  It can be shown using operator techniques that the non-Hermitian
$PT$-symmetric quantum mechanical Hamiltonian with a "wrong-sign" quartic
potential $-gx^4$ is equivalent to a Hermitian Hamiltonian with a positive
quartic potential together with a linear term. A naive derivation of the same
result in the path-integral approach misses this linear term. In a recent paper
by Bender et al. it was pointed out that this term was in the nature of a
parity anomaly and a more careful, discretized treatment of the path integral
appeared to reproduce it successfully. However, on re-examination of this
derivation we find that a yet more careful treatment is necessary, keeping
terms that were ignored in that paper. An alternative, much simpler derivation
is given using the additional potential that has been shown to appear whenever
a change of variables to curvilinear coordinates is made in a functional
integral.
