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Equation 66 · Part 6 · No Particle Without a Cosigner

Ending index or upper bound: infty

∫0∞F˙inertial(Ω′) dΩ′\int_0^\infty \dot{\mathcal F}_{\rm inertial}(\Omega')\,d\Omega'
∞\infty

What this part means

This label says where the repeated addition, multiplication, or accumulation stops. It sets the last term or end of the range.

Its job in the formula

infty appears in the bound of this integral. The bound states where the repeated operation starts, ends, or which values it includes.

The passage around this formula

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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Learn the underlying idea

Σ adds a collection of terms. Π multiplies them. The lower and upper labels tell you which terms belong to the collection.

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Sources cited in the surrounding passage

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