Manuscript XVIII-B@2 · XVIII-B-v2.tex · proposition · 27ed4f4b

Proposition: A compact spectral detector

The exact two-particle spectral calculation › Spectral detection, not observation · lines 594–637 · proof included or linked

XVIII-B@2#prop:detector, XVIII-B@2#eq:vacnorm

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For every nonzero $f \in C_{c}^{\infty} (M)$ , take $j^{ab} = η^{ab} f$ . Then

∥ T (j) Ω ∥^{2} = \int \frac{d ^{4} p}{( 2 π ) ^{4}} j^{ab} (p) W_{ab c d}^{>} (p) j^{c d} (p) = 6 π ℏ^{2} \int_{p^{0} > 0 p^{2} > a} \frac{d ^{4} p}{( 2 π ) ^{4}} (p^{2})^{2} r_{0} (p^{2}) ∣ f (p) ∣^{2} > 0.

This detects the spectral coefficient and is independent of the finite retarded contact prescription.

It is not a finite measurement instrument, an Enter destructive readout, a classical record, or a decoder.

Proof. The norm follows by vacuum positivity and the Fourier representation of $W^{>}$ . Since $η^{ab} η^{c d} Q_{2}_{ab c d} = 0$ and $η^{ab} η^{c d} Q_{0}_{ab c d} = 3 (p^{2})^{2}$ , the second line follows.

The Fourier transform of a nonzero compact smooth function is real analytic on real momentum space and cannot vanish on the entire open future threshold region. The positive integrand is therefore nonzero on an open set. Rapid Fourier decay makes the integral finite.

A finite change in the retarded splitting does not change the already fixed vacuum Wightman distribution. No apparatus or selective map is constructed by this norm.

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XVIII-B@2#eq:vacnorm · line 598
$∥ T (j) Ω ∥^{2} = \int \frac{d ^{4} p}{( 2 π ) ^{4}} j^{ab} (p) W_{ab c d}^{>} (p) j^{c d} (p) = 6 π ℏ^{2} \int_{p^{0} > 0 p^{2} > a} \frac{d ^{4} p}{( 2 π ) ^{4}} (p^{2})^{2} r_{0} (p^{2}) ∣ f (p) ∣^{2} > 0.$

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section: Spectral detection, not observation