Manuscript IX-C@1 · IX-C.tex · section · 03ce8e6b

The two-edge expressions

The six placements and the nonconstant outputs › The two-edge expressions · lines 722–748

IX-C@1#eq:g1223, IX-C@1#eq:g1213, IX-C@1#eq:g1323

The following are labelled candidate expressions:

G_{12, 23}^{lab} = G_{12, 13}^{lab} = G_{13, 23}^{lab} = 2 \int f (x) g (y) f^{'} (x^{'}) K (x, y) K (x^{'}, y) \partial_{t} ϕ (x) \partial_{t^{'}} ϕ (x^{'}) d^{4} x d^{4} y d^{4} x^{'}, 2 \int f (x) g (y) f^{'} (x^{'}) K (x, y) S_{0 ν} (x, x^{'}) h_{00} (y) w^{ν} (x^{'}) d^{4} x d^{4} y d^{4} x^{'}, 2 \int f (x) g (y) f^{'} (x^{'}) K (x^{'}, y) S_{μ 0} (x, x^{'}) w^{μ} (x) h_{00} (y) d^{4} x d^{4} y d^{4} x^{'} .

The superscript “lab” means that the expressions identify the desired contraction placements. It does not assert the existence of the indicated distributional products.

Displayed formulas in this passage

IX-C@1#eq:g1223 · line 725
$G_{12, 23}^{lab} = G_{12, 13}^{lab} = G_{13, 23}^{lab} = 2 \int f (x) g (y) f^{'} (x^{'}) K (x, y) K (x^{'}, y) \partial_{t} ϕ (x) \partial_{t^{'}} ϕ (x^{'}) d^{4} x d^{4} y d^{4} x^{'}, 2 \int f (x) g (y) f^{'} (x^{'}) K (x, y) S_{0 ν} (x, x^{'}) h_{00} (y) w^{ν} (x^{'}) d^{4} x d^{4} y d^{4} x^{'}, 2 \int f (x) g (y) f^{'} (x^{'}) K (x^{'}, y) S_{μ 0} (x, x^{'}) w^{μ} (x) h_{00} (y) d^{4} x d^{4} y d^{4} x^{'} .$
IX-C@1#eq:g1213 · line 725
$G_{12, 23}^{lab} = G_{12, 13}^{lab} = G_{13, 23}^{lab} = 2 \int f (x) g (y) f^{'} (x^{'}) K (x, y) K (x^{'}, y) \partial_{t} ϕ (x) \partial_{t^{'}} ϕ (x^{'}) d^{4} x d^{4} y d^{4} x^{'}, 2 \int f (x) g (y) f^{'} (x^{'}) K (x, y) S_{0 ν} (x, x^{'}) h_{00} (y) w^{ν} (x^{'}) d^{4} x d^{4} y d^{4} x^{'}, 2 \int f (x) g (y) f^{'} (x^{'}) K (x^{'}, y) S_{μ 0} (x, x^{'}) w^{μ} (x) h_{00} (y) d^{4} x d^{4} y d^{4} x^{'} .$
IX-C@1#eq:g1323 · line 725
$G_{12, 23}^{lab} = G_{12, 13}^{lab} = G_{13, 23}^{lab} = 2 \int f (x) g (y) f^{'} (x^{'}) K (x, y) K (x^{'}, y) \partial_{t} ϕ (x) \partial_{t^{'}} ϕ (x^{'}) d^{4} x d^{4} y d^{4} x^{'}, 2 \int f (x) g (y) f^{'} (x^{'}) K (x, y) S_{0 ν} (x, x^{'}) h_{00} (y) w^{ν} (x^{'}) d^{4} x d^{4} y d^{4} x^{'}, 2 \int f (x) g (y) f^{'} (x^{'}) K (x^{'}, y) S_{μ 0} (x, x^{'}) w^{μ} (x) h_{00} (y) d^{4} x d^{4} y d^{4} x^{'} .$

The two-edge expressions

Displayed formulas in this passage

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