Manuscript XX-B@1 · XX-B.tex · lemma · 56d708a5

Lemma: Self-adjointness and invariant gauge kernel

The coupled Euler operator and the normalized action Hessian › Full linearization · lines 554–569 · proof included or linked

XX-B@1#lem:H, XX-B@1#eq:HK

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The operator $H$ is formally self-adjoint with respect to $(\cdot, \cdot)_{0}$ , and

H K = 0, K^{†} H = 0.

Proof. Self-adjointness is symmetry of the second variation on compact variations. Covariance of the Euler equations and the on-shell background give $H K = 0$ . Taking the formal adjoint gives the second identity.

Displayed formulas in this passage

XX-B@1#eq:HK · line 558
$H K = 0, K^{†} H = 0.$

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section: Full linearization
section: Full linearization