Manuscript VII-C@1 · VII-C.tex · proposition · 8b2ade29

Proposition: Tensor cancellation

Cauchy covariances and the full tensor audit › The vanishing ww contraction · lines 642–683 · proof included or linked

VII-C@1#prop:tensor, VII-C@1#eq:CH, VII-C@1#eq:CHC

For the matrix (eq:Hhh),

C_{x}^{μ} H_{hh ρ σ} (x, y) C_{x}^{μ} C_{y}^{ν} H_{hh} (x, y) = δ^{μ}_{(ρ} \partial^{x}_{σ)} Δ_{0}^{+} (x - y), = - \frac{1}{2} η^{μν} □_{x} Δ_{0}^{+} (x - y) = 0.

The induced free Cauchy contractions satisfy

H_{w w} = 0.

Proof. In four dimensions,

η^{α β} P_{α β ρ σ} = - η_{ρ σ} .

The trace contribution in $C_{x}^{μ}$ cancels the trace term in the first divergence of $P$ , leaving (eq:CH). Applying $C_{y}^{ν}$ , using $\partial^{y} = - \partial^{x}$ on $Δ_{0}^{+} (x - y)$ , cancels the two mixed second-derivative terms and gives (eq:CHC).

To justify the Cauchy operations without multiplying singular unsmeared kernels, use (eq:positive). On each massless momentum mode, the twice-contracted tensor numerator in (eq:CHC) vanishes before either zero-Cauchy solution operator is applied. The remaining mode factors have polynomial growth and locally integrable infrared behavior. Thus the smeared contraction is zero. Equivalently, the Cauchy covariance construction based on (eq:Cauchymatrix) gives the zero distribution.

Displayed formulas in this passage

VII-C@1#eq:CH · line 645
$C_{x}^{μ} H_{hh ρ σ} (x, y) C_{x}^{μ} C_{y}^{ν} H_{hh} (x, y) = δ^{μ}_{(ρ} \partial^{x}_{σ)} Δ_{0}^{+} (x - y), = - \frac{1}{2} η^{μν} □_{x} Δ_{0}^{+} (x - y) = 0.$
VII-C@1#eq:CHC · line 645
$C_{x}^{μ} H_{hh ρ σ} (x, y) C_{x}^{μ} C_{y}^{ν} H_{hh} (x, y) = δ^{μ}_{(ρ} \partial^{x}_{σ)} Δ_{0}^{+} (x - y), = - \frac{1}{2} η^{μν} □_{x} Δ_{0}^{+} (x - y) = 0.$
unlabeled · line 657
$H_{w w} = 0.$
unlabeled · line 664
$η^{α β} P_{α β ρ σ} = - η_{ρ σ} .$

Nearby in this section

remark: Remark: Three limitations