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

Lemma: Projector identities

The exact affine gauge propagator · lines 1287–1321 · proof included or linked

XVI-O1-ST@5#lem:projector

For $k^{2} \neq = 0$ ,

A^{2} = I, FR = \frac{k ^{2}}{2} I, H = \frac{2}{k ^{2}} F^{*} R^{*}, H^{2} = H .

Also

R R^{*} = S_{4} (k),

where

S_{4} (k)_{ab, c d} = \frac{1}{4} (k_{a} k_{c} η_{b d} + k_{a} k_{d} η_{b c} + k_{b} k_{c} η_{a d} + k_{b} k_{d} η_{a c}) .

Proof. Since $J^{2} = 4 J$ , $A^{2} = I$ . For a covector $v$ ,

R^{*} R v = \frac{1}{2} (k^{2} v + k (k \cdot v)), Tr_{η} (R v) = k \cdot v .

This gives $FR = k^{2} I /2$ . Furthermore, $R^{*} F^{*} = R^{*} AR = FR$ , which proves the projector identity. Expanding $R R^{*}$ gives the four terms.

Displayed formulas in this passage

unlabeled · line 1290
$A^{2} = I, FR = \frac{k ^{2}}{2} I, H = \frac{2}{k ^{2}} F^{*} R^{*}, H^{2} = H .$
unlabeled · line 1297
$R R^{*} = S_{4} (k),$
unlabeled · line 1301
$S_{4} (k)_{ab, c d} = \frac{1}{4} (k_{a} k_{c} η_{b d} + k_{a} k_{d} η_{b c} + k_{b} k_{c} η_{a d} + k_{b} k_{d} η_{a c}) .$
unlabeled · line 1311
$R^{*} R v = \frac{1}{2} (k^{2} v + k (k \cdot v)), Tr_{η} (R v) = k \cdot v .$

Lemma: Projector identities

Displayed formulas in this passage

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