In this paper we investigated the causality problem present in the recent published work about the Doplicher–Fredenhagen–Roberts–Amorim (DFRA) noncommutative framework which analyzed the complex scalar field. We described the noncommutative Klein–Gordon equation with a source term. Its solution was accomplished through the retarded, advanced and causal Green functions constructed in this noncommutative ten-dimensional DFRA space–time. We believe that this solution constitutes the first step in the elaboration of a quantum field theory using the DFRA formalism where the well known noncommutative parameter from now on is an ordinary coordinate of the system and therefore has a canonical conjugate momentum. We discussed the dimensional reduction of this noncommutative space–time (D = 10) in order to recover commutativity in D = 4 space–time, which is not the usual one in noncommutativity literature.

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The standard electroweak interaction is here re-assessed to accommodate two different situations in Particle Physics. The first one is a Z ′ -model at the TeV-scale physics. The second one tackles the recent discussion of a possible fifth force mediated by a 17-MeV X-boson associated with an electron-positron emission in the transition of an excited 8-Beryllium to its ground state. The anomaly-free model that provides these two scenarios is based on an SUL(2) × UR(1)J × U (1)Ksymmetry. It yields a new massive neutral boson, an exotic massive neutral fermion, right-neutrinos and an additional neutral Higgs particle, which stems from a supplementary Higgs field, introduced along with the usual Higgs doublet responsible for the electroweak breaking and the masses of W ± and Z 0 . Yukawa interactions of the two scalars generate the masses of the Standard Model leptons, neutrinos and a new exotic fermion of the model. The vacuum expectation values of the Higgses fix up two independent energy scales. One of them is the well-confirmed electroweak scale, 246 GeV, whereas the other one is set up by adopting an experimental estimate for the Z ′ -mass.

We obtain the κ-deformed versions of the retarded and advanced Green functions and show that their causality properties are blurred in a time interval of the order of a length parameter q=1/(2κ). The functions also indicate a smearing of the light cone. These results favor the interpretation of q as a fundamental length scale below which the concept of a point in space-time should be substituted by the concept of a fuzzy region of radius q, as proposed long ago by Heisenberg.

We obtained the Feynman propagators for a noncommutative (NC) quantum mechanics defined in the recently developed Doplicher-Fredenhagen-Roberts-Amorim (DFRA) NC background that can be considered as an alternative framework for the NC spacetime of the early Universe. The operators formalism was revisited and we applied its properties to obtain a NC transition amplitude representation. Two examples of DFRA's systems were discussed, namely, the NC free particle and NC harmonic oscillator. The spectral representation of the propagator gave us the NC wave function and energy spectrum. We calculated the partition function of the NC harmonic oscillator and the distribution function. Besides, the extension to NC DFRA quantum field theory is straightforward and we used it in a massive scalar field. We had written the scalar action with self-interaction φ 4 using the Weyl-Moyal product to obtain the propagator and vertex of this model needed to perturbation theory. It is important to emphasize from the outset is that the formalism demonstrated here will not be constructed introducing a NC parameter in the system, as usual. It will be generated naturally from an already NC space. In this extra dimensional NC space, we presented also the idea of dimensional reduction to recover commutativity.