The Standard Model is a quantum field theory. In Chapter 4 we discussed the classical electromagnetic field. The transition to a quantum field will be made in Chapter 8. In this chapter we begin our discussion of the Dirac equation, which was invented by Dirac as an equation for the relativistic quantum wave function of a single electron. However, we shall regard the Dirac wave function as a field, which will subsequently be quantised along with the electromagnetic field. The Dirac equation will be regarded as a field equation. The transition to a quantum field theory is called second quantisation. The field, like the Dirac wave function, is complex. We shall show how the Dirac field transforms under a Lorentz transformation, and find a Lorentz invariant Lagrangian from which it may be derived.

On quantisation, the electromagnetic fields Aμ(x), Fμν(x) become space- and time-dependent operators. The expectation values of these operators in the environment described by the quantum states are the classical fields. The Dirac fields ψ(x) also become space- and time-dependent operators on quantisation. However, there are no corresponding measurable classical fields. This difference reflects the Pauli exclusion principle, which applies to fermions but not to bosons. In this chapter and in the following two chapters, the properties of the Dirac fields as operators are rarely invoked: for the most part the manipulations proceed as if the Dirac fields were ordinary complex functions, and the fields can be thought of as single-particle Dirac wave functions.

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.

Calculations in QCD have been made in two ways: lattice simulations at low energies, and perturbative calculations at high energies. In this chapter we outline some of the results obtained.

Lattice QCD and confinement

It was pointed out in Section 16.1 that, at low energies, a non-perturbative approach to QCD is needed. ‘Lattice QCD’ is such an approach. The gluon fields are defined on a four-dimensional lattice of points (nμ, n)a, where a is the lattice spacing and the nμ are integers. Field derivatives are replaced by discrete differences. This gives a ‘lattice regularised’ QCD. The lattice spacing corresponds to an ultraviolet cut-off, since wavelengths < 2a cannot be described on the lattice. A lattice does not have full rotational symmetry in space, but it is believed that nevertheless continuum QCD corresponds to the limit a → 0. Current computing power allows lattices of ∼(36)4 points. The range of the strong nuclear force is ∼ 1 fm. To fit such a distance comfortably on the lattice, we can anticipate that we shall not want a to be much less than (2fm)/36 = 0.056fm (and ℏc/a > 3.5 GeV).

In the high energy perturbation theory described in Section 16.3, the renormalisation parameter λ and the dimensionless coupling parameter g are combined to give a single physical parameter, Λ, having the dimensions of energy. The relationship between the effective coupling constant αs(Q2) and Λ in the lowest order of perturbation theory is given by (16.25).