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.

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.

Light-cone Hamiltonian and commutators

We are at long last in a position to quantize the relativistic string. We have acquired considerable intuition for the dynamics of classical relativistic strings, and we have examined in detail how to quantize the simpler, but still nontrivial, relativistic point particle. Moreover, having taken a brief look into the basics of scalar, electromagnetic, and gravitational quantum fields in the light-cone gauge, we will be able to appreciate the implications of quantum open string theory. In this chapter we will deal with open strings. We will assume throughout the presence of a space-filling D-brane. In the next chapter we will quantize the closed string.

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.

Maxwell fields coupling to open strings

Among the quantum states of open strings attached to a D-brane we found photon states with polarizations and momentum along the D-brane directions. We thus deduced that a Maxwell field lives on the world-volume of a D-brane. The existence of this Maxwell field was in fact necessary to preserve the gauge invariance of the term that couples the Kalb–Ramond field to the string in the presence of a D-brane. We also learned that the endpoints of open strings carry Maxwell charge.

Since any D-brane has a Maxwell field, it is physically reasonable to expect that background electromagnetic fields can exist: there may be electric or magnetic fields that permeate the D-brane.

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.

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 22 and the similar looking string diagrams were all tree diagrams. This means that the graphs (see, for example, Figure 22.2) 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 23.1. This diagram represents an incoming particle which splits into two particles that rejoin to form an outgoing particle.

Area functional for spatial surfaces

The action for a relativistic string must be a functional of the string trajectory. Just as a particle traces out a line in spacetime, a string traces out a surface. The line traced out by the particle in spacetime is called the world-line. The two-dimensional surface traced out by a string in spacetime will be called the world-sheet. A closed string, for example, will trace out a tube, while an open string will trace out a strip. These two-dimensional world-sheets are shown in the spacetime diagram of Figure 6.1. The lines of constant x0 in these surfaces are the strings. These are the objects an observer sees at the fixed time x0.

Under the rubric of synchrotron radiation we understand the electromagnetic waves emitted by a charge moving with relativistic velocity and undergoing a transverse acceleration. It is characterized by a small opening angle and a high frequency caused by the velocity of the charge being close to that of light. Owing to the relatively simple motion of the charge, the radiation has clear polarization properties. Ordinary synchrotron radiation is emitted by a charge moving on a circular arc determined by a deflecting magnetic field. It has a broad spectrum, a typical frequency being γ3 times higher than the Larmor frequency of the charge. This spectrum can be modified by varying the curvature of the trajectory 1/ρ within a distance smaller than the formation length of the radiation, as is realized in undulators.

Synchrotron radiation has been investigated theoretically for over a century and experimentally for about half this time. Thanks to its unique properties, this radiation has become a research tool for many fields of science and electron-storage rings serving as radiation sources are spread over the whole globe.

This book tries to explain synchrotron radiation from basic principles and to derive its main properties. It is divided into four parts. First the general case of the electromagnetic fields created by an accelerated relativistic charge is investigated.