Ok, but OP of the post above starts with “Suppose we have a system S with eigenfunctions {φi}”, so I don’t see why (or how) they should depend on the observer. I’m not claiming these are just arbitrary functions. The point is that requiring the the time-evolution on pure states of the form ψ⊗φi to map to pure states of the same kind is arbitrary choice that distinguishes the eigenfunctions. Why can’t we chose any other orthonormal basis at this point, say some ONB (wi)i, and require that wi⊗ψ↦ESwi⊗ψi, where ψi is defined so that this makes sense and is unitary? (I guess this is what you mean with “diagonalization”, but I dislike the term because if we chose a non-eigenfunction orthonormal basis the construction still “works”, the representation just won’t be diagonal in the first component).
Ok, but OP of the post above starts with “Suppose we have a system S with eigenfunctions {φi}”, so I don’t see why (or how) they should depend on the observer. I’m not claiming these are just arbitrary functions. The point is that requiring the the time-evolution on pure states of the form ψ⊗φi to map to pure states of the same kind is arbitrary choice that distinguishes the eigenfunctions. Why can’t we chose any other orthonormal basis at this point, say some ONB (wi)i, and require that wi⊗ψ↦ESwi⊗ψi, where ψi is defined so that this makes sense and is unitary? (I guess this is what you mean with “diagonalization”, but I dislike the term because if we chose a non-eigenfunction orthonormal basis the construction still “works”, the representation just won’t be diagonal in the first component).