Stephen: Thanks. First, not everything corresponding to a length or such obeys that particular rule... consider the Lorenz metric... any "lightlike" vector has a norm of zero, for instance, and yet that particular matric is rather useful physically. :) (admittedly, you get that via the minus sign, and if your norm is such that it treats all the components in some sense equivalently, you don't get that... well, what about norms involving cross terms?)

More to the subject... why is any norm preserved? That is, why only allow norm preserving transforms?

Which brings be to Eliezer:

So? Why does the universe "choose" rules that say "no outcome pump"? That's way up the ladder of stuff built out of other stuff. (as far as communicating faster than light, I'd think "outcome pump" type things are the main 'crazy' result of FTL in the first place)

Actually, I think I didn't communicate my question accurately. You derived it would be an outcome pump by noting it would change the Born derived probabilities (At least, that's my understanding of the significance of you noting that the ratios of the squared magnitudes changing.) But the Born probabilities are already the "odd rule out"... so I wanted to know if there was any other reason/argument you could think of as to why we have norm preservation without appealing to the Born rule. (Does that clarify my question?)

I mean, if I was letting myself use the Born rule, I could just say that the probabilities have to sum to 1, and that hands me the unitaryness. But my whole point was "the restriction to unitary transforms _itself_ seems to be related to squared magnitude stuff. So by understanding why that restriction exists in reality, maybe I'd have a better idea where the Born rule is coming from"