Manuscript III-B@1 · III-B-v1.tex · lemma · d2c98f62

Lemma: Differential identities

Operators, signs, and smooth evolution · lines 514–560 · proof included or linked

III-B@1#lem:identities, III-B@1#eq:commutator

For smooth fields,

D K ξ = □ ξ, G^{(1)} (K ξ) = 0, \partial^{μ} G^{(1)} (h)_{μν} = 0,

and

[P, X_{ξ}] u = (□ ξ^{ν}) \partial_{ν} u + 2 (\partial^{μ} ξ^{ν}) \partial_{μ} \partial_{ν} u, L_{K ξ} = - [P, X_{ξ}] .

Also,

□_{η + ε h} - m^{2} = P + ε L_{h} (mod ε^{2}) .

Proof. In $D K ξ$ , the derivatives of $\partial_{μ} ξ^{μ}$ cancel. Substitution into (eq:Glin) gives $G^{(1)} K = 0$ . Taking a divergence in (eq:Glin) cancels the wave and divergence terms pairwise.

Apply the product rule twice to $□ (ξ^{ν} \partial_{ν} u)$ ; the mass term commutes with $X_{ξ}$ . Substitution of $K ξ$ into (eq:Lh) gives the negative commutator.

Finally use

(η + ε h)^{μν} = η^{μν} - ε h^{μν}, ∣ det (η + ε h) ∣ = 1 + \frac{ε}{2} tr_{η} h (mod ε^{2})

in the divergence formula for the scalar wave operator.

Displayed formulas in this passage

III-B@1#eq:commutator · line 523
$[P, X_{ξ}] u = (□ ξ^{ν}) \partial_{ν} u + 2 (\partial^{μ} ξ^{ν}) \partial_{μ} \partial_{ν} u, L_{K ξ} = - [P, X_{ξ}] .$
unlabeled · line 517
$D K ξ = □ ξ, G^{(1)} (K ξ) = 0, \partial^{μ} G^{(1)} (h)_{μν} = 0,$
unlabeled · line 532
$□_{η + ε h} - m^{2} = P + ε L_{h} (mod ε^{2}) .$
unlabeled · line 551
$(η + ε h)^{μν} = η^{μν} - ε h^{μν}, ∣ det (η + ε h) ∣ = 1 + \frac{ε}{2} tr_{η} h (mod ε^{2})$

Lemma: Differential identities

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