Canonical representation of the Snyder-de Sitter algebra with correct flat and commutative limits
The Snyder--de Sitter algebra provides a Lorentz-covariant deformation of phase-space geometry characterized by a curvature parameter $α$ and a noncommutativity parameter $β$. We construct an explicit canonical (Darboux) representation of this algebra that is regular in both parameters. Starting from the symplectic structure associated with the Snyder--de Sitter Poisson brackets, we derive the canonical transformation between the physical phase-space variables and Darboux coordinates and obtain its inverse in closed form to all orders in $α$ and $β$. The resulting representation has well-defined flat ($α\to0$) and commutative ($β\to0$) limits, reducing respectively to the Snyder and de Sitter phase-space algebras, while the simultaneous limit $(α,β)\to(0,0)$ yields the standard canonical phase-space coordinates. We then apply this representation to the construction of Poisson gauge transformations on Snyder--de Sitter phase space. In particular, we derive the corresponding gauge transformation matrix and Poisson field strength, both of which are regular in $α$ and $β$. Our construction provides a convenient framework for investigating gauge theories and other physical systems formulated on Snyder--de Sitter phase space.