String Theory – part 6: The Basic Principles

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The Standard Model of particle physics describes all elementary particles as point-like objects. Hence, they neither extend in any direction nor possess any internal structure. Despite its tremendous success in predicting any physical phenomena in particle physics, we can be sure that it is incomplete since it describes only three interactions, leaving gravitational interaction behind. Moreover, all attempts to include gravity into the picture have failed due to the fierce quantum fluctuations at the micro-scale of the very space-time fabric itself. This contradiction has led physicists to search for a deeper understanding of Nature. In the last several decades String Theory has been on the theoretical frontier of the search for a unified theory.

String Theory proposes a unique way of describing the Universe on the tiniest of scales. The change of description, as physicists figured out, helps both General Relativity and Quantum Mechanics to be peacefully unified under a new overarching framework. String Theory suggests that the elementary components of matter are not point-like particles like in the Standard Model, but rather they are one-dimensional fibres, which could be thought of as infinitely thin strands perpetually vibrating each with a particular pattern. These strands are called strings. But unlike the strings on a guitar or violin, which consist of molecules and atoms, the strings of String Theory do not consist of anything, being essentially a fundamental and indivisible component of matter. According to String Theory all the elementary particles of the Standard Model – quarks, electrons, neutrinos, photons and others – are represented by one fundamental entity – vibrating string. These strings, however, are so small that they appear to be point-like objects even if we examine them with the most energetic particle accelerators to date, such as LHC (Large Hadron Collider).

But a simple theoretical change of point-like objects to strings already leads to profound consequences. Firstly, it seems to resolve the contradiction between GR and QM. Secondly, as I’ve mentioned above, all the matter particles and all the force-carriers are represented by one fundamental entity, so that we could say that String Theory is a unified theory of all matter and interactions that we know of. Finally, as we shall see in this and the following articles, String Theory once again dramatically changes our understanding of the physical world surrounding us. The description that you’ll see is taken from Brian Greene’s book “The Elegant Universe”.

A Brief History

In 1968th a young theoretical physicist Gabriele Veneziano was analysing the experimental results of the strong nuclear interaction. He had spent several years on that task until eventually he realised that a particular exotic mathematical formula called Euler’s beta-function seemed to be capable of explaining all the characteristics of particles involved in the strong interaction. This Veneziano’s work led to a multitude of other works which used the beta-function to describe the vast arrays of data. Veneziano’s realisation, however, was incomplete, since everybody understood that this function works, but nobody could explain why it is so. By 1970th Leonard Susskind, Yoichiro Nambu and Holger Bech Nielsen had managed to find the physical reason behind the Euler beta-function. They showed that if we replace dimensionless particles with tiny vibrating one-dimensional strings, then the strong nuclear interaction would be precisely described by the beta-function.

But even though this theory was simple and intuitively straight-forward, it was soon found to be flawed. In the 1970s experimentalists were able to look deeper into the subatomic world and what they found was that some predictions of the theory based on strings instead of point-like particles was in a direct contradiction with experimental data. At the same time a part of quantum field theory – Quantum Chromodynamics – was being developed intensively. This theory, which is based on the model of point-like particles, was extremely successful in explaining the characteristics of the strong interaction, which led the majority of physicists to abandon string theory.

Some researchers, however, did not want to scrap the theory since its mathematical structure was so beautiful that they believed that it had to point to something profound. Initially, one of the problems with String Theory was that it had too wide a range of characteristics for the force-carriers of the strong interaction. Some of these did describe the behaviour of gluons, but some predicted the behaviour of some other particles which had nothing to do with the strong interaction. Now here was a surprise for physicists. In 1974th two physicists, John Schwarz and Joel Scherk, realised that this drawback was actually a huge benefit for String Theory. They examined those strange modes of string vibrations and found out that one of these modes coincided strikingly with the characteristics of a long-sought particle responsible for the gravitational interaction – graviton. Although these particles are still beyond detection, physicists can explicitly describe some of their characteristics. Schwarz and Scherk found that these characteristics are precisely realised for some modes of vibration. Based on that they concluded that the first step of String Theory into physics was unsuccessful just because physicists narrowed down its domain of usage too much, looking only for the description of the strong interaction. What Schwarz and Scherk showed was that String Theory is not just the theory of the strong interaction, but rather it is a theory that includes the force of gravity apart from everything else.

