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

The six placements and the nonconstant outputs

The six placements and the nonconstant outputs · lines 679–696

IX-C@1#eq:self

Put

S_{μν} (x, x^{'}) = \partial_{μ}^{x} \partial_{ν}^{x^{'}} Δ_{m}^{+} (x - x^{'}) .

IX-A admits the scalar-contracted sharp self-product

T_{f, f^{'}} [h] = \int f (x) f^{'} (x^{'}) S_{μν} (x, x^{'}) w^{μ} (x) w^{ν} (x^{'}) d^{4} x d^{4} x^{'} .

This is inherited, not a second-stage closure theorem IXA (prop:knownoutputs, item 2).

Displayed formulas in this passage

IX-C@1#eq:self · line 688
$T_{f, f^{'}} [h] = \int f (x) f^{'} (x^{'}) S_{μν} (x, x^{'}) w^{μ} (x) w^{ν} (x^{'}) d^{4} x d^{4} x^{'} .$
unlabeled · line 683
$S_{μν} (x, x^{'}) = \partial_{μ}^{x} \partial_{ν}^{x^{'}} Δ_{m}^{+} (x - x^{'}) .$