Manuscript V@1 · V.tex · section · e7ddf37d

Exact switched bilinear and unswitched remainder

The one-oscillator mixed insertion › Exact switched bilinear and unswitched remainder · lines 2734–2793

V@1#eq:unswitchedvectors, V@1#eq:remainder, V@1#eq:missingswitch

For the unswitched fixed-time pairing choose $T$ later than $supp f$ , and set

q_{a}^{0} (t, x) = - I_{[t_{0}, T]} (t) \partial_{i} (a^{ij} \partial_{j} u_{f}), v_{a}^{0} = K q_{a}^{0} .

Although $q_{a}^{0}$ may jump at $t_{0}$ , it has compact spacetime support and is uniformly spatially smooth on the finite time slab. Its spatial Fourier transform decreases rapidly with all spatial derivatives. Momentum derivatives of the mass-shell exponential introduce bounded-time and polynomially controlled factors, so $v_{a}^{0} \in S (R^{3})$ . This defines

B_{f}^{0, t_{0}} = Q \otimes Φ (v_{e}^{0}) + Π \otimes Φ (v_{r}^{0}) .

It is a mode-integral definition for these specific nonsmooth tests, not an extension of the free field to arbitrary distributions.

The exact remainder is

R_{g, f} - B_{f}^{0, t_{0}} = Q \otimes Φ (v_{e} - v_{e}^{0}) + Π \otimes Φ (v_{r} - v_{r}^{0}),

where

v_{a} - v_{a}^{0} = K (I_{[t_{0}, T]} Q_{(g - 1) a} u_{f}) .

Equivalently it is the quantization of $\int f S_{t_{0}} ((g - 1) h) φ$ . This is a switching and initial-time remainder, not a renormalization ambiguity.

If one incorrectly substitutes $g L_{a}$ for $L_{g a}$ , the missing contribution is exactly

- a = e, r \sum O_{a} \otimes \int u_{f} (\partial_{i} g) a^{ij} \partial_{j} ϕ, O_{e} = Q, O_{r} = Π.

Constancy of $g$ merely near $supp f$ does not generally make this past-sampling term vanish.

Displayed formulas in this passage

V@1#eq:unswitchedvectors · line 2738
$q_{a}^{0} (t, x) = - I_{[t_{0}, T]} (t) \partial_{i} (a^{ij} \partial_{j} u_{f}), v_{a}^{0} = K q_{a}^{0} .$
V@1#eq:remainder · line 2763
$R_{g, f} - B_{f}^{0, t_{0}} = Q \otimes Φ (v_{e} - v_{e}^{0}) + Π \otimes Φ (v_{r} - v_{r}^{0}),$
V@1#eq:missingswitch · line 2785
$- a = e, r \sum O_{a} \otimes \int u_{f} (\partial_{i} g) a^{ij} \partial_{j} ϕ, O_{e} = Q, O_{r} = Π.$
unlabeled · line 2754
$B_{f}^{0, t_{0}} = Q \otimes Φ (v_{e}^{0}) + Π \otimes Φ (v_{r}^{0}) .$
unlabeled · line 2772
$v_{a} - v_{a}^{0} = K (I_{[t_{0}, T]} Q_{(g - 1) a} u_{f}) .$

Exact switched bilinear and unswitched remainder

Displayed formulas in this passage

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