Manuscript VII-H@1 · VII-H.tex · section · d20d2a9f

Relative and joint test spaces

Definition sheet for the quantum benchmark › Relative and joint test spaces · lines 571–601

VII-H@1#eq:joint-relative, VII-H@1#eq:correlation-test

Take

E = D (M; R) .

The response kernel in relative coordinates is $R \in D^{'} (M; R)$ . Its joint kernel is the pullback

R (x, y) = R (x - y) .

The difference map $(x, y) \mapsto x - y$ is a submersion, so this pullback exists.

For $H \in D (M \times M; R)$ , define

F_{H} (z) = \int H (y + z, y) d^{4} y .

This is a compact smooth test, and

⟨ R, H ⟩ = ⟨ R, F_{H} ⟩