The working principle and fields of an antenna can often be shown by considering a parallel LC circuit. The plates can be moved to opposite ends and the inductor can be stretched out, both of which decreases the capacitance and inductance of the circuit and increases the resonant frequency.
In the limiting case of a half-wave dipole, the antenna must be center-fed (if fed by a current source) and connected to the transmission line. To accomplish this, the inductor in the middle needs to be cut and there is no current flow directly through the inductor:
As can be seen in the picture above, the center is insulated, while this point in the LC circuit corresponds to the center of the inductor, which is conducting.
While in a regular LC circuit a current flows through the inductor, it now moves through the whole transmission line at each cycle. It seems to me that these circuits are not equivalent anymore, since that introduces all the impedance of the transmission line into the circuit.
It's not so clear to me what changes if the inductor is still connected (apart from presenting a short circuit to the transmission line), and does this "cutting" change the inductance of the antenna itself?