Equation 62 · No Particle Without a Cosigner

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e^{-i\Omega\Delta\tau}

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e^{-i\Omega\Delta\tau}

Symbol e^-iOmegaΔτ

$e^-$ iOmegaΔτ 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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change

Capital delta attached to a quantity marks a difference between two values of that quantity; the article’s sign convention determines the order.

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superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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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…

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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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[6] Quantum Fields in Curved Space ↗

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