Manuscript II@2 · II.tex · section · 50d64def

Polarized Einstein identities

Ward reduction and coupling factorization › Polarized Einstein identities · lines 743–811

II@2#eq:ward3, II@2#eq:ward4, II@2#eq:tree

For a gauge parameter $ζ$ , write

G_{ζ} = \partial ζ + (\partial ζ)^{T}

and

(R_{ζ} h)_{μν} = ζ^{ρ} \partial_{ρ} h_{μν} + h_{ρ ν} \partial_{μ} ζ^{ρ} + h_{μ ρ} \partial_{ν} ζ^{ρ} .

Under the coordinate variation scaled by $κ$ ,

δ h = G_{ζ} + κ R_{ζ} h .

The Einstein density varies by a total derivative. The first two coefficient identities are

δ_{0} E_{3} + δ_{1} S_{2} = 0, δ_{0} E_{4} + δ_{1} E_{3} = 0.

Polarizing them gives

v_{3} (G_{ζ}, u, v) v_{4} (G_{ζ}, u, v, w) = - ⟨ K_{0} u, R_{ζ} v ⟩ - ⟨ K_{0} v, R_{ζ} u ⟩, = - v_{3} (R_{ζ} u, v, w) - v_{3} (u, R_{ζ} v, w) - v_{3} (u, v, R_{ζ} w) .

Here pairings include spacetime integration or, for waves, the corresponding stripped conserved-momentum coefficient. The factors agree with the definition of $v_{n}$ : taking two or three distinct field derivatives introduces no extra factorial.

Define the cubic current by

⟨ J (v, w), z ⟩ = v_{3} (z, v, w) .

If $K_{0} v = K_{0} w = 0$ , equation (eq:ward3) with arbitrary $ζ$ implies conservation of $J (v, w)$ . Thus Lemma lem:inverse applies at the physical end of each exchange.

The stripped four-point prescription is

T_{4} = V_{4} - {ij ∣ k l} \in {12∣34, 13∣24, 14∣23} \sum V_{3} (i, j, \cdot) D (k_{i} + k_{j}) V_{3} (\cdot, k, l) .

The common momentum delta, $g^{2}$ , and overall Feynman factor $i$ have been removed.

For free $u, v, w$ ,

v_{3} (G_{ζ}, u, D J (v, w)) = - ⟨ K_{0} D J (v, w), R_{ζ} u ⟩ = - v_{3} (R_{ζ} u, v, w) .

Together with the exchange minus sign in (eq:tree), this cancels the corresponding term of (eq:ward4).

Displayed formulas in this passage

II@2#eq:ward3 · line 768
$v_{3} (G_{ζ}, u, v) v_{4} (G_{ζ}, u, v, w) = - ⟨ K_{0} u, R_{ζ} v ⟩ - ⟨ K_{0} v, R_{ζ} u ⟩, = - v_{3} (R_{ζ} u, v, w) - v_{3} (u, R_{ζ} v, w) - v_{3} (u, v, R_{ζ} w) .$
II@2#eq:ward4 · line 768
$v_{3} (G_{ζ}, u, v) v_{4} (G_{ζ}, u, v, w) = - ⟨ K_{0} u, R_{ζ} v ⟩ - ⟨ K_{0} v, R_{ζ} u ⟩, = - v_{3} (R_{ζ} u, v, w) - v_{3} (u, R_{ζ} v, w) - v_{3} (u, v, R_{ζ} w) .$
II@2#eq:tree · line 794
$T_{4} = V_{4} - {ij ∣ k l} \in {12∣34, 13∣24, 14∣23} \sum V_{3} (i, j, \cdot) D (k_{i} + k_{j}) V_{3} (\cdot, k, l) .$
unlabeled · line 746
$G_{ζ} = \partial ζ + (\partial ζ)^{T}$
unlabeled · line 750
$(R_{ζ} h)_{μν} = ζ^{ρ} \partial_{ρ} h_{μν} + h_{ρ ν} \partial_{μ} ζ^{ρ} + h_{μ ρ} \partial_{ν} ζ^{ρ} .$
unlabeled · line 757
$δ h = G_{ζ} + κ R_{ζ} h .$
unlabeled · line 762
$δ_{0} E_{3} + δ_{1} S_{2} = 0, δ_{0} E_{4} + δ_{1} E_{3} = 0.$
unlabeled · line 785
$⟨ J (v, w), z ⟩ = v_{3} (z, v, w) .$
unlabeled · line 804
$v_{3} (G_{ζ}, u, D J (v, w)) = - ⟨ K_{0} D J (v, w), R_{ζ} u ⟩ = - v_{3} (R_{ζ} u, v, w) .$