Wait... I think I've got it (note, am going on no sleep here, so I apologise in advance if this isn't as clear as it could be)

Pretty much any "reasonable" transform of a quantum state is unitary, right?

That is, the time evolution would be a repeated unitary transform (if considered in discrete steps) or an integrated one (if considered continuously)

Unitary transforms have inverses, right?

So pretty much no matter _what_ the "true" way of slicing states and evolving them is, there will be some transformation to a different set of orthognal states that acts local, right?

non-local-operator*state-vector = local-operator*transformation-matrix*transformed-state-vector, right?

So just like the majority of states don't factor into nice conditionally independant components, majority of "slicings" don't have local behavior, in this case, there's, no matter what, a transform into an orthognal basis such that the corresponding operator is local.

Maybe I ought take a page out of GR and think of the transformation matricies/operators as geometric rather than arithmetical objects? ie, same transformer/same state vector, just different basis. (or alternately, rotated)

So maybe the position space is simply the space (or one of them) that the appropriate transfrom "just happens" to be local.

Or is this completely way off?