Quantum Probability Geometrically Realized in Projective Space
Stephen Bruce Sontz
TL;DR
The paper advances a geometric program for quantum probability by realizing quantum events as projective subspaces of $\mathbb{C}P(\mathcal{H})$, and by pushing standard probability formulas (including consecutive, conditional, and interference probabilities) down to this projective setting. It introduces geometric observables, covariant symbols, and geometric density matrices, showing how Born’s rule and Wigner’s rule acquire a purely geometric interpretation while preserving their probabilistic content. A central contribution is the family of invariant functions $A_{n}$ on projective configurations, along with an operator-symbol framework that ties self-adjoint operators to bounded functions on phase space, enabling a Hamiltonian picture on $\mathbb{C}P(\mathcal{H})$. The work frames quantum collapse and entanglement as geometric properties and sketches extensions to broader von Neumann-algebra settings, offering a bridge between geometry and operator-algebraic quantum theory with potential insights for infinite-dimensional systems.
Abstract
The principal goal of this paper is to pass all quantum probability formulas to the projective space associated to the complex Hilbert space of a given quantum system, providing a more complete geometrization of quantum theory. Quantum events have consecutive and conditional probabilities, which have been used in the author's previous work to clarify `collapse' and to generalize the concept of entanglement by incorporating it into quantum probability theory. In this way all of standard textbook quantum theory can be understood as a geometric theory of projective subspaces without any special role for the zero-dimensional projective subspaces, which are also called pure states. The upshot is that quantum theory is the probability theory of projective subspaces, or equivalently, of quantum events. For the sake of simplicity the ideas are developed here in the context of a type I factor, but comments will be given about how to adopt this approach to more general von Neumann algebras.