Quantum Probability Geometrically Realized in Projective Space

The principal goal of this paper and its originality consist in passing all formulas for quantum probability to the projective space associated to the complex Hilbert space of a given quantum system, thereby providing a geometric foundation of quantum probability, which should be considered as a step towards an eventual axiomization. Quantum events have consecutive and conditional probabilities, which have been used in the author’s work to clarify ‘collapse of the state’ and to generalize the concept of entanglement by incorporating it into quantum probability theory. In this way much of standard textbook quantum theory can be understood in the setting of the geometry of a projective space and its subspaces. The ultimate, future goal is to formulate all of quantum theory as 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.