Proposition: Off-diagonal equal-time singularities

On $\mathbf{x}\ne \mathbf{y}$,

At the full diagonal there are, in addition, conormal directions from the spatial Newton singularity. In particular, the kernel cannot be treated as smooth merely because its spatial endpoints are distinct.

Proof. Away from the spatial diagonal the Newton coefficient is smooth and nonzero. Multiplication by it preserves the wavefront set of $\delta (\tau -{\tau }^{\prime })$, which is the conormal set displayed in (eq:instantWF).

At coincidence, the relative-coordinate kernel is the tensor product of a temporal delta and the spatial Newton distribution. The latter is singular at its spatial origin. Its Fourier transform has inverse-quadratic spatial behavior rather than rapid decay; the temporal delta supplies arbitrary temporal frequencies. These produce conormal directions at the full diagonal.