Negative Energy Densities And Quantum Inequalities In The Dirac Field

In classical physics, the energy density is non-negative. However, in quantum field theory, there exist states for which the local energy density can be negative. Such negative energy densities bring about a series of serious problems, including the failure of the classical energy conditions, the violation of the second law of thermodynamics or of the cosmic censorship, and superluminal travel.The laws of quantum field theory place serious constraints, such as Ford-Roman quantum inequalities, on negative energies at the same time they introduce them. The quantum inequality limits the magnitude and duration of negative energy. It indicates that there can not exist negative energy with arbitrary magnitude for an arbitrarily long time. Quantum inequalities have been derived for scalar and electromagnetic fields in flat as well as curved spacetimes. As for the Dirac field the result is not perfect.In this paper we will explore the negative energy in the Dirac field. Firstly, we will examine the negative energy densities for states that are the superposition of two multi-electron-positron states. 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. It is demonstrated that the negative energy densities satisfy two quantum inequalities. Our results also reveal that the sign of energy density for a quantum state may be a coordinate-dependent quantity in quantum theory. Secondly, we investigate the energy density produced by a state vector which is the superposition of three single electron states in the Dirac field in the four-dimensional Minkowski spacetime. We derive the conditions on which the energy density can be negative. Then we show that the energy density satisfies two quantum inequalities in the ultrarelativistic limit. Lastly, a proof is given to the quantum inequality for negative energy densities in the massive Dirac field produced by the superposition of two single particle electron states.Our results indicate that negative energy densities can not persist with an arbitrarily large magnitude for an arbitrarily long time and the amount of negative energy is less than the quantum energy uncertainty in quantum theory. That is to say, the negative energy is submerged by the normal fluctuation of energy in quantum field, so that thebizarre effects, such as the violation of the second law of thermodynamics or of the cosmic censorship, and superluminal travel, will not occur.