Manuscript X-B@3 · X-B-v3.tex · section · 2d415d58

One-contraction quantum verification

B2: one retarded map, its sign, support, and quantum order › One-contraction quantum verification · lines 1002–1040

X-B@3#eq:contraction-check

To verify the quantum interpretation without constructing a second product programme, introduce the auxiliary kernel

Δ_{F} = Δ_{+} + i Δ_{S}^{A} = Δ_{+} - i E_{m}^{A} .

It is symmetric with these conventions. Both $Δ_{F} c$ and $Δ_{+} c$ are smooth. For the pair (A1)–(A2), consider only the auxiliary contraction expression

C_{F} (V, F_{c}) = V F_{c} + ℏ ⟨ V^{'}, Δ_{F} c ⟩ .

It is well defined because pairing $V^{'}$ with a smooth function requires only compact vertex coefficients.

The Wick expression terminates exactly:

V ⋆ F_{c} = V F_{c} + ℏ ⟨ V^{'}, Δ_{+} c ⟩ .

Consequently

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

For $V^{'} = - L φ$ , this is exactly $- F_{L E_{m}^{A} c}$ . The auxiliary expression supplies the usual one-contraction retarded interpretation DF; it does not constitute a claimed family of time-ordered maps.

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X-B@3#eq:contraction-check · line 1027
$\frac{i}{ℏ} (C_{F} (V, F_{c}) - V ⋆ F_{c}) = - ⟨ V^{'}, Δ_{S}^{A} c ⟩ = - ⟨ c, Δ_{S}^{R} V^{'} ⟩ .$
unlabeled · line 1006
$Δ_{F} = Δ_{+} + i Δ_{S}^{A} = Δ_{+} - i E_{m}^{A} .$
unlabeled · line 1015
$C_{F} (V, F_{c}) = V F_{c} + ℏ ⟨ V^{'}, Δ_{F} c ⟩ .$
unlabeled · line 1023
$V ⋆ F_{c} = V F_{c} + ℏ ⟨ V^{'}, Δ_{+} c ⟩ .$

One-contraction quantum verification

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