2
Electromagnetism as a Gauge Theory
2.1 Introduction
The previous chapter introduced the basic ideas of the Standard Model of
particle physics, in which quarks and leptons interact via the exchange of
gauge field quanta. We must now look more closely into what is the main
concern of this book – namely, the particular nature of these ‘gauge
theories’.
One of the relevant forces – electromagnetism – has been well understood in
its classical guise for many years. Over a century ago, Faraday, Maxwell and
others developed the theory of electromagnetic interactions, culminating in
Maxwell’s paper of 1864 (Maxwell 1864). Today Maxwell’s theory still stands
– unlike Newton’s ‘classical mechanics’ which was shown by Einstein to require
modifications at relativistic speeds, approaching the speed of light. Moreover,
Maxwell’s electromagnetism, when suitably married with quantum mechanics,
gives us ‘quantum electrodynamics’ or QED. We shall see in chapter 10 that
this theory is in truly remarkable agreement with experiment. As we have
already indicated, the theories of the weak and strong forces included in the
Standard Model are generalizations of QED, and promise to be as successful
as that theory. The simplest of the three, QED, is therefore our paradigmatic
theory.
From today’s perspective, the crucial thing about electromagnetism is that
it is a theory in which the dynamics (i.e. the behaviour of the forces) is
intimately related to a symmetry principle. In the everyday world, a symmetry
operation is something that can be done to an object that leaves the object
looking the same after the operation as before. By extension, we may consider
mathematical operations – or ‘transformations’ – applied to the objects in our
theory such that the physical laws look the same after the operations as they
did before. Such transformations are usually called invariances of the laws.
Familiar examples are, for instance, the translation and rotation invariance
of all fundamental laws: Newton’s laws of motion remain valid whether or
not we translate or rotate a system of interacting particles. But of course –
precisely because they do apply to all laws, classical or quantum – these two
invariances have no special connection with any particular force law. Instead,
41
Electromagnetism as a Gauge Theory
2.1 Introduction
The previous chapter introduced the basic ideas of the Standard Model of
particle physics, in which quarks and leptons interact via the exchange of
gauge field quanta. We must now look more closely into what is the main
concern of this book – namely, the particular nature of these ‘gauge
theories’.
One of the relevant forces – electromagnetism – has been well understood in
its classical guise for many years. Over a century ago, Faraday, Maxwell and
others developed the theory of electromagnetic interactions, culminating in
Maxwell’s paper of 1864 (Maxwell 1864). Today Maxwell’s theory still stands
– unlike Newton’s ‘classical mechanics’ which was shown by Einstein to require
modifications at relativistic speeds, approaching the speed of light. Moreover,
Maxwell’s electromagnetism, when suitably married with quantum mechanics,
gives us ‘quantum electrodynamics’ or QED. We shall see in chapter 10 that
this theory is in truly remarkable agreement with experiment. As we have
already indicated, the theories of the weak and strong forces included in the
Standard Model are generalizations of QED, and promise to be as successful
as that theory. The simplest of the three, QED, is therefore our paradigmatic
theory.
From today’s perspective, the crucial thing about electromagnetism is that
it is a theory in which the dynamics (i.e. the behaviour of the forces) is
intimately related to a symmetry principle. In the everyday world, a symmetry
operation is something that can be done to an object that leaves the object
looking the same after the operation as before. By extension, we may consider
mathematical operations – or ‘transformations’ – applied to the objects in our
theory such that the physical laws look the same after the operations as they
did before. Such transformations are usually called invariances of the laws.
Familiar examples are, for instance, the translation and rotation invariance
of all fundamental laws: Newton’s laws of motion remain valid whether or
not we translate or rotate a system of interacting particles. But of course –
precisely because they do apply to all laws, classical or quantum – these two
invariances have no special connection with any particular force law. Instead,
41