Dynamics in a noncommutative phase space

TL;DR

This paper develops a framework for dynamics in a noncommutative phase space invariant under quantum group symmetries, formulating a $q$-deformed calculus that preserves conventional symmetries like rotation and Lorentz invariance.

Contribution

It introduces a $q$-deformed differential calculus on noncommutative phase space, enabling Hamiltonian and Lagrangian dynamics that maintain traditional symmetries.

Findings

Formulated $q$-deformed differential calculus.

Constructed Hamiltonian and Lagrangian dynamics.

Preserves rotational and Lorentz invariance.

Abstract

Dynamics has been generalized to a noncommutative phase space. The noncommuting phase space is taken to be invariant under the quantum group $GL_{q,p}(2)$. The $q$-deformed differential calculus on the phase space is formulated and using this, both the Hamiltonian and Lagrangian forms of dynamics have been constructed. In contrast to earlier forms of $q$-dynamics, our formalism has the advantage of preserving the conventional symmetries such as rotational or Lorentz invariance.