Manuscript En01-B@2 · En01-B-v2.tex · section · f226d1eb

Convention and rigid-body specification

Declared regularized model and its dynamics › Convention and rigid-body specification · lines 959–1000

En01-B@2#eq:DP

Let $μ$ be a positive finite intrinsic mass measure on $R^{3}$ , with total mass $M$ , and let

g_{r_{c}} (r) = (2 π r_{c}^{2})^{- 3/2} e^{- ∣ r ∣^{2} / (2 r_{c}^{2})}, f_{r_{c}} = g_{r_{c}} * μ, r_{c} > 0.

The width $r_{c}$ is the standard deviation in each Cartesian direction. A fixed-orientation rigid body moves along a unit vector $e$ , with centre-of-mass position operator $x$ on $L^{2} (R)$ :

μ_{r_{c}} (r) = f_{r_{c}} (r - x e) .

We declare the generator

G_{r_{c}} (ρ) = - \frac{G}{2ℏ} \int \int \frac{[ μ _{r_{c}} ( r ) , [ μ _{r_{c}} ( r ^{'} ) , ρ ]]}{∣ r - r ^{'} ∣} d r d r^{'} .

The alternative adds $λ G_{r_{c}}$ , with $λ \geq 0$ . Unit strength means this normalization, not every convention called Díosi–Penrose in the literature. Regularization and normalization are physical model specifications Bahrami2014,TilloyDiosi2017.

Our Fourier convention is

f (k) = \int e^{- ik \cdot r} f (r) d r, f (r) = (2 π)^{- 3} \int e^{ik \cdot r} f (k) d k .

Thus $f_{r_{c}} (k) = e^{- r_{c}^{2} ∣ k ∣^{2} /2} μ (k)$ .

Displayed formulas in this passage

En01-B@2#eq:DP · line 978
$G_{r_{c}} (ρ) = - \frac{G}{2ℏ} \int \int \frac{[ μ _{r_{c}} ( r ) , [ μ _{r_{c}} ( r ^{'} ) , ρ ]]}{∣ r - r ^{'} ∣} d r d r^{'} .$
unlabeled · line 963
$g_{r_{c}} (r) = (2 π r_{c}^{2})^{- 3/2} e^{- ∣ r ∣^{2} / (2 r_{c}^{2})}, f_{r_{c}} = g_{r_{c}} * μ, r_{c} > 0.$
unlabeled · line 973
$μ_{r_{c}} (r) = f_{r_{c}} (r - x e) .$
unlabeled · line 993
$f (k) = \int e^{- ik \cdot r} f (r) d r, f (r) = (2 π)^{- 3} \int e^{ik \cdot r} f (k) d k .$

Convention and rigid-body specification

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