Manuscript V@1 · V.tex · theorem · e7ddf37d

Theorem: Switched identity

The one-oscillator mixed insertion › Exact switched bilinear and unswitched remainder · lines 2712–2734 · proof included or linked

V@1#thm:switched, V@1#eq:switchedidentity

For compact real $g, f$ ,

B_{f}^{g, t_{0}} = R_{g, f} on D_{mix} .

This equality uses $L_{g a}$ , including derivatives of the switch.

Proof. On free scalar matrix elements, $Q_{g a} ϕ = L_{g a} ϕ$ . Since $g a$ is supported after $t_{0}$ ,

- G_{R} Q_{g a} ϕ = S_{t_{0}} (g a) ϕ .

Smearing and using Theorem thm:mixed proves the identity. The retarded product was constructed before this Duhamel comparison.

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