  We introduced a new formulation for the path integral formalism for a
noncommutative (NC) quantum mechanics defined in the recently developed
Doplicher-Fredenhagen-Roberts-Amorim (DFRA) NC framework that can be considered
an alternative framework for the NC spacetime of the early Universe. The
operators formalism was revisited and we apply its properties to obtain a NC
transition amplitude representation. Two DFRA's systems were discussed, the NC
free particle and NC harmonic oscillator. Some temperature concepts in this NC
space are also considered. The extension to NC DFRA quantum field theory is
straightforward and we apply it to a massive scalar field. We construct the
generating functional and the effective action to give rise
one-particle-irreducible diagrams. As an example, we set the basis for a
$n\;(n\geq 3)$ self-interaction $\phi^{n}$ to obtain the correction of the
perturbation theory to the propagator and vertex of this model. The main
concept that we would like 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.
