The effect of space-time curvature on Hilbert space

Vitaly Kuyukov

Jincheng Zhang

This paper introduces Quantum State Curvature Response Theory (QSCRT), a novel framework proposing that curvature within Hilbert space induces measurable physical responses in quantum systems. The core concept is expressed as [Rψ → Fψ], drawing an analogy to Einstein's theory of General Relativity, where spacetime curvature dictates gravitational effects. QSCRT posits that changes in the curvature of the quantum state space—represented by the curvature tensor Rψ—will generate a corresponding force Fψ. This theory suggests a dynamic nature of Hilbert space itself, moving beyond a purely geometric description. The research explores various parameter spaces of quantum systems to investigate this curvature response and its potential implications. The work aims to establish a foundation for understanding the inherent dynamics of quantum space and its influence on quantum phenomena, with potential implications for future research and theoretical advancements. ---

Chia-Yi Ju, Adam Miranowicz, Yueh-Nan Chen +2 more

Studies have shown that the Hilbert spaces of non-Hermitian systems require nontrivial metrics. Here, we demonstrate how evolution dimensions, in addition to time, can emerge naturally from a geometric formalism. Specifically, in this formalism, Hamiltonians can be interpreted as a Christoffel symbol-like operators, and the Schroedinger equation as a parallel transport in this formalism. We then derive the evolution equations for the states and metrics along the emergent dimensions and find that the curvature of the Hilbert space bundle for any given closed system is locally flat. Finally, we show that the fidelity susceptibilities and the Berry curvatures of states are related to these emergent parallel transports.

G. Papini

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.