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One-mode time-ordered correlator versus retarded response

Physical scope

One exact oscillator mode with the stated core, vacuum, and conventions. The negative-time correlation differs from a retarded response; this makes no spatial-locality, signalling, interacting-field, or empirical claim.

Example

For the exact single oscillator mode of One-mode oscillator, Feynman boundary prescription, and retarded contrast, let $ω > 0$ in $s^{- 1}$ and choose $t = - π / (2 ω)$ . The normalized time-ordered vacuum correlator is $- i / (2 ω)$ , while the retarded response kernel is $0$ . The correlator at negative time is not a response to a future source. This fixed-mode calculation does not establish a spatial-causality or superluminal-signalling result.

Facts & Assumptions

Given: ${A C}_{ω}$ , $m_{o} > 0$ in kilograms, $ω > 0$ in $s^{- 1}$ , $ℏ > 0$ in $J s$ , and the one-mode oscillator result of One-mode oscillator, Feynman boundary prescription, and retarded contrast.

[F1]

One-mode oscillator, Feynman boundary prescription, and retarded contrast: for real $t$ , $C_{F} (t) = e^{- i ω ∣ t ∣} / (2 ω)$ and $G_{R} (t) = 1_{[0, \infty)} (t) \sin (ω t) / ω$ ; the statement assumes ${A C}_{ω}$ and the exact oscillator basis and domain.

[F2]

QFT example units, metric, gauge, and pairing conventions: the frequency $ω$ is in $s^{- 1}$ , time is in seconds, and the Hilbert pairing convention is the stated first-variable-linear one.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ is assumed only because the exact oscillator supplier invokes it.

[A2]

Physical interpretation of a supplied constructed quantum model: a physical reading is limited to an actually constructed state and dynamics; the postulate supplies no spatial-locality or interacting-field conclusion.

Verification

1.1givenF1F2A1algebra

The chosen time has units seconds because $[ω] = s^{- 1}$ , and $ω ∣ t ∣ = π / 2$ [F2]. Therefore [F1] gives $C_{F} (t) = e^{- i π / 2} / (2 ω) = - i / (2 ω)$ .

2.1F1A2step 1.1algebra∎

Since $t < 0$ , the indicator in [F1] is zero, so $G_{R} (t) = 0$ . Thus the two values differ, and the nonzero time-ordered correlation is not the retarded response at this negative time. The physical interpretation remains within the supplied one-mode construction described by [A2].

One-mode time-ordered correlator versus retarded response

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources