  The recognition that the eigenvalues of a non-Hermitian Hamiltonian could all
be real if the Hamiltonian had an antilinear symmetry such as $PT$ stimulated
new insight into the underlying structure of quantum mechanics. Specifically,
it lead to the realization that Hilbert space could be richer than the
established Dirac approach of constructing inner products out of ket vectors
and their Hermitian conjugate bra vectors. With antilinear symmetry one must
instead build inner products out of ket vectors and their antilinear
conjugates, and it is these inner products that would be time independent in
the non-Hermitian but antilinearly symmetric case even as the standard Dirac
inner products would not be. Moreover, and in a sense quite remarkably,
antilinear symmetry could address not only the temporal behavior of the inner
product but also the issue of its overall sign, with antilinear symmetry being
capable of yielding a positive inner product in situations where the standard
Dirac inner product is found to have ghostlike negative signature. Antilinear
symmetry thus solves the ghost problem in quantum field theory by showing that
when a theory has ghost states it is being formulated in the wrong Hilbert
space, with antilinear symmetry providing a Hilbert space that is ghost free.
Antilinear symmetry does not actually get rid of the ghost states. Rather, it
shows that the reasoning that led one to think that ghosts were present in the
first place is faulty.
