If the transporter transports the original brain particles from the departure pod, then the person who comes out of the arrival pod may, or may not have the original brain’s PI. I don’t personally think they will be the same even if they use coherent quantum teleportation, because the intermittent transport of particles will break the heretofore unbroken process of self-awareness, thereby nessesitating a re-boot, spawing a new PI. But, at least it doesn’t break any physical laws that I’m aware of, so who knows.
But, let’s look at it a different way. If you accept that there is nothing unique about any elemental particle, then there should be nothing unique about an assemblage of particles that are identical to your brain’s particles, arranged identically, correct? There is no difference between the two brains. They should have the exact same memories, and the exact same mental processes. No outside observer will be able to tell the difference between the two, and each will believe that they are the same person.
Theoretically there should be no barrier to assembling any number of exact copies of brains using the same type, number and arrangement of particles, anywhere, anytime. It would simply take having an exceptionally detailed blueprint and an exceptionally high-tech particle assembler to get the job done.
So, set this assembler to assemble a copy of you and your brain as it will be 10 minutes from now on a habitable planet a million light years away, a million years in the past (don’t ask how your going to get the assembler there…that’s not relevant to this thought experiment, nor how you activate it).
So, in ten minutes when this other you pops up on some extra-terrestial planet a million years ago, does he have any affect you you? If the you here and now is asked, “I’ve got to kill one of you”, do you care which one? You should. You should say, “kill the one who will appear a million miles away/ a million years ago” because you have no vested interest in him. You will not become him. The confusion comes in asking the extra-terrestrial you if he believes he was you on Earth 10 minutes ago. He will say, “yes.” He will be virtually correct, but not exactly correct. He has your memories, he is conscious, he has a PI, but he was not you 11 minutes ago; he didn’t even exist then.
Some believe that you are your memories in any conscious mind. I don’t. In fact, I believe you are still you with no memories at all, or even with someone else’s memories…on a very basic level, just a feeling of being alive. As such, there can be only one you—one PI per customer; more than one is paradoxical. How can you have a unique PI in 2 brains that are not unique? It must be due to an unbroken process that continues over time and is unique only in the fact that it is not the same process as any other. Equal, but separate, and always have been.
Let’s explore some other scenarios. Consider concert pianist Valentina Lisitsa, age 40. Erase all memories from the first 10 years of her life. Is she still the same person? I think so. Erase the next 10 years; still the same? Can she still play the piano at concert level? She played Royal Albert Hall in August, and she remembers every piece exactly and the muscle memory is intact. But, as far as she’s concerned, she never learned how to play the piano, or read music. Let’s replace her first 20 years of memories with those of Stephen Hawking. Is she now a young physicist and a middle aged pianist? Let’s combine all of Valentina’s memories with all of Stephen Hawking’s. Is she now Valentina, Steven, or two people? Does she have two PI’s. If only one, whose?
I think she’s only one person, the same person she ever was, only with someone else’s memories. Should the wheel-chaired Stephen off himself in order to live on in Valentina’s body. No.