Manuscript I-B-VP@3 · I-B-VP-v3.tex · theorem · 93c04eb4

Theorem: Convergent observable integrals

Scale-dependent diagonal nonnegative extensions · lines 2177–2237 · proof included or linked

I-B-VP@3#thm:diagonal

For one irregular deficient mode of charge $q$ , put $ε = M r$ , $K = K_{ν} (x)$ . Then

δ F = \frac{ν}{4 π ^{2} α r ^{2}} Φ_{ν} (ε), Φ_{ν} (ε) = \frac{c _{0}}{ν} \int_{0}^{\infty} \frac{x ^{1 + 2 ν} K _{ν} ( x ) ^{2}}{x ^{2 ν} + ε ^{2 ν}} d x,

δ j_{φ} = - 2 e q δ F, δ j_{t} = δ j_{r} = δ j_{z} = 0,

and

δ T^{a}_{b} = \frac{1}{4 π ^{2} α r ^{4}} \int_{0}^{\infty} x d x \frac{c _{0} x ^{2 ν}}{x ^{2 ν} + ε ^{2 ν}} t^{a}_{b} (x),

where

t^{t}_{t} = t^{z}_{z} t^{r}_{r} t^{φ}_{φ} = x^{2} K^{2} + (4 ξ - 1) (ν^{2} K^{2} + x^{2} K^{'2} + x^{2} K^{2}), = x^{2} K^{'2} - (x^{2} + ν^{2}) K^{2} + 4 ξ x K K^{'}, = 2 ν^{2} K^{2} - 4 ξ x K K^{'} + (4 ξ - 1) (ν^{2} K^{2} + x^{2} K^{'2} + x^{2} K^{2}) .

All off-diagonal entries vanish. Corrections from independent diagonal modes add.

Proof. Differentiate (eq:dG) before coincidence. The Euclidean momentum angular average gives $[\partial_{τ} \partial_{τ^{'}}] \mapsto p^{2} /2$ . Radial differentiation and

K_{ν}^{''} = - K_{ν}^{'} / x + (1 + ν^{2} / x^{2}) K_{ν}

reduce the point-split tensor to the displayed expressions. The exponential and small-momentum bounds of Theorem thm:nonnegative-state justify differentiation.

At fixed $p$ , the relevant stress integrand is proportional to $r^{- 2} t (p r)$ . The Bessel equation gives

x (t^{r}_{r})^{'} - t^{r}_{r} = t^{φ}_{φ},

which is radial conservation before momentum integration. The trace follows from Lemma lem:ward.

Displayed formulas in this passage

unlabeled · line 2203
$t^{t}_{t} = t^{z}_{z} t^{r}_{r} t^{φ}_{φ} = x^{2} K^{2} + (4 ξ - 1) (ν^{2} K^{2} + x^{2} K^{'2} + x^{2} K^{2}), = x^{2} K^{'2} - (x^{2} + ν^{2}) K^{2} + 4 ξ x K K^{'}, = 2 ν^{2} K^{2} - 4 ξ x K K^{'} + (4 ξ - 1) (ν^{2} K^{2} + x^{2} K^{'2} + x^{2} K^{2}) .$
unlabeled · line 2181
$δ F = \frac{ν}{4 π ^{2} α r ^{2}} Φ_{ν} (ε), Φ_{ν} (ε) = \frac{c _{0}}{ν} \int_{0}^{\infty} \frac{x ^{1 + 2 ν} K _{ν} ( x ) ^{2}}{x ^{2 ν} + ε ^{2 ν}} d x,$
unlabeled · line 2190
$δ j_{φ} = - 2 e q δ F, δ j_{t} = δ j_{r} = δ j_{z} = 0,$
unlabeled · line 2195
$δ T^{a}_{b} = \frac{1}{4 π ^{2} α r ^{4}} \int_{0}^{\infty} x d x \frac{c _{0} x ^{2 ν}}{x ^{2 ν} + ε ^{2 ν}} t^{a}_{b} (x),$
unlabeled · line 2222
$K_{ν}^{''} = - K_{ν}^{'} / x + (1 + ν^{2} / x^{2}) K_{ν}$
unlabeled · line 2232
$x (t^{r}_{r})^{'} - t^{r}_{r} = t^{φ}_{φ},$

Theorem: Convergent observable integrals

Displayed formulas in this passage

Nearby in this section