This chapter provides an elementary introduction to quantum field theory, which is the established ‘language’ of the Standard Model of particle physics. Mechanical systems are usefully characterized by the number of degrees of freedom they possess: thus a one-dimensional pendulum has one degree of freedom, two coupled one-dimensional pendulums have two degrees of freedom – which may be taken to be their angular displacements, for example. The system under discussion had just two degrees of freedom. The stage in our programme is to treat such systems quantum mechanically, as we should certainly have to for a real solid. It is still true that – if the potential energy is a quadratic function of the displacements – the transformation allows us to write the total energy as a sum of N mode energies, each of which has the form of a harmonic oscillator. The essential idea – quantizing independent modes – can be applied to an enormous variety of ‘oscillations’.