PIRSA:09080016

Jordan algebras and spectrality as tools for axiomatic characterization

APA

Barnum, H. (2009). Jordan algebras and spectrality as tools for axiomatic characterization. Perimeter Institute. https://pirsa.org/09080016

MLA

Barnum, Howard. Jordan algebras and spectrality as tools for axiomatic characterization. Perimeter Institute, Aug. 15, 2009, https://pirsa.org/09080016

BibTex

          @misc{ pirsa_09080016,
            doi = {},
            url = {https://pirsa.org/09080016},
            author = {Barnum, Howard},
            keywords = {Quantum Foundations},
            language = {en},
            title = {Jordan algebras and spectrality as tools for axiomatic characterization},
            publisher = {Perimeter Institute},
            year = {2009},
            month = {aug},
            note = {PIRSA:09080016 see, \url{https://pirsa.org}}
          }
          

Abstract

The normalized-state spaces of finite-dimensional Jordan algebras constitute a relatively narrow class of convex sets that includes the finite-dimensional quantum mechanical and classical state spaces. Several beautiful mathematical characterizations of Jordan statespaces exist, notably Koecher's characterization as the bases of homogeneous self-dual cones, and Alfsen and Shultz's characterization based on the notion of spectral convex sets plus additional axioms.  I will review the notion of spectral convex set and the Alfsen-Shultz characterization and discuss how these mathematical characterizationsof Jordan state spaces might be useful in developing accounts of quantum theory based on more operational principles, for example ones concerning information processing.  If time permits, I will present joint work with Cozmin Ududec in which we define analogues of multiple-slit experiments in systems described by spectral convex state spaces, and obtain results on Sorkin's notion of higher-level interference in this setting.  For example, we show that, like the finite-dimensional quantum systems which are a special case, Jordan state spaces exhibit only lowest-order (I_2 in Sorkin's hierarchy) interference.