Manuscript XVI-J@1 · XVI-J.tex · section · 6c1f306d

The source polynomial

Finite source and density definitions › The source polynomial · lines 542–617

XVI-J@1#eq:C2, XVI-J@1#eq:source, XVI-J@1#eq:Qmatrix

For a symmetric matrix $H$ , define

N = η H, u_{0} = η, u_{1} = - η H η, u_{2} = η H η H η,

b_{1} = \frac{1}{2} tr N, b_{2} = \frac{1}{8} (tr N)^{2} - \frac{1}{4} tr (N^{2}) .

The tensor $C_{2}$ can be specified compactly and completely by the coefficient identity

C_{2}^{μν α β} (H) = [λ^{2}] (1 + λ b_{1} + λ^{2} b_{2}) (U^{μν} U^{α β} - U^{μ (α} U^{β) ν}), U = u_{0} + λ u_{1} + λ^{2} u_{2} .

Here $[λ^{2}]$ means polynomial coefficient extraction.

For a symmetric covariant source jet $J = (J_{α β})$ , set

Q (H; J)^{μν} = 2 C_{2}^{μν α β} (H) J_{α β}, s (X, Y; J) = \frac{1}{2} (Q (X + Y; J) - Q (X; J) - Q (Y; J)) .

Thus $Q (H; J) = s (H, H; J)$ .

A second, matrix form useful for hand calculations is the following. Write

V = η H η, W = η H η H η, c = tr (η J), k = tr (V J), j = tr (W J) .

Then

\frac{1}{2} Q (H; J) = b_{2} (η c - η J η) + b_{1} (- V c - η k + V J η + η J V) + W c + η j + V k - W J η - η J W - V J V .

Equation (eq:Qmatrix) follows by expanding

[λ^{2}] - det (η + λ H) (U_{exact} tr (U_{exact} J) - U_{exact} J U_{exact}), U_{exact} = (η + λ H)^{- 1} .

It agrees term by term with (eq:C2).

The actual source jet is always

J_{α β} (y) = \partial_{α} \partial_{β} σ_{ε} (y) .

Displayed formulas in this passage

XVI-J@1#eq:C2 · line 557
$C_{2}^{μν α β} (H) = [λ^{2}] (1 + λ b_{1} + λ^{2} b_{2}) (U^{μν} U^{α β} - U^{μ (α} U^{β) ν}), U = u_{0} + λ u_{1} + λ^{2} u_{2} .$
XVI-J@1#eq:source · line 574
$Q (H; J)^{μν} = 2 C_{2}^{μν α β} (H) J_{α β}, s (X, Y; J) = \frac{1}{2} (Q (X + Y; J) - Q (X; J) - Q (Y; J)) .$
XVI-J@1#eq:Qmatrix · line 592
$\frac{1}{2} Q (H; J) = b_{2} (η c - η J η) + b_{1} (- V c - η k + V J η + η J V) + W c + η j + V k - W J η - η J W - V J V .$
unlabeled · line 545
$N = η H, u_{0} = η, u_{1} = - η H η, u_{2} = η H η H η,$
unlabeled · line 551
$b_{1} = \frac{1}{2} tr N, b_{2} = \frac{1}{8} (tr N)^{2} - \frac{1}{4} tr (N^{2}) .$
unlabeled · line 586
$V = η H η, W = η H η H η, c = tr (η J), k = tr (V J), j = tr (W J) .$
unlabeled · line 602
$[λ^{2}] - det (η + λ H) (U_{exact} tr (U_{exact} J) - U_{exact} J U_{exact}), U_{exact} = (η + λ H)^{- 1} .$
unlabeled · line 613
$J_{α β} (y) = \partial_{α} \partial_{β} σ_{ε} (y) .$