Manuscript Oai365-A@1 · Oai365-A.tex · section · a3840e3d

A Ward identity is not a source rule

Regularity, metric jets, and dynamics separation › A Ward identity is not a source rule · lines 1618–1654

For any smooth scalar $χ$ , the tensor

I_{μν} = \partial_{μ} \partial_{ν} χ - η_{μν} □ χ

is identically conserved:

\partial^{μ} I_{μν} = 0.

This follows by commuting flat derivatives. Conservation therefore does not select a unique stress tensor.

Also $H$ and $H + c I$ induce identical state evolution, since their propagators differ by a global phase. An operational process table does not thereby determine an absolute energy observable or a physical source normalization.

A dynamics comparison must separately specify:

J : D \to {metric two-jets}, S : D \to {source tensors},

and then prove, or disprove on a stated domain,

G^{(1)} (J (D)) = κ_{cmp} S (D) .

Defining the source to be the left side is not an independent operational derivation.

These are the geometric and source gaps isolated in XXII-E CheungXXIIE. A connection-Laplacian kinetic comparator does not close them.

Displayed formulas in this passage

unlabeled · line 1621
$I_{μν} = \partial_{μ} \partial_{ν} χ - η_{μν} □ χ$
unlabeled · line 1626
$\partial^{μ} I_{μν} = 0.$
unlabeled · line 1638
$J : D \to {metric two-jets}, S : D \to {source tensors},$
unlabeled · line 1644
$G^{(1)} (J (D)) = κ_{cmp} S (D) .$