λEnter Calculus CompanionRegistered manuscripts, kept distinct

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

External first retarded coefficient

The external-metric first retarded product › Representation and the external coefficient · lines 2308–2339

V@1#fig:external-contraction

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metricscalarrepresented quantumdefect / unresolvedsupport / scope
Time-ordered product
V(g;h)=12∫ghμν:Tμν:V(g;h)=\tfrac12\int gh^{\mu\nu}:T_{\mu\nu}:
normal three-scalar block
one cross-contraction WmFW_m^{\mathrm F}
Ordinary product
ϕ(f)\phi(f)
the same normal block
one cross-contraction WmW_m
retarded subtraction
R1,1=i(T2−Vϕ)R_{1,1}=i(T_2-V\phi) — normal blocks cancel
ϕg,h(1)=−GRQghϕ,R1,1=ϕ(Qghuf)\phi^{(1)}_{g,h}=-G_R\mathsf Q_{gh}\phi,\qquad R_{1,1}=\phi(\mathsf Q_{gh}u_f)

At most one cross-contraction. No double-contraction vacuum block. Off-shell labels stay until the product is constructed. Not a loop diagram.

External first retarded coefficient. The two products have the same completely normal-ordered three-scalar block, which cancels in the retarded subtraction. Their single cross-contractions differ by , leaving a linearly smeared free field with compact test . The off-shell insertion label is retained until the product is constructed.

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