The physics community, however, reacted on this suggestion quite composedly. String Theory failed on its attempt to describe the strong nuclear force, and the majority of physicists thought that trying to apply this theory to such a global issue as combining all the forces is just a waste of time. Further analysis showed that String Theory has its own inconsistencies with Quantum Theory, so that at the start of 1980s it seemed that the gravitational interaction still keeps itself not incorporable into the quantum picture. This was until 1984, when Schwarz and Michael Green – another pioneer of String Theory – showed that those inconsistencies with Quantum Theory could be resolved. Moreover, they showed that String Theory does have sufficient broadness to include all four interactions and all kinds of matter. This news spread across the entire physics community and this time it had tremendous success. A lot of physicists, including even undergraduate students, immediately started working with the theory with huge passion.

The period 1984 – 1986 has since been known as the “first superstring revolution”, where superstring refers to the title which the theory got at that time – Superstring theory. The prefix ‘super’ has to do with one of the main characteristics of the theory – supersymmetry – which we shall consider in the later articles. In that period there were thousands of scientific articles written by a huge number of physicists. These works showed that many characteristics of the Standard Model naturally result from the String Theory system. Moreover, many of those characteristics obtain a more complete description in String Theory than they do in the Standard Model. These achievements convinced many physicists that the theory may eventually become the grand unified theory for all matter and interactions.

However, after all those successes physicists continued facing significant obstacles. In theoretical physics obtaining precise solutions to the given equations may occasionally be very difficult. Usually physicists are trying to obtain approximate solutions, which could make a ton of sense, in this case. And this strategy works very well when you have a complete form of analysed equations. In String Theory, however, the situation is far more complicated. Here we have a situation where even the very derivation of equations has become so astonishingly complex that physicists have managed to derive only their approximate form. Thus in String Theory we have to search for approximate solutions to approximate equations. After the first superstring revolution physicists faced the situation where these approximate equations were incapable of giving the right answers to some crucial questions. After many unsuccessful attempts to deal with this situation many physicists, feeling frustration, have stopped working on String Theory and returned to their previous works. For those who remained in the camp it became clear that they have to develop new methods that would allow them to extravagate beyond those approximate solutions that had been obtained.

The end of this stagnation was provided by one of the strongest physicists of our era, Edward Witten, in his presentation at a string theory conference in 1995. In that presentation Witten showed the way to overcome the problem with approximate equations and laid the foundation for the second superstring revolution. In this and the next five articles we shall explore the achievements made in the first superstring revolution, and then Witten’s work with the achievements of the second superstring revolution will be considered.

What are Strings?

As we have previously seen, String Theory suggests that all particles – if examined with an extraordinary level of precision, many times that of the contemporary instruments – would be seen as tiny vibrating filaments of energy. As we shall see later, the length of a typical string is close to the Planck length, which is one hundred billion billion (10 to the 20th power) times smaller than the size of an atomic nucleus. So this is not surprising that contemporary accelerators cannot test the string nature of matter. Thus currently we have to rely on our theoretical investigations. We will describe the astonishing conclusions based on these investigations, but first let us consider the question about the nature of strings themselves. What are they made of? How could we be sure that they are truly a fundamental ingredient of matter? From the time of The Ancient Greeks the atom was believed for a long time to be this fundamental ingredient. Then, when the existence of atoms was confirmed, we discovered that these atoms themselves consist of protons, neutrons and electrons. Yet again this was not the end of the story, and eventually we found out that protons and neutrons consist of even smaller particles – quarks. So aren’t strings just a layer in that puzzle?

