8 - Quantising fields: QED

Summary

We turn now to the quantisation of the electrodynamic fields introduced in Chapter 7. So far we have treated the electromagnetic field and the Dirac field as classical fields (though we were compelled in Chapter 7 to recognise that Dirac fields anticommute). On quantisation, these fields become operator fields, acting on the states of a system. The classical total field energy becomes the Hamiltonian operator, which determines the dynamics of the system. We shall use the formalism of annihilation and creation operators; this formalism is reviewed briefly in Appendix C for readers not already familiar with it.

Quantum electrodynamics, or QED, is an important component of the Standard Model. It is also the foundation of our understanding of the material world at the atomic level. However, we do not wish to enter into the technical complications of electrons in atoms or in material media. In this chapter we shall only consider more simple situations of a few interacting photons, electrons and positrons, at energies sufficiently high for bound systems of electrons and positrons to be ignored. In these situations, the free field approximation to QED provides a sound basis for understanding the interactions of particles as perturbations on their free behaviour.

This is not a text on quantum field theory, and our outline of perturbation theory in this chapter is necessarily sketchy. But our intention is to try to give some insight into how the results of calculations, presented in later chapters, are arrived at.