Manuscript Em02-E@1 · Em02-E-v1.tex · section · f3795050

An explicit Laplace operator expansion

Clock slices and harmonic spatial labels › An explicit Laplace operator expansion · lines 496–524

Em02-E@1#eq:b, Em02-E@1#eq:A

Set

Q_{1} = a^{- 2} q_{1}, Q_{2} = a^{- 2} q_{2},

where spatial indices on $Q_{1}, Q_{2}$ are raised using $δ_{ij}$ . Define

b_{1} b_{2} A_{1} A_{2} = \frac{1}{2} tr Q_{1}, = \frac{1}{2} tr Q_{2} + \frac{1}{4} (tr Q_{1})^{2} - \frac{1}{2} tr (Q_{1}^{2}), = b_{1} 1 - Q_{1}, = b_{2} 1 - 2 b_{1} Q_{1} + 2 Q_{1}^{2} - Q_{2} .

These matrices are unrelated to the component maps denoted by $A$ in Em02-A–D.

For a scalar $F$ , the slice Laplacian is

Δ_{q_{λ}} F = (det q_{λ})^{- 1/2} \partial_{i} ((det q_{λ})^{1/2} q_{λ}^{ij} \partial_{j} F) .

Displayed formulas in this passage

Em02-E@1#eq:b · line 504
$b_{1} b_{2} A_{1} A_{2} = \frac{1}{2} tr Q_{1}, = \frac{1}{2} tr Q_{2} + \frac{1}{4} (tr Q_{1})^{2} - \frac{1}{2} tr (Q_{1}^{2}), = b_{1} 1 - Q_{1}, = b_{2} 1 - 2 b_{1} Q_{1} + 2 Q_{1}^{2} - Q_{2} .$
Em02-E@1#eq:A · line 504
$b_{1} b_{2} A_{1} A_{2} = \frac{1}{2} tr Q_{1}, = \frac{1}{2} tr Q_{2} + \frac{1}{4} (tr Q_{1})^{2} - \frac{1}{2} tr (Q_{1}^{2}), = b_{1} 1 - Q_{1}, = b_{2} 1 - 2 b_{1} Q_{1} + 2 Q_{1}^{2} - Q_{2} .$
unlabeled · line 499
$Q_{1} = a^{- 2} q_{1}, Q_{2} = a^{- 2} q_{2},$
unlabeled · line 517
$Δ_{q_{λ}} F = (det q_{λ})^{- 1/2} \partial_{i} ((det q_{λ})^{1/2} q_{λ}^{ij} \partial_{j} F) .$