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 space–time of the early universe. The operators' formalism was revisited and we applied its properties to obtain an 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 ϕ4 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, that the formalism demonstrated here will not be constructed by introducing an NC parameter in the system, as usual. It will be generated naturally from an already existing NC space. In this extra dimensional NC space, we presented also the idea of dimensional reduction to recover commutativity.
Chaotic dynamics of a free particle interacting linearly with a harmonic oscillator