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Equation 10 · Part 2 · No Particle Without a Cosigner

Symbol Omega

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],
Ω\Omega

What this part means

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

Its job in the formula

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

The passage around this formula

…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 worth fixing in the reader’s mind before anything else happens to…

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A function assigns an output to each allowed input. The expression f(x) means “apply f to x”.

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Sources cited in the surrounding passage

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