There are two answers to this question. The answer which naturally emerges from the mathematical apparatus of String Theory suggests that strings are truly fundamental, similar to the notion of atoms of The Ancient Greeks. So the question as to what they are made of is devoid of any meaning. Here we can think of a linguistic analogy, where letters represent the fundamental layer in any text. The text itself consists of paragraphs, paragraphs consist of sentences, sentences of words, words of letters, and there we have it; the question as to what letters consist of does not make any sense. Likewise, strings in String Theory are considered fundamental, albeit having some structure. So they do not consist of anything smaller. This is the first answer.

The second possibility is rather different. Despite all its successes we still don’t know whether String Theory is actually correct, and whether it would represent the final unified theory if it is confirmed. If String Theory is correct but fails to represent the final theory, strings might be just another layer in the Nature’s nest-doll. This is also quite possible.

In this series of articles, apart from the last few chapters, we shall be considering strings in the sense given by the first answer above, i.e. regard them as the truly fundamental component of matter.

The Unification through String Theory

Apart from the absence of gravitational interaction in its description, the Standard Model has another problem. It gives no explanation of the nature of some parameters it works with. For example, why those particles which we described in the first and fourth articles of this series have the characteristics we observe? Why the parameters of the particles such as their mass, electric charge, and others have the particular values we’ve measured? And finally, why does Nature have these particles arranged in three families? Did these properties emerge just by chance in our universe or do they hide some profound physical meaning?

The Standard Model is not capable of giving answers to these questions since it takes all of these experimentally obtained parameters as its input data. Without this input data the Standard Model would be incapable of making testable predictions. You might say that one of the pieces listed above – the mass of particles – has a mechanism which explains the experimentally observed values. Indeed, this mechanism was confirmed in July of 2012 when a team working at CERN announced the discovery of the long-sought Higgs boson. This particle is the smallest pocket of the Higgs field, just as a photon is the smallest pocket of the electromagnetic field. This Higgs field represents the mechanism by which elementary particles acquire mass. The intensity of interaction between any type of particle and this field gives the particular masses to different particles. But here the question about one parameter (mass) just translates to the question about other parameters – the characteristics of the Higgs field (or of the Higgs boson). Again, the Standard Model gives no answer to this question, so although the importance of the Higgs boson discovery is hard to overestimate, we still have the list of parameters in the Standard Model whose values are taken as the input data.

In String Theory all these characteristics are defined by only one parameter – the mode of a string vibration. We can first consider an analogy with a simple violin string. Any string can perform essentially an infinite number of different resonance oscillations. You can see a short list of such oscillations in figure 1 below.

strings violin

The term resonance oscillation means that it has a certain frequency (which you can think of as the number of periods per unit time) and that the number of oscillations between the two ends of a string could have only an integer value (similar to the properties of waves that we were considering in the article regarding the principles of Quantum Mechanics). Your ear perceives resonance oscillations of strings as different musical notes. Likewise, strings in String Theory also have these characteristics. Some examples of strings with different modes of vibration are shown in the figure 2.

modes of vibration

String Theory suggests that like strings of a violin being exposed to different resonance oscillations produce different musical tones, tiny fundamental strings being exposed to different modes of vibration produce different masses and different coupling constants. This implies that all those particle characteristics that the Standard Model takes as input data are determined by the modes of vibration being implemented by the strings sitting inside particles. We can conceptualise this by considering the mass of a particle. The energy of a particular mode of vibration is determined by two parameters: the amplitude (the distance between the midline and the peaks), and the frequency (the number of periods per unit time). The larger these two parameters are – the higher the energy. This means that more intense vibrations cause a string to be of higher energy, while less intense vibrations are associated with strings having lower energy, which makes intuitive sense.

strings energy

Now recall that according to the Special Theory of Relativity mass and energy are two sides of the same coin. The higher the energy – the more massive an object is, and vice versa. And here we can draw a conclusion: according to String Theory the mass of a particle is determined by the vibrational energy of the inner string. Therefore the strings inside heavy particles vibrate intensely, while the vibration of a string inside a light particle is quite composed. And because an object’s mass determines its gravitational characteristics, we have a direct connection between the mode of vibration of a string and the response to gravity of the corresponding particle.

