Manuscript IV@1 · IV.tex · lemma · 351cb2d7

Lemma: Source matrix elements

Model 3: scalar stress form and exterior record › Normal-ordered source and the reconstructed form · lines 2628–2664 · proof included or linked

IV@1#lem:stress3

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For $Ψ, X \in D_{3}$ ,

t_{μν}^{Ψ, X} (x) = ⟨ Ψ, : T_{μν} (x) : X ⟩

belongs to the complexified $Q$ , and $t^{Ψ, X} = t^{X, Ψ}$ .

Proof. Expand the finite creation-operator products defining the two vectors. Commuting the annihilators leaves finitely many products of scalar wave packets and their derivatives, multiplied by one-particle inner products. For a Schwartz label the wave packet is Schwartz in space, uniformly with all derivatives on finite time intervals: spatial weights are obtained by momentum integrations by parts, and $m > 0$ makes all derivatives of $ω_{p}$ smooth and polynomially bounded.

For polarized solutions define

T_{μν} (v, w) = \partial_{(μ} v \partial_{ν)} w - \frac{1}{2} η_{μν} (\partial^{α} v \partial_{α} w + m^{2} v w) .

Its divergence is

\partial^{μ} T_{μν} (v, w) = \frac{1}{2} ((P v) \partial_{ν} w + (P w) \partial_{ν} v) = 0.

This proves conservation term by term. Hermitian symmetry follows from the normal-ordered expression.

Displayed formulas in this passage

unlabeled · line 2631
$t_{μν}^{Ψ, X} (x) = ⟨ Ψ, : T_{μν} (x) : X ⟩$
unlabeled · line 2651
$T_{μν} (v, w) = \partial_{(μ} v \partial_{ν)} w - \frac{1}{2} η_{μν} (\partial^{α} v \partial_{α} w + m^{2} v w) .$
unlabeled · line 2658
$\partial^{μ} T_{μν} (v, w) = \frac{1}{2} ((P v) \partial_{ν} w + (P w) \partial_{ν} v) = 0.$

Lemma: Source matrix elements

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