In a recent paper it was shown that all the Hilbert space formulas for quantum probabilities can be realized as functions of geometric properties of the associated projective space, but those functions were expressed using the structures of the associated Hilbert space. In this paper a direct description of all these probabilities is given as formulas involving only the geometric properties of the projective space itself without referring to the associated Hilbert space theory. In large part this depends on a projection theorem for complex projective space which is analogous to the projection theorem for Hilbert spaces. The importance of this is that this exhibits quantum probability in terms of the geometry of a Riemannian metric in a non-linear Kähler manifold without any reference to a linear Hilbert space. As such this is a part of a larger program of the geometrization of physics. This opens the possibility of generalizations of quantum theory in other similar geometric settings. The theory presented includes projective spaces of both finite and infinite dimension. Some comments explain how quantum theory based on a von Neumann algebra is compatible with this approach.
Presented is a geometric reformulation of the Born rule for finite-dimensional quantum systems. The state space is identified with complex projective space equipped with its canonical Fubini–Study geometry. Three structural axioms — locality in projective distance, invariance under projective unitaries, and additivity/non-contextuality for orthogonal decompositions — are shown to reduce the problem of transition probabilities to the framework of Gleason's theorem, thereby uniquely determining the transition probability P ([ψ] → [ϕ]) = |⟨ϕ|ψ⟩|2. The argument provides a transparent geometric reformulation and interpretation of Gleason’s theorem. I then show that any relativistic Dirac spinor theory automatically realizes this geometry locally, so the Born rule is inherited without further assumption. Finally, I formulate a precise conjecture for an octonionic analogue involving the exceptional group G2 (or F4/E8), then illustrate the idea with a simple finite toy model derived from the seven imaginary octonions and the Fano plane.
The quantum formula for the spin correlation of the bipartite singlet spin state,
CQ(a,b), is derived on the basis of a probability distribution
ρ(ϕ) that is generic, i. e., independent of
(a,b). In line with a previous result obtained within the framework of the quantum formalism, the probability space is partitioned according to the sign of the product
A=αβ of the individual spin projections
α and
β onto
a and
b. A specific partitioning and a corresponding set of realizations {$ ϕ$} are associated with every measurement setting
(a,b); this precludes the transfer of
α or
β from
CQ(a,b) to
CQ(a,b′), for
b′=b. A geometric model that reproduces the spin correlation serves to validate our approach, giving a concrete meaning to the quantum result in terms of a (local random variable) probability distribution.