  A new proof is given for why the non-Hermitian, PT-Invariant cubic oscillator
with imaginary coupling has real eigenvalues. The proof consists of two steps.
In the first step, it is shown that for many PT-Invariant Hamiltonians, one can
define corresponding Hermitian Hamiltonians that have the same eigenvalues when
quantized using indefinite metric. The second step is to show that the
indefinite-metric eigenvalues of the Hermitian cubic oscillator are real since
the norms of its eigenstates do not vanish. The correspondence between
PT-Invariant Hamiltonians and indefinite-metric Hermitian Hamiltonians is
further discussed as a way to determine vacuum stability for certain
supersymmetric gauge theories.
