Simplifying $D$-Dimensional Physical-State Sums in Gauge Theory and Gravity

We provide two independent systematic methods of performing $D$-dimensional physical-state sums in gauge theory and gravity in such a way so that spurious light-cone singularities are not introduced. A natural application is to generalized unitarity in the context of dimensional regularization or theories in higher spacetime dimensions. Other applications include squaring matrix elements to obtain cross sections, and decompositions in terms of gauge-invariant tensors.