Equation 64 · No Particle Without a Cosigner

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\dot{\mathcal F}_{\rm inertial}(\Omega) = 0

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Inputs and operations0

Result or conditiondotmathcal F_rm inertial(Omega)

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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\dot{\mathcal F}_{\rm inertial}

Symbol dotmathcal F_rm inertial

dotmathcal $F_r$ m inertial has a dot, marking the rate of change of the underlying indexed quantity with respect to the article’s time variable.

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\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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=

The expressions on both sides represent the same quantity under the stated assumptions.

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subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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What the article says around this equation

An inertial worldline’s Wightman function is the flat-space, two-point function of the free vacuum, whose Fourier transform against $e^{-i\Omega\Delta\tau}$ has support only for $\Omega$ <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 $\dot{\mathcal F}_{\rm inertial}(\Omega)$ = 0 identically for every $\Omega$ >0 , not approximately, not in some limit — exactly zero. The normalization integral $\int_0^\infty$ $\dot{\mathcal F}_{\rm inertial}(\Omega')$ \,d $\Omega$ ' is therefore also exactly zero, and $p_{\rm inertial}(\Omega)$ is…

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An inertial worldline’s Wightman function is the flat-space, two-point function of the free vacuum, whose Fourier transform against $e^{-i\Omega\Delta\tau}$ has support only for $\Omega$ <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 $\dot{\mathcal F}_{\rm inertial}(\Omega)$ = 0 identically for every $\Omega$ >0 , not approximately, not in some limit — exactly zero. The normalization integral $\int_0^\infty$ $\dot{\mathcal F}_{\rm inertial}(\Omega')$ \,d $\Omega$ ' is therefore also exactly zero, and $p_{\rm inertial}(\Omega)$ is 0/0 : undefined.

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Sources cited in the surrounding passage

[6] Quantum Fields in Curved Space ↗

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