I assume someone has written a long treatise on the subject, but I have not considered the subject pressing enough to actually research it.
http://lesswrong.com/lw/r5/the_quantum_physics_sequence/
It's long and hard to understand, but easier to understand than other explanations of quantum physics.
You can skip right to the answer, though you may not accept it without having understood the background.
http://lesswrong.com/lw/qx/timeless_identity/
>If you can see the moments of now braided into time, the causal dependencies of future states on past states, the high-level pattern of synapses and the internal narrative as a computation within it - if you can viscerally dispel the classical hallucination of a little billiard ball that is you, and see your nows strung out in the river that never flows - then you can see that signing up for cryonics, being vitrified in liquid nitrogen when you die, and having your brain nanotechnologically reconstructed fifty years later, is actually less of a change than going to sleep, dreaming, and forgetting your dreams when you wake up.
EDIT: Useful thought experiment from philosophy, devised during the time Ronald Reagan was still alive. Suppose someone told you that they would take you, erase your memories from your brain, upload Ronald Reagan's memories, and then torture the inhabitant of your body. How willing are you to say, "go ahead! You'll just be torturing Ronald Reagan, not me!"? Personally, I'm not so willing.
Peter F. Hamilton explores this in "The Reality Dysfunction".
Although it's Sci-Fi and a space-opera it's it illustrates the problem pretty well.
Then I'd quite rightly get thrown out of the thought experiment for involving the Golden Rule without a licence.
The question is whether, before the fact, you would go into minor surgery without anesthetic at all knowing perfectly well that you would forget all of the pain afterwards. Maybe you would if it was the best option, but I would wince in anticipation.
One approach is to say "it doesn't matter" because self-awareness is illusory. It doesn't exist in the same sense as matter or fields. What does exist in your mind is memory of all state transitions ever happened in your brain. It's like a linked list going back, where each element is a state of the brain. That chain is mistaken for something that exists, whereas it's just a logically linked list of some data. You are nothing but information.