The book "The Little Book of the Great String Theory"

 image String theory is often called the "theory of everything", because its goal is to describe all the fundamental forces of interaction in the universe, including gravity, quantum Mechanics and the theory of relativity. This revolutionary concept presents a new understanding of space and time, it seeks to explain the relationship of phenomena such as black holes and quark-gluon plasma, additional dimensions and quantum fluctuations.

Despite the complexity of the topic under consideration, Professor Stephen Gabser of Princeton University offers a capacious, affordable And an entertaining introduction to this one of the most widely discussed areas of physics today. With a minimum of mathematics, using interesting analogies, the author explains the essence of supersymmetry, duality, curvature of space-time so that it will be clear to any reader with knowledge of high school.

While the provisions of string theory are not completely proved, The secrets that have already been revealed to us allow us to admire the harmonious harmony of the universe and discuss the practical application of future discoveries in high-energy physics.

Gravitation Against Quantum Mechanics

Quantum mechanics and the General Theory of Relativity-two triumphal physical theories that emerged at the beginning of the 20th century-turned out to be inconsistent with each other. The difficulty arises when applying a method called renormalization. I will talk about the renormalization using the example of photons and gravitons, which we have already discussed in previous chapters. The point of inconsistency is that the photons lead us to a renormalizable theory (which means "good theory"), whereas gravitons lead to an unrenormalizable theory, and this actually means that we do not have a general theory describing photons and gravitons

Photons interact with electrical charges, but they are themselves electrically neutral. For example, an electron having an electric charge in a hydrogen atom, jumping from one energy level to another, emits a photon. This is what I mean when I say that photons interact with charges. The statement that the photon itself does not have an electric charge is equivalent to the statement that light does not conduct electricity. If it was not so, then you would receive an electric shock each time, grabbing for any object that had lain long enough in the sunlight. Photons do not interact with each other; They interact only with electric charges.

Gravitons react not to charges, but to mass, energy and momentum. And because they carry energy, they interact with each other. It may seem that this is not a particular problem, but it is precisely because of this that we are faced with difficulties. Quantum mechanics teaches us that gravitons behave both as waves and as particles. Particles are hypothetical point objects. And the point graviton will draw you the stronger, the closer you are to it. Its gravitational field can be described as the emission of other gravitons. We will refer to the test graviton as the parent graviton, and the gravitons emitted by it as the daughter gravitons. The gravitational field near the mother graviton is very strong. So, its daughter gravitons have huge energies and impulses. This follows directly from the uncertainty principle: the daughter gravitons are observed at a very small distance Δx from the parent graviton, and therefore, according to the uncertainty relation, Δp × Δx ≥ h / 4π, the uncertainty of their momentum, Δp, is very large. The trouble is that gravitons are also sensitive to the momentum. The daughter gravitons themselves will emit gravitons. The whole process branches and incredibly quickly diverges: you can not take into account all the consequences of the interaction of all gravitons.

In fact, something similar happens near the electron. If you try to measure an electric field very close to an electron, then you will provoke it to emit a photon with a very large momentum. This seems innocuous because, as we know, photons do not emit other photons. The trouble is that a photon can give birth to an electron-positron pair, which then emits more photons that will generate new electrons and positrons … Complete mess! The most surprising thing is that in the case of electrons and photons, you can nevertheless completely describe the whole multitude of particles cascaded from each other. Sometimes they talk about clothes, or "coat", from the offspring in which the electron is wrapped. Physicists use the term "virtual particles" to describe the electronic progeny. Renormalization is a mathematical method that makes it possible to trace all this mess.


The idea of ​​renormalization is that the "bare" electron is assumed to have an infinite charge and an infinite mass, but as soon as we "dressed" the electron, its charge and mass acquire finite values.

The problem with gravitons is that we are not in State to renormalize the cloud of virtual gravitons surrounding it. The general theory of relativity – the theory of gravitation – is non-renormalizable. This may seem like an intricate technical problem: there remains a weak chance that we simply look at the problem from the wrong side. There is also an even weaker chance that the theory, called the Maximum Supergravity Theory, turns out to be renormalizable. However, I and most string theorists are sure that there are fundamental difficulties in combining quantum mechanics and gravity.

