Justifying the identification of observers’ Hilbert spaces in applying Wigner’s theorem

Clarify and justify the assumption that the Hilbert spaces H and H' associated with different observers can be canonically identified (H = H') in the application of Wigner’s theorem to spacetime transformations, so that a single projective unitary representation U(A) acts on one Hilbert space. Determine precise conditions on the observers’ state spaces and transition probabilities that warrant this identification.

Background

In presenting Wigner’s theorem, the paper considers two observers O and O' with Hilbert spaces H and H', respectively, and uses the invariance of transition probabilities to infer the existence of a unitary or antiunitary map between their state spaces. The standard route to a single projective unitary representation U(A) acting on a single Hilbert space requires identifying H and H'.

The author notes that while this identification is often assumed (e.g., in Weinberg’s treatment), it is not obvious why it is justified. A rigorous account of when and why H = H' can be assumed is needed to ground the construction of projective unitary representations from Wigner’s theorem.