Manuscript X-I-ST@1 · X-I-ST.tex · proposition · 9745fd40

Proposition: Next-order density diagnostic

The finite lift and its explicit next residual › Why a covariant density does not erase all switching terms · lines 1346–1368 · proof included or linked

X-I-ST@1#prop:density, X-I-ST@1#eq:cutoffresidual

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Modulo compact divergences,

B_{a} + s_{0} V_{1} = - \frac{1}{4 !} \int (\partial_{μ} a) c^{μ} ϕ^{4} d^{4} x .

Proof. Since $s_{0} (tr r /2) = \partial_{μ} c^{μ}$ ,

s_{0} V_{1} = \frac{1}{4 !} \int a (\partial_{μ} c^{μ}) ϕ^{4} d^{4} x .

Also $B_{a} = \int a c^{μ} \partial_{μ} (ϕ^{4} /4!)$ . One integration by parts proves (eq:cutoffresidual).

Displayed formulas in this passage

X-I-ST@1#eq:cutoffresidual · line 1349
$B_{a} + s_{0} V_{1} = - \frac{1}{4 !} \int (\partial_{μ} a) c^{μ} ϕ^{4} d^{4} x .$
unlabeled · line 1360
$s_{0} V_{1} = \frac{1}{4 !} \int a (\partial_{μ} c^{μ}) ϕ^{4} d^{4} x .$

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section: Why a covariant density does not erase all switching terms
section: Why a covariant density does not erase all switching terms