  Ordinary Hermitian $\lambda \phi^4$ theory is known to exist in $d<4$
dimensions when $\lambda>0$. For negative values of the coupling, it has been
suggested that a physical meaningful definition of the interacting theory can
be given in terms of ${\cal PT}$-symmetric field theory. In this work, we
critically re-examine the relation between analytically continued Hermitian
field theory with quartic interaction, and ${\cal PT}$-symmetric field theory,
including $O(N)$ models. We find that in general ${\cal PT}$-symmetric field
theory does not correspond to the analytic continuation of the Hermitian
theory, except at high temperature where the instanton contribution present in
the analytically continued theory can be neglected.
