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Is Star Trek’s Warp Drive Possible?
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The "Ship of Theseus" example is how our bodies work normally, the transporter makes a new you out of separate matter in another place. Nothing implies a transfer of consciousness, just an exact copy of you is there. Of course, I realize that for the purposes of the shows and movies, none of this is a concern, but a real version of this would be ethically fucked up.
In practical terms, the transporter in order to preserve quantum identity, would need to be reproducing you at the same time as it destroys the old you. To be widely accepted by society, it would need to preserve consciousness continuity, so you'd briefly feel being in two places at the same time, then just at the destination.
Now, a power failure mid transfer... wouldn't be pretty.