Using more abstract reasoning researchers found a similar correspondence between the particles’ response to other interactions and other characteristics of strings’ vibration. For example, the coupling constants of the strong, weak, and electromagnetic interactions and electric charge of a particle are determined by the mode of vibration. Moreover, a similar principle holds for the force-carrier particles as well. Photons, gluons, weak gauge bosons, and – what’s particularly important – gravitons represent nothing but other vibrational modes of the same fundamental physical entity – string.

Thus in String Theory vibration determines everything! The measured characteristics of all elementary particles are determined by certain modes of vibration of inner strings. This view is radically different from the view we had before String Theory, where it was assumed that the difference between elementary particles is dictated by the fact that these particles are made, in a sense, of different ‘material’. Conversely, String Theory suggests that the material of all matter and all fundamental forces is essentially one. Each different particle represents one particular string, and all these strings are essentially equivalent to one another. And the difference between particles is dictated by the different modes of strings’ vibration. What we thought of as different material appears to represent the different tones performed by fundamental strings. The Universe consisting of countless strings works like a cosmic symphony.

Cosmic Symphony

As we’ve just seen, the characteristics of elementary particles are determined by the modes of vibration of corresponding strings. This suggests that if we are to determine precisely what modes are allowed, then we would be able to describe those elementary particles’ characteristics and see if the theory’s predictions match the experimental results. To figure out the list of allowed modes we have to ‘push’ a string by every possible means. But as we’ve seen, these strings are way too small for us to carry out such an experiment. Instead what we can do is to use our mathematical apparatus in order to ‘push’ strings theoretically. In the 1980s many researchers felt that this list would shortly be found and that the theory of everything was already in our hands. As was later figured out, however, the elation was premature. String Theory might eventually become the theory of everything, but it has some obstacles not allowing researchers to determine the spectrum of vibrational modes to a desired degree of accuracy. Therefore, despite some truly outstanding achievements, String Theory is incapable of providing the necessary testable details yet. So even though the theory is, in principle, capable of describing all the elementary particles’ characteristics, a lot of work needs to be done in order to achieve this goal.

In the following articles we shall see the problems which the experts in String Theory are facing, but first let us briefly familiarise ourselves with some of them. We are all familiar with what tension is. The objects around us could experience quite different levels of tension load. For example, the strings of a violin have much lower level of tension than the strings of a grand piano. You can feel it when you play such a musical instrument: you have to exert more force to play a melody on a grand piano than on a violin. That is because the grand piano’s strings with their high tension require more external energy to move. We can measure the tension of the aforementioned strings because we know their stiffness. In String Theory, again, we can’t conduct a direct experiment on a string since it is too small. In 1974, however, when Schwarz and Scherk found out that one pattern of string vibration corresponds to the graviton, by using an indirect method they also managed to determine its typical tension. What they found was shocking. Their calculations showed that the intensity of an interaction (in this case – gravitational) is inversely proportional to the tension of the corresponding string, and since the gravitational interaction is so extraordinarily weak, the derived value was colossal: one thousand billion billion billion billion (10 to the 39th) tons – the so called Planck tension. Thus the fundamental strings are much stiffer than ordinary strings. This result has three important consequences.

The Consequences of Stiff Strings

Firstly, the strings of a violin or a grand piano are anchored, which guarantees that their length remains constant. Conversely, there is nothing to restrict the shortening of fundamental strings. As a result, the colossal tension forces these strings to be squeezed up to ultramicroscopic size. The calculations demonstrate that a typical string being exposed to the Planck tension squeezes to the Planck length – 10 to the negative 35th meters.

Secondly, because of such strong tension the energy of a typical string takes on an extraordinarily large value. Recall that you have to exert more force on a string of a grand piano to make it vibrate than on a string of a violin because of different tension. Therefore, two strings vibrating with the exact same pattern but having different tension will possess different amounts of energy. The string with a higher tension will possess more energy than the string with a lesser tension. This tells us that the amount of energy of a string depends on two parameters: the particular mode of vibration and the particular level of tension. With this description you might think that if we decrease the frequency of a string and its amplitude in a continuous fashion, its energy would decrease correspondingly, until it reaches zero. But recall the quantum mechanical picture that we were discussing in the fourth chapter of this series. After that description we have known that according to quantum mechanics any fluctuations and wave-like perturbations – including the vibration of strings – can have only discrete amounts of energy. Thus the amount of energy in possession of a particular string is represented by the product of an integer and the minimal potential value of energy. This minimal value is proportional to the string’s tension and its frequency, and the integer number is defined by the amplitude.

