Manuscript Em01-F@3 · Em01-F-v3.tex · section · 225e59af

Exact cone-seed transposition

Exact cone-seed transposition · lines 2301–2420

Em01-F@3#eq:Irr, Em01-F@3#eq:Iresp, Em01-F@3#eq:P47, Em01-F@3#eq:P5p, Em01-F@3#eq:P5m, Em01-F@3#eq:P6, Em01-F@3#eq:P8, Em01-F@3#eq:P9d, Em01-F@3#eq:P9m

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For compact smooth $V$ , define

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 .

Both are ordinary absolutely convergent integrals. The cone-tip weights are locally integrable.

The determinant-one charts are

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) .

Write $W_{j} = W \circ κ_{j}$ .

In the $47$ -chart,

B_{μ} = \partial_{c^{μ}} + \partial_{s^{μ}} + \partial_{t^{μ}} .

Set

P_{47}^{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) .

For the other two-cone charts,

P_{5, +}^{I, J} W P_{5, -}^{I, J} 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) .

In the response chart let

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

They represent original $\partial_{z}, \partial_{a}$ , respectively. Define

P_{6}^{I, J} W = - D^{α μ, I, J} \partial_{s^{α}} (f (c + s) \partial_{s^{μ}} W),

P_{8}^{I, J} W = α, β, M, L \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),

P_{9, der}^{I, J} W = α, β, M, L \sum q_{α β}^{M, I, J} P^{L}_{M} A^{μν}_{L} \times Z^{α} A^{β} (g (c + s) (\partial_{μ} F) (c + s) \partial_{s^{ν}} W),

P_{9, mass}^{I, J} W = - m^{2} α, β, M, L \sum q_{α β}^{M, I, J} P^{L}_{M} b_{L} \times Z^{α} A^{β} (g (c + s) F (c + s) W) .

Every derivative acts on the entire parenthesis.

Displayed formulas in this passage

Em01-F@3#eq:Irr · line 2305
$I_{RR} (V) = \frac{1}{16 π ^{2}} \int \frac{V ( c , ( ∣ x ∣ , x ) , ( ∣ x ^{'} ∣ , x ^{'} ) , y )}{∣ x ∣ ∣ x ^{'} ∣} d^{4} c d^{4} y d^{3} x d^{3} x^{'},$
Em01-F@3#eq:Iresp · line 2316
$I_{resp} (V) = \frac{1}{4 π} \int \frac{V ( c , ( ∣ x ∣ , x ) , 0 , y )}{∣ x ∣} d^{4} c d^{4} y d^{3} x .$
Em01-F@3#eq:P47 · line 2343
$P_{47}^{I, J} W = - A^{ρ μ, J} A^{σ ν, I} \partial_{s^{σ}} \partial_{t^{ν}} \partial_{t^{ρ}} (f (c) B_{μ} W) - δ^{μ}_{ν} A^{ρ α, J} A^{σ ν, I} \partial_{s^{σ}} \partial_{t^{ρ}} ((\partial_{α} f) (c) B_{μ} W) + δ^{μ}_{ν} A^{ρ α, J} A^{σ ν, I} \partial_{s^{σ}} \partial_{t^{α}} \partial_{t^{ρ}} (f (c) B_{μ} W) .$
Em01-F@3#eq:P5p · line 2359
$P_{5, +}^{I, J} W P_{5, -}^{I, J} W = A^{α β, I} A^{ρ μ, J} \partial_{s^{α}} \partial_{t^{β}} \partial_{t^{ρ}} (f (c + s) \partial_{s^{μ}} W), = A^{α β, J} A^{ρ μ, I} \partial_{s^{α}} \partial_{t^{β}} \partial_{t^{ρ}} (f (c + s) \partial_{s^{μ}} W) .$
Em01-F@3#eq:P5m · line 2359
$P_{5, +}^{I, J} W P_{5, -}^{I, J} W = A^{α β, I} A^{ρ μ, J} \partial_{s^{α}} \partial_{t^{β}} \partial_{t^{ρ}} (f (c + s) \partial_{s^{μ}} W), = A^{α β, J} A^{ρ μ, I} \partial_{s^{α}} \partial_{t^{β}} \partial_{t^{ρ}} (f (c + s) \partial_{s^{μ}} W) .$
Em01-F@3#eq:P6 · line 2381
$P_{6}^{I, J} W = - D^{α μ, I, J} \partial_{s^{α}} (f (c + s) \partial_{s^{μ}} W),$
Em01-F@3#eq:P8 · line 2388
$P_{8}^{I, J} W = α, β, M, L \sum q_{α β}^{M, I, J} P^{L}_{M} A^{ρ μ}_{L} \frac{s _{ρ}}{2} Z^{α} A^{β} (f (c + s) \partial_{s^{μ}} W),$
Em01-F@3#eq:P9d · line 2397
$P_{9, der}^{I, J} W = α, β, M, L \sum q_{α β}^{M, I, J} P^{L}_{M} A^{μν}_{L} \times Z^{α} A^{β} (g (c + s) (\partial_{μ} F) (c + s) \partial_{s^{ν}} W),$
Em01-F@3#eq:P9m · line 2409
$P_{9, mass}^{I, J} W = - m^{2} α, β, M, L \sum q_{α β}^{M, I, J} P^{L}_{M} b_{L} \times Z^{α} A^{β} (g (c + s) F (c + s) W) .$
unlabeled · line 2328
$κ_{47} (c, s, t, y) κ_{5, +} (c, s, t, y) κ_{5, -} (c, s, t, y) κ_{resp} (c, s, r, y) = (c - s, c, c - t, y), = (c, c + s, c - t, y), = (c - t, c + s, c, y), = (c, c + s, c + r, y) .$
unlabeled · line 2339
$B_{μ} = \partial_{c^{μ}} + \partial_{s^{μ}} + \partial_{t^{μ}} .$
unlabeled · line 2375
$Z_{μ} = \partial_{r^{μ}}, A_{μ} = \partial_{c^{μ}} - \partial_{s^{μ}} - \partial_{r^{μ}} .$

Nearby in this section

theorem: Theorem: Exact seed action