Equation 66 · No Particle Without a Cosigner

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\int_0^\infty \dot{\mathcal F}_{\rm inertial}(\Omega')\,d\Omega'

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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 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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d

Symbol d

d 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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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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0

Starting index or lower bound: 0

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

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\infty

Ending index or upper bound: infty

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

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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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