You do in fact live for a while after the head is chopped off. There's the famous account of the 18th century philosopher who was sentenced to the guillotine during the French Revolution, because you know, rich. He wanted to test it, and told them to count the number of times he blinked after he was killed, and he got 13 out before he stopped. And you know, this makes sense. Just because your heart isn't connected to your brain anymore doesn't mean your brain isn't doing anything. Otherwise, you'd die every time your heart beats, because blood doesn't always move through your body. It moves in pulses every second or so. So your brain isn't getting new oxygen for a large chunk of time. And just because it's all removed from your body doesn't really mean anything, your brain is still fine. The reason you die is actually asphyxiation. On a cellular level, because your brain is no longer getting oxygen. So you can live for quite a long time relatively with no body. There's been a scientist who's removed a dog's head and attached it to another body, and the head survived for a bit (until the whole system failed, essentially).