5
Quantum Field Theory I: The Free Scalar
Field
In this chapter we shall give an elementary introduction to quantum field
theory, which is the established ‘language’ of the Standard Model of particle
physics. Even so long after Maxwell’s theory of the (classical) electromagnetic
field, the concept of a ‘disembodied’ field is not an easy one; and we are
going to have to add the complications of quantum mechanics to it. In such a
situation, it is helpful to have some physical model in mind. For most of us, as
for Lord Kelvin, this still means a mechanical model. Thus in the following two
sections we begin by considering a mechanical model for a quantum field. At
the end, we shall – like Maxwell – throw away the ‘mechanism’ and have simply
quantum field theory. Section 5.1 describes this programme qualitatively;
section 5.2 presents a more complete formalism, for the simple case of a field
whose quanta are massless, and move in only one spatial dimension. The
appropriate generalizations for massive quanta in three dimensions are given
in section 5.3.
5.1 The quantum field: (i) descriptive
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. A
scalar field φ(x, t) corresponds to a system with an infinite number of degrees
of freedom, since at each continuously varying point x an independent ‘displacement’ φ(x, t), which also varies with time, has to be determined. Thus
quantum field theory involves two major mathematical steps: the description
of continuous systems (fields) which have infinitely many degrees of freedom,
and the application of quantum theory to such systems. These two aspects are
clearly separable. It is certainly easier to begin by considering systems with
a discrete – but possibly very large – number of degrees of freedom, for example a solid. We shall treat such systems first classically and then quantum
mechanically. Then, returning to the classical case, we shall allow the number
115
Quantum Field Theory I: The Free Scalar
Field
In this chapter we shall give an elementary introduction to quantum field
theory, which is the established ‘language’ of the Standard Model of particle
physics. Even so long after Maxwell’s theory of the (classical) electromagnetic
field, the concept of a ‘disembodied’ field is not an easy one; and we are
going to have to add the complications of quantum mechanics to it. In such a
situation, it is helpful to have some physical model in mind. For most of us, as
for Lord Kelvin, this still means a mechanical model. Thus in the following two
sections we begin by considering a mechanical model for a quantum field. At
the end, we shall – like Maxwell – throw away the ‘mechanism’ and have simply
quantum field theory. Section 5.1 describes this programme qualitatively;
section 5.2 presents a more complete formalism, for the simple case of a field
whose quanta are massless, and move in only one spatial dimension. The
appropriate generalizations for massive quanta in three dimensions are given
in section 5.3.
5.1 The quantum field: (i) descriptive
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. A
scalar field φ(x, t) corresponds to a system with an infinite number of degrees
of freedom, since at each continuously varying point x an independent ‘displacement’ φ(x, t), which also varies with time, has to be determined. Thus
quantum field theory involves two major mathematical steps: the description
of continuous systems (fields) which have infinitely many degrees of freedom,
and the application of quantum theory to such systems. These two aspects are
clearly separable. It is certainly easier to begin by considering systems with
a discrete – but possibly very large – number of degrees of freedom, for example a solid. We shall treat such systems first classically and then quantum
mechanically. Then, returning to the classical case, we shall allow the number
115