Quantum Mechanics and Deformations of Geometrical Dynamics

Ivan G. Avramidi, Roberto Niardi

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.

Jincheng Zhang

This paper introduces a novel approach to geometry modeling utilizing nonlinear quantum adaptive geometry transformations. We propose a method that transforms complex geometric models into adaptable geometric models, enabling self-adaptive deformation within the model. The core mechanism involves a quantum process that dynamically adjusts geometric parameters based on a feedback loop, resulting in a highly flexible and responsive model. This approach offers significant advantages over traditional methods, particularly in handling intricate geometries and complex deformations. This research addresses the limitations of existing techniques by introducing a fundamentally new method that leverages quantum mechanics to achieve a level of adaptability not previously attainable. This work demonstrates the feasibility and potential of this approach through mathematical analysis and illustrative examples.

Abhay Ashtekar, Troy A. Schilling

Jincheng Zhang

Quantum state dynamics, a novel approach to quantum system modeling, offers the potential to achieve precise system representation and prediction by incorporating dynamic geometric transformations. This research investigates the application of such transformations to quantum state modeling, identifying correlations between geometric structure and computational performance. The core mechanism centers around establishing a geometric deformation model, utilizing computational techniques to reveal these relationships. The study demonstrates the efficacy of this approach in enhancing the accuracy of quantum state simulations.