AI 中文总结
本文通过表示-符号对应关系构建量子平面形式q-哈密顿力学与杰克逊动力学的过渡，建立q→1时各相关量到经典哈密顿动力学的一阶收敛，为二者提供严格桥梁。
AI 中文摘要
量子平面上的形式q-哈密顿力学通过非对易坐标与协变q-导数表述，其可计算实现通常写为包含杰克逊有限差分的常微分系统。本文通过表示-符号对应关系构建这两个层面之间的过渡：量子平面坐标代数及其协变微分演算由光滑交换函数空间上的乘法、伸缩与杰克逊算子实现，正规序将坐标代数与带有显式结合星积的多项式符号空间等同；在此框架内，形式q-导数与对应杰克逊算子精确交织，形式q-哈密顿作用降为符号上的精确星-杰克逊作用，坐标可观测量的星积修正消失，可精确恢复可计算的杰克逊坐标方程，一般可观测量将星积替换为普通乘法会产生可控一阶误差；还证明哈密顿符号的乘法是所表示算子哈密顿的主导交换近似，最后研究所得欧几里得杰克逊向量场，建立算子、符号、向量场与有限时间轨迹形式在q→1时到经典哈密顿动力学的一阶收敛性，这些结果为形式量子平面哈密顿力学与q-变形哈密顿蒙特卡洛所用动力学之间提供了严格桥梁。
英文摘要
We develop a representation--symbol correspondence for $q$-Hamiltonian mechanics on the quantum plane. The coordinate algebra and its covariant differential calculus are realized simultaneously on a smooth commutative function space by multiplication, dilation, and Jackson operators. Normal ordering identifies the coordinate algebra with a polynomial symbol space and transports operator composition to an explicit associative star product. Under this correspondence, the induced covariant $q$-derivatives intertwine exactly with the Jackson operators, and the ordered Hamiltonian action descends to an exact star-Jackson action. For the coordinate observables, the relevant Jackson derivatives are constants, so the star factors reduce to the unit and the Jackson coordinate equations are exact symbol images of the formal quantum-plane equations. For nonlinear observables, pointwise multiplication produces a genuine commutativization error. We quantify this distinction through a first-order expansion of the operator quantization map, study the derivation, divergence, and energy defects of the induced pointwise Jackson dynamics, and prove convergence of the represented Hamiltonian, the symbol actions, the Jackson vector field, and its finite-time trajectories to their classical counterparts as $q\to1$. The resulting framework connects covariant quantum-plane differential calculus, deformation products, and Hamiltonian dynamics while keeping exact algebraic statements separate from commutative and classical approximations.
