Locally negative energy due to quantum coherence effects in quantum field theory is discussed. In a previous work, it was argued that a beam carrying negative energy must satisfy an uncertainty-principle-type inequality of the form \ensuremath{\Vert}\ensuremath{\Delta}E\ensuremath{\Vert}\ensuremath{\Delta}t\ensuremath{\le}1, where \ensuremath{\Vert}\ensuremath{\Delta}E\ensuremath{\Vert} is the magnitude of the negative energy that may be transmitted in a time \ensuremath{\Delta}t. This conclusion applied only to two-dimensional spacetime, and was based on an examination of particular classes of quantum states. In the present work, we give more precise formulations of this type of inequality for a free massless scalar field in both two- and four-dimensional flat spacetime. These inequalities are proven to hold for all quantum states. The physical interpretation of these inequalities is also discussed, and it is argued that they are likely to prevent negative energy from producing such large-scale effects as violations of the second law of thermodynamics or of cosmic censorship.