Physics Friday 130

Part 10: Electrodynamic Potentials and Gauge Symmetry

As noted here, the introduction of Faraday’s Law means that in electrodynamics, the electric field is not irrotational, and we no longer have the scalar electric potential φ with . But we still have no magnetic monopoles, so we can still use a magnetic vector potential A, with .
Next, we examine Faraday’s Law:
;
moving the right-hand term over, we can put this as
.
Rewriting in terms of A,
.
The spatial and time derivatives commute, so we have

.
So, while the electric field is no longer irrotational, the quantity is, and so can be written as the gradient of a scalar function:
,
or, solving for the electric field,
.
This and thus satisfy the two of Maxwell’s equations that are homogeneous: Faraday’s Law and the absence of magnetic monopoles.
Note that when the magnetic field is static, , and the above reduces to our electrostatic potential.

Recall that when we first discussed the vector potential, it was noted that the vector potential is not unique, and that adding the gradient of any scalar field to A does not change the resulting field. We called such a transformation a gauge transformation, and noted that we could choose a potential such that , calling that choice the Coulomb gauge.
Now, for electrodynamics, we can again add the gradient of a scalar function Λ to A without changing the magnetic field. However, if we perform the transformation from A to , we see that
. With , we then see that to keep the electric field unchainged, we must simulateously change the scalar field from φ to (so that we have cancelling terms).
This is the gauge transformation for electrodynamics, and the invariance of the electric and magnetic fields under these is called gauge invariance.
As before, the Coulomb gauge (also known as the “transverse gauge”) is that where the potentials are chosen so that . In this case, we plug our expression for the electric field due to potentials into Gauss’ Law:

.
.
Since the time and space terms commute, and noting that the divergence of the gradient is the Laplacian,
.
In the Coulomb gauge, the second term on the left is zero, and so our scalar potential obeys the Poisson’s equation
,
the same equation the electric potential obeys in electrostatics; though, we can no longer compute the electric field from this potential alone, but must find the magnetic vector potential. As in electrostatics, the solution to this is given by
, the instantaneous Coulomb potential; hence the name.

Another convienient choice of gauge is done by expressing both Gauss’ Law and Ampère’s law in terms of the potentials. For Gauss’ Law, we have
,
as noted above. Rearranging Ampère’s law so that the field terms are on the same side, we have
.;
using the fact that ,
.
Plugging in and ,
.
Now, the formula for curl of a curl tells us that
; and our time and space derivatives commute, so
, and so we have

.
.
Now, to remove φ from the second equation, we use the freedom of gauge transformations to choose potentials that obey
,
this is known as the Lorenz gauge condition, and the resulting gauge the Lorenz gauge. It is named for Ludvig Lorenz (not to be confused with Hendrik Lorentz).
If we make this choice, the second equation (Ampère’s Law) becomes
,
an inhomogeneous (vector) wave equation with wave velocity c. For the first equation (Gauss’ law) , the Lorenz condition means
; making this substitution,

,
and we have an inhomogeneous wave equation for the scalar potential; this choice has uncoupled the equations for φ and A.
Note that if we have φ and A obeying the Lorenz condition, and we make the gauge transform

,
with a scalar field Λ, then
,
so if Λ obeys the homogeneous wave equation , then the transformed fields φ‘ and A‘ also obey the Lorenz condition. Thus, the Lorenz gauge is not a single choice of potentials, as is the Coulomb gauge, but instead a restricted class of choices, with all potentials in it belonging to the Lorenz gauge.
The Lorenz gauge places φ and A on more equal and symmetric footing, and is also independent of the choice of coordinate system, making it the natural choice for dealing with electromagnetism in special relativity.

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