Geometric reconstruction of quantum theory

          @misc{ pirsa_PIRSA:25100168,
            doi = {10.48660/25100168},
            url = {https://pirsa.org/25100168},
            author = {Medina S{\'a}nchez, Nicol{\'a}s},
            keywords = {Quantum Foundations},
            language = {en},
            title = {Geometric reconstruction of quantum theory},
            publisher = {Perimeter Institute},
            year = {2025},
            month = {oct},
            note = {PIRSA:25100168 see, \url{https://pirsa.org}}
          }
          

Abstract

We ask why finite-dimensional quantum states can be represented as mixtures of pure states on complex projective space, with reversible transitions given by unitary transformations up to phase. We derive this from two principles. First, determinism of probabilities: the evolution on the state space is smooth and reversible, and each state specifies observable probabilities together with auxiliary variables that drive the deterministic trajectory. Second, invariance: every reversible transformation preserves a physical quantity, so trajectories are tangent to level sets of a real function. From these assumptions we show that the state space is isomorphic to complex projective space and the reversible transformations form a subgroup of the projective unitary group. This yields a minimal reconstruction of quantum theory and a route to more transparent physical interpretations.