In the years 1952 and 1965, H.-C. Wang and U. Hirzebruch showed that the two-point homogeneous compact spaces with convex metrics are isometric to the spheres, the real, complex, octonion projective spaces and the Moufang plane and as well to the sets of the minimal idempotents or pure states in the simple Euclidean Jordan algebras. Here we reveal the physical meaning of these mathematical achievements for the quantum mechanical transition probability. We show that this transition probability features a maximum degree of homogeneity in all simple Euclidean Jordan algebras, which includes common finite-dimensional Hilbert space quantum theory. The atomic parts of these algebras or, equivalently, the extreme boundaries of their state spaces can be characterized by purely topological means. This is an important difference to many other recent approaches that aim to distinguish the entire state spaces among the convex compact sets. An interesting case with non-homogeneous transition probability arises, when the
E6-symmetric bioctonionic projective plane is used as quantum logic.
The common structure of the space of pure states ℘ of a classical or a quantum mechanical system is that of a Poisson space with a transition probability. This is a topological space equipped with a Poisson structure, as well as with a function p:℘×℘→[0,1], with certain properties. The Poisson structure is connected with the transition probabilities through unitarity (in a specific formulation intrinsic to the given context). In classical mechanics, where p(ρ,σ)=δρσ, unitarity poses no restriction on the Poisson structure. Quantum mechanics is characterized by a specific (complex Hilbert space) form of p, and by the property that the irreducible components of ℘ as a transition probability space coincide with the symplectic leaves of ℘ as a Poisson space. In conjunction, these stipulations determine the Poisson structure of quantum mechanics up to a multiplicative constant (identified with Planck's constant). Motivated by E. M. Alfsen, H. Hanche-Olsen and F. W. Shultz (Acta Math.144 (1980) 267–305) and F.W. Shultz (Commun. Math. Phys.82 (1982) 497–509), we give axioms guaranteeing that ℘ is the space of pure states of a unital C*-algebra. We give an explicit construction of this algebra from ℘.
Max Born's statistical interpretation made probabilities play a major role in quantum theory. Here we show that these quantum probabilities and the classical probabilities have very different origins. While the latter always result from an assumed probability measure, the first include transition probabilities with a purely algebraic origin. Moreover, the general definition of transition probability introduced here comprises not only the well-known quantum mechanical transition probabilities between pure states or wave functions, but further novel cases. A transition probability that differs from 0 and 1 manifests the typical quantum indeterminacy in a similar way as Heisenberg's and others' uncertainty relations and, furthermore, rules out deterministic states in the same way as the Bell-Kochen-Specker theorem. However, the transition probability defined here achieves a lot more beyond that: it demonstrates that the algebraic structure of the Hilbert space quantum logic dictates the precise values of certain probabilities and it provides an unexpected access to these quantum probabilities that does not rely on states or wave functions.