On the homogeneity of the quantum transition probability

TL;DR

The work links the quantum transition probability to the geometry of pure-state spaces arising from simple Euclidean Jordan algebras, leveraging Hirzebruch–Wang results that atomic parts $E_A$ form two-point homogeneous convex spaces. It shows that in all simple EJAs the transition probability is maximally homogeneous, with $P(q|p)=trace(p∘q)$ for atoms and invariance under automorphisms; Hilbert-space quantum theory is included as a special case with $P(q|p)=|\langle \varphi|\psi\rangle|^2$. Reducible algebras break homogeneity, and the exotic $E_6$-symmetric bioctonionic plane yields a non-homogeneous but highly symmetric model, illustrating the limits of homogeneity as a structural principle. Overall, the paper argues that quantum probability originates from a purely topological structure of pure states, framed by finite-dimensional compactness, with clear boundaries on extending these ideas to infinite dimensions.

Abstract

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 $E_6$-symmetric bioctonionic projective plane is used as quantum logic.