LocalizedMode · section of the QIQT-H book

QIQTH.LocalizedMode

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LocalizedMode · entries 490–490 of 1000

Lemma 490 (localized_mode_hTkk).  source ↗

The localization mode from the field — hTkk from one universal calibration. With the mode taken to be the field’s directional derivative D = ∑ₐ vₐ ∂ₐφ(x) times a reference profile g₀ (and ff' = D·g₀'), the transparent hTkk identity holds for the generator (x,v) as soon as g₀ satisfies the single calibration (−2π ∫ conj(g₀)·g₀').im = 2π/ℏ. The boost charge scales as D² (quadratic in the mode), so the classical (∂φ)² null energy is reproduced. No regularity of g₀ is needed for the identity itself — only the value of its boost-charge integral.

(−(2⋅π⋅∫(θ:R),(starRingEnd C) (g θ)⋅g′ θ)).im=2⋅π/ℏ→2⋅π/ℏ⋅(∑bv b⋅∂b(φ)(x))2=(−(2⋅π⋅∫(θ:R),(starRingEnd C) ((∑bv b⋅∂b(φ)(x))⋅g θ)⋅((∑bv b⋅∂b(φ)(x))⋅g′ θ))).im(-(2 \cdot \pi \cdot \int (\theta : \mathbb{R}), (\mathrm{starRingEnd}\,\mathbb{C})\,(\mathrm{g}\,\theta) \cdot \mathrm{g}^{\prime}\,\theta)).\mathrm{im} = 2 \cdot \pi / \hbar \to 2 \cdot \pi / \hbar \cdot {(\sum_{b} v\,b \cdot \href{/browser/qiqth-curvature#d-qiqth-curvature-pd}{\partial_{{b}}({\varphi})({x})})}^{2} = (-(2 \cdot \pi \cdot \int (\theta : \mathbb{R}), (\mathrm{starRingEnd}\,\mathbb{C})\,((\sum_{b} v\,b \cdot \href{/browser/qiqth-curvature#d-qiqth-curvature-pd}{\partial_{{b}}({\varphi})({x})}) \cdot \mathrm{g}\,\theta) \cdot ((\sum_{b} v\,b \cdot \href{/browser/qiqth-curvature#d-qiqth-curvature-pd}{\partial_{{b}}({\varphi})({x})}) \cdot \mathrm{g}^{\prime}\,\theta))).\mathrm{im}

Proof. Immediate from the definitions. □\square

Used by qiqt_gr_freefield_complete, qiqt_gr_freefield_gaussian.

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