Manuscript XX-E@1 · XX-E.tex · proposition · cf7a8f40

Proposition: Necessary centrality condition

Why the free input does not supply a quantum Ward operation › The central-second-jet issue · lines 827–850 · proof included or linked

XX-E@1#prop:central, XX-E@1#eq:centralcondition

A derivation implementing these two formulas on the central-parameter free algebra can exist only if

E_{B} (A_{ξ}^{†} a, g) = 0 for all relevant a, g .

Whenever this pairing is nonzero, the proposed derivation is inconsistent with centrality.

Proof. The relation $[M_{a}, Z_{g}]_{⋆_{W}} = 0$ must remain zero under a derivation. Its proposed variation is

[2 Z_{A_{ξ}^{†} a}, Z_{g}]_{⋆_{W}} + [M_{a}, (K ξ, g)_{0} id]_{⋆_{W}} = 2 i ℏ E_{B} (A_{ξ}^{†} a, g) id .

This gives the necessary condition and the stated incompatibility when it fails.

Displayed formulas in this passage

XX-E@1#eq:centralcondition · line 831
$E_{B} (A_{ξ}^{†} a, g) = 0 for all relevant a, g .$
unlabeled · line 843
$[2 Z_{A_{ξ}^{†} a}, Z_{g}]_{⋆_{W}} + [M_{a}, (K ξ, g)_{0} id]_{⋆_{W}} = 2 i ℏ E_{B} (A_{ξ}^{†} a, g) id .$