Where is Captain Kirk during the time of transmission? What is his status?
Clearly, it takes some amount of time to transport as evidenced by the "transporter jingle" (at least a second or two). Apparently, we are defined by physical/biological information which is copied by the transporter and by the present "state" of the brains neural pathways (also copied and transported). Or is it dismantled and reassembled at the receiving station?
This brings up an interesting scenario.
Okay you've gone to the back of the screen and they put you on a table next to a "clone" who looks exactly like you when you were young. They put an "electronic helmet" on you and connect it to one on the clone. Hit the switch --- bzzzzzz --- crackle --- bzzzzz. Now the clone wakes up and looks at you -- "Hey it works! This is great!". You look at him in astonishment while one of the technicians slits your throat. "We can't have two of you around."
Now would you still choose to do it? No? WHY?? Would you (as the clone) stop the technician from slitting your throat (the old you)?
The fact that the previous scene is as yet impossible does not alter the kinds of philosophical questions it raises.
Alright, you're a new man now. You emerge from behind the screen and are greeted by your anxious wife. She asks you many questions and thereby verifies that it is indeed you her husband (this occurs while she hears a "little scuffle" behind the screen --- the 'throat slitting' remember?). But everything is alright. Will you allow her to "take the plunge" tomorrow? No? Why not? You'd then be a matched pair again.
Obviously, since no information (except the throat slitting) is lost, it should be perfectly acceptable to anybody. Shouldn't it?
The issue here is ...