47
2.3. The Maxwell equations: Lorentz covariance and gauge invariance
the form (2.18). The ‘Lorentz-covariant and gauge-invariant field equations’
satisfied by A
μ then follow from equations (2.18) and (2.19):
❗A
ν
− ∂
ν (∂ μ A
μ ) = j
ν .
(2.22)
em
Since gauge transformations turn out to be of central importance in the
quantum theory of electromagnetism, it would be nice to have some insight
into why Maxwell’s equations are gauge invariant. The all-important ‘fourth’
equation (2.8) was inferred by Maxwell from local charge conservation, as
expressed by the continuity equation
∂ μ j
μ = 0.
(2.23)
em
The field equation
∂ μ F
μν
j
ν
=
(2.24)
em
then of course automatically embodies (2.23). The mathematical reason it
does so is that F
μν is a four-dimensional kind of ‘curl’
F
μν
≡ ∂
μ A
ν
− ∂
ν A
μ
(2.25)
which (as we have seen in (2.21)) is unchanged by a gauge transformation
A
μ
→ A
′μ = A
μ
− ∂
μ χ.
(2.26)
Hence there is the suggestion that the gauge invariance is related in some way
to charge conservation. However, the connection is not so simple. Wigner
(1949) has given a simple argument to show that the principle that no physical quantity can depend on the absolute value of the electrostatic potential, when combined with energy conservation, implies the conservation of
charge. Wigner’s argument relates charge (and energy) conservation to an
invariance under transformation of the electrostatic potential by a constant:
charge conservation alone does not seem to require the more general space–
time-dependent transformation of gauge invariance.
Changing the value of the electrostatic potential by a constant amount is
an example of what we have called a global transformation (since the change
in the potential is the same everywhere). Invariance under this global transformation is related to a conservation law: that of charge. But this global
invariance is not sufficient to generate the full Maxwellian dynamics. However, as remarked by ’t Hooft (1980), one can regard equations (2.12) and
(2.13) as expressing the fact that the local change in the electrostatic potential V (the ∂χ/∂t term in (2.13)) can be compensated – in the sense of leaving
the Maxwell equations unchanged – by a corresponding local change in the
magnetic vector potential A. Thus by including magnetic effects, the global
2.3. The Maxwell equations: Lorentz covariance and gauge invariance
the form (2.18). The ‘Lorentz-covariant and gauge-invariant field equations’
satisfied by A
μ then follow from equations (2.18) and (2.19):
❗A
ν
− ∂
ν (∂ μ A
μ ) = j
ν .
(2.22)
em
Since gauge transformations turn out to be of central importance in the
quantum theory of electromagnetism, it would be nice to have some insight
into why Maxwell’s equations are gauge invariant. The all-important ‘fourth’
equation (2.8) was inferred by Maxwell from local charge conservation, as
expressed by the continuity equation
∂ μ j
μ = 0.
(2.23)
em
The field equation
∂ μ F
μν
j
ν
=
(2.24)
em
then of course automatically embodies (2.23). The mathematical reason it
does so is that F
μν is a four-dimensional kind of ‘curl’
F
μν
≡ ∂
μ A
ν
− ∂
ν A
μ
(2.25)
which (as we have seen in (2.21)) is unchanged by a gauge transformation
A
μ
→ A
′μ = A
μ
− ∂
μ χ.
(2.26)
Hence there is the suggestion that the gauge invariance is related in some way
to charge conservation. However, the connection is not so simple. Wigner
(1949) has given a simple argument to show that the principle that no physical quantity can depend on the absolute value of the electrostatic potential, when combined with energy conservation, implies the conservation of
charge. Wigner’s argument relates charge (and energy) conservation to an
invariance under transformation of the electrostatic potential by a constant:
charge conservation alone does not seem to require the more general space–
time-dependent transformation of gauge invariance.
Changing the value of the electrostatic potential by a constant amount is
an example of what we have called a global transformation (since the change
in the potential is the same everywhere). Invariance under this global transformation is related to a conservation law: that of charge. But this global
invariance is not sufficient to generate the full Maxwellian dynamics. However, as remarked by ’t Hooft (1980), one can regard equations (2.12) and
(2.13) as expressing the fact that the local change in the electrostatic potential V (the ∂χ/∂t term in (2.13)) can be compensated – in the sense of leaving
the Maxwell equations unchanged – by a corresponding local change in the
magnetic vector potential A. Thus by including magnetic effects, the global