Manuscript X-L-ST@1 · X-L-ST.tex · proof · 20a32a22

Proof: Verification of the displayed identities

The normalized free auxiliary–ghost package › Fields, pairings, and operators · lines 372–388

Proof. Trace reversal is involutive in dimension four. Substitution into $C K v$ cancels the two divergence-gradient terms and leaves $□ v$ . Compact integration by parts gives the transpose formula.

Consequently,

P_{gr} K = - \frac{1}{4} P_{tr} K □ = \frac{1}{2} C^{*} □,

and hence

A K = \frac{1}{2} C^{*} □ - \frac{1}{2} C^{*} C K = 0.

Displayed formulas in this passage

unlabeled · line 379
$P_{gr} K = - \frac{1}{4} P_{tr} K □ = \frac{1}{2} C^{*} □,$
unlabeled · line 384
$A K = \frac{1}{2} C^{*} □ - \frac{1}{2} C^{*} C K = 0.$

Nearby in this section

section: Fields, pairings, and operators
section: Fields, pairings, and operators