Symmetry, Self-Duality and the Jordan Structure of Quantum Theory

          @misc{ pirsa_PIRSA:11050037,
            doi = {10.48660/11050037},
            url = {https://pirsa.org/11050037},
            author = {Wilce, Alexander},
            keywords = {Quantum Foundations},
            language = {en},
            title = {Symmetry, Self-Duality and the Jordan Structure of Quantum Theory },
            publisher = {Perimeter Institute},
            year = {2011},
            month = {may},
            note = {PIRSA:11050037 see, \url{https://pirsa.org}}
          }
          

Abstract

This talk reviews recent and on-going work, much of it joint with Howard Barnum, on the origins of the Jordan-algebraic structure of finite-dimensional quantum theory. I begin by describing a simple recipe for constructing highly symmetrical probabilistic models, and discuss the ordered linear spaces generated by such models. I then consider the situation of a probabilistic theory consisting of a symmetric monoidal *-category of finite-dimensional such models: in this context, the state and effect cones are self-dual. Subject to a further ``steering" axiom, they are also homogenous, and hence, by the Koecher-Vinberg Theorem, representable as the cones of formally real Jordan algebras. Finally, if the theory contains a single system with the structure of a qubit, then (by a result of H. Hanche-Olsen), each model in the category is the self-adjoint part of a C*-algebra.