  A representation of finite-dimensional probabilistic models in terms of
formally real Jordan algebras is obtained, in a strikingly easy way, from
simple assumptions. This provides a framework in which real, complex and
quaternionic quantum mechanics can be treated on an equal footing, and allows
some (but not too much) room for other alternatives. This is based on earlier
work (arXiv:1206:2897), but the development here is further simplified, and
also extended in several ways. I also discuss the possibilities for organizing
probabilistic models, subject to the assumptions discussed here, into symmetric
monoidal categories, showing that such a category will automatically have a
dagger-compact structure. (Recent joint work with Howard Barnum and Matthew
Graydon (arXiv:1507.06278) exhibits several categories of this kind.)
