Extended Probability and Hermiticity in the C3-Hilbert Space: Visible and Hidden Axes in Quantum Geometry
AKTAŞ, BORA
Abstract We propose a geometric extension of quantum mechanics based on the C3C_{3}C3-Hilbert space, where scalar elements satisfy j3=−1j^{3} = -1j3=−1. This algebra introduces two additional phase channels, jjj and j2j^{2}j2, naturally separating quantum states into visible and hidden components. The visible axis (j−j2)(j - j^{2})(j−j2) encodes measurable space–time curvature differences responsible for observable quantum behavior, while the hidden axis (j+j2)(j + j^{2})(j+j2) encapsulates non-observable phase accumulations linked to internal curvature balance. A new probability measure is introduced as the real trace of the C3C_{3}C3-inner product, yielding a positive-definite and physically meaningful norm ∥ψ∥phys2=∥ψ1∥2+∥ψj∥2+∥ψj2∥2.\|\psi\|^{2}_{\mathrm{phys}} = \|\psi_{1}\|^{2} + \|\psi_{j}\|^{2} + \|\psi_{j^{2}}\|^{2}.∥ψ∥phys2=∥ψ1∥2+∥ψj∥2+∥ψj2∥2. This generalization preserves Born’s rule while extending Hermiticity and unitarity to multi-channel operators via the C3C_{3}C3-adjoint A^†3\hat{A}^{\dagger_{3}}A^†3. The resulting framework establishes a covariant link between geometric curvature and quantum probability, offering a route toward curvature-based interpretations of uncertainty and measurement. The model recovers standard complex Hilbert mechanics as the classical limit (ψj,ψj2→0)(\psi_{j}, \psi_{j^{2}} \to 0)(ψj,ψj2→0), while allowing curvature-induced phase entanglement when these channels are active. This approach provides a natural embedding for time operators, curved metrics, and multi-carrier extensions of quantum dynamics.
•Nov 24, 2025•0 citations