Manuscript VII-A-ST@1 · VII-A-ST.tex · section · c39a67a3

Direct second-order cancellation

Direct second-order cancellation · lines 2201–2242

Write

d^{μ} = w^{ν} \partial_{ν} w^{μ} - v^{μ} .

From (eq:intertwiner-coeff),

d^{' μ} = d^{μ} + w^{ν} \partial_{ν} ξ^{μ} + \frac{1}{2} ξ^{ν} \partial_{ν} ξ^{μ} - ζ^{μ} .

Substitute this, $w^{'} = w + ξ$ , and the scalar transformations in (eq:A12).

Terms differentiating $u_{1, g}$ cancel between $X_{ξ} u_{1, g}$ and the changed transport term. Terms proportional to $\partial_{μ} ϕ_{0}$ have coefficient

ζ^{μ} + \frac{1}{2} ξ^{ν} \partial_{ν} ξ^{μ} - (w^{ν} + ξ^{ν}) \partial_{ν} ξ^{μ} + w^{ν} \partial_{ν} ξ^{μ} + \frac{1}{2} ξ^{ν} \partial_{ν} ξ^{μ} - ζ^{μ} = 0.

The remaining Hessian coefficient is the symmetric contraction of

\frac{1}{2} ξ^{μ} ξ^{ν} - (w^{μ} + ξ^{μ}) ξ^{ν} + \frac{1}{2} (w^{μ} ξ^{ν} + ξ^{μ} w^{ν} + ξ^{μ} ξ^{ν}),

which vanishes against $\partial_{μ} \partial_{ν} ϕ_{0}$ . Thus $A_{2}^{(g)'} = A_{2}^{(g)}$ directly.

This check requires both the inverse-coordinate displacement $w \cdot \partial w - v$ and the quadratic Taylor term. Omitting either would invalidate the cancellation.

Displayed formulas in this passage

unlabeled · line 2205
$d^{μ} = w^{ν} \partial_{ν} w^{μ} - v^{μ} .$
unlabeled · line 2209
$d^{' μ} = d^{μ} + w^{ν} \partial_{ν} ξ^{μ} + \frac{1}{2} ξ^{ν} \partial_{ν} ξ^{μ} - ζ^{μ} .$
unlabeled · line 2219
$ζ^{μ} + \frac{1}{2} ξ^{ν} \partial_{ν} ξ^{μ} - (w^{ν} + ξ^{ν}) \partial_{ν} ξ^{μ} + w^{ν} \partial_{ν} ξ^{μ} + \frac{1}{2} ξ^{ν} \partial_{ν} ξ^{μ} - ζ^{μ} = 0.$
unlabeled · line 2227
$\frac{1}{2} ξ^{μ} ξ^{ν} - (w^{μ} + ξ^{μ}) ξ^{ν} + \frac{1}{2} (w^{μ} ξ^{ν} + ξ^{μ} w^{ν} + ξ^{μ} ξ^{ν}),$