This thread is about what makes a person "them". In each of the following scenarios the question is whether or not the original person is still alive/existent, or whether the original person is dead and replaced by an imposter. Or, to put it another way, which of the following types of teleporters would you be willing to actually use?
1. Suppose I wish to travel from one planet to another (or just across the room, or whatever). I step into a machine that scans every bit of information about every particle in my body to within the limits of the uncertainty principle. During the scanning process the machine breaks my body up into a swarm of individual nuclei and electrons, and sends them across space at nearly the speed of light. When they reach their destination they are all perfectly reassembled, with each atom placed exactly where it was before. My reconstructed body has all the same memories etc as before, and I wasn't able to perceive the transport process. As far I as can tell, there's simply a flash of light and I suddenly appear somewhere else.
2. The same scenario as 1, but this time particles of the same type are all randomly substituted for one another. The machine only notes "there was a particle of type X at location Y" and makes sure that the appropriate particle type ends up in the appropriate place, but doesn't make any effort to ensure that the individual particles end up in their original places. I'm still made of the same material, but now the material has been "shuffled around."
3. The same as scenario 1, but this time the machine doesn't send my original particles across space. Instead it simply sends the information about where the particles should be located, and the receiving machine assembles a perfect copy of the original from a stockpile of elements that it keeps on hand. The resulting "product" is completely indistinguishable from the result of scenario 1 or 2, but now my "new" body is made from completely new material and the remains of my old body is left as a cloud of dissipating particles on the other planet.
4. The same as scenario 3, except this time the machine doesn't disintegrate my original body during the scanning process. Instead I am placed in a deeply unconscious state while my body is scanned and the new copy created. When the process is finished, my original body is atomized as before and my new body is revived. (Did the technician who destroyed my original body commit murder?) As far as I can tell, I simply fell asleep in one place and woke up in another. Again, my new body is completely indistinguishable from my old body.
5. Same as scenario 4, except this time I'm not unconscious during the scanning process. There is a fraction of a second period in which the original and the copy are both conscious before the original is atomized.