Now let's take the string theory. The initial assumption underlying it is that the particles are not pointlike. Instead, the particles are represented in the form of vibrational modes of the string. According to the generally accepted idea of ​​string theory, strings are infinitely thin, but having finite length (of the order of 10-34 meters) objects interacting with each other in the manner of gravitons. "Stop-stop! – you will protest. "But is not the general problem with a cloud of virtual particles – in this case virtual strings – leading us to the same inability to trace the whole process of interaction, as in the case of gravitons?" The fact that strings are not point objects kills the described problem in the bud. The source of the difficulty in the case of gravitons is the assumption that they, in accordance with the term "point particle", have infinitesimal dimensions. Replacing gravitons with oscillating strings smooths the "sharp angles" of their interaction with each other. "On the fingers" it can be explained this way: when a graviton generates another virtual graviton, you can precisely specify the place and time, where it happened. But when the string branches, it looks like a branch of a water pipe.


At the branch point, there is no point at which a fracture occurs, the Y-shaped figure illustrating this process looks like a smooth continuous piece of pipe, only of an unusual shape. All this leads to the fact that the division of the string turns out to be a more "gentle" process than the division of a particle. Physicists say that strings interact by their nature "soft", while particles interact by their nature "hard". It is this softness that provides the best behavior of string theory, than the general theory of relativity, with respect to the applicability of the quantum-mechanical description.

Strings in space-time

Let us recall briefly what we said about the vibrations of the piano string. If you tighten the string between two pins and hit it with a hammer, it vibrates with a certain frequency. The frequency is the number of vibrations per second. In addition to the fundamental frequency, the piano string vibrates also on the overtones – oscillations of higher frequencies, giving the piano sound a characteristic coloration. I cited this analogy in describing the behavior of an electron in a hydrogen atom: it also has a fundamental vibrational mode corresponding to the ground state with minimal energy, and additional modes corresponding to higher energy levels.

The analogy described may not completely satisfy you: " Well, what relation does an electron in a hydrogen atom have to a standing wave on a piano string? "You ask. The analogy with an infinitesimal planetoid circling in orbit around a tiny sun – the atomic nucleus, is closer to the majority, is not it? Is such an analogy good? Yes and no. Quantum mechanics claims that the concept of an electron as a particle and the concept of an electron as a wave are so deeply intertwined that the quantum-mechanical motion of an electron-particle around a proton can really be described as a standing wave.

Comparison of a piano string with The strings that appear in string theory are in fact a very correct method. To avoid confusion with different kinds of strings, I will refer to those strings that string theory deals with as "relativistic strings". This term has a very deep didactic meaning, because string theory includes the theory of relativity, both special and general. Now I want to talk about one string theory design, which is so similar to a piano string, as far as a string can at all be like a string. Relativistic strings can end in objects that are called D-branes. If we omit the effects associated with the interaction of strings, then the D-branes can be regarded as infinitely heavy. Details on the D-branes will be discussed in the next chapter, and now I will make only a small digression, so to speak, as a "crutch". The simplest D-brane is called the D0-brane (pronounced "de-zero brane"). This is a point particle. I already hear the indignation of some readers about the return to point particles: "Did not the author recently state that string theory aims to get rid of point particles?" Well, that was until the middle of the 1990s, and then the point particles again Returned to the theory of strings, and not alone, but brought behind a whole zoo of unknown animals. But I'm getting ahead of myself. All I want is to bring a string-theoretical analog of piano pins that hold the string in a tight state – and the D0-branes are so relevant in this role that they can not resist telling them. In short, we will stretch the relativistic string between two D0-branes, like a piano string between two pins. The D0-branes themselves are not attached to anything, but they remain immobile, because they have infinite mass. Funny, is not it? So, okay. About D0-branes – in the next chapter, and now – only about a strung string.

The lowest energy level of a stretched string corresponds to the absence of oscillations. Well … almost to the absence, because small quantum oscillations are always present, and this fact is important. It is most correct to imagine the lower energy level as having a small vibrational energy within the limits of quantum mechanics permitted. The excited levels of the relativistic string correspond to its oscillations either at the fundamental frequency or on the overtones of the fundamental frequency, and it can vibrate at several frequencies simultaneously, as well as the piano string. But, like an electron in a hydrogen atom, a relativistic string can not vibrate at an arbitrary frequency. An electron can select energy levels from a discrete set. In relativistic strings, everything is exactly the same. Different vibrational levels have different energies, and since mass and energy are related by the relation E = mc2, different masses correspond to different vibrational states.

