  In this work, we investigate a very important but unstressed result in the
work of Carl M. Bender, Jun-Hua Chen, and Kimball A. Milton (
J.Phys.A39:1657-1668, 2006). In this article, Bender \textit{et.al} have
calculated the vacuum energy of the $i\phi^{3}$ scalar field theory and its
Hermitian equivalent theory up to $g^{4}$ order of calculations. While all the
Feynman diagrams of the $i\phi^{3}$ theory are finite in $0+1$ space-time
dimensions, some of the corresponding Feynman diagrams in the equivalent
Hermitian theory are divergent. In this work, we show that the divergences in
the Hermitian theory originate from superrenormalizable, renormalizable and
non-renormalizable terms in the interaction Hamiltonian even though the
calculations are carried out in the $0+1$ space-time dimensions. Relying on
this interesting result, we raise the question, is the superficial degree of
divergence of a theory is representation dependent? To answer this question, we
introduce and study a class of non-Hermitian quantum field theories
characterized by a field derivative interaction Hamiltonian. We showed that the
class is physically acceptable by finding the corresponding class of metric
operators in a closed form. We realized that the obtained equivalent Hermitian
and the introduced non-Hermitian representations have coupling constants of
different mass dimensions which may be considered as a clue for the possibility
of considering non-Renormalizability of a field theory as a non-genuine
problem. Besides, the metric operator is supposed to disappear from path
integral calculations which means that physical amplitudes can be fully
obtained in the simpler non-Hermitian representation.
