From Quantum-Plane Hamiltonians to Jackson Dynamics: Dilation Representations, Normal Symbols, and Euclidean Limits

Formal $q$-Hamiltonian mechanics on the quantum plane is expressed through noncommuting coordinates and covariant $q$-derivatives, whereas its computable realization is usually written as an ordinary differential system involving Jackson finite differences.

In this paper, we formulate the passage between these two levels through a representation--symbol correspondence.

The quantum-plane coordinate algebra and its covariant differential calculus are realized by multiplication, dilation, and Jackson operators on a smooth commutative function space.

Normal ordering then identifies the coordinate algebra with a polynomial symbol space endowed with an explicit associative star product.

Within this framework, the formal $q$-derivatives intertwine exactly with the corresponding Jackson operators, and the formal $q$-Hamiltonian action descends to an exact star-Jackson action on symbols.

For the coordinate observables, the star-product corrections vanish, so the computable Jackson coordinate equations are recovered exactly.

For general observables, replacing the star product by ordinary multiplication produces a controlled first-order error.

We also show that multiplication by the Hamiltonian symbol is the leading commutative approximation of the represented operator Hamiltonian.

Finally, we study the resulting Euclidean Jackson vector field and establish first-order convergence of the operator, symbol, vector-field, and finite-time trajectory formulations to classical Hamiltonian dynamics as $q\to1$.

These results provide a rigorous bridge between formal quantum-plane Hamiltonian mechanics and the dynamics used in $q$-deformed Hamiltonian Monte Carlo.