> in classical mechanics there is this notion of a path parameterized on time, never of a change of time parameterized by other variablesIf the solution of your classical equations give a monotonous x(t) in R, then you can of course formally also solve it for t(x), but that might be besides the point.
>Same guy. So you say time is special and not just another variable?Well I asked you what variable (or privilage for that matter) means to you. A non-relativistic QM state amplitude in position space will come as a function of the spectrum (say R or R^3) of X. Is that a variable to you?
It's different than what time is, the parameter introduced with the equation governing the dynamics of the system. That's a t-indexed family of vectors in the same one vector space. For each time, all x "variables" are there, because X is not informed about time (even if in the Heisenberg picture, you may also write X(t):=exp(-iH)·X·exp(-iH)).
PS in case you don't know it, google "Heisenberg picture", which is related to your question, even if only tangentially.
https://en.wikipedia.org/wiki/Heisenberg_pictureBut I wrote that line about periodicity
|t>=|t+s>
for you to think about how it contrasts with X
At least in the standard setup, you don't really make use of a "time-operator" T such that it applied to a generic state in the "static, unknowing" Hilbert space would know about it, in that
"T|psi>=tv|psi>"
Which is not to say I can't imagine you bundle all thing up and keep track of t.
>The Schrodinger equation implies recognition that state vectors exists as snapshots that change, not that a state can be "already" defined across all time and space, just across space, which then "evolves".>I mean i have heard of attempts to treat time as just another operator, which changes according to some unphysical variable, so that everything is treated on equal footing.https://en.wikipedia.org/wiki/Wheeler%E2%80%93DeWitt_equationI think that's enough to bite for now
