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PREFACE This book titled ‘STRINGS IN THE COSMOS’ is inspired from the lectures given by Professor Leonard Susskind on string theory. It is an independent research work done by, the author ‘RISHAB SINGHA’ to meet the needs of those readers who wants to learn string theory from scratch. With the availability of loads of information on string theory, it becomes very chaotic for a beginner to study string theory in a systematic and an organized way. Therefore, this book has been specifically, designed to help understand a beginner the very basics of string theory. The author does not claim any originality in this work. However, the subject matter of the book has been arranged systematically keeping continuity as far as possible. Different topics have been, discussed in easy and simple language and illustrated with suitable diagrams where necessary.

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CONTENTS

Chapter 1: Origin of string theory Chapter 2: Relativistic vs Non- Relativistic kinematics Chapter 3: What is a particle? Chapter 4: The Tachyon problem Chapter 5: Theory of closed strings Chapter 6: Dimensional analysis of strings Chapter 7: Scattering of strings Chapter 8: Conformal mapping Chapter 9: Strings on a surface Chapter 10: Compactification of strings

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INTRODUCTION Big bang theory tells us that the expanding universe, we see today was very hot, dense and compact to a very tiny size. The universe started from the space-time singularity and the so, called beginning of time. However, there is a problem with the theory. We can only explain things after one minute from the big bang. As we travel back in time by rewinding the clock using the reverse calculations of general theory of relativity, we can explain the universe upto a point of 10−32 sec. If we travel further back in time to 𝑡 = 0 (space-time singularity) to explain how was the universe that’s, where the problem starts. Beyond 10−32 sec, the universe was too hot for the forces to exist in a way that we see today, if we go further back to 10−40 sec then we can see that the strong nuclear force also unites with the electroweak force. We have to keep in mind that we have moved back in time by using the reverse calculations of general theory of relativity. However, if we travel further back in time then the general theory of relativity breaks down and comes in serious conflict with quantum mechanics. This is, known as the singularity problem. Hence, we need a grand unified theory.

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1 ORIGIN OF STRING THEORY The four fundamental pillars of physics which are known as Newtonian mechanics, General and Special theory of relativity and Quantum mechanics forms the basic foundation of modern physics, but the problem which physicists have been trying to solve for decades is to fit Quantum mechanics and General relativity together although quantum mechanics fits perfectly with special theory of relativity. As a result, we have Quantum field theory and Quantum chromo dynamics, but the problem of a “GRAND UNIFIED THEORY” or “ THE THEORY OF EVERYTHING” still remains i.e. we need a new theory which could unify all the four fundamental pillars of physics in one theory and “STRING THEORY” is considered to be a promising contender. In other words, string theory can explain the quantum theory of gravity. However, the origin of string theory is not from the quantum theory of gravity or we can say that string theory has not been developed keeping in mind of the gravity problem rather it has its roots from a very different branch of physics, it was originally developed from particle physics more particularly from hadrons physics. What are hadrons? If we break an atom, we get a proton, neutron and an electron further if we break a neutron or an electron we get a quark. The subatomic particles, formed by the combination of these quarks are hadrons. Baryons and mesons are examples of hadrons, the baryons comprised of three quarks whereas the mesons contain two quarks and an anti-quark. Furthermore, mesons can be of two types: ∏ (pie) meson and 𝞺 (rho) meson. More emphasis, have been given to mesons for explaining string theory but during the time of 60’s and 70’s very little was known, it was not even known that mesons had a quark content to it and the idea of gluons didn’t even existed at that time. A gluon is an elementary particle that acts as an exchange particle for the strong force between quarks. It is analogous to the exchange of photons in the electromagnetic force between charged particles. In other words, they glue quarks together, forming hadrons such as protons and neutrons. For example, because of the electromagnetic force, the hydrogen nuclei form a ball of gas. If the force, have not been acted upon the nucleus,

5 the ball would not have existed. It is because of the force the sun exists by fusing hydrogen to form helium and because of the existence of sun we get light, which is interacting with us in the form of particles called photons, or we can say that the electromagnetic force is interacting with us in the form of photons or exchange particles. Similar concept have been used for gluons, when two quarks combine, there is some strong force which holds the quarks together, that strong force must interact with the help of some exchange particles, those particles are termed as gluons. However, it was not a part of the study. What was part of the study was the number of states of the particles and the number of particle states was large. There was proton, neutron another particle that was similar to them with greater spin and another particle with heavier mass and larger spin. Therefore, people drew diagrams and started plotting these particles, which were later on called Chew-Frautschi plots.

