ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Worked example B03 — nonretarded one-mode correlator

Example

For the one-mode oscillator in One-mode oscillator, Feynman boundary prescription, and retarded contrast, choose $t_{*} = - π / (2 ω)$ , where $ω > 0$ is in $s^{- 1}$ . Then the normalized Feynman correlator is $- i / (2 ω)$ , while the explicitly defined kernel $G_{R}$ is zero. This compares the two functions in the stated one-mode model and makes no claim about a response to an external source.

Facts & Assumptions

Given: ${A C}_{ω}$ , $ω > 0$ in $s^{- 1}$ , $t_{*} = - π / (2 ω) < 0$ , and the fixed-mode definitions of the Feynman correlator and retarded kernel.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ is assumed by the oscillator supplier.

[F1]

QFT example units, metric, gauge, and pairing conventions: $ω t$ is dimensionless, and these formulas carry no spatial-locality claim.

[F2]

One-mode oscillator, Feynman boundary prescription, and retarded contrast: $C_{F} (t) = e^{- i ω ∣ t ∣} / (2 ω)$ and $G_{R} (t) = 1_{[0, \infty)} (t) \sin (ω t) / ω$ .

Verification

1.1givenA1F1F2algebra

Since $ω ∣ t_{*} ∣ = π / 2$ , the oscillator formula gives $C_{F} (t_{*}) = e^{- i π / 2} / (2 ω) = - i / (2 ω)$ . The units are seconds on both sides.

2.1givenF2step 1.1algebra

Since $t_{*} < 0$ , the support indicator in $G_{R}$ is zero, so $G_{R} (t_{*}) = 0$ . Thus the correlator is nonzero at negative time while the retarded response vanishes there.

3.1givenF1F2step 2.1∎

This distinguishes the two kernels in this single oscillator mode only; it proves no spatial signal property or interacting-field statement.

Worked example B03 — nonretarded one-mode correlator

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources