The closest cousins of quantum theory from three simple principles

          @misc{ pirsa_PIRSA:12080007,
            doi = {10.48660/12080007},
            url = {https://pirsa.org/12080007},
            author = {Ududec, Cozmin},
            keywords = {Mathematical physics},
            language = {en},
            title = {The closest cousins of quantum theory from three simple principles},
            publisher = {Perimeter Institute},
            year = {2012},
            month = {aug},
            note = {PIRSA:12080007 see, \url{https://pirsa.org}}
          }
          

Abstract

A very general way of describing the abstract structure of quantum theory is to say that the set of observables on a quantum system form a C*-algebra.  A natural question is then, why should this be the case - why can observables be added and multiplied together to form any algebra, let alone of the special C* variety?  I will present recent work with Markus Mueller and Howard Barnum, showing that the closest algebraic cousins to standard quantum theory, namely the Jordan-algebras, can be characterized by three principles having an informational flavour, namely: (1) a generalized spectral decomposition, (2) a high degree of symmetry, and (3) a requirement on conditioning on the results of observations.   I'll then discuss alternatives to the third principle, as well as the possibility of dropping it as a way of searching for natural post-quantum theories.