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 Cavendish were performed in the period from 1771 to 1773. They were followed by the investigations of 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 Oersted discovered that the electric current on a wire can deflect the needle of a compass placed nearby. Shortly thereafter, Biot-Savart (1820) and 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 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.

Duality symmetries and Hamiltonians

Duality symmetries are some of the most interesting symmetries in physics. The term “duality” is generally used by physicists to refer to the relationship between two systems that have very different descriptions but identical physics. The main subject of this chapter is one such situation that arises in closed string theory. You may think that a world where one dimension is curled up into a circle of radius R could easily be distinguished from a world in which the circle has radius α′/R (recall that α′ has units of length-squared), but in closed string theory these two worlds are indistinguishable for any value of R. There is a duality symmetry that relates them to each other. This symmetry is called T-duality, where the T stands for toroidal.

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.

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.

String theory is one of the most exciting fields in theoretical physics. This ambitious and speculative theory offers the potential of unifying gravity and all the other forces of nature and all forms of matter into one unified conceptual structure.

String theory has the unfortunate reputation of being impossibly difficult to understand. To some extent this is because, even to its practitioners, the theory is so new and so ill understood. However, the basic concepts of string theory are quite simple and should be accessible to students of physics with only advanced undergraduate training.

I have often been asked by students and by fellow physicists to recommend an introduction to the basics of string theory. Until now all I could do was point them either to popular science accounts or to advanced textbooks. But now I can recommend to them Barton Zwiebach's excellent book.

Zwiebach is an accomplished string theorist, who has made many important contributions to the theory, especially to the development of string field theory. In this book he presents a remarkably comprehensive description of string theory that starts at the beginning, assumes only minimal knowledge of advanced physics, and proceeds to the current frontiers of physics. Already tested in the form of a very successful undergraduate course at MIT, Zwiebach's exposition proves that string theory can be understood and appreciated by a wide audience.

T-duality and D-branes

Let us consider the propagation of open strings in a spacetime in which one spatial dimension has been curled up into a circle. Assume that we have a space-filling D25-brane, so that the open string endpoints are free to move all over space. As before, we choose the x25 dimension to be compactified:

All open string coordinates, including X25, satisfy Neumann boundary conditions at both endpoints, so they are all of NN type. In the presence of a compact dimension, closed strings exhibit fundamentally new states: closed strings can wrap around the compact dimension so that they cannot be shrunk to a point.

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.

Introduction

In this book, the quantization of strings was carried out using light-cone coordinates and the light-cone gauge. String theory is a Lorentz invariant theory, but Lorentz symmetry is not manifest in the light-cone quantum theory. Indeed, the choice of a particular coordinate X+ for special treatment hides from plain view the Lorentz symmetry of the theory. While hidden, the Lorentz symmetry is still a symmetry of the quantum theory, as we demonstrated by the construction of the Lorentz generator M−I. This generator has the expected properties when the spacetime has the critical dimension.

Since Lorentz symmetry is of central importance, it is natural to ask if we can quantize strings preserving manifest Lorentz invariance. It is indeed possible to do so. The Lorentz covariant quantization has some advantages over the light-cone quantization. Our light-cone quantization of open strings did not apply to D0-branes because the light-cone gauge requires that at least one spatial open string coordinate have Neumann boundary conditions. Covariant quantization applies to D0-branes.

Dp-branes and boundary conditions

A Dp-brane is an extended object with p spatial dimensions. In bosonic string theory, where the number of spatial dimensions is 25, a D25-brane is a space-filling brane. The letter D in Dp-brane stands for Dirichlet. In the presence of a D-brane, the endpoints of open strings must lie on the brane. As we will see in more detail below, this requirement imposes a number of Dirichlet boundary conditions on the motion of the open string endpoints.

Not all extended objects in string theory are D-branes. Strings, for example, are 1-branes because they are extended objects with one spatial dimension, but they are not D1-branes. Branes with p spatial dimensions are generically called p-branes. A 0-brane is some kind of particle.