Klein-Gordon Theory in Noncommutative Phase Space

We extend the three-dimensional noncommutative relations of the position and momentum operators to those in the four dimension. Using the Seiberg-Witten (SW) map, we give the Heisenberg representation of these noncommutative algebras and endow the noncommutative parameters associated with the Planck constant, Planck length and cosmological constant. As an analog with the electromagnetic gauge potential, the noncommutative effect can be interpreted as an effective gauge field, which depends on the Plank constant and cosmological constant. Based on these noncommutative relations, we give the Klein-Gordon (KG) equation and its corresponding current continuity equation in the noncommutative phase space including the canonical and Hamiltonian forms and their novel properties beyond the conventional KG equation. We analyze the symmetries of the KG equations and some observables such as velocity and force of free particles in the noncommutative phase space. We give the perturbation solution of the KG equation