Averaged energy conditions for free scalar fields in flat spacetime

This paper initiates a research program to determine whether, and in what situations, quantum field theory enforces averaged energy conditions on the renormalized stress-energy tensors of quantum fields. This program is motivated by the important roles of averaged energy conditions in general-relativistic singularity theorems, and in preventing the existence of classical, traversable wormholes and wormhole-induced closed timelike curves. As a first step in this research program, this paper shows that a quantized, free scalar field in Minkowski space-time has the following properties: The weak-energy condition is satisfied for a wide class of states when averaged along a complete null geodesic, but it can be violated when averaged along a nongeodesic curve. If the curvature coupling constant in the scalar wave equation is restricted to a certain range, which includes conformal coupling, then the strong-energy condition is satisfied for the same wide class of states when averaged along a complete timelike geodesic. It is shown, further, that this enforcement of energy conditions is not universally true in all spacetimes: by closing up Minkowski spacetime in a spatial direction (e.g., by identifying x=0 with x=L), one can produce quantum states of a free scalar field that violate the averaged weak-energy condition.