A very important implication here is the following. Since the minimal energy value of a string is proportional to its enormous tension, this value would also be enormous compared to the energy of elementary particles that we are familiar with. It would be equal to the value known as the Planck energy. If we translate this value into mass by using E = mc^2, we obtain the mass roughly 10 billion billion (10 to the 19th) that of a proton. This value, you guessed it, is known in physics as the Planck mass (as you can see, almost everything in String Theory is ‘Plancktized’). This means that the typical mass of a string is equal to the product of an integer value and the Planck mass.

Here an important question emerges: if the natural scale of string theory has such overwhelmingly huge values for both energy and mass, how could it be used for much lighter particles such as protons, electrons and neutrinos?

The answer comes out from the laws of quantum mechanics again. Heisenberg’s Uncertainty Principle guarantees that there is no such thing as rest state for an elementary particle (recall the fourth chapter again). All particles perpetually experience quantum fluctuations. This is also true for strings: regardless of how ‘calm’ a string seems to be, it is always subject to quantum fluctuations. A relevant fact that the string theory researchers found in the 1970s shows that string vibrations and quantum fluctuations cancel each other out to a large degree, if we look at them from the energy perspective. This turns out to be possible if we take into account that the energy of quantum fluctuations is negative in value from the quantum mechanical point of view. Moreover, the value of this energy is approximately equal to the Planck energy value, hence it significantly reduces the positive energy of string vibration. This implies that the minimal energy (which we thought was equal to the Planck energy) in many cases cancels out to such a degree that the string’s mass gets close to the mass of elementary particles which the LHC and other contemporary accelerators deal with. Consequently, these very modes with the minimal value of energy correspond to the elementary particles which we have been aware of to date. For example, when Schwarz and Scherk were investigating that particular mode of vibration corresponding to the graviton, they found out that the energy of this mode cancels out entirely leading to a massless particle. And since it has been experimentally proven that the force of gravity propagates with the speed of light, and only massless particles can move with that speed, this provided credence for Schwarz’s and Scherk’s work.

However, such low-energy modes are the exception rather than the rule in String Theory, and typical vibrational patterns are much heavier than these. This suggests that all the particles that we were considering in the first article represent just a tiny island in the ocean full of high-energy strings. Even such heavy particles at the t-quark and the Higgs boson (with the masses around 187 and 125 that of a proton respectively) are detected in experiments because the enormous energy of their inner strings is significantly reduced by quantum fluctuations.

This leads us directly to the third consequence which is particularly significant in String Theory. There is, literally, an infinite number of possible modes of vibration. Does that not imply that there should be an infinite number of elementary particles which would certainly contradict the experimental data? The answer to this question is positive, but it does not necessarily imply the contradiction. String Theory does suggest that each possible vibrational pattern should correspond to an elementary particle. Our previous analysis, however, shows that the vast majority of vibrational modes corresponds to extremely heavy particles, many times that of the Planck mass. And since the most powerful particle accelerator to date (LHC) is capable of achieving energy roughly one million billion (10 to the 15th) times lower than the Planck mass, we can conclude that our possibility to thoroughly test such energies is long way off. However, this is only the start of our investigations on String Theory and later we will see that there could be other methods to test some of the theory’s predictions.

Gravity and Quantum Mechanics in String Theory

The unified picture in String Theory which we’ve just seen looks really tantalising. But the most astonishing achievement of the theory is surely the resolution of the contradiction between General Relativity and Quantum Mechanics. Recall from the previous chapter that the problem of their unification lies in the inconsistency between the main principles of the two. GR’s main principle requires the fabric of space to be nice and smooth, while one of the main principles of QM – Heisenberg’s Uncertainty Principle – says that such a picture is unapproachable if we consider this fabric on the Planck scale. On that scale the fierce quantum fluctuations of that very fabric lead to a disruption of the smooth geometric shape of space.

