  Energy densities of the quantum states that are superposition of two
multi-electron-positron states are examined. It is shown that the energy
densities can be negative only when two multi-particle states have the same
number of electrons and positrons or when one state has one more
electron-positron pair than the other. In the cases in which negative energy
could arise, we find that the energy is that of a positive constant plus a
propagating part which oscillates between positive and negative, and the energy
can dip to negative at some places at for a certain period of time if the
quantum states are properly manipulated. It is demonstrated that the negative
energy densities satisfy the quantum inequality. Our results also reveal that
for a given particle content, the detection of negative energy is an operation
that depends on the frame where any measurement is to be performed. This
suggests that the sign of energy density for a quantum state may be a
coordinate-dependent quantity in quantum theory.
