  The action for gravity and the standard model includes, as well as the
positive energy fermion and boson fields, negative energy fields. The
Hamiltonian for the action leads through a positive and negative energy
symmetry of the vacuum to a cancellation of the zero-point vacuum energy and a
vanishing cosmological constant in the presence of a gravitational field
solving the cosmological constant problem. To guarantee the quasi-stability of
the vacuum, we postulate a positive energy sector and a negative energy sector
in the universe which are identical copies of the standard model. They interact
only weakly through gravity. As in the case of antimatter, the negative energy
matter is not found naturally on Earth or in the universe. A positive energy
spectrum and a consistent unitary field theory for a pseudo-Hermitian
Hamiltonian is obtained by demanding that the pseudo-Hamiltonian is ${\cal
P}{\cal T}$ symmetric. The quadratic divergences in the two-point vacuum
fluctuations and the self-energy of a scalar field are removed. The finite
scalar field self-energy can avoid the Higgs hierarchy problem in the standard
model.
