42
2. Electromagnetism as a Gauge Theory
they constrain the form of the allowed laws to a considerable extent, but by
no means uniquely determine them. Nevertheless, this line of argument leads
one to speculate whether it might in fact be possible to impose further types
of symmetry constraints so that the forms of the force laws are essentially
determined. This would then be one possible answer to the question: why are
the force laws the way they are? (Ultimately of course this only replaces one
question by another!)
In this chapter we shall discuss electromagnetism from this point of view.
This is not the historical route to the theory, but it is the one which generalizes
to the other two interactions. This is why we believe it important to present
the central ideas of this approach in the familiar context of electromagnetism
at this early stage.
A distinction that is vital to the understanding of all these interactions
is that between a global invariance and a local invariance. In a global invariance the same transformation is carried out at all space–time points: it
has an ‘everywhere simultaneously’ character. In a local invariance different
transformations are carried out at different individual space–time points. In
general, as we shall see, a theory that is globally invariant will not be invariant under locally varying transformations. However, by introducing new force
fields that interact with the original particles in the theory in a specific way,
and which also transform in a particular way under the local transformations,
a sort of local invariance can be restored. We will see all these things more
clearly when we go into more detail, but the important conceptual point to be
grasped is this: one may view these special force fields and their interactions
as existing in order to permit certain local invariances to be true. The particular local invariance relevant to electromagnetism is the well-known gauge
invariance of Maxwell’s equations: in the quantum form of the theory this
property is directly related to an invariance under local phase transformations
of the quantum fields. A generalized form of this phase invariance also underlies the theories of the weak and strong interactions. For this reason they are
all known as ‘gauge theories’.
A full understanding of gauge invariance in electrodynamics can only be
reached via the formalism of quantum field theory, which is not easy to master – and the theory of quantum gauge fields is particularly tricky, as we
shall see in chapter 7. Nevertheless, many of the crucial ideas can be perfectly adequately discussed within the more familiar framework of ordinary
quantum mechanics, rather than quantum field theory, treating electromagnetism as a purely classical field. This is the programme followed in the rest
of part I of this volume. In the present chapter we shall discuss these ideas in
the context of non-relativistic quantum mechanics; in the following two chapters, we shall explore the generalization to relativistic quantum mechanics,
for particles of spin-0 (via the Klein–Gordon equation) and spin1 (via the
2
Dirac equation). While containing substantial physics in their own right, these
chapters constitute essential groundwork for the quantum field treatment in
parts II–IV.
2. Electromagnetism as a Gauge Theory
they constrain the form of the allowed laws to a considerable extent, but by
no means uniquely determine them. Nevertheless, this line of argument leads
one to speculate whether it might in fact be possible to impose further types
of symmetry constraints so that the forms of the force laws are essentially
determined. This would then be one possible answer to the question: why are
the force laws the way they are? (Ultimately of course this only replaces one
question by another!)
In this chapter we shall discuss electromagnetism from this point of view.
This is not the historical route to the theory, but it is the one which generalizes
to the other two interactions. This is why we believe it important to present
the central ideas of this approach in the familiar context of electromagnetism
at this early stage.
A distinction that is vital to the understanding of all these interactions
is that between a global invariance and a local invariance. In a global invariance the same transformation is carried out at all space–time points: it
has an ‘everywhere simultaneously’ character. In a local invariance different
transformations are carried out at different individual space–time points. In
general, as we shall see, a theory that is globally invariant will not be invariant under locally varying transformations. However, by introducing new force
fields that interact with the original particles in the theory in a specific way,
and which also transform in a particular way under the local transformations,
a sort of local invariance can be restored. We will see all these things more
clearly when we go into more detail, but the important conceptual point to be
grasped is this: one may view these special force fields and their interactions
as existing in order to permit certain local invariances to be true. The particular local invariance relevant to electromagnetism is the well-known gauge
invariance of Maxwell’s equations: in the quantum form of the theory this
property is directly related to an invariance under local phase transformations
of the quantum fields. A generalized form of this phase invariance also underlies the theories of the weak and strong interactions. For this reason they are
all known as ‘gauge theories’.
A full understanding of gauge invariance in electrodynamics can only be
reached via the formalism of quantum field theory, which is not easy to master – and the theory of quantum gauge fields is particularly tricky, as we
shall see in chapter 7. Nevertheless, many of the crucial ideas can be perfectly adequately discussed within the more familiar framework of ordinary
quantum mechanics, rather than quantum field theory, treating electromagnetism as a purely classical field. This is the programme followed in the rest
of part I of this volume. In the present chapter we shall discuss these ideas in
the context of non-relativistic quantum mechanics; in the following two chapters, we shall explore the generalization to relativistic quantum mechanics,
for particles of spin-0 (via the Klein–Gordon equation) and spin1 (via the
2
Dirac equation). While containing substantial physics in their own right, these
chapters constitute essential groundwork for the quantum field treatment in
parts II–IV.