As always, Wait But Why has an excellent dicussion on
What Makes You You.
I also found this comic on a
Teleportation Machine that talks for instance about how the consciousness interruption might be no different from what happens when you go to sleep.
Anyway, what I really wanted to talk about is how "teleportation" works in real life, that is,
Quantum Teleportation. In this process what is moved is not matter, but the information on the exact quantum state of a system of particles, that can then be reproduced on another identical system.
Basically one needs a particle (or many) whose state one wishes to teleport plus an additional entangled pair of particles. One particle of the entangled pair is kept at the original location and the other is sent to the receiver. The entanged pair of particles are in crude terms, like a pair of socks, if you know one is right, the other must be left, so even though they are separated by a large distance, learning the state of one of them means you also learn the state of the other.
In order to teleport the state of a particle, the person who wishes to do so makes a measurement on its state and also on his or her half of the entangled pair. Because of the entanglement, what happens is that the other particle of the pair, that is in the location you wish to teleport to will be put in a particular state, which is not yet the state you wanted to teleport, but is related to it. The first person then beams the result of their measurement to the the receiver (this is the step where actual transfer of information happens, and it is necessarily not faster than light), and this is enough information for them to reconstruct completely the state of the particle you wanted to teleport, by simple operations.
I'm sorry if the explanation was too confusing, you may find many more details (as well as a precise mathematical description) in the Wiki article. What I wanted to say that is relevant to this debate is that the processes of learning the state of the system you wanted to teleport necessarily destroys it, in an irreversible measurement. That means that at no point are there two copies of the system, as at the moment the measurement is made, the original is effectively destroyed, but all the information that specified its state is preserved. The result that you cannot make a copy without destroying the original is amusingly enough called the
No-cloning Theorem
As others have said elementary particles in nature are not distinguishable, not even in principle, so if you put two identical systems in the same state, they are completely identical, and no measurement can tell them apart. Since there is no remaining copy, per the last paragraph, that means that for all intents and purposes, the person on the other side is as much "you" as it can be, and there' nothing you can do to disprove that.
That said, I agree that it still
feels as though you died and then where recreated, this seems to just be an icky feeling we have, not a rational conclusion. As all the information that makes you you (and there should be nothing else) is preserved, and "printed" in a identical body. I certainly would be still be afraid to use such a machine even though logically there's nothing to fear, because it seems to be an innate thing, a fear of losing continuity or something.
Anyways, at this point I'm just babbling so I will just stop here and hopefully someone finds something interesting in this wall of text haha.