44
2. Electromagnetism as a Gauge Theory
This can only be true in situations where the charge density is constant in
time. For the general case, Maxwell modified Amp` ere’s law to read
∂E
∇ × B = j +
(2.8)
em
∂t
which is now consistent with (2.5). Equations (2.1)–(2.3), together with (2.8),
constitute Maxwell’s equations in free space (apart from the sources).
It is worth spending a moment on the vitally important continuity equation
(2.5) – note the Feynman quotation at the start of this section. Let us integrate
this equation over any arbitrary volume Ω, and write the result as
∫
∫
∂
ρ em dV = −
∇ · j em dV.
(2.9)
∂t Ω
Ω
Equation (2.9) states that the rate of decrease of charge in any arbitrary
volume Ω is due precisely and only to the flux of current out of its surface;
that is, no net charge can be created or destroyed in Ω. Since Ω can be
made as small as we please, this means that electric charge must be locally
conserved : a process in which charge is created at one point and destroyed at a
distant one is not allowed, despite the fact that it conserves the charge overall
or ‘globally’. The ultimate reason for this is that the global form of charge
conservation would necessitate the instantaneous propagation of signals (such
as ‘now, create a positron over there’), and this conflicts with special relativity
– a theory which, historically, flowered from the soil of electrodynamics. The
extra term introduced by Maxwell – the ‘electric displacement current’ – owes
its place in the dynamical equations to a local conservation requirement.
We remark at this point that we have just introduced another local/global
distinction, similar to that discussed earlier in connection with invariances. In
this case the distinction applies to a conservation law, but since invariances
are related to conservation laws in both classical and quantum mechanics, we
should perhaps not be too surprised by this. However, as with invariances,
conservation laws – such as charge conservation in electromagnetism – play a
central role in gauge theories in that they are closely related to the dynamics.
The point is simply illustrated by asking how we could measure the charge
of a newly created subatomic particle X. There are two conceptually different
ways:
(i) We could arrange for X to be created in a reaction such as
A + B → C + D + X
where the charges of A, B, C and D are already known. In this case
we can use charge conservation to determine the charge of X.
(ii) We could see how particle X responded to known electromagnetic
fields. This uses dynamics to determine the charge of X.
2. Electromagnetism as a Gauge Theory
This can only be true in situations where the charge density is constant in
time. For the general case, Maxwell modified Amp` ere’s law to read
∂E
∇ × B = j +
(2.8)
em
∂t
which is now consistent with (2.5). Equations (2.1)–(2.3), together with (2.8),
constitute Maxwell’s equations in free space (apart from the sources).
It is worth spending a moment on the vitally important continuity equation
(2.5) – note the Feynman quotation at the start of this section. Let us integrate
this equation over any arbitrary volume Ω, and write the result as
∫
∫
∂
ρ em dV = −
∇ · j em dV.
(2.9)
∂t Ω
Ω
Equation (2.9) states that the rate of decrease of charge in any arbitrary
volume Ω is due precisely and only to the flux of current out of its surface;
that is, no net charge can be created or destroyed in Ω. Since Ω can be
made as small as we please, this means that electric charge must be locally
conserved : a process in which charge is created at one point and destroyed at a
distant one is not allowed, despite the fact that it conserves the charge overall
or ‘globally’. The ultimate reason for this is that the global form of charge
conservation would necessitate the instantaneous propagation of signals (such
as ‘now, create a positron over there’), and this conflicts with special relativity
– a theory which, historically, flowered from the soil of electrodynamics. The
extra term introduced by Maxwell – the ‘electric displacement current’ – owes
its place in the dynamical equations to a local conservation requirement.
We remark at this point that we have just introduced another local/global
distinction, similar to that discussed earlier in connection with invariances. In
this case the distinction applies to a conservation law, but since invariances
are related to conservation laws in both classical and quantum mechanics, we
should perhaps not be too surprised by this. However, as with invariances,
conservation laws – such as charge conservation in electromagnetism – play a
central role in gauge theories in that they are closely related to the dynamics.
The point is simply illustrated by asking how we could measure the charge
of a newly created subatomic particle X. There are two conceptually different
ways:
(i) We could arrange for X to be created in a reaction such as
A + B → C + D + X
where the charges of A, B, C and D are already known. In this case
we can use charge conservation to determine the charge of X.
(ii) We could see how particle X responded to known electromagnetic
fields. This uses dynamics to determine the charge of X.