The scalar-switch defect: a separate classical identity

This section is not a hypothesis of Theorem thm:mismatch. It records why VI's switch defect is a different question, and proves the identity directly.

Use ${\eta }_{-}$, fix $m>0$, and put

Let ${\varphi }_0$ satisfy $P{\varphi }_0=0$. For a symmetric tensor $a$, define the density variation

If

then direct expansion gives

For a compact smooth vector field $\xi$ supported in ${M_T}^{\circ }$, set

Calculation gives

Consequently,

for the unswitched coordinate problem, by zero-data uniqueness.

Let $g\in {C_c}^{\infty }({M_T}^{\circ };\mathbb{R})$ be a fixed scalar interaction switch. Let $G_{m,0}$ be the smooth zero-data inverse of $P$, and define

The massive Cauchy inverse is the standard smooth wave Cauchy construction; its local uniqueness uses the preceding energy proof with the additional nonnegative term $m^2{z}^2$ BGP.