Manuscript XII-C@1 · XII-C.tex · lemma · dead1004

Lemma: Derivative cancellation and cubic divergence

C2: the changed fixed-background Noether identity › An explicit ramp defect with no third derivatives · lines 912–955 · proof included or linked

XII-C@1#lem:divergence, XII-C@1#eq:original-cubic-noether, XII-C@1#eq:Zdivergence

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The following identities hold off shell:

N_{0, ν} (J_{3}) + Q_{ν} (A γ) N_{0, ν} (Z_{σ}_{ε}) = 0, = - w Q_{ν} (A γ) + R_{σ_{ε}, ν} .

In particular, differentiating the actual source introduces no surviving third derivative of $σ_{ε}$ .

Proof. The contracted Bianchi identity, written for $- q G^{μν}$ , gives $N_{ν} (2 - q G / κ) = 0$ . Its coefficient of $κ$ is (eq:original-cubic-noether).

Now differentiate (eq:Z). The terms from $w J_{3}$ are

w N_{0, ν} (J_{3}) + 2 η_{ρ ν} (\partial_{μ} σ_{ε}) J_{3}^{μ ρ} .

Differentiation of (eq:actualsource) gives the remaining terms of (eq:Rampdef) and an apparent term

4 η_{ρ ν} C_{2}^{μ ρ α β} \partial_{μ} \partial_{α} \partial_{β} σ_{ε} .

For every symmetric three-covector tensor $S_{μα β}$ ,

C^{μ ρ α β} (q) S_{μα β} = 0 :

the two products of inverse metrics become equal after permuting the three contracted indices. Every coefficient $C_{r}$ has the same property. The apparent third derivative therefore vanishes. Substitute (eq:original-cubic-noether).

Displayed formulas in this passage

XII-C@1#eq:original-cubic-noether · line 915
$N_{0, ν} (J_{3}) + Q_{ν} (A γ) N_{0, ν} (Z_{σ}_{ε}) = 0, = - w Q_{ν} (A γ) + R_{σ_{ε}, ν} .$
XII-C@1#eq:Zdivergence · line 915
$N_{0, ν} (J_{3}) + Q_{ν} (A γ) N_{0, ν} (Z_{σ}_{ε}) = 0, = - w Q_{ν} (A γ) + R_{σ_{ε}, ν} .$
unlabeled · line 934
$w N_{0, ν} (J_{3}) + 2 η_{ρ ν} (\partial_{μ} σ_{ε}) J_{3}^{μ ρ} .$
unlabeled · line 940
$4 η_{ρ ν} C_{2}^{μ ρ α β} \partial_{μ} \partial_{α} \partial_{β} σ_{ε} .$
unlabeled · line 947
$C^{μ ρ α β} (q) S_{μα β} = 0 :$

Nearby in this section

section: An explicit ramp defect with no third derivatives