It would be wonderful if I could say that the frequency of oscillation of a string is connected with its Energy by a simple relation of the type E = hν, as was the case for photons. Unfortunately, everything is not so simple. The total mass of the string consists of several components. The first of these is the rest mass of the string, which corresponds to the tension of the string between two D0-branes. The second is the mass corresponding to the vibrational energy, which in turn is composed of the vibrational energies of all overtones. Let me remind you that the energy and mass are related by the relation E = mc2. Finally, the third component is a mass corresponding to the energy of unremovable quantum fluctuations, called zero-point oscillations. The term "zero-point oscillations" makes us remember the fundamental non-elimination of quantum fluctuations. So: the contribution of zero-point energy to the string mass … is negative! I agree, it's strange. Very strange. To show how strange this is, I will give an example. If we confine ourselves to one vibrational mode of the string, we see that the zero-point energy of this mode is positive. Each of the higher overtones separately gives an even greater positive contribution to the energy of the string. But if we appropriately sum up the contributions of all overtones, then we get a negative number. If you think that this is not bad enough, then here's even more bad news: I hid part of the truth, saying that the contribution of zero-point energy is negative. All these effects-the rest mass, the oscillation energy, and the zero-point energy-enter the expression of the total mass by the squares of their quantities. And if in this sum the energy of zero-point oscillations prevails, then the square of the total mass turns out to be negative, and this means that the mass itself turns out to be imaginary, like the root from minus one.


Before you reject this nonsense indignantly, let me add that in the string theory the whole direction of research has been devoted to the elimination of the described problem. In a nutshell, the problem is that the square of the mass of the relativistic string in its lowest energy state is negative. Strings in this state are called tachyons. Yes-yes, these are the same tachyons, which in each series are opposed to the heroes of the "Star Trek". This is certainly bad news. In the model I described, it would be possible to get rid of the negative square of mass by pulling the D0-branes to which the ends of the string are attached, far enough so that the tension of the string becomes greater than the zero-point energy. But when there are no D0-branes nearby, the string itself remains. Deprived of the ability to attach to anything, it can close itself to itself. Now it is not stretched between something and something and can fluctuate, and maybe not. The only thing that it can not stop doing is to fluctuate at the quantum level. And, as before, quantum oscillations transform such a string into a tachyon, which is very, very bad for the theory. According to modern ideas, tachyons are unstable, they are akin to a pencil balancing on the point. You can try to balance such a pencil, but any slight blow will overturn it. The theory of strings containing tachyons resembles a theory describing the millions of pencils that fill the space at the tip.

However, I have too thickened the colors. There is a saving solution for tachyons. Suppose that the ground state of a tachyon string corresponds to the imaginary mass and its square: m2 <0. The vibrational energy also gives a definite contribution to the squared mass. Using the right deck and handing the cards in the right way, you can achieve that the total mass of the string will be exactly zero. This is encouraging, because, as we know, in the real world there are massless particles, for example, photons or gravitons. Therefore, if the strings really describe the real world, then they must be massless or, more strictly, at least some quantum states of the strings should be massless.

Note that you need to take the correct deck of cards. With this metaphor I wanted to say that we need 26-dimensional space-time. Perhaps you already guessed that everything will come to this disgrace, so I will not apologize. There are several arguments in favor of 26 measurements, but most of them are purely mathematical, and I'm afraid that they will not seem overwhelming to the bulk of readers. The argument I give is more physical. We would like to obtain massless quantum states of strings. We know that quantum zero-point oscillations "push" m2 in the negative direction. We also know that vibrational modes "push" m2 in the opposite direction. The minimum possible value of the vibration energy does not depend on the dimensionality of space, while the magnitude of the quantum zero-point oscillations depends. Let's look at it from which side: when something is vibrating – a piano string or anything else – it does it in some specific direction. The piano string oscillates in the direction in which the hammer struck it; For example, the piano string oscillates up and down, but not right-left. The oscillation selects one direction and ignores the others. В противоположность этому квантово-механические нулевые колебания происходят во всех возможных направлениях, и добавление каждого нового измерения добавляет квантовой флуктуации еще одно направление, в котором могут происходить колебания. Больше возможных направлений колебаний, или, как их называют, степеней свободы, означает большее количество флуктуаций, что приводит к большему отрицательному вкладу в m2. Остается лишь подсчитать, как правильно подобрать вклады в общую массу колебательных мод и нулевых колебаний. Получается, что одну колебательную моду с минимальным значением энергии компенсирует одно 26-мерное квантовое нулевое колебание. Смотрите на это с оптимизмом, ведь количество необходимых измерений могло оказаться нецелым! Что бы мы делали, например, с двадцатью шестью с половиной измерениями?