There are two different answers to the question as to how String Theory resolves this problem. One of those answers is rough, but it helps to build up a conceptual picture, and the second one is more accurate, though it is a bit more complicated. We shall consider both subsequently.

The rough answer could be interpreted as follows. Although this might sound a bit naive, one way to investigate the structure of objects is in throwing other objects at them and looking at how the thrown objects behave after being reflected off the object under consideration. As a simple example you can recall that the way we see any objects at all is because the photons of light reflect off them and are imposed upon our retina, and the information about them is transmitted into our brains. Particle accelerators use a similar principle: two particles are accelerated close to the speed of light and are smashed onto each other. After that collision scientists analyse the debris leftover and obtain the information about the structure of those particles.

The basic rule for such research is that the size of thrown objects (particles in this case) defines the resolution limit of the particle accelerator. To better understand this statement suppose you’ve got a peculiar task. You have some rigid object – say, a peach stone – which you can’t see, and an apparatus that strikes smaller rigid spheres at it. You need to depict the stone just looking at the trajectories of the reflected spheres. Of course, this might seem to be an impossible task, but we are considering the situation in principle, rather than in practice.

At the first try your apparatus strikes quite large spheres at the stone, say only two times smaller than the stone itself. In this case even if you are an expert in this game, the trajectories of reflected spheres would only be capable of providing you the information about nothing but the overall shape of the stone. This is because the resolution of the spheres which have a large size in comparison with the stone is insufficient for the small details in the stone’s structure to leave a noticeable fingerprint on the trajectory of reflected spheres.

peach stone 1

Next time, the apparatus is charged with much smaller spheres – 5 millimeters in diameter. In this case the resolution becomes a lot better since much smaller details in the stone’s structure leave a sufficient fingerprint on the sphere trajectory for a much more accurate depiction.

peach stone 2

Finally, on the last try the apparatus is charged with even tinier spheres – only ½ millimeter in diameter. Here the very subtle details in the structure of the stone influence the behaviour of the reflected spheres, so that the picture depicted by you could be considered a masterpiece.

Peach fruit stone on black
Peach fruit stone on black

The idea behind this imaginary situation is simple. The size of the measuring probe must be sufficiently small compared to the investigated physical characteristics; otherwise their resolution would be insufficient to study the structures of interest.

The same conclusions are also applicable, of course, if we decide to investigate the structure of our stone on molecular, atomic, and subatomic scales. The resolution capacity of the ½ millimeter spheres will tell us nothing about such tiny structures; they are too large to explore molecular scales. That’s why particle accelerators use particles, such as protons and electrons, as their measuring probes. On the subatomic scales, where the laws of quantum theory change our everyday notions, the most appropriate resolution is given by the quantum wavelength, which defines the uncertainty in the location of a particle. If this sounds strange, you might want to review the explanation of the Uncertainty Principle given in the fourth article. We established there that the minimal uncertainty in a particle’s location is approximately equal to the wavelength of the particle which is used as the measuring probe. In that chapter, however, we were also told that the wavelength of a particle is inversely proportional to its momentum which is defined, roughly speaking, by the particle’s energy. Thus, by increasing the energy of a measuring probe we can reduce its wavelength, and hence amplify its resolution capacity. Which makes intuitive sense: particles with higher energy have higher penetrating capacity and can be used to investigate finer and finer details.

This leads us to the obvious difference between dimensionless particles and string fibres. Since a string does have some length, it cannot be used to investigate structures shorter than its length (whose value, as we’ve seen, is approximately equal to the Planck length). In 1988 two string theorists – David Gross and Paul Mende – showed that the continuous increase of a string’s energy does not lead to the continuous increase of its resolution capacity, which would be the case for point-like particles. These physicists demonstrated that for a string, increases in energy initially lead to increases in its resolution capacity, but eventually, when it reaches some value, further increases in energy amplify the size of the string instead of increasing its resolution capacity. And as we’ve seen, when the string gets larger, its resolution capacity diminishes. The typical size of a string is close to the Planck length, but if we were to pump it up with an enormous amount of energy – the amount we can hardly imagine, but which was typical right after the Big Bang – then the string could be inflated to the macroscale. This would be a horrible instrument to explore the microworld! This implies that whatever method you use, the physical size of strings would not allow you to use them for digging into the sub-planck scales.

