Manuscript Em01-G@5 · Em01-G-v5.tex · section · 3fbfe951

Frozen rows and complete seed transpositions

Frozen rows and complete seed transpositions · lines 2318–2402

Use the charts

κ_{47} (c, s, t, y) κ_{5, +} (c, s, t, y) κ_{5, -} (c, s, t, y) κ_{resp} (c, s, r, y) = (c - s, c, c - t, y), = (c, c + s, c - t, y), = (c - t, c + s, c, y), = (c, c + s, c + r, y),

and fields

B_{μ} = \partial_{c^{μ}} + \partial_{s^{μ}} + \partial_{t^{μ}}, Z_{μ} = \partial_{r^{μ}}, A_{μ} = \partial_{c^{μ}} - \partial_{s^{μ}} - \partial_{r^{μ}} .

The complete transposed operators are

P_{47}^{I, J} W = P_{5, +}^{I, J} W = P_{5, -}^{I, J} W = P_{6}^{I, J} W = P_{8}^{I, J} W = P_{9 d}^{I, J} W = P_{9 m}^{I, J} W = - A^{ρ μ, J} A^{σ ν, I} \partial_{s^{σ}} \partial_{t^{ν}} \partial_{t^{ρ}} (f (c) B_{μ} W) - δ^{μ}_{ν} A^{ρ α, J} A^{σ ν, I} \partial_{s^{σ}} \partial_{t^{ρ}} ((\partial_{α} f) (c) B_{μ} W) + δ^{μ}_{ν} A^{ρ α, J} A^{σ ν, I} \partial_{s^{σ}} \partial_{t^{α}} \partial_{t^{ρ}} (f (c) B_{μ} W), A^{α β, I} A^{ρ μ, J} \partial_{s^{α}} \partial_{t^{β}} \partial_{t^{ρ}} (f (c + s) \partial_{s^{μ}} W), A^{α β, J} A^{ρ μ, I} \partial_{s^{α}} \partial_{t^{β}} \partial_{t^{ρ}} (f (c + s) \partial_{s^{μ}} W), - D^{α μ, I, J} \partial_{s^{α}} (f (c + s) \partial_{s^{μ}} W), \sum q_{α β}^{M, I, J} P^{L}_{M} A^{ρ μ}_{L} \frac{s _{ρ}}{2} Z^{α} A^{β} (f (c + s) \partial_{s^{μ}} W), \sum q_{α β}^{M, I, J} P^{L}_{M} A^{μν}_{L} Z^{α} A^{β} (g (c + s) \partial_{μ} F (c + s) \partial_{s^{ν}} W), - m^{2} \sum q_{α β}^{M, I, J} P^{L}_{M} b_{L} Z^{α} A^{β} (g (c + s) F (c + s) W) .

Every derivative acts on its full indicated parenthesis.

The measures are

I_{RR} (V) = \frac{1}{16 π ^{2}} \int \frac{V ( c , ( ∣ x ∣ , x ) , ( ∣ x ^{'} ∣ , x ^{'} ) , y )}{∣ x ∣ ∣ x ^{'} ∣} d^{4} c d^{4} y d^{3} x d^{3} x^{'},

I_{resp} (V) = \frac{1}{4 π} \int \frac{V ( c , ( ∣ x ∣ , x ) , 0 , y )}{∣ x ∣} d^{4} c d^{4} y d^{3} x .

The exact actions are

⟨ K_{j}^{I, J}, W ⟩ = {I_{RR} (P_{j}^{I, J} [W \circ κ_{j}]), I_{resp} (P_{j}^{I, J} [W \circ κ_{resp}]), j = 47, 5 +, 5 -, j = 6, 8, 9 d, 9 m .

They follow by integration by parts against locally finite seed measures. The two $K_{5}$ line signs cancel; the outer derivative supplies the final transposition sign. In $K_{6}$ , the original attachment derivative acts only on the retarded line. The $K_{8}$ multiplier remains outside the source-transposed derivatives. The derivative-root outer sign becomes positive, whereas the mass-root sign stays negative.

Displayed formulas in this passage

unlabeled · line 2336
$P_{47}^{I, J} W = P_{5, +}^{I, J} W = P_{5, -}^{I, J} W = P_{6}^{I, J} W = P_{8}^{I, J} W = P_{9 d}^{I, J} W = P_{9 m}^{I, J} W = - A^{ρ μ, J} A^{σ ν, I} \partial_{s^{σ}} \partial_{t^{ν}} \partial_{t^{ρ}} (f (c) B_{μ} W) - δ^{μ}_{ν} A^{ρ α, J} A^{σ ν, I} \partial_{s^{σ}} \partial_{t^{ρ}} ((\partial_{α} f) (c) B_{μ} W) + δ^{μ}_{ν} A^{ρ α, J} A^{σ ν, I} \partial_{s^{σ}} \partial_{t^{α}} \partial_{t^{ρ}} (f (c) B_{μ} W), A^{α β, I} A^{ρ μ, J} \partial_{s^{α}} \partial_{t^{β}} \partial_{t^{ρ}} (f (c + s) \partial_{s^{μ}} W), A^{α β, J} A^{ρ μ, I} \partial_{s^{α}} \partial_{t^{β}} \partial_{t^{ρ}} (f (c + s) \partial_{s^{μ}} W), - D^{α μ, I, J} \partial_{s^{α}} (f (c + s) \partial_{s^{μ}} W), \sum q_{α β}^{M, I, J} P^{L}_{M} A^{ρ μ}_{L} \frac{s _{ρ}}{2} Z^{α} A^{β} (f (c + s) \partial_{s^{μ}} W), \sum q_{α β}^{M, I, J} P^{L}_{M} A^{μν}_{L} Z^{α} A^{β} (g (c + s) \partial_{μ} F (c + s) \partial_{s^{ν}} W), - m^{2} \sum q_{α β}^{M, I, J} P^{L}_{M} b_{L} Z^{α} A^{β} (g (c + s) F (c + s) W) .$
unlabeled · line 2321
$κ_{47} (c, s, t, y) κ_{5, +} (c, s, t, y) κ_{5, -} (c, s, t, y) κ_{resp} (c, s, r, y) = (c - s, c, c - t, y), = (c, c + s, c - t, y), = (c - t, c + s, c, y), = (c, c + s, c + r, y),$
unlabeled · line 2330
$B_{μ} = \partial_{c^{μ}} + \partial_{s^{μ}} + \partial_{t^{μ}}, Z_{μ} = \partial_{r^{μ}}, A_{μ} = \partial_{c^{μ}} - \partial_{s^{μ}} - \partial_{r^{μ}} .$
unlabeled · line 2373
$I_{RR} (V) = \frac{1}{16 π ^{2}} \int \frac{V ( c , ( ∣ x ∣ , x ) , ( ∣ x ^{'} ∣ , x ^{'} ) , y )}{∣ x ∣ ∣ x ^{'} ∣} d^{4} c d^{4} y d^{3} x d^{3} x^{'},$
unlabeled · line 2379
$I_{resp} (V) = \frac{1}{4 π} \int \frac{V ( c , ( ∣ x ∣ , x ) , 0 , y )}{∣ x ∣} d^{4} c d^{4} y d^{3} x .$
unlabeled · line 2385
$⟨ K_{j}^{I, J}, W ⟩ = {I_{RR} (P_{j}^{I, J} [W \circ κ_{j}]), I_{resp} (P_{j}^{I, J} [W \circ κ_{resp}]), j = 47, 5 +, 5 -, j = 6, 8, 9 d, 9 m .$

Nearby in this section

section: Frozen rows and complete seed transpositions