Continuous Systems

Abstract

If one considers the motions of a continuous system — a stretched string or the air in a room — it might at first seem that one would have to deal with the infinite limit of an N-partiele system and that vast complexities would arise. This would of course be true if one wanted to understand everything the individual molecules are doing, but we will not be so ambitious. The right approach is to forget all about discreteness and deal from the beginning with continuous functions of the coordinates. This does not always work. If for example we want to study the elastic vibrations of a crystal we must start from the intermolecular forces. Arguments of this kind go back to Cauchy in 1822 and are of a much higher order of difficulty than any to be attempted here. We shall consider a few examples of mechanical vibrations — stretched strings, vibrating membranes, sound waves — and then an example of a field that can be treated in much the same way — the matter field ψ. Finally, a simple matter field will be quantized.