受约束量子粒子通过引力对偶引起的二维曲面几何动力学：$\boldsymbol{\boldsymbol{\textbf{S}^2}}$解析模型计算
Geometrodynamics of a 2D Curved Surface due to a Constrained Quantum Particle via its Gravitational Dual: $\mathbf{\mathcal{S}^2}$ Analytical Model Calculations
AI总结：
本研究扩展da Costa的受约束量子粒子计算，通过构建二维作用量原理并结合引力对偶分析量子态对曲面几何的微扰反作用，在二维球面模型中验证形状演化，还引入外场揭示新曲面物理。
摘要：
我们对da Costa的量子力学受约束粒子计算进行了独特且新颖的扩展。这是通过分析量子受限粒子的本征态和能谱对曲面本身几何结构的微扰反作用实现的，从而解决该模型中的形状优化问题。首先，我们构建了量子受约束粒子的二维作用量原理，通过变分波函数可重现包含da Costa曲面曲率诱导势的薛定谔方程。我们进一步证明，推导的作用量原理与二维 dilation引力理论对偶，通过对嵌入二维逆度量变分其泛函，得到相应的几何动力学爱因斯坦方程。我们微扰求解该爱因斯坦方程：先解da Costa的薛定谔方程得到初始本征系统，将其作为初始输入数据代入含微扰度量的爱因斯坦方程。作为概念验证，我们在二维球面（$\textbf{S}^2$）上进行计算，展示其首次迭代微扰形状演化；还引入外电磁场，推导完整场论场方程以供未来研究。外场通过二维对偶引力理论中曲面诱导的拉回$U(1)$耦合体现，从而揭示该特定范式下有趣且丰富的新曲面物理。
英文摘要：
We provide a unique and novel extension of da Costa's calculation of a quantum mechanically constrained particle. This is achieved by analyzing the perturbative back reaction of the quantum confined particle's eigenstates and spectra upon the geometry of the curved surface itself, thereby addressing the problem of shape optimization in this model. We do this by first formulating a two-dimensional action principle of the quantum constrained particle, which upon variation of the wave function reproduces Schrödinger's equation including da Costa's surface curvature-induced potentials. We further demonstrate that our derived action principle is dual to a two-dimensional dilation gravity theory and we vary its functional with respect to the embedded two-dimensional inverse-metric to obtain the respective geometrodynamical Einstein equation. We solve this resulting Einstein equation perturbatively by first solving the da Costa's Schrödinger equation to obtain an initial eigensystem, which is used as initial-input data for a perturbed metric inserted into the derived Einstein equation. As a proof of concept, we perform this calculation on a two-sphere and show its first iterative perturbed shape evolution. We also turn on external electromagnetic fields and formulate the full field theoretic field equations for future investigation. The external fields manifest themselves via a surface induced, pulled-back $U(1)$ coupling in our two-dimensional dual gravity theory, thereby revealing interesting and rich new surface physics in this specific paradigm.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
