Manuscript En01-C@1 · En01-C.tex · lemma · 00a2161c

Lemma: Exact second-moment action

Model boundary and common physical controls › The declared gravitational contribution · lines 755–773 · proof included or linked

En01-C@1#lem:momentsG

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The full regularized generator and its quadratic comparison have the same action on $x^{2}, (x p + p x) /2, p^{2}$ :

G_{r_{c}}^{*} (x^{2}) = 0, G_{r_{c}}^{*} ((x p + p x) /2) = 0, G_{r_{c}}^{*} (p^{2}) = 2 ℏ^{2} A_{r_{c}} 1.

Proof. Conjugation of $p$ by a position kick shifts it by $ℏ k_{e}$ , up to a sign convention irrelevant after integration. Evenness of $Q_{r_{c}}$ cancels the linear term. The quadratic term integrates to $2 ℏ^{2} A_{r_{c}}$ . The position is unchanged. The same commutator calculation proves the quadratic-model identities.

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unlabeled · line 759
$G_{r_{c}}^{*} (x^{2}) = 0, G_{r_{c}}^{*} ((x p + p x) /2) = 0, G_{r_{c}}^{*} (p^{2}) = 2 ℏ^{2} A_{r_{c}} 1.$

Lemma: Exact second-moment action

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