Quantum states with negative energy density in the Dirac field and quantum inequalities

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 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 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.