No other sci-fi series gave us the theory that by stepping into a transporter beam you are killed, then resurrected into the same person
TOS never got into any of the technology in that kind of detail.
The Making of Star Trek quotes Roddenberry as wanting the audience to simply accept whatever they saw at face value. At most, McCoy grumped about scattering a man's molecules all over the cosmos. But no TV episode got into the mechanics of it.
Post-TOS spin-offs, books and fan discussion groups have taken a stab at it, though. I think the two basic theories are "ultra-fax" and "wormhole." The wormhole idea is common sci-fi handwavium of a "shortcut" through hyperspace. So the traveler is never altered, merely pushed through this shortcut "tunnel" to the other side. "Ultra-fax" is where it gets complicated.
With this approach, matter itself is not transported, only information is sent. And there is typically a sender and receiver, like a telephone. Thus, matter is "transposed" into a signal that is transmitted and reconstituted into matter at the receiving side, like the pressure waves of sound into a phone converted to electrical pulses, and back into sound. Here's the catch: this method is like a paper fax in that a copy is made at the far end. If the local original is not destroyed, you then have two of a thing. Some sci-fi novels have addressed this situation.
But
Star Trek has no remote receiver! True, but the "fax copy" situation still applies. The Trek transporter must disintegrate an original while buffering the information to make it. Energy, perhaps like phasers, is then projected to a remote locus and re-assembled. James P. Hogan's
The Genesis Machine describes a Trek transporter-like technology that transposes while sending the information through a faster-than-light hyperspace realm.
But the "copy" angle has been addressed in sci-fi having nothing to do with transporters. The Schwarzenegger movie
Sixth Day posits an advanced form of cloning that also covers "syncordings" of the subject's mind. The "bad guy" pushing this technology pitches it as immortality. The truth is that the copy only thinks it is the same person and has all the original's memories. Yet everyone, original and copies alike, will one day come to know death
personally. That's not immortality.
Dennis E. Taylor's "Bobiverse" books approached from yet another angle. The original Bob was one of our contemporaries, a man who dies in a car accident in the early 21st century, and his head is cryogenically preserved. He "wakes up" a little over a century later to find out that he is a "replicant," a scan of Bob's brain now supported in computer simulation. This software engineer and sci-fi fan is given a dream come true: he will be the "machine intelligence" to guide an interstellar Von Neumann probe. ("Artificial" intelligences weren't up to the task, but this replicant approach is more flexible.)
The Von Neumann concept is a machine that will duplicate itself, and thus Bob becomes "Bobs" spreading out to other stars and paving the way for humanity to follow. Yet each new copy is a distinctly new person. The Bobs ponder this variance, making philosophical guesses about subtle differences in equipment, or quantum hand-wavium. The first clue comes in one of the later books.
Being computer sims, the Bobs make backups through their FTL network. If a Bob is destroyed, they can "resurrect" him. And the new Bob carries on indistinguishable from the lost one. Then one Bob makes the jump across their vast region of explored space, rather than taking the long way and traveling all those tedious years. The surprise is that the transmitted Bob is not a new Bob, like when they build new ones. The difference is that the transmitted Bob shut himself down before transmitting—a risk, of course. But the fact that he comes out "the same Bob" rather than a new person, they start speculating about souls... (Maybe new personalities only show up if a copy is made while the original is still active?)
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