Figure 1: Chew- Frautschi plots. Here ‘l’ is the angular momentum. However, the question that arises is that why we chose 𝑚2 . Was it arbitrary or it had a very deep meaning? The questions, which we will find answers to when we proceed further. 1

Firstly, we plot the proton and the neutron with 𝑚2 =1and l= then there was another 2

3

1

2

2

particle with l= these were fermions and l is quantized by

integers and the order of

the particles goes on in a similar fashion remarkably forming a straight line. The ∏ (pie) mesons also exist in these trajectories. This trajectories, is known as Regge trajectories. ∏ (pie) mesons are mass less with zero (0) angular momentum. Now, if we put 𝞺 (rho) meson with l=1all of these trajectories are parallel to each other. The thing

6 to observe here is that whenever we increase the angular momentum by1unit mass is been increased by 𝑚2 . why? What is the connection here? Let us take an example, if we take a basketball and spin it because the basketball is at rest it will have some energy and hence mass. We can plot it somewhere in the graph and if we keep on increasing its angular momentum at some point centrifugal force will be so large that it will be torn apart ending it somewhere. Similarly if we consider an atom, and increase, its ‘l’ then the energy will also increase as well as mass where energy and mass being the same thing. At very high ‘l’, the mass is also very high, atoms will ionize, and we get a graph similar to fig 1. The main point taken from this was that the hadrons were very different from electron, proton or neutron. If we go on increasing the angular momentum of hadrons, it will go on increasing i.e. hadrons can have excited states with increased angular momentum but in case of electrons, it cannot exist in excited states with increased angular momentum, if we do so at some point it will always start decreasing unlike hadrons. This discrepancy became a matter of study. In addition, as we go further we will come to know that the hadrons have a stretching ability as it spins.

1.1 PION- PION SCATTERING:

Figure 2: Conventional Feynman diagram. What is happening here is that a ∏ (pie) meson is scattering another ∏ (pie) meson. Now from the above diagram we can see two ∏ mesons come together and form a 𝞺 meson. Although the 𝞺 meson is not composite here but just for the sake of addressing the particle, we call it 𝞺 meson and again it breaks down into ∏ mesons.

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Figure 3: According to quantum field theory, fig 3 is the same diagram as shown in fig 2 just upside down. Similar to fig 2 here also a 𝞺 meson is been exchanged between two∏ meson scattering but the 𝞺 meson which is formed may be or is in an excited state. However, the 𝞺 meson which is been formed is not unique what it means is that from the Chew-Frautschi plots, looking at Regge trajectories we can see that the 𝞺 meson can have excited states. Therefore, the 𝞺 meson formed in ∏-∏ scattering, can exist in any state. Quantum field theory suggests that we add all the excited states into the 𝞺 mesons, but here we will separate the excited states in the ∏-∏ scattering unlike quantum field theory suggests. Therefore, the 𝞺 meson formed will be in the first excited state, the next state, or the next state giving rise to a string like idea.

Figure 4: Anilation process Here 𝞺 ρ′ ρ′′ and so on are the probable excited states that tend to exist, however numerically it was very peculiar and each time they ended up by adding all the states.

8 For this reason, Leonard Susskind and other scientists started to draw diagrams.

Figure 5:

Figure 6:

If we cut the topology shown in diagram 5 horizontally into halves it will look like a Feynman diagram, shown in figure 6. Similarly if we cut the topology vertically into halves we will obtain a Feynman diagram, which is shown in figure 8.

Figure 7:

Figure 8:

This is basically, the origin. One more thing is been added out of curiosity and fun, it was that if they are quarks then what is holding those quarks together obviously some kind of force. We know, also mentioned earlier that force interacts by exchange of particles, so ‘gluons’ are exchanged by the force which keeps the quarks together as shown in figure 9.

