## Of The Nonrelativistic String

This book will discuss how nonrelativistic quantum mechanics can be extended to describe:

• relativistic systems and

• systems in which particles can be created and annihilated.

The key to both of these extensions is field theory, and we therefore begin with an introduction to this topic. In this chapter we will discuss the quantization of the nonrelativistic, one-dimensional string. This is a many-body system which is also simple and familiar. Quantization of this many-body system leads directly to the (new) concept of a quantum field, and many of the properties of quantum fields can be introduced and illustrated using the nonrelativistic one-dimensional string ?s an example. The goal of this chapter is to use this simple system to develop an intuition and understanding of the meaning and properties of quantized fields. In subsequent chapters some of these ideas will be developed again in a more general, abstract way, and it is hoped that the intuition gained in this chapter will remove much of the mystery which might otherwise surround those more abstract discussions.

The discussion of relativistic systems begins in the next chapter, where the ideas developed here are immediately extended to the electromagnetic field.

### 1.1 THE ONE-DIMENSIONAL CLASSICAL STRING

We will approach the treatment of a continuous string by first considering a system of point masses connected together by "springs" and then letting the number of point masses go to infinity, and the distance between them go to zero, in such a way that a continuous system with a uniform density and tension emerges.

Start, then, with a "lumpy" string of overall length L made up of N points, each with mass m, coupled together by springs with a spring constant k. Assume that the oscillators move about their equilibrium positions in a periodic pattern, which is best realized by thinking of the string as closed on itself in a circle, as

shown in Fig. 1.1. The oscillators are constrained to vibrate along the circumference of the ring (which has a radius very much greater than the equilibrium separation I so that the system will be treated as a linear system with periodic boundary conditions). The Oth and iVth oscillators are identical, so that if fa is the displacement of the zth oscillator from equilibrium, then

d(j> o ~dt dt periodic boundary conditions.

The kinetic energy (KE) and potential energy (PE) are

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