Lower Bounds on Negative Energy Densities for the Scalar Field in Flat Spacetime

We obtain a lower bound on the spacetime-weighted average of the energy density for the scalar field in four-dimensional flat spacetime. The bound takes the form of a quantum inequality. The inequality does not rely on the quantum state and its form is only related to the weights, namely the spacetime sampling functions which are assumed to be smooth, positive and compactly supported. It is found that the inequality is just equal to the temporal quantum energy inequality. When the characteristic length of the temporal sampling function tends to zero, the lower bound becomes divergent. This is consistent with the fact that the spatial restriction on negative energy density does not exist in four-dimensional spacetime.