Effects as Functions on Projective Hilbert Space

Davide Pastorello

Roberto Giuntini, Antonio Ledda, Francesco Paoli

Arun Kumar Pati, S. V. Lawande

We show that the quadratic short time behavior of transition probability is a natural consequence of the inner product of the Hilbert space of the quantum system. We provide a relation between the survival probability and the underlying geometric structure such as the Fubini-Study metric defined on the projective Hilbert space of the quantum system. This predicts the quantum Zeno effect even for systems described by nonlinear and nonunitary evolution equations, within the collapse mechanism of the wave function during measurement process.

Stephen Bruce Sontz

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.

Paul Busch, PEKKA J. LAHTI, Juha-Pekka Pellonpää +1 more

Borros Arneth

We develop a mathematical foundation for the Diagram Hilbert Space (DHS) framework by constructing the projection map from an underlying Hilbert space of relational diagram states to emergent macroscopic geometry. Using standard results from functional analysis and operator theory, we define closed observable subspaces, orthogonal projectors, coarse-grained states, entropy functionals, and induced observables. The formalism shows how classical configuration spaces may arise from subspace selection and information loss. We further derive effective projected dynamics and an overlap-based notion of emergent distance. The resulting framework provides a rigorous basis for phenomenological applications involving emergent gravity, galactic dynamics, and weak lensing.