Manuscript XX-F@1 · XX-F.tex · theorem · de0bd24f

Theorem: Complete first-order Leibniz defect

Coupled gauge jets and an explicit quantum-algebra test › A compact regular comparison algebra · lines 594–631 · proof included or linked

XX-F@1#thm:Leibniz, XX-F@1#eq:Leibniz

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On $P_{reg}$ ,

V_{ξ} (F ⋆_{W} G) - (V_{ξ} F) ⋆_{W} G - F ⋆_{W} (V_{ξ} G) = ℏ T_{ξ} (F, G) (mod ℏ^{2}),

where

T_{ξ} (F, G) = - 2 ⟨ F_{ζ}^{(1)}, A_{ξ} W G_{z}^{(1)} ⟩ - 2 ⟨ F_{z}^{(1)}, W A_{ξ}^{†} G_{ζ}^{(1)} ⟩ .

In particular,

T_{ξ} (M_{a}, Z_{g}) = - 2 W (A_{ξ}^{†} a, g), T_{ξ} (Z_{g}, M_{a}) = - 2 W (g, A_{ξ}^{†} a) .

Proof. The operation $V_{ξ}$ is a derivation of ordinary polynomial multiplication. Differentiating its action with respect to $z$ gives

(V_{ξ} F)_{z}^{(1)} = V_{ξ} (F_{z}^{(1)}) + 2 A_{ξ}^{†} F_{ζ}^{(1)} .

In (eq:star), $W$ is fixed. The terms in which $V_{ξ}$ differentiates $F_{z}^{(1)}$ or $G_{z}^{(1)}$ cancel between the three expressions. The two extra derivative terms are exactly (eq:Leibniz).

Displayed formulas in this passage

XX-F@1#eq:Leibniz · line 604
$T_{ξ} (F, G) = - 2 ⟨ F_{ζ}^{(1)}, A_{ξ} W G_{z}^{(1)} ⟩ - 2 ⟨ F_{z}^{(1)}, W A_{ξ}^{†} G_{ζ}^{(1)} ⟩ .$
unlabeled · line 597
$V_{ξ} (F ⋆_{W} G) - (V_{ξ} F) ⋆_{W} G - F ⋆_{W} (V_{ξ} G) = ℏ T_{ξ} (F, G) (mod ℏ^{2}),$
unlabeled · line 611
$T_{ξ} (M_{a}, Z_{g}) = - 2 W (A_{ξ}^{†} a, g), T_{ξ} (Z_{g}, M_{a}) = - 2 W (g, A_{ξ}^{†} a) .$
unlabeled · line 621
$(V_{ξ} F)_{z}^{(1)} = V_{ξ} (F_{z}^{(1)}) + 2 A_{ξ}^{†} F_{ζ}^{(1)} .$

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section: A compact regular comparison algebra