In this paper we consider the massive and massless action for relativistic particle in D-dimensional flat space–time. We show that the Poincare´ space–time algebra in the commutator version, and the Killing field provides the generators of the Poincare´ algebra. We apply the non-commutative version to action, which is not Poincare´ invariant. This leads us to consider the twisted Poincare´ 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 invariance 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, <arXiv:hep-th/0309032>]. 2006 Elsevier Ltd. All rights reserved.
