Quantization of Relativistic Fields

Abstract

This chapter is dedicated to relativistic quantum fields. We shall begin by investigating a system of coupled oscillators for which the quantization properties are known. The continuum limit of this oscillator system yields the equation of motion for a vibrating string in a harmonic potential. This is identical in form to the Klein--Gordon equation. The quantized equation of motion of the string and its generalization to three dimensions provides us with an example of a quantized field theory. The quantization rules that emerge here can also be applied to non-material fields. The fields and their conjugate momentum fields are subject to canonical commutation relations. One thus speaks of ``canonical quantization''. In order to generalize to arbitrary fields, we shall then study the properties of general classical relativistic fields. In particular, we will derive the conservation laws that follow from the symmetry properties (Noether's theorem).