  A Hamiltonian operator $\hat H$ is constructed with the property that if the
eigenfunctions obey a suitable boundary condition, then the associated
eigenvalues correspond to the nontrivial zeros of the Riemann zeta function.
The classical limit of $\hat H$ is $2xp$, which is consistent with the
Berry-Keating conjecture. While $\hat H$ is not Hermitian in the conventional
sense, ${\rm i}{\hat H}$ is ${\cal PT}$ symmetric with a broken ${\cal PT}$
symmetry, thus allowing for the possibility that all eigenvalues of $\hat H$
are real. A heuristic analysis is presented for the construction of the metric
operator to define an inner-product space, on which the Hamiltonian is
Hermitian. If the analysis presented here can be made rigorous to show that
${\hat H}$ is manifestly self-adjoint, then this implies that the Riemann
hypothesis holds true.
