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Equation 10 · Part 6 · No Particle Without a Cosigner

Symbol e^-iOmega(τ-τ')

Fx(Ω)=∫dτ dτ′ χ(τ)χ(τ′) e−iΩ(τ−τ′) W[x(τ),x(τ′)],\mathcal F_x(\Omega) = \int d\tau\, d\tau'\, \chi(\tau)\chi(\tau')\, e^{-i\Omega(\tau-\tau')}\, W\big[x(\tau), x(\tau')\big],
e−iΩ(τ−τ′)e^{-i\Omega(\tau-\tau')}

What this part means

e−e^-iOmega(τ-τ') is one of the signed contributions combined to compute the quantity on the left.

Its job in the formula

e−e^-iOmega(τ-τ') is one of the signed contributions combined to compute the quantity on the left.

The passage around this formula

The detector, not the mode decomposition, is the object this article works with, for a reason that matters later: a detector’s response is what an experiment can actually report, while “the Bogoliubov coefficients of the true vacuum” is not the kind of thing any apparatus reads off a dial. The response functional for a pointlike, linearly coupled two-level (Unruh-DeWitt) detector on worldline x(τ\tau) , parametrized by its own proper time τ\tau , with energy gap Ω\Omega and switching function χ(τ)\chi(\tau) controlling when the interaction is on, is Fx(Ω)=∫dτ dτ′ χ(τ)χ(τ′) e−iΩ(τ−τ′) W[x(τ),x(τ′)]\mathcal F_x(\Omega) = \int d\tau\, d\tau'\, \chi(\tau)\chi(\tau')\, e^{-i\Omega(\tau-\tau')}\, W\big[x(\tau), x(\tau')\big]. where W is the field’s two-point Wightman function evaluated along the trajectory [ 5 ] . Three things in this expression are…

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Sources cited in the surrounding passage

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