Manuscript IX-A@1 · IX-A.tex · lemma · 55aa6f6d

Lemma: Fixed tensor identities

Selected old coefficients and the admission test › Covariance identities used in the tests · lines 1408–1457 · proof included or linked

IX-A@1#lem:tensor, IX-A@1#eq:K

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For the covariance (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} □_{η} Δ_{0}^{+} (x - y) = 0.

The induced $w w$ covariance is zero. The mixed component selected by $h_{00}$ is

K (x, y) := H_{w h}^{0}_{00} (x, y) = \int \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot (x - y)} [(- \frac{t}{4 r} + \frac{i}{8 r ^{2}}) e^{- i r (t - s)} - \frac{i}{8 r ^{2}} e^{i r (t + s)}],

where $r = ∣ p ∣$ .

Proof. The trace of $P$ in four dimensions is $- η_{ρ σ}$ . Substitution into $C$ cancels the trace term and gives the first identity. Applying $C_{y}$ cancels the mixed second derivatives and gives the second.

For the $w w$ statement, the contracted massless momentum numerator vanishes before either zero-Cauchy operator is applied. The smeared covariance is therefore zero.

For $K$ , solve the zero-data equation on a mode:

E_{0} (e^{- i r t}) = - \frac{i t}{2 r} e^{- i r t} + \frac{i sin ( r t )}{2 r ^{2}} .

Multiply by the derivative $- i r$ and the spectral density $1/ (2 r)$ , and separate the two sine frequencies. This gives (eq:K). The infrared factors $r^{- 1}$ and $r^{- 2}$ are locally integrable in three dimensions.

Displayed formulas in this passage

unlabeled · line 1411
$C_{x}^{μ} H_{hh ρ σ} (x, y) C_{x}^{μ} C_{y}^{ν} H_{hh} (x, y) = δ^{μ}_{(ρ} \partial^{x}_{σ)} Δ_{0}^{+} (x - y), = - \frac{1}{2} η^{μν} □_{η} Δ_{0}^{+} (x - y) = 0.$
IX-A@1#eq:K · line 1422
$K (x, y) := H_{w h}^{0}_{00} (x, y) = \int \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot (x - y)} [(- \frac{t}{4 r} + \frac{i}{8 r ^{2}}) e^{- i r (t - s)} - \frac{i}{8 r ^{2}} e^{i r (t + s)}],$
unlabeled · line 1447
$E_{0} (e^{- i r t}) = - \frac{i t}{2 r} e^{- i r t} + \frac{i sin ( r t )}{2 r ^{2}} .$

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section: Covariance identities used in the tests