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Figure 9: Now if we slice the two-dimensional space time diagram into halves we will see a pair of quarks connected together by a string of gluons something as shown in figure 10.

Figure 10: We would obtain similar diagrams if we cut it horizontally. This was a very crude and important idea of string theory. Now once a hadron becomes, a string we can spin it something, which we cannot do with an electron and then we can try to find out the nature of the string like elasticity and other various properties. We can also try to find out what kind of energy the string has. Firstly, it has kinetic energy (because of rotation) and stretching energy i.e. how energy increases as a function of angular momentum. If we recall the Chew- Frautschi plots and the Regge trajectories, we will see that l versus 𝑚2 is not as arbitrary as it seems it has a very deep connection. For example if we attach a golf ball at the two ends of a shoe lace and rotate it i.e. if we increase its energy at some point the lace will break but this seems unlikely in the case of the particles of Chew- Frautschi diagrams. The Regge trajectories of the particles behaving as strings seems to increase infinitely with the increase in the energy which suggests that l versus 𝑚2 does not seem so random instead it has a very deep meaning to it which in turn

10 gives us a string like behavior of the particles. Further the gluon field acts as a Maxwell field and the quark and the anti quark as the poles just as the poles of a bar magnet.

Figure 11: In the above diagram comparing the analogy to a bar magnet we have dipoles with magnetic flux lines spreading out. The connection between these flux lines and the gluon field as of our current understanding (quantum mechanically) is that fields can be described as particles or fields. If we take the field description, the gluon field between a quark and an anti quark where the field lines are spreading out will be similar to the field configuration of a positive and negative charge. According to classical electrodynamics as we separate the two charged particles the field lines spread out and the area of the field diminishes but as of our current understanding due to non-linearity of quantum chromo dynamics the field lines tend to attract each other and the effect of which the field lines acts as strings. If we stretch, the field lines, does not spread in fact it becomes longer and longer with each stretch. We can think of the field lines as gluons behaving as or forming strings. Nevertheless, these strings are very different from the real or ordinary strings. If we stretch a rubber band, the number of molecules in the rubber band is fixed and if we stretch hard, the atoms move away from each other and at some point, it will break. However, it does not happen in the case of gluon strings. When we stretch, the gluon strings, the atoms move away from each other but here as the atoms moves away the gap between the atoms are, filled up with newly formed atoms, which, makes it possible to stretch the gluon string to infinity without breaking it. Therefore, we can say that hadron physics gives rise to the basic idea of string theory and at the very heart of nature the particles behaves as strings. Hence, we can say that the origin of string theory is from the hadron physics.

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2 RELATIVISTIC VS NON-RELATIVISTIC KINEMATICS Let us now discuss some ideas about particles, energy and momentum. The mathematics that is been used to study hadrons was quite not right. Physicists kept getting particles with zero mass and spin two i.e. a graviton. Therefore, it occurred to the physicists that maybe the mathematics that is been used was actually the mathematics of quantum gravity instead of theory of hadrons.

2.1 How can we describe a relativistic string? Let us consider the energy of a point particle (non- relativistically). So the expression of energy for many non–interacting particles, can be written as E = ∑𝑖

𝑝2 2𝑚

+ 𝐵𝑖 ……………… (1)

Here 𝐵𝑖 is a constant that we added which behaves as an internal energy of the system, let us call it the binding energy. The only difference is that the internal energy is independent of 𝑃2 (momentum) and for many purposes, we can drop the added constant because it does not make any difference. Now the question is how we can obtain the non-relativistic expression of energy from relativity. We know, that the relativistic expression for the energy of a particle is given by E = ∑ √𝑃2 𝐶 2 + 𝑚2 𝐶 4 .... …. . (2) Where P=𝑃𝑥 + 𝑃𝑦 + 𝑃𝑧 .The non-relativistic limit is appropriate for particles moving very slowly which means its momentum is very small i.e. 𝑃2 𝐶 2 ≪ 𝑚2 𝑐 4 . Therefore, E =√𝑚2 𝑐 4 (1 +

𝑃2 𝐶 2 𝑚2 𝐶 4

)

12 E = m𝑐 2 √(1 +

𝑃2 𝐶 2 𝑚2 𝐶 4

)

