I would like to discuss the consequences of the concept of an internal observer in quantum theory. If we assume that we have a universe that evolves unitarily at a global scale and an observer is defined as a subsystem of that universe, what can this observer learn about the universe?

Some assumptions are that the evolution of the universe is generated by interaction. Interactions are local and the observer gathers information by interacting with his environment. But most importantly, there is no measurement postulate that the observer could use to perform a quantum measurement.

I would argue that everything the observer learns about the universe is contained in the state history of the subsystem he has interacted with. If one assumes that he can derive a law for the evolution of this state history, then he can only reconstruct the state history up to isomorphisms. Meaning, if he finds plausible state history psi(t) which evolves unitarily like $\psi(t) = U(t,t\_0)[\psi(t\_0)]$, then any $\phi(t)$ for which a bijection $f$ exists so that $\phi(t) = f(\psi(t))$ and $\psi(t) = f^{-1}(\phi(t))$ and which evolves with the same law $\phi(t) = U(t,t\_0)[\phi(t\_0)]$ is an equivalent description of the state history of the system he is part of. (Note that psi is not necessarily a ket. I would like to stay agnostic of the actual representation for now)

A very simple example for such a bijection would be the multiplication with a nonzero complex number. Another would be a spacetime symmetry transformation.

Do you think this argumentation is correct so far? I will continue the argument to the point where it actually gets more interesting, but I would like to see some confirmation so that we can later discuss the consequences.