And the conflict between GR and QM appears exactly on the sub-planck scales. But if strings are elementary and fundamental components of matter, and they cannot be used to investigate the sub-planck scales, then nothing consisting of them could experience the effects of quantum fluctuations on those scales even in principle. We can draw an analogy by thinking of what we feel when we stroke the surface of a table. Even though the table appears completely smooth to our hand, we know that it actually consists of discrete objects, and in fact is granular and rough. Our fingers are just too large objects to notice that. Likewise, strings appear to be too large to experience the destructive effects of quantum fluctuations, and since they are the smallest objects, nothing would be subject to those effects whatsoever. Particularly, String Theory wipes off the fatal infinite solutions in physical equations. I think I should repeat this conclusion once again just to hit the point home. Strings are considered the most fundamental components of matter in String Theory. Yet, they appear to be too large to notice the destructive quantum fluctuations on the sub-planck scales. This suggests that there is no method which would allow anything at all to experience the effects of those fluctuations, even if they exist on the sub-planck scales.

Have We Solved Anything At All?

The explanation given above might seem unsatisfying to you at this point. Instead of resolving the conflict on the sub-planck scales we seem to use the non-zero size of strings just to get round the problem. Have we solved anything at all? Yes, in fact we have. This part would allow us to better understand this.

An important thing to comprehend from the above explanation is that the conflict between GR and QM lies in our very assumption that the structure of space can be defined on the sub-planck scale. And this assumption is based on point-like nature of elementary particles. This implies that the main conflict of the XX century physics has been begotten by ourselves. And since we assumed that all elementary particles should have no physical dimensionality, we had to consider the structure of space on arbitrarily small scales. And there we faced the contradiction which seemed almost impossible to resolve. String Theory suggests that we bumped up against this problem just because our assumption was incorrect. According to the rules of String Theory, there is a limit to which the definition of space can make any sense at all. The fatal inconsistencies between GR and QM emerge out of our unawareness of this limit.

Here I imagine a question that might pop up in your mind. If String Theory proposes such a simple answer to this important problem, why had so much time passed before physicists came up to the idea of elementary components of matter having some physical extension? You might be surprised to hear that this idea had already been up for several decades. Some of the greatest minds of the XX century such as Heisenberg, Dirac, Pauli and Feynman did have a conjecture that the elementary components of matter might have a form of a tiny droplet instead of being point-like objects. When they were trying to develop a theory with such objects as its elementary components, however, they run into a problem of building up a theory which would be consistent with the prevailing principles of physics, such as the conservation of information (required by Quantum Mechanics), and the incapability of information to travel faster than the speed of light (one of the main principles of the theories of relativity). Each of their attempts, as well as the attempts of other physicists, encountered the non-compliance with at least one of these principles. That’s why it was thought for a long time that it was impossible to construct a theory based not on point-like particles. The researchers on String Theory, however, have been showing again and again that the theory is consistent with all the fundamental principles of physics.

The More Accurate Answer

The answer given above has familiarised us with the basic concept of how String Theory could cope with the devastating quantum fluctuations on the sub-planck scales, and we could easily jump right to the next article at this point. But as long as we are familiar with the basic concepts behind the Special theory of Relativity, we now have the means to describe how String Theory has resolved the contradiction in a more accurate way.

Firstly, let us consider how two dimensionless particles would interact with each other, and how they would be used as measuring probes if they existed. If we consider a pair of particles as some sort of billiard balls, and let them follow two crossing paths, eventually they would collide, which in turn would influence the direction of their motion.

