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Quantum theory's Hilbert space apparatus in its finite-dimensional version isnearly reconstructed from four simple and quantum-mechanically motivatedpostulates for a quantum logic. The reconstruction process is not complete,since it excludes the two-dimensional Hilbert space and still includes theexceptional Jordan algebras, which are not part of the Hilbert space apparatus.Options for physically meaningful potential generalizations of the apparatusare discussed.

A simple and quantum-mechanically motivated characterization of the formally real Jordan algebras