Manuscript X-B-ST@1 · X-B-ST.tex · proposition · d03c150c

Proposition: Symmetry and the one-contraction identity

C2: the one-contraction boundary › The one pair and its causal requirements · lines 1762–1789 · proof included or linked

X-B-ST@1#prop:check, X-B-ST@1#eq:check

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For (A1)–(A2),

C_{F} (V, F_{c}) = C_{F} (F_{c}, V),

and

\frac{i}{ℏ} (C_{F} (V, F_{c}) - V ⋆ F_{c}) = ⟨ V^{'}, E_{A} c ⟩ = - F_{L E_{A} c} .

For $L = L_{1, g} [h]$ , its free class is exactly (eq:Rone).

Proof. Transpose symmetry of $Δ_{F}$ proves the first claim. The Wick expansion terminates at one contraction because $F_{c}^{(2)} = 0$ . Thus the difference is

ℏ ⟨ V^{'}, (Δ_{F} - Δ_{+}) c ⟩ = - i ℏ ⟨ V^{'}, E_{A} c ⟩ .

Multiply by $i /ℏ$ , then use $V^{'} = - L ψ$ and formal self-adjointness of $L$ .

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X-B-ST@1#eq:check · line 1769
$\frac{i}{ℏ} (C_{F} (V, F_{c}) - V ⋆ F_{c}) = ⟨ V^{'}, E_{A} c ⟩ = - F_{L E_{A} c} .$
unlabeled · line 1765
$C_{F} (V, F_{c}) = C_{F} (F_{c}, V),$
unlabeled · line 1783
$ℏ ⟨ V^{'}, (Δ_{F} - Δ_{+}) c ⟩ = - i ℏ ⟨ V^{'}, E_{A} c ⟩ .$

Proposition: Symmetry and the one-contraction identity

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