  In an earlier paper it was argued that the conventional double-scaling limit
of an O(N)-symmetric quartic quantum field theory is inconsistent because the
critical coupling constant is negative and thus the integral representing the
partition function of the critical theory does not exist. In this earlier paper
it was shown that for an O(N)-symmetric quantum field theory in
zero-dimensional spacetime one can avoid this difficulty if one replaces the
original quartic theory by its PT-symmetric analog. In the current paper an
O(N)-symmetric quartic quantum field theory in one-dimensional spacetime [that
is, O(N)-symmetric quantum mechanics] is studied using the Schroedinger
equation. It is shown that the global PT-symmetric formulation of this
differential equation provides a consistent way to perform the double-scaling
limit of the O(N)-symmetric anharmonic oscillator. The physical nature of the
critical behavior is explained by studying the PT-symmetric quantum theory and
the corresponding and equivalent Hermitian quantum theory.