We get the expression of energy with a correction term

𝑃2 𝐶 2

𝑚2 𝐶 4

, which is been described as ƹ

a small quantity. Now expanding the expression using the approximation √1 + ƹ=1+ . 2 Where, the term ƹ is equal to a small quantity. We get, E = m𝑐 2 + E=

𝑝2 2𝑚

𝑃2 𝐶 2 2𝑚2 𝐶 4

m𝑐 2

+ m𝑐 2 …. …. . (3)

Therefore, we got a non-relativistic expression, which is an approximation, and it is good when everything or all the particles are moving slowly. It is just an approximation if the whole system is moving slowly. However, we can have a box of particles where the box is moving slowly but inside the box, the particles may be moving with great velocity close to the velocity of light. In such cases, the approximate non-relativistic limit is suitable only when the system moves very slowly. However, the particles inside the box will have relative motion. Hence, we can show that non-relativistic physics is an exact description of relativistic physics. Therefore, instead of thinking of the system in its rest frame we will consider the system to boost up to have a huge momentum along one axis. Here we consider the axis to be z -axis. Now re writing the expression of relativistic energy we get E = √𝑃2 + 𝑚2 = √𝑃𝑧2 + 𝑃𝑥2 + 𝑃𝑦2 + 𝑚2 …. …. …. . (4) Where, C is equal to1. Since, we boosted the entire system along z –axis, 𝑃𝑧 will be very large and ‘m’ here is the rest mass and the components of momentum perpendicular to the boost does not change. Therefore here ‘𝑃𝑧 ’ is the big quantity and𝑃𝑥 , 𝑃𝑦 , m are kept fixed. So by expanding equation 4 we get, E = 𝑃𝑧 √1 +

𝑃𝑥2 +𝑃𝑦2 +𝑚2 𝑃𝑧2

Further by binomial expansion, E = 𝑃𝑧 (1+

𝑃𝑥2 +𝑃𝑦2 +𝑚2 2𝑃𝑧2

)

13 E = ∑ 𝑃𝑧 +∑

𝑃2 2𝑃𝑧

+

𝑚2 2𝑃𝑧

. From momentum, conservation ∑ 𝑃𝑧 is the total momentum of

the system and does not change. Adding it or subtracting it to the system will make no difference. Hence, we can drop it. Therefore, E=∑

𝑃2 2𝑃𝑧

+

𝑚2 2𝑃𝑧

…. …. . (5)

If we think of the energy of the system to be Hamiltonian, we can see that the Hamiltonian according to quantum mechanics is been related with time evolution. H= 𝜕

iħ . We can see in equation (5) that 𝑃𝑧 is in the denominator, which means that the 𝜕𝑡

energies are very small and the system is changing very slowly. From the Hamiltonian equation, we can see that for large ‘H’ we have an increasing 𝑃𝑧 and we will get a decreasing t. So what is happening here? The answer is we are having time dilation. More we boost the system up in our reference frame the slower things takes place. Comparing equation (5) with equation (1) we can see that the quantity 𝑃2 in equation (5), which represents motion in X-Y plane, is the energy equal to the square of X-Y momentum, which is also similar in the case of non- relativistic expression of a particle. Now the role of mass in the relativistic expression as compared to the non–relativistic expression is not rest mass, it is the momentum along the z –axis. However, what is mass, it is inertia and the momentum along the z- axis is also acting as a kind of inertia w.r.t the forces in the perpendicular direction and the whole quantity reduces to the nonrelativistic expression of energy in two dimensions. Now what about the term

𝑚2 2𝑃𝑧

.Since

we considered 𝑃𝑧 independent to the state of motion (𝑃2 ) in two dimensions just as in the case of non-relativistic expression we have considered the binding energy or internal energy to be independent of momentum 𝑃2 . Therefore, the term

𝑚2 2𝑃𝑧

plays the

role of binding energy or internal energy in the relativistic expression of energy. This analogy is incredibly useful for studying particle dynamics and absolutely, central to the study of strings. From the above analogy, we can conclude in a very precise and exact way the motion of a relativistic system when it is been boosted to a large momentum where it behaves completely non-relativistically with respect to the motion in the plane perpendicular to the boost. For this reason string theory is often referred, to as the mathematics of non-relativistic strings.

