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The framework Doplicher-Fredenhagen-Roberts (DFR) of a noncommutative (NC)space-time is considered as a alternative approach to study the NC space-timeof the early Universe. In this formalism, the parameter of noncommutative$\theta^{\mu\nu}$ is promoted to a coordinate of the space-time, andconsequently, we are describing a field theory in a space-time withextra-dimension. Consequently, there is a canonical momentum associated to thisnew coordinate in which the effects of a new physics can emerge in thepropagation of the fields along the extra-dimension. The Fourier space of thisframework is automatically extended by the addition of new momenta components.The main concept that we would like to emphasize from the outset is that theformalism demonstrated here will not be constructed introducing a NC parameterin the system, as usual. It will be generated naturally from an already NCspace. When the components of the new momentum are zero, the DFR approach isreduced to the usual NC case, in which $\theta^{\mu\nu}$ is a antisymmetricconstant matrix. We study a scalar field action with self-quartic interaction$\phi^{4}\star$ defined in the DFR NC spacetime, obtaining the Feynman rules inthe Fourier space for the scalar propagator and vertex of the model. With theserules we are able to build out the radiative corrections to one loop order forthe model propagator. The influence of the NC scale, as well as the propagationof the field in the extra-dimension, are analyzed in the ultravioletdivergences scenario. We investigate the actual possibility if this$\theta^{\mu\nu}$ conjugate momentum has the property of healing the mixingIR/UV divergences that emerges in this recently new NC spacetime quantum fieldtheory.

Self-quartic interaction for a scalar field in an extended DFR noncommutative spacetime