Relativistic particles and commutator algebras with twisted Poincaré transformation

In this paper we consider the massive and massless action for relativistic particle in D-dimensional flat space–time. We show that the Poincaré space–time algebra in the commutator version, and the Killing field provides the generators of the Poincaré algebra. We apply the non-commutative version to action, which is not Poincaré invariant. This leads us to consider the twisted Poincaré transformation, finally by using this transformation, we see that the action is invariant. By using the non-commutative space in massless action, in contrast to the commutative case the scale and conformal in-variance is broken by massive term [For an interesting discussion of the cosmological constant problem see A. Zee, Dark energy and the nature of the graviton, ].