chuck wrote:
Is it not true that if the currents on the transmission line are unbalanced
(i.e., unequal on the two conductors) then the transmitter must already be
connected to ground? If not, what is the path of the differential current?

Your thinking is correct. If you had a small, battery operated rig, you
could theoretically force current balance by completely isolating it
from ground. You might burn yourself when you touch it, though.

In practice, differential current can flow to ground via capacitive
coupling from the rig to ground, or by flowing onto any connecting wires
such as power supply, speaker, or key leads, which will radiate.

Common mode feedline current is generally sinusoidally distributed. So
you can have a relatively small common mode current at the rig but a
much larger one at places on the feedline, if the rig happens to fall at
a common mode current node. When the current is due to coupling, a
single balun will be ineffective if placed at a natural current node, or
in some other cases will only change the distribution by moving the
node, rather than reducing the overall current. In those cases, two
baluns placed about a quarter wavelength apart are effective.

Roy Lewallen, W7EL