Manuscript Em01-F@3 · Em01-F-v3.tex · lemma · 225e59af

Lemma: Complete tensor-vertex simplification

Exact localized crossed-line and tensor regrouping › Trace-reversal simplification · lines 2722–2762 · proof included or linked

Em01-F@3#lem:tensorsimplify, Em01-F@3#eq:Ntrace, Em01-F@3#eq:verticesimplify

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The endpoint factor satisfies

N^{μν}_{I} = (e^{I})^{μν} - \frac{1}{2} η^{μν} ϑ_{I}, η_{μν} N^{μν}_{I} = - ϑ_{I} .

For every smooth scalar $d$ ,

Z_{I}^{ν} (N d) = f^{*} ((e^{I})^{μν} \partial_{2, μ} d), M_{I} (N d) = - ϑ_{I} f^{*} d .

These identities retain the complete kinetic, trace-kinetic, and mass vertex.

Proof. Contract the tensor numerator in (eq:Nfactor) with the endpoint maps. This gives the first identity in (eq:Ntrace); taking its trace in dimension four gives the second.

Substitute into $Z$ :

\partial_{2, μ} N^{μν}_{I} d - \frac{1}{2} η_{μα} \partial_{2}^{ν} N^{μα}_{I} d = (e^{I})^{μν} \partial_{2, μ} d .

The two trace contributions cancel exactly. The mass identity is the second trace formula.

Displayed formulas in this passage

Em01-F@3#eq:Ntrace · line 2725
$N^{μν}_{I} = (e^{I})^{μν} - \frac{1}{2} η^{μν} ϑ_{I}, η_{μν} N^{μν}_{I} = - ϑ_{I} .$
Em01-F@3#eq:verticesimplify · line 2734
$Z_{I}^{ν} (N d) = f^{*} ((e^{I})^{μν} \partial_{2, μ} d), M_{I} (N d) = - ϑ_{I} f^{*} d .$
unlabeled · line 2753
$\partial_{2, μ} N^{μν}_{I} d - \frac{1}{2} η_{μα} \partial_{2}^{ν} N^{μα}_{I} d = (e^{I})^{μν} \partial_{2, μ} d .$

Nearby in this section

section: Trace-reversal simplification