Why does energy flow between a high voltage transmission line and linemen approaching it on a helicopter? The video High power line workers shows linemen servicing a live high voltage transmission line. As the helicopter approaches the line, the lineman reaches out with a metallic wand that is conductively connected to the helicopter and a standing discharge appears between the line and the wand.


What is going on? A significant amount of energy appears to be flowing between the line and the helicopter, but a complete circuit does not exist because the two bodies are only connected by a single conductor. Some answers to related questions imply that a single conductor can't electrocute a person:

AC or DC, you only get electrocuted if current passes through
  your body... Touching just one wire at a time gives the current
  nowhere much to go. (Source)
You will not get a shock unless you complete the circuit to ground.
  This is why power lines can be worked on while live, from a
  helicopter. (Source)

However, the video makes it clear that a significant amount of energy is flowing. Another answer claims that a body (e.g. bird) approaching a high-voltage line would only experience a transient current:

[I]nitially there's a potential difference between the wire and the
  bird and there would be a (very short lived) transient current which
  is similar to electrostatic discharge you feel when you touch an
  object with electrostatic charge build-up. (Source)

But the discharge in the video appears to be sustained instead of short-lived. Another poster states that the discharge is deadly and the conductive wand is essential for the linemen's safety:

[T]he initial equalization process [between the helicopter and line]
  would kill [the linemen] if the arc went through [his] heart instead
  of the conducting [wand]. (Source)

Some answers speak of capacitance, high electric fields and corona discharge. How can one understand the flow of electromagnetic energy and charged particles between the two bodies as the helicopter approaches the line?
 A: I understand that the question is very old but I think it is a really good question and deserves a clarification in the form of another answer.
There is another video that highlights the process better: https://www.youtube.com/watch?v=DPNK7bc2qvM
The sequence of events in the video is as follows:


*

*The wand is placed in the proximity of the powerline so that an arc is initiated.
https://www.youtube.com/watch?v=DPNK7bc2qvM&t=128s

*The arc disappears only when the wand is placed on the powerline, also a separate wire is clamped to the powerline while the wand is still on the powerline1 

*Once the work is done, the wand is on the powerline1  and the clamp is now being disconnected.

*As the helicopter and the wand leave the powerline, the wand creates an arc again!
https://www.youtube.com/watch?v=DPNK7bc2qvM&t=172s
Certain facts are also clear:


*

*Powerlines are AC so the potential on the powerline is always alternating

*Before helicopter connects to the powerline, the potential on the helicopter is constant (whatever the static value is).

*The arc is present in both cases: when the helicopter arrives and when the helicopter leaves.

*Workers wear Faraday cages
So, what is going on is the following:


*

*As the helicopter approaches the powerline, there is a potential difference between the two (one is AC while another one is constant).

*This voltage induces a current. That is, a huge number of electrons are sucked into the powerline from the helicopter in one half-cycle of AC. Then, on the second half-cycle of AC, the huge number of excess electrons are shoved into the helicopter by the powerline.

*This current is always there while the helicopter is connected and the work is being done. There is no true "equalizing" between the helicopter and the powerline. This current goes either through the wand or through the clamp.
This is obvious when the helicopter is departing and the clamp is disconnected since the same arc is produced again. 

*However, the worker wears a Faraday cage so this current flows through the Faraday cage rather than through the worker's body. In other words, while the powerline shoves huge number of electrons back and forth, the left arm and the right arm of the worker is still exposed to the same potential so the electrons don't actually move through the worker's body.

Footnotes:
1 In the video the worker makes the utmost effort to keep the wand right on the powerline when he is connecting or disconnecting the clamp.
A: The helicopter and the power lines are at different potentials, the difference being so great as to cause the air in between to become a conductor.  If you applied such a potential difference across a line worker it would probably result in death.  You will note that the line worker is holding a metal stake which has a "pointed" end.  This increases the induced E-field around the point and thus increases the chances of the air becoming a conductor around the pointed end.
You can think of the helicopter as a capacitor (a device which stores electric charge)  whose capacitance is much greater than that of a bird and so it takes much more charge to make the voltage of the helicopter to rise than for the bird.
As the charge flows between the wand (connected to the helicopter by a conducting cable) and the power line the helicopter charges up so that its potential moves towards that of the power line. No significant charge through the line worked because it looks as though non-conducting gloves are worn. 
Whilst this is happening any sharp edges or points on the helicopter will possibly start making the air a conductor.  Charge from the helicopter would be sprayed out from these edges and points and this is called corona or silent discharge.  It is what you hear when you walk under a power line and it is what can produce interference on your car radio.
Eventually the potential difference between the power line and the helicopter is low enough for the helicopter to come close enough so that the line worker can connect a conducting wire between the helicopter and the power line so that they and the line worker, stay at the same potential.