Quantum field theory shows that a similar thing happens in its domain: the two particles collide with each other and change the direction of their motion. But the details of this process in quantum field theory are different. Let’s imagine that our pair of particles is an electron and its antiparticle – a positron. When a particle and its antiparticle collide they annihilate giving off pure energy, which then translates into a virtual photon. This photon, in turn, travels a short distance and releases the energy it contains in the form of another electron-positron pair. These particles then continue following the same path as they would if they were two billiard balls (you can see it in the figure 7 below). What’s of particular interest to us here is the point at which two initial particles collide with each other and annihilate. As we shall see, this point can be precisely determined if we work with point-like objects.

particle interaction

What would happen if we replace point-like particles with tiny oscillating loops (strings)? The basic properties of the interaction would remain unchanged, unless we examine the interaction on the Planck scales. Suppose we have two strings with the modes of vibration corresponding to an electron and positron. The process would seem similar to that described above. Two strings would follow crossing paths, collide and annihilate releasing a virtual photon (which is nothing but another string), that would travel a short distance and would again be splitted up into two strings. Since the photon is just another string, we would see that two initial strings, in a sense, merged together and became one for a fraction of a second. You can see it in the figure 8 below. Here we have a picture of so-called world surface, which we can split by 1-dimensional vertical slices and obtain the position of the strings at each point of time. As we can see, at the centre of this world surface our strings collide and merge together into one string representing a virtual photon.

strings interaction

As we have emphasised above, the interaction between two point-like particles happens at a point which can be precisely determined. Any observer, no matter how fast they are moving, would agree on where the interaction has occurred. For the interaction between one-dimensional strings, however, this is not the case.

Suppose we have two observers – John and Kate – moving relative to each other. Now we return to our world surface shown above. By slicing it we can rebuild the process of string interaction moment after moment. Firstly, we shall take John’s perspective and see how the interaction would appear to him. Below we can see how the world surface would be sliced from John’s point of view. Each slice is showing the simultaneous events – or simultaneous positions of strings in this case – from John’s frame of reference. As always, here we have only two-dimensional array of events, but the argument that we are drawing would be applicable for a 3-dimensional world surface as well. The particular importance is in the third slice where the strings come in contact and merge.

john

Now let’s take Kate’s perspective and repeat the process once again. As we were discussing in the second article, Kate and John would disagree on the question of simultaneity because of their relative motion. From Kate’s reference frame simultaneous events would lie on our world surface at a different angle.

kate

Comparing two frames of reference – John’s and Kate’s – we see that their opinions on where and when the strings came in contact are different. This implies that there is no sharply defined point in space and moment in time where the strings did come in contact! Both these characteristics are once again dependent on the observer’s frame of reference. So according to String Theory the point of interaction has, in a sense, been spread out over the entire surface represented by two figures above on slices (c).

Conversely, if we are to consider the interaction between point-like particles, we will again conclude that there is a definite point in space and moment in time where the interaction has occurred.

Interactions between dimensionless particles happen at such definite points of spacetime. When the particle under consideration is a graviton, this leads to a catastrophic outcome – an equation gives an infinite answer. Conversely, the non-zero length of strings smears that point of spacetime where the interaction takes place. And since different observers register the interaction occurring at different points of spacetime, the actual place where the interaction takes place is literally being smeared-out throughout this entire area. This magnification, if applied to the gravitational interaction, helps us to get rid of infinite answers, thus it resolves the contradiction between GR and QM. This is the more accurate description of the rough answer given above.

The details on the sub-planck scales, which we would be able to examine with the point-like objects, appear just not reachable in String Theory. And if the theory does represent the ultimate description of Nature, there is no way to go into the realms of sub-planck scales by any means. We can steer clear of the conflict between GR and QM in the Universe where there is a limit to how deep we can get in terms of scales. This is a universe of String Theory, in which the laws of General Relativity peacefully coexist with the laws of Quantum Mechanics.

I thank everyone for taking some time to read this article. The next time we shall be considering a very important concept in String Theory, and will familiarise ourselves with why the theory changed its name to “Superstring Theory” after the first superstring revolution.

 

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