Let's consider the following picture (source):
When current passes the half-wavelength matching section it changes the phase to the opposite, this is why we get +V and -V in the feed point, 2V difference in total. Also I understand that the current splits to 2 x I/2, this is why we get ( R = 2V/[I/2] ) 1:4 impedance matching.
What I'm not sure about is what happens to the current in the shield of the half-wavelength coax. The current in the shield of the feed line should flow somewhere.
Does it flow to the half-wavelength coax and cancels itself because of the phase shift? What prevents it from jumping over the beginning of the half-wavelength section and flowing from the end to the beginning of this section? Could you please explain this part in a little more details.
Update: Or there is no current in the shield at all? After all it has no connection to the center part of the cable.