Stephen: Is the point you’re making basically along the lines of “vector as geometric object rather than list of numbers”?
Sure, I buy that. Heck, I’m naturally inclined toward that perspective at this time. (In part because have been studying GR lately)
Aaanyways, so I guess basically what you’re saying is that all operators corresponding to time evolution or whatever are just rotations or such in the space? And why the 2-norm instead of, say, the 1-norm? why would the universe “prefer” to preserve the sum of the squared magnitudes rather than the sum of the magnitudes? ie, why is the rule “unitary” rather than “stochastic”, for instance? (Well, I have a partial answer for that myself… reversibility. Stochastic isn’t necessarally reversible, right? unitary is though, so there is that...)
If I’m understanding what you’re trying to say, basically you’re saying “it’s as if you use any ole transform, then just divide by the factor the norm’s been changed by, so you may as well have that ‘already in’ the transform”… But if the transform isn’t some multiple of a unitary transform, then there won’t be any single scalar value that takes care of that, right? Why instead of “norm preserving” isn’t the rule “any invertable linear transform”?
Or did I completely and utterly misunderstand what you were trying to say?
Stephen: Is the point you’re making basically along the lines of “vector as geometric object rather than list of numbers”?
Sure, I buy that. Heck, I’m naturally inclined toward that perspective at this time. (In part because have been studying GR lately)
Aaanyways, so I guess basically what you’re saying is that all operators corresponding to time evolution or whatever are just rotations or such in the space? And why the 2-norm instead of, say, the 1-norm? why would the universe “prefer” to preserve the sum of the squared magnitudes rather than the sum of the magnitudes? ie, why is the rule “unitary” rather than “stochastic”, for instance? (Well, I have a partial answer for that myself… reversibility. Stochastic isn’t necessarally reversible, right? unitary is though, so there is that...)
If I’m understanding what you’re trying to say, basically you’re saying “it’s as if you use any ole transform, then just divide by the factor the norm’s been changed by, so you may as well have that ‘already in’ the transform”… But if the transform isn’t some multiple of a unitary transform, then there won’t be any single scalar value that takes care of that, right? Why instead of “norm preserving” isn’t the rule “any invertable linear transform”?
Or did I completely and utterly misunderstand what you were trying to say?