All Quantum Probability viewed in Complex Projective Geometry

• URL: http://arxiv.org/abs/2605.17578v2
• Date: Sat, 23 May 2026 19:16:41 GMT
• Title: All Quantum Probability viewed in Complex Projective Geometry
• Abstract summary: 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.<n>In this paper a direct description of all these probabilities is given as formulas involving only the geometric properties of the projective space itself.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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 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.

Related papers

• Conceptual and Geometric Foundations for a Teleparallel Approach to Quantum Gravity [45.88028371034407]
  We revisit quantum field theory in curved spacetime as a semi-classical framework for quantum matter on classical geometries.<n>We outline a teleparallel framework based on coframe and spin-connection variables, where gravity is encoded in torsion rather than string curvature.
  arXiv  Detail & Related papers  (2026-06-08T15:02:26Z)
• Complexity of Satisfiability in Kochen-Specker Partial Boolean Algebras [0.0]
  We show that the satisfiability problem for the class of non-variable partial Boolean algebras is NP-complete.<n>We also show that the satisfiability problem for the class of all Hilbert spaces and all finite-dimensional Hilbert spaces is undecidable.
  arXiv  Detail & Related papers  (2026-02-27T16:43:13Z)
• "It from Bit": The Hartle-Hawking state and quantum mechanics for de Sitter observers [8.482707797354104]
  We argue that the one-state property of closed universes is fully compatible with the finite-dimensional quantum mechanics experienced by observers inside de Sitter space.<n>We show that the quantum mechanics experienced by an observer inside de Sitter space emerges from the classical statistics encoded in the baby-universe Hilbert space.
  arXiv  Detail & Related papers  (2026-02-05T17:50:15Z)
• Towards entropic uncertainty relations for non-regular Hilbert spaces [44.99833362998488]
  The Entropic Uncertainty Relations (EUR) result from inequalities that are intrinsic to the Hilbert space and its dual with no direct connection to the Canonical Commutation Relations.<n>The analysis of these EUR in the context of singular Hilbert spaces has not been addressed.
  arXiv  Detail & Related papers  (2025-03-24T23:41:50Z)
• Evidence for Exceptional Points as Topological Defects [0.0]
  We examine how an arbitrary quantum state evolves when transported along closed parameter loops.<n>Our results demonstrate that nontrivial holonomy can emerge in the presence of exceptional points.
  arXiv  Detail & Related papers  (2024-12-09T14:58:26Z)
• Hilbert space geometry and quantum chaos [39.58317527488534]
  We consider the symmetric part of the QGT for various multi-parametric random matrix Hamiltonians. We find for a two-dimensional parameter space that, while the ergodic phase corresponds to the smooth manifold, the integrable limit marks itself as a singular geometry with a conical defect.
  arXiv  Detail & Related papers  (2024-11-18T19:00:17Z)
• Quantum Probability Geometrically Realized in Projective Space [0.0]
  This paper aims to pass all quantum probability formulas to the projective space associated to the complex Hilbert space of a given quantum system.<n>The upshot is that quantum theory is the probability theory of projective subspaces, or equivalently, of quantum events.
  arXiv  Detail & Related papers  (2024-10-23T20:29:15Z)
• Strict area law entanglement versus chirality [19.441221681703468]
  Chirality is a gapped phase of matter in two spatial dimensions that can be manifested through non-zero thermal or electrical Hall conductance.<n>We prove two no-go theorems that forbid such chirality for a quantum state in a finite dimensional local Hilbert space with strict area law entanglement entropies.
  arXiv  Detail & Related papers  (2024-08-19T18:00:01Z)
• Connecting classical finite exchangeability to quantum theory [45.76759085727843]
  Exchangeability is a fundamental concept in probability theory and statistics.<n>It allows to model situations where the order of observations does not matter.<n>It is well known that both theorems do not hold for finitely exchangeable sequences.
  arXiv  Detail & Related papers  (2023-06-06T17:15:19Z)
• Continuous percolation in a Hilbert space for a large system of qubits [58.720142291102135]
  The percolation transition is defined through the appearance of the infinite cluster. We show that the exponentially increasing dimensionality of the Hilbert space makes its covering by finite-size hyperspheres inefficient. Our approach to the percolation transition in compact metric spaces may prove useful for its rigorous treatment in other contexts.
  arXiv  Detail & Related papers  (2022-10-15T13:53:21Z)
• Constraint Inequalities from Hilbert Space Geometry & Efficient Quantum Computation [0.0]
  Useful relations describing arbitrary parameters of given quantum systems can be derived from simple physical constraints imposed on the vectors in the corresponding Hilbert space. We describe the procedure and point out that this parallels the necessary considerations that make Quantum Simulation of quantum fields possible. We suggest how to use these ideas to guide and improve parameterized quantum circuits.
  arXiv  Detail & Related papers  (2022-10-13T22:13:43Z)
• Why and whence the Hilbert space in quantum theory? [0.0]
  We explain why and how the Hilbert space comes about in quantum theory. An issue of deriving the norm topology may no have a short-length solution but is likely solvable in the affirmative.
  arXiv  Detail & Related papers  (2021-10-07T13:55:47Z)
• Generalized Probabilistic Theories in a New Light [0.0]
  A new answer to the question of why our universe is quantum mechanical rather than classical will be presented. This paper shows that there is still a possibility that there might be a deterministic level from which our universe emerges.
  arXiv  Detail & Related papers  (2021-03-08T21:28:19Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.