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2.2 STRING HYPOTHESIS Now we are in a condition to think of particles as strings using the above mentioned two dimensional, analogies with non-relativistic physics. To explore these strings as for our convenience something close to a rubber band can do. What we will do now is we will portray the mathematical description of a two dimensional rubber band, which is moving around in two dimensions. Let us take our rubber band to be an open rubber band. Let us say that someone took a scissor and cut it open. Firstly, let us try to understand what open string is and ask ourselves a question, what is the energy stored in these strings? We can think of the strings as a collection of point particles and eventually we will assign limits to it. By assigning a limit, what I mean is we will consider the mass of the string to be zero that is because the whole string has a finite mass and we are thinking of the string as being a collection of a virtual infinity of point masses. Therefore, it is for our own good and for simplicity, we consider the mass of the string to be zero. Hence, the energy of the string is been given by E = 𝑚𝑖 ∑

𝑥̇ 𝑖2 +𝑦̇ 𝑖2 2

…. …. …. … …. (6)

I.e. the energy is proportional to its kinetic energy where 𝑥𝑖 , 𝑦𝑖 represents sum of all the points also; we can say that the point particles are attracting each other. If it were not, so they would fly apart and would not be forming the string. In addition to the points, we also add little springs that connect the particles like a chain of little balls and little 𝑥̇ 2

springs. Rewriting equation (6) we get: E=𝑚𝑖 ∑ 𝑖 . What will be its potential energy? 2

The potential energy is the sum over all the neighboring pairs. Hence, the formula becomes (from Hooks law) E = 𝑚𝑖 ∑

𝑥̇ 𝑖2 2

+K

(𝑥𝑖 −𝑥𝑖+1 )2 2

…. …. . (7)

Where K is spring’s constant. Now what will happen if we go to a continuum limit, in simple words the points get more and more dense, we have to do two things firstly, we consider the mass to be zero for reasons, which is been mentioned earlier and the spring constant to be big. Nevertheless, why we consider the spring constant to be big?

15 Suppose we take a rubber band and stretch it, the rubber band will be very easy to stretch but if we take two points very close to the each other then it will be very hard to stretch. It means that spring constant gets big as mass gets small. Now replacing the sum by an integral, we notice now that the integral is over a parameter along the string for which we will have to introduce a mathematical parameter along the string and we call that parameter sigma (σ). Sigma goes from one end of the string where we can arbitrarily say zero i.e. σ is zero at one end and at other end we can arbitrarily say σ to be ∏. However, the question the reader should ask is why ∏ and not anything else? Because it will be useful to call it ∏ for reasons which we will get to know as we move forward. We will also study closed string, which goes all the way round in a loop for such situations it will be convenient to say that it goes from zero to 2∏. However, in the case of open strings it goes from zero to ∏. Therefore, this sum over the mass points is going to be an integral from zero to ∏ i.e. ∏

E = ∫0 𝑑𝜎[

𝑋̇ 2 (𝜎) 2

𝜕𝑥

+ ( )2 ] …. …. .. (8) 𝜕𝜎

X (σ) is a small element of the string. This is the energy of the string and we can also, express it in the form of Lagrangian as ∏

L = ∫0 𝑑𝜎[

𝑋̇ 2 (𝜎) 2

1 𝜕𝑥

− ( )2 ]…. …. . (9) 2 𝜕𝜎

Similarly, we can also express it in the form of Hamiltonian as ∏

H = ∫0 𝑑𝜎[

𝑋̇ 2 (𝜎) 2

1 𝜕𝑥

+ ( )2 ]…. …. (10) 2 𝜕𝜎

2.3 STRINGS WITH NO CENTRE OF MASS Now let us focus on a string, which happens to have no overall centre of mass motion in the two dimensional X-Y plane. What we are going to do is, use a model for a relativistic string, which is been based on infinite momentum thinking where we boost the system and the two axis move perpendicular to the direction of the boost. Here 𝑋̇ 2 = 𝑋 2 + 𝑌 2 𝜕𝑥̇

𝜕𝑥

𝜕𝑦

𝜕𝜎

𝜕𝜎

𝜕𝜎

( )2 = ( )2 + ( )2 …. …. …. .. (11)