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Published equation contexts

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]

Why this formula appears here

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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Ω\Omega

Symbol Omega

Omega is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

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e−iΩ(τ−τ′)e^{-i\Omega(\tau-\tau')}

Symbol e^-iOmega(τ-τ')

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

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How to interpret it

The result depends on every term or point included by the summation or integral. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

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],

Equation 10 · Evolutionary Physics

No Particle Without a Cosigner

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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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