Geometric Deformation of Quantum Mechanics

André Lichnerowicz

Binayak Dutta-Roy, Gautam Kumar Ghosh

Hoshang Heydari

We present a geometric formulation of quantum mechanics based on the symplectic structure of the projective Hilbert space. Building upon the standard Kähler framework, we introduce an extension in which the symplectic structure is allowed to couple to a metric-affine background geometry, leading to a deformation of the Hamiltonian flow on the state space. We show that, under suitable conditions, the deformed structure remains symplectic and defines a well-posed Hamiltonian system. The formulation reduces to standard Schrödinger dynamics in the limit where the geometric deformation vanishes. Explicit analytical examples are constructed to illustrate the effect of the deformation. In particular, curvature-dependent deformations lead to a rescaling of Hamiltonian flows, while torsion-induced contributions produce direction-dependent corrections. In addition, geometric phases acquire corrections determined by the deformed symplectic structure. These results provide a mathematically consistent framework for exploring geometric modifications of quantum evolution induced by background curvature and affine structure.

Enrico Santamato

Jincheng Zhang

This paper investigates the potential of quantum computing to validate the quantum properties of matrix operations, leveraging geometric deformation techniques. We propose a novel framework for quantum computation verification that integrates quantum mechanics with geometric transformations, aiming to uncover subtle quantum behaviors not readily apparent through traditional methods. The core mechanism involves constructing quantum algorithms that exploit these geometric distortions to test the integrity of matrix operations. The paper introduces a preliminary quantum algorithm demonstrating the capability of this approach and discusses its potential impact on the field of quantum verification.

Jincheng Zhang

Quantum entanglement, a bizarre phenomenon in quantum mechanics, offers a potentially revolutionary approach to geometric manipulation. This paper explores the design of a novel geometric deformation algorithm based on quantum entanglement, aiming to generate subtle and complex transformations with high fidelity. We detail the core mechanism, outlining the mathematical framework and potential applications in areas ranging from image processing and 3D modeling to fundamental physics research. The algorithm's core claim focuses on leveraging entanglement to achieve a level of precision unattainable through traditional geometric methods. This work addresses the limitations of classical approaches by introducing a fundamentally different methodology rooted in quantum properties.