detail. Nevertheless, it is possible to evaluate the assumptions that are required to support the Quantum Theories of Gravity which we have today, and evaluate the predictions that they make, and whether or not they have been proven experimentally.

String Theory seems, at frst, like a hopeful candidate. In spite of this, as there is no experimental evidence, we cannot accept it as a true theory. Although the maths in

String Theory is beautiful, and many believe that mathematics is the language of the universe, it is possible that String Theory is nothing more than a dream of a Unifed

Theory. Without any evidence, this will not change. Many of the concepts which are key to String Theory, or M-theory, are similar: they may simply just not exist. In my opinion, I believe that our current theories of Quantum Gravity should not be explored in any more depth, as the world they describe is experimentally proven to be dissimilar to the world which we live in (at the energies we have reached so far). It may be useful to analyse data from future particle accelerators to look for super-particles, as it could be possible that the reason they have not yet been detected is because the current reachable energies are too low, but this is improbable.

Furthermore, it may be possible that the universe is not algebraically compressible on a fundamental scale, and so a Theory of Everything may not even exist. This is contradicted by ideas surrounding reductionism, as when levels of understanding are reduced to the fundamental, it seems certain that there must be a theory which incorporates the four fundamental forces. There is defnite evidence that gravity does afect quanta, which is demonstrated in the Neutron Interferometer experiments.

Hence, I believe that if a Quantum Theory of Gravity does exist, a new approach is needed to fnd it, which is dissimilar to Kaluza-Klein like theories, as extra dimensions have not yet been detected.

In general, this evidence suggests that today, a Theory of Quantum Gravity is certainly required, but a new approach to discovering it is needed. It is also conceivable that a

Theory of Quantum Gravity may only be an aesthetic issue, as without it we have a relatively clear picture of the universe, but just without a single equation to describe it.

Simultaneously, the two theories we have today do describe the world which we witness with incredible accuracy, but they also contradict one another when not applied in their ideal circumstances. In this sense, a theory of Quantum Gravity would not actually be that useful, as we can already describe the world with Relativity and

Quantum Mechanics.

Therefore, in my opinion, I believe that a Theory of Quantum Gravity is relevant in physics today, but there are more relevant areas of research that should be prioritised over studies like String Theory which is purely theoretical. It is important to physics as discoveries to time spent on research is far too low to allow Quantum Gravity to be a viable motive for research.

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