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Ω<0\Omega<0

Why this formula appears here

Fx(Ω)\mathcal F_x(\Omega) , viewed as a function of Ω\Omega for fixed everything else, has a natural reading as a spectrum: it says how much excitation probability accumulates at each possible gap, for a whole notional bank of detectors sharing one trajectory, one switching profile, and one field state, differing only in Ω\Omega . Restricting to the excitation branch Ω\Omega>0 — “did this detector behave as though it absorbed a quantum” is the only branch this article treats as bearing on particle content, since the decay branch Ω\Omega<0 mixes genuine field structure with detector-specific spontaneous-emission physics that is present even in flat, empty spacetime with no field excitation at all —…

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Ω\Omega

Symbol Omega

Omega is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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Published contexts (2)

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Ω<0\Omega<0

Equation 20 · Evolutionary Physics

No Particle Without a Cosigner

This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

Fx(Ω)\mathcal F_x(\Omega) , viewed as a function of Ω\Omega for fixed everything else, has a natural reading as a spectrum: it says how much excitation probability accumulates at each possible gap, for a whole notional bank of detectors sharing one trajectory, one switching profile, and one field state, differing only in Ω\Omega . Restricting to the excitation branch Ω\Omega>0 — “did this detector behave as though it absorbed a quantum” is the only branch this article treats as bearing on particle content, since the decay branch Ω\Omega<0 mixes genuine field structure with detector-specific spontaneous-emission physics that is present even in flat, empty spacetime with no field excitation at all —…

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Ω<0\Omega<0

Equation 63 · Evolutionary Physics

No Particle Without a Cosigner

This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

An inertial worldline’s Wightman function is the flat-space, two-point function of the free vacuum, whose Fourier transform against e−iΩΔτe^{-i\Omega\Delta\tau} has support only for Ω\Omega<0 — this is nothing more exotic than vacuum stability, the same textbook fact that says an unaccelerated atom in its ground state, coupled to the electromagnetic vacuum, never spontaneously jumps up in energy [ 6 ] . Consequently F˙inertial(Ω)\dot{\mathcal F}_{\rm inertial}(\Omega) = 0 identically for every Ω\Omega>0 , not approximately, not in some limit — exactly zero. The normalization integral ∫0∞\int_0^\infty F˙inertial(Ω′)\dot{\mathcal F}_{\rm inertial}(\Omega')\,dΩ\Omega' is therefore also exactly zero, and pinertial(Ω)p_{\rm inertial}(\Omega) is…

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