Если вы еще не вполне освоились с разными типами колебаний, не переживайте. Они очень похожи. Единственное различие между колебательными модами и квантовыми нулевыми колебаниями состоит в том, что колебательные моды могут присутствовать, а могут и не присутствовать, в то время как нулевые колебания присутствуют всегда. Нулевые колебания — это те минимальные движения, наличия которых требует принцип неопределенности. Помимо основной моды, в колебаниях струны присутствуют и обертоны, придающие струне новые квантово-механические свойства. Я предпочитаю представлять себе различные моды в виде простых механических моделей, например круговых колебаний, колебаний в форме листа клевера или крутильных колебаний. Каждая форма соответствует отдельной частице. Другими словами, одна и та же струна может выступать в роли различных частиц в зависимости от формы происходящих на ней колебаний. Но говорить о форме колебаний все же не совсем корректно, потому что эти колебания не механические, а квантово-механические. Правильнее говорить, что каждой частице соответствует своя квантовая мода. Геометрическая форма — это лишь удобный способ визуализации квантово-механических свойств.


Итак, мы имеем: хорошую новость, плохую новость и очень плохую новость. Струны, обладая разными колебательными модами, способны вести себя как фотоны или как гравитоны. Это хорошая новость. Они могут делать это только в 26-мерном пространстве. Это плохая новость. Кроме того, существуют колебательные моды, приводящие к мнимым массам и превращающие струны в тахионы, которые привносят в теорию нестабильность. Ужаснее этой новости быть не может.

Переход к суперструнам позволяет излечить теорию от тахионов, а заодно снизить количество необходимых измерений с 26 до 10. К тому же суперструны допускают новый тип колебательных мод, заставляющий их вести себя как электроны. Это уже по-настоящему круто. А если бы еще удалось придумать такие супер-пупер-струны, которые бы позволили сократить число измерений до четырех, можно было бы открывать собственный бизнес по их продаже. В этой шутке присутствует лишь доля шутки. В действительности существует вариант супер-пупер-струнной теории, называемый «теория струн с расширенной локальной суперсимметрией», сокращающий число измерений до четырех. К сожалению, эти измерения могут существовать только парами, то есть получаются либо четыре пространственных измерения и ни одно-го временнˆого, либо два пространственных измерения и два временнˆых. Словом, ничего хорошего. Нам-то нужно три пространственных и одновременное измерение. Из десяти измерений суперструнной теории — девять пространственных и одно временное. Нужно каким-то образом избавиться от шести лишних пространственных измерений, чтобы соотнести теорию с реальным миром.

Я много чего хотел бы рассказать о суперструнах, но этот рассказ ожидает своей очереди в следующих главах. Сейчас же я предпочту остановиться на вопросе лечения теории от тахионов. Суперструны флуктуируют не просто в пространстве-времени, а значительно более сложным и абстрактным образом. Эти особые виды флуктуаций позволяют решить проблему тахионов, но не так, как вы, возможно, подумали. Тахионы попрежнему остаются в теории как одно из решений для колебательных мод, обладающих мнимой массой, но фишка в том, что если вы будете рассматривать моды, отвечающие за поведение суперструны как фотона, гравитона, электрона или какой-то другой реальной частицы, то, как бы вы ни сталкивали эти частицы, каким бы образом они между собой ни взаимодействовали, они никогда не порождают тахионы. Тахионы как бы возможны, но они никогда не возникают. И это означает, что теория по-прежнему балансирует на лезвии ножа, но существует особый тип симметрии, помогающий сохранять это хрупкое равновесие. Такой тип симметрии называется суперсимметрией. Физики надеются найти экспериментальные доказательства существования суперсимметрии в ближайшие годы. Если они их найдут, многие из нас поверят в суперструны. Но об этом — в седьмой главе.

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