Manuscript XVI-O1-ST@5 · XVI-O1-ST-v5.tex · theorem · dd45e0d1

Theorem: Exact affine inverse under the declared normalization

The exact affine gauge propagator · lines 1323–1351 · proof included or linked

XVI-O1-ST@5#thm:propagator, XVI-O1-ST@5#eq:propagator

For

K_{ξ} = k^{2} A + (ξ^{- 1} - 1) G,

the rational inverse at $k^{2} \neq = 0, ξ \neq = 0$ is

P_{ξ} = \frac{A}{k ^{2}} + 2 (ξ - 1) \frac{S _{4} ( k )}{( k ^{2} ) ^{2}} .

Proof. Write $K_{1} = k^{2} A$ , $P_{1} = A / k^{2}$ , and $a_{ξ} = ξ^{- 1} - 1$ . Then

K_{ξ} = (I + a_{ξ} H) K_{1}, (I + a_{ξ} H)^{- 1} = I + (ξ - 1) H .

Multiplying by $P_{1}$ gives

P_{ξ} = \frac{A}{k ^{2}} + \frac{2 ( ξ - 1 )}{( k ^{2} ) ^{2}} R R^{*} .

Displayed formulas in this passage

XVI-O1-ST@5#eq:propagator · line 1330
$P_{ξ} = \frac{A}{k ^{2}} + 2 (ξ - 1) \frac{S _{4} ( k )}{( k ^{2} ) ^{2}} .$
unlabeled · line 1326
$K_{ξ} = k^{2} A + (ξ^{- 1} - 1) G,$
unlabeled · line 1342
$K_{ξ} = (I + a_{ξ} H) K_{1}, (I + a_{ξ} H)^{- 1} = I + (ξ - 1) H .$
unlabeled · line 1347
$P_{ξ} = \frac{A}{k ^{2}} + \frac{2 ( ξ - 1 )}{( k ^{2} ) ^{2}} R R^{*} .$