Manuscript Psy02-A@1 · Psy02-A.tex · section · d92ce402

Regularized mass data and units

A separately declared reduction model › Regularized mass data and units · lines 2272–2309

Psy02-A@1#eq:mass-Gram, Psy02-A@1#eq:EG

Let $H_{c} = C^{d}$ , with basis $∣ a ⟩$ , $a = 1, \dots, d$ . Assign smooth compactly supported real mass-density profiles $μ_{a} (x)$ . They may be required to be nonnegative and to have a fixed total mass; the mathematical positivity below does not require equal total mass.

Define

K_{ab} E_{G, ab} = \frac{G}{ℏ} \int_{R^{3} \times R^{3}} \frac{μ _{a} ( x ) μ _{b} ( y )}{\abs x - y} d^{3} x d^{3} y, = \frac{G}{2} \int_{R^{3} \times R^{3}} \frac{[ μ _{a} ( x ) - μ _{b} ( x )] [ μ _{a} ( y ) - μ _{b} ( y )]}{\abs x - y} d^{3} x d^{3} y .

Here $G$ has its gravitational units, $E_{G, ab}$ has energy units, and $K_{ab}$ has inverse-time units. The smoothing profiles are explicit regulators, not parameter-free point masses.

Displayed formulas in this passage

Psy02-A@1#eq:mass-Gram · line 2283
$K_{ab} E_{G, ab} = \frac{G}{ℏ} \int_{R^{3} \times R^{3}} \frac{μ _{a} ( x ) μ _{b} ( y )}{\abs x - y} d^{3} x d^{3} y, = \frac{G}{2} \int_{R^{3} \times R^{3}} \frac{[ μ _{a} ( x ) - μ _{b} ( x )] [ μ _{a} ( y ) - μ _{b} ( y )]}{\abs x - y} d^{3} x d^{3} y .$
Psy02-A@1#eq:EG · line 2283
$K_{ab} E_{G, ab} = \frac{G}{ℏ} \int_{R^{3} \times R^{3}} \frac{μ _{a} ( x ) μ _{b} ( y )}{\abs x - y} d^{3} x d^{3} y, = \frac{G}{2} \int_{R^{3} \times R^{3}} \frac{[ μ _{a} ( x ) - μ _{b} ( x )] [ μ _{a} ( y ) - μ _{b} ( y )]}{\abs x - y} d^{3} x d^{3} y .$

Nearby in this section

lemma: Lemma: Positive mass-density Gram matrix