Representation--Symbol Correspondence for $q$-Hamiltonian Mechanics on the Quantum Plane

Abstract

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.