All Quantum Probability viewed in Complex Projective Geometry

TL;DR

This paper reformulates quantum probabilities purely in terms of the geometry of complex projective spaces, avoiding Hilbert space references, and explores implications for the geometrization of physics.

Contribution

It provides a direct geometric description of quantum probabilities using only the properties of complex projective space, extending to infinite dimensions.

Findings

Quantum probabilities expressed via complex projective geometry.

Projection theorem for complex projective space analogous to Hilbert spaces.

Compatibility with quantum theory based on von Neumann algebras.

Abstract

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 K\"ahler 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.