Manuscript VII-I@1 · VII-I.tex · proposition · 907e96ab

Proposition: Positive marginal state evaluation

Interactions, normalization, and a controlled diagnostic › A positive scalar diagnostic with a fixed classical metric · lines 2032–2065 · proof included or linked

VII-I@1#prop:scalar-state, VII-I@1#eq:positive-diagnostic

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In the Minkowski scalar vacuum,

ω_{0} (ϕ (d_{κ})^{*} ⋆ ϕ (d_{κ})) = ℏ \int_{R^{3}} \frac{∣ d _{κ} ( ω _{m} ( p ) , p ) ∣ ^{2}}{( 2 π ) ^{3} 2 ω _{m} ( p )} d^{3} p .

This is finite coefficientwise. There are real compact $f$ for which its leading coefficient is strictly positive. It is a scalar marginal diagnostic on a fixed classical metric, not a state evaluation of an interacting quantum metric observable.

Proof. The scalar two-point spectral formula gives (eq:positive-diagnostic). Smooth compact smearings have rapidly decreasing Fourier transforms, and the massless infrared factor is locally integrable in three dimensions.

For example, at $m = 0$ , take a real nonnegative compact bump $f$ with nonzero integral. Its Fourier transform is nonzero near zero, including a nonempty portion of the positive mass shell. The leading spectral integral is therefore strictly positive.

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VII-I@1#eq:positive-diagnostic · line 2035
$ω_{0} (ϕ (d_{κ})^{*} ⋆ ϕ (d_{κ})) = ℏ \int_{R^{3}} \frac{∣ d _{κ} ( ω _{m} ( p ) , p ) ∣ ^{2}}{( 2 π ) ^{3} 2 ω _{m} ( p )} d^{3} p .$

Proposition: Positive marginal state evaluation

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