  The scalar field theory with potential $V(\varphi)=\textstyle{\frac{1}{2}}
m^2\varphi^2-\textstyle{\frac{1}{4}} g\varphi^4$ ($g>0$) is ill defined as a
Hermitian theory but in a non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric
framework it is well defined, and it has a positive real energy spectrum for
the case of spacetime dimension $D=1$. While the methods used in the literature
do not easily generalize to quantum field theory, in this paper the
path-integral representation of a $\mathcal{P}\mathcal{T}$-symmetric
$-g\varphi^4$ theory is shown to provide a unified formulation for general $D$.
A new conjectural relation between the Euclidean partition functions
$Z^{\mathcal{P}\mathcal{T}}(g)$ of the non-Hermitian
$\mathcal{P}\mathcal{T}$-symmetric theory and $Z_{\rm Herm}(\lambda)$ of the
$\lambda \varphi^4$ ($\lambda>0$) Hermitian theory is proposed: $\log
Z^{\mathcal{P}\mathcal{T}}(g)=\textstyle{\frac{1}{2}} \log Z_{\rm Herm}(-g+{\rm
i} 0^+)+\textstyle{\frac{1}{2}}\log Z_{\rm Herm}(-g-{\rm i} 0^+)$. This
relation ensures a real energy spectrum for the non-Hermitian
$\mathcal{P}\mathcal{T}$-symmetric $-g\varphi^4$ field theory. A closely
related relation is rigorously valid in $D=0$. For $D=1$, using a semiclassical
evaluation of $Z^{\mathcal{P}\mathcal{T}}(g)$, this relation is verified by
comparing the imaginary parts of the ground-state energy
$E_0^{\mathcal{P}\mathcal{T}}(g)$ (before cancellation) and $E_{0,\rm
Herm}(-g\pm {\rm i} 0^+)$.
