Units and parameters

Units are nothing other than fixed quantities that we use for purposes of reference. A measurement involves finding the unit-free ratio of an observable quantity to the appropriate unit. Consider, for example, the definition of a second in the international system of units (SI system). The SI second (s) is defined to be the duration of 9 192 631 770 periods of the radiation emitted in the transition between the two hyperfine levels of the cesium-133 atom. When we measure the time elapsed between two events, we are really counting a unit-free, or dimensionless, number: the number that tells us how many seconds fit between the two events or, alternatively, how many periods of the cesium radiation fit between the two events. The same goes for length. The unit called the meter (m) is nowadays defined as the distance traveled by light in a certain fraction of a second (1/299 792 458 of a second, to be precise). Mass introduces a third unit, the prototype kilogram (kg), kept safely in Sèvres, France.

Introduction

In our investigation of classical string motion we had a great deal of freedom in choosing the coordinates on the world-sheet. This freedom was a direct consequence of the reparameterization invariance of the action, and we exploited it to simplify the equations of motion tremendously. Reparameterization invariance is an example of a gauge invariance, and a choice of parameterization is an example of a choice of gauge. We saw that the light-cone gauge – a particular parameterization in which τ is related to the light-cone time X+ and σ is chosen so that the p+-density is constant – was useful to obtain a complete and explicit solution of the equations of motion.

Classical field theories sometimes have gauge invariances. Classical electrodynamics, for example, is described in terms of gauge potentials Aµ. The gauge invariance of this description is often used to great advantage. The classical theory of a scalar field is simpler than classical electromagnetism.

A review of statistical mechanics

Our study of string thermodynamics will make use of both the microcanonical and canonical ensembles. Recall that the microcanonical ensemble consists of a collection of copies of a particular system A, one for each state accessible to A at a particular fixed energy E. In the canonical ensemble we consider the system A in thermal contact with a reservoir at a temperature T. This ensemble contains copies of the system A together with the reservoir, one copy for each allowed state of the combined system. In the canonical ensemble the energy of system A varies among members of the ensemble.

Loop diagrams and ultraviolet divergences

When calculating scattering amplitudes in particle physics, one typically uses an approximation scheme in which the strength of the interactions is assumed to be small. The amplitude is then written in terms of a perturbative series expansion in powers of this small interaction parameter. The Feynman diagrams we considered in Chapter 25 and the similar looking string diagrams were all tree diagrams. This means that the graphs (see, for example, Figure 25.2) contain no nontrivial closed paths, or loops. Tree diagrams give the first term in the perturbative expansion of scattering amplitudes. To go beyond this lowest-order approximation, one must consider Feynman diagrams with loops.

Consider the Feynman diagram with a loop shown in Figure 26.1. This diagram represents an incoming particle which splits into two particles that rejoin to form an outgoing particle.

It has been almost five years since I finished writing the first edition of A First Course in String Theory. I have since taught the undergraduate string theory course at MIT three times, and I have received comments and suggestions from colleagues all over the world. I have learned what parts of the book are most challenging for the students, and I have heard requests for extra material.

As in the first edition, the book is broadly divided into Part I (Basics) and Part II (Developments). In this second edition I have improved the clarity of many arguments and the general readability of Part I. This part is studied by the largest number of readers, many of them independently and outside of the classroom setting. The changes should make study easier. There are more figures and the number of problems has been increased to better cover the range of ideas developed in the text. Part I has five new sections and one new chapter. The new sections discuss the classical motion of closed strings, cosmic strings, and orbifolds. The new chapter, Chapter 14, is the last one of Part I. It explains the basics of superstring theory.

Part II has changed as well. The ordering of chapters has been altered to bring T-duality earlier into the book. The material relevant to particle physics has been collected in Chapter 21 and includes a new section on moduli stabilization and the landscape.

Fundamental string charge

As we have seen before, a point particle can carry electric charge because there is an interaction which allows the particle to couple to a Maxwell field. The world-line of the point particle is one-dimensional and the Maxwell gauge field Aµ carries one index. This matching is important. The particle trajectory has a tangent vector dxµ(τ)/dτ, where τ parameterizes the world-line. Because it has one Lorentz index, the tangent vector can be multiplied by the gauge field Aµ to form a Lorentz scalar.

The road to unification

Over the course of time, the development of physics has been marked by unifications: events when different phenomena were recognized to be related and theories were adjusted to reflect such recognition. One of the most significant of these unifications occurred in the nineteenth century.

For a while, electricity and magnetism had appeared to be unrelated physical phenomena. Electricity was studied first. The remarkable experiments of Henry Cavendish were performed in the period from 1771 to 1773. They were followed by the investigations of Charles Augustin de Coulomb, which were completed in 1785. These works provided a theory of static electricity, or electrostatics. Subsequent research into magnetism, however, began to reveal connections with electricity. In 1819 Hans Christian Oersted discovered that the electric current on a wire can deflect the needle of a compass placed nearby. Shortly thereafter, Jean-Baptiste Biot and Felix Savart (1820) and André-Marie Ampère (1820–1825) established the rules by which electric currents produce magnetic fields. A crucial step was taken by Michael Faraday (1831), who showed that changing magnetic fields generate electric fields. Equations that described all of these results became available, but they were, in fact, inconsistent. It was James Clerk Maxwell (1865) who constructed a consistent set of equations by adding a new term to one of the equations.

Introduction

We have so far studied bosonic string theories, both open and closed. These string theories live in 26-dimensional spacetime, and all of their quantum states represent bosonic particle states. Among them we found important bosonic particles, such as the photon and the graviton. Non-Abelian gauge bosons, needed to transmit the strong and weak forces, also arise in bosonic string theory, as we will see in Chapter 15.

Realistic string theories, however, must also contain the states of fermionic particles. You may recall that a quantum state of identical bosonic particles is symmetric under the exchange of any two of the particles. A quantum state of identical fermionic particles, on the other hand, is antisymmetric under the exchange of any two of the particles. Quarks and leptons are fermionic particles. To obtain them we need superstring theories.

Introduction

Interactions and the forces that mediate them make the world interesting. If the electron and the proton did not interact, there would be no hydrogen atom. The fine structure constant α = e2/(4πħc) quantifies the strength of electromagnetic interactions and determines the interaction potential between the electron and the proton (see Section 13.4). The hot filament of a light bulb emits photons, some of which are absorbed by your eye. Emission and absorption processes are also interactions. A neutron can turn into a proton, an electron, and an antineutrino. This process, called β-decay, is the result of a weak interaction.

In string theory the strength of interactions is parameterized by the string coupling g. The value of this dimensionless number is determined by the expectation value of the dilaton field, as we discussed in Section 13.4. The string coupling g, together with the slope parameter α′, determines the value of Newton's constant. The constants g and α′ also determine the tension of D-branes.

The idea of having a serious string theory course for undergraduates was first suggested to me by a group of MIT sophomores sometime in May of 2001. I was teaching Statistical Physics, and I had spent an hour-long recitation explaining how a relativistic string at high energies appears to approach a constant temperature (the Hagedorn temperature). I was intrigued by the idea of a basic string theory course, but it was not immediately clear to me that a useful one could be devised at this level.

A few months later, I had a conversation with Marc Kastner, the Physics Department Head. In passing, I told him about the sophomores' request for a string theory course. Kastner's instantaneous and enthusiastic reaction made me consider seriously the idea for the first time. At the end of 2001, a new course was added to the undergraduate physics curriculum at MIT. In the spring term of 2002 I taught String Theory for Undergraduates for the first time. This book grew out of the lecture notes for that course.

When we think about teaching string theory at the undergraduate level the main question is, “Can the material really be explained at this level?”. After teaching the subject two times, I am convinced that the answer to the question is a definite yes. Although a complete mastery of string theory requires a graduate-level physics education, the basics of string theory can be well understood with the limited tools acquired in the first two or three years of an undergraduate education.

Equations of motion for transverse oscillations

We will begin our study of strings with a look at the transverse fluctuations of a stretched string. The direction along the string is called the longitudinal direction, and the directions orthogonal to the string are called the transverse directions. We consider, for notational simplicity, the case when there is only one transverse direction – the generalization to additional transverse directions is straightforward.

Working in the (x, y) plane, let the classical nonrelativistic string have its endpoints fixed at (0, 0), and (a, 0). In the static configuration the string is stretched along the x axis between these two points. In a transverse oscillation, the x-coordinate of any point on the string does not change in time. The transverse displacement of a point is given by its y coordinate. The x direction is longitudinal, and the y direction is transverse. To describe the classical mechanics of a homogeneous string, we need two pieces of information: the tension T0 and the mass per unit length µ0. The total mass of the string is then M = µ0a.