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In this work, we study the vacuum stability of the classical unstable $\left(-\phi^{4}\right) $ scalar field potential. Regarding this, we obtained theeffective 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 effectivepotential is bounded from below, which proves the vacuum stability of thetheory in space-time dimensions higher than the previously studied $0+1$ case.In our calculations, we used the canonical quantization regime in which onedeals with operators rather than classical functions used in the path integralformulation. Therefore, the non-Hermiticity of the effective field theory isobvious. Moreover, the method we employ implements the canonical equal-timecommutation relations and the Heisenberg picture for the operators. Thus, themetric operator is implemented in the calculations of the transitionamplitudes. Accordingly, the method avoids the very complicated calculationsneeded in other methods for the metric operator. To test the accuracy of ourresults, we obtained the exponential behavior of the vacuum condensate forsmall coupling values, which has been obtained in the literature using othermethods. We assert that this work is interesting, as all the studies in theliterature advocate the stability of the $\left( -\phi^{4}\right) $ theory atthe quantum mechanical level while our work extends the argument to the levelof field quantization.

Vacuum Stability of the $\mathcal{PT}$-Symmetric $\left( -\phi^{4}\right) $ Scalar Field Theory