  In this work, we study the vacuum stability of the classical unstable $\left(
-\phi^{4}\right) $ scalar field potential. Regarding this, we obtained the
effective potential, up to second order in the coupling, for the theory in
$1+1$ and $2+1$ space-time dimensions. We found that the obtained effective
potential is bounded from below, which proves the vacuum stability of the
theory in space-time dimensions higher than the previously studied $0+1$ case.
In our calculations, we used the canonical quantization regime in which one
deals with operators rather than classical functions used in the path integral
formulation. Therefore, the non-Hermiticity of the effective field theory is
obvious. Moreover, the method we employ implements the canonical equal-time
commutation relations and the Heisenberg picture for the operators. Thus, the
metric operator is implemented in the calculations of the transition
amplitudes. Accordingly, the method avoids the very complicated calculations
needed in other methods for the metric operator. To test the accuracy of our
results, we obtained the exponential behavior of the vacuum condensate for
small coupling values, which has been obtained in the literature using other
methods. We assert that this work is interesting, as all the studies in the
literature advocate the stability of the $\left( -\phi^{4}\right) $ theory at
the quantum mechanical level while our work extends the argument to the level
of field quantization.
