Manuscript XX-B@1 · XX-B.tex · proposition · 56d708a5

Proposition: Exact contraction normalization

Normalized Hadamard contraction data › Bosonic and ghost contractions · lines 1307–1335 · proof included or linked

XX-B@1#prop:W, XX-B@1#eq:Wfield, XX-B@1#eq:WCCR, XX-B@1#eq:WWF

The bidistributions in (eq:WP) satisfy

P_{x} W_{P} W_{P}^{*} WF^{'} (W_{P}) = 0, W_{P} P = 0, = W_{P}, W_{P} - W_{P}^{T} = i E_{P}, \subset N_{+} .

Proof. The difference of two exact inverses is a bisolution. Since $F_{P}^{T} = F_{P}$ and $(G_{P}^{-})^{T} = G_{P}^{+}$ ,

W_{P, 0} - W_{P, 0}^{T} = i (G_{P}^{+} - G_{P}^{-}) = i E_{P} .

Hermitian averaging preserves that identity because $E_{P}$ is real and transpose-antisymmetric. It also preserves the bisolution equations.

The Feynman orientation fixes the wavefront of $i (F_{P} - G_{P}^{-})$ to the positive-frequency relation. Hermitian kernel adjunction reverses both frequency and slot order, preserving that relation.

Displayed formulas in this passage

XX-B@1#eq:Wfield · line 1310
$P_{x} W_{P} W_{P}^{*} WF^{'} (W_{P}) = 0, W_{P} P = 0, = W_{P}, W_{P} - W_{P}^{T} = i E_{P}, \subset N_{+} .$
XX-B@1#eq:WCCR · line 1310
$P_{x} W_{P} W_{P}^{*} WF^{'} (W_{P}) = 0, W_{P} P = 0, = W_{P}, W_{P} - W_{P}^{T} = i E_{P}, \subset N_{+} .$
XX-B@1#eq:WWF · line 1310
$P_{x} W_{P} W_{P}^{*} WF^{'} (W_{P}) = 0, W_{P} P = 0, = W_{P}, W_{P} - W_{P}^{T} = i E_{P}, \subset N_{+} .$
unlabeled · line 1323
$W_{P, 0} - W_{P, 0}^{T} = i (G_{P}^{+} - G_{P}^{-}) = i E_{P} .$

Proposition: Exact contraction normalization

Displayed formulas in this passage

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