In the Star Trek universe, how do we know that the transporter didn't kill you and create a replicant?

One of the reasons I would personally never get in a transporter is that there is no evidence that the Real Me isn't just vaporized and a Copy Me created at the other location. The Copy Me might have all my memories and be indistinguishable from the original, but it isn't the Real Me.

I know that some early characters (especially in ST:Enterprise) had problems with transporters -- but these mostly seemed to concern safety. No one discussed the metaphysics of the technology.

Was there ever proof -- or even discussion -- of the question of whether the transporter really was transporting, or just creating a heck of a good copy?

Note 1: Yes, the transporter tech in ST sends the 'atoms' as well as the 'bits' to reconstruct the atoms, so it's not quite a 23rd century fax machine. But you're still being vaporized.....

Note 2: And yes, I do wake up each morning wondering if I am the same person who went to sleep the previous night, thank you very much. Why do you ask?

Answer

Yes there is. In STE-Daedalus there is the following dialog with Emory Erickson, the inventor of the transporter:

ARCHER: I have to confess, given a choice, I'd much rather use a good old-fashioned shuttlepod.
EMORY: I'll never forget the protests when the transporter was first approved for bio-matter.
DANICA: Oh, God. Here we go.
EMORY: People said it was unsafe, that it caused brain cancer, psychosis, and even sleep disorders. And then there was all that metaphysical chatter about whether or not the person who arrived after the transport was the same person who left, and not some weird copy.
TUCKER: Which would make all of us copies.
EMORY: I had to fight all of that nonsense, and I'm not going to tell you there weren't costs. I'm living proof of that, but I won. Mankind is better off. Makes everything I've fought for worthwhile.
TUCKER: Here's to a successful experiment.

Obviously Erickson was successful enough to allay the metaphysical fears, since - as we know - most people are comfortable with the transporter.