Isometry-Preserving Flows and Unitary Structure in Linear Dynamical Systems

This work investigates structural constraints imposed on linear dynamical systems by the preservation of inner-product geometry. In particular, we analyze transformations that preserve transition probabilities and show that continuous isometry-preserving flows must be unitary. Closely related to Wigner’s theorem, the paper emphasizes the rigidity of Hilbert-space geometry and the restrictive consequences that follow from requiring distinguishability-preserving evolution. The results are intended as a structural analysis rather than a reconstruction of quantum theory, highlighting how modest geometric assumptions constrain admissible dynamics.