Talk given on the 13th of December 2021 for the Algebra, Particles, and Quantum Theory seminar series. Organized by Nichol (Cohl) Furey, Humboldt-Universität zu Berlin Abstract: This talk will focus on the mathematical properties of Euclidean Jordan Algebras (EJAs) viewed as potential models for the state and observable spaces of physical systems, and two new characterizations of these algebras in terms of such properties, by myself and collaborators. EJAs were introduced and investigated early in the development of quantum theory, as abstractions of the algebra of Hermitian operators on a Hilbert space, initially in finite dimension. Jordan, von Neumann, and Wigner classified them: they are real, complex, and quaternionic quantum theory, systems whose state spaces are balls (``spin factors"), and one exceptional case (which may be thought of as three-state octonionic quantum theory). The "general probabilistic theories" (GPT) framework formulates potential physical theories in terms of systems having convex, compact state spaces, on which the probabilities of measurement outcomes are given by affine functionals. Dynamics and composite systems are also described in the framework. A major part of the GPT research program has been to find principles, mathematically natural, of physical or information-processing significance, or all three, that narrow down the very wide landscape of possibilities available in the GPT framework to the familiar spaces of quantum density matrices (states) and POVM elements (measurement outcomes). Such characterizations often proceed by first characterizing the finite-dimensional EJAs. After summarizing important mathematical properties of the EJAs, I will describe several such characterizations, including the Koecher-Vinberg system relating EJAs to homogeneous self-dual cones (which can be interpreted as ``unnormalized states", the positive semidefinite Hermitian matrices being an example), but focusing on two new results. Joachim Hilgert and I [1,2] characterized EJAs by two properties: a generalized spectral decomposability formulated entirely in terms of convexity, and a ``strong symmetry" property of the state space, also formulated in convex terms: transitive action of the symmetry group on sets of simultaneously perfectly distinguishable pure states. Work of HB with C. Ududec [4] characterizes them via homogeneity of their cones of unnormalized states (whose mathematical, physical, and information-processing significance I will discuss) and transitive action of the symmetry group on pure states. Further physical principles characterizing the usual, complex, quantum systems within the class of EJAs will be described: tomographic locality, or energy observability [3] meaning that the generators of continuous symmetries of the state space [3, but close to ideas of Connes (orientation) and of Alfsen and Shultz (dynamical correspondence)]: the generators of continuous symmetries of the state space are observables. If time permits, I will also discuss the possibilities for forming composites of such systems, focusing on my work with Matthew Graydon and Alex Wilce [5]. [1] H. Barnum and J. Hilgert, "Strongly symmetric spectral convex bodies are Jordan algebra state spaces", arxiv.org/abs/1904.03753 [2] H. Barnum and J. Hilgert, "Spectral properties of convex sets", Journal of Lie Theory 30 (2020) 315-344. Preprint close to this available at: winephysicssong.com/2021/09/01/strongly-symmetric-spectral-convex-sets-are-jordan-algebra-state-spaces/ [3] H. Barnum, M. Mueller and C. Ududec, "Higher-order interference and single-system postulates characterizing quantum theory", New Journal of Physics 16 (2014) 123029. arXiv:1403.4147 [4] H. Barnum and C. Ududec, in preparation. [5] Composites and Categories of Euclidean Jordan Algebras, Quantum 4, 359 (2020). arxiv.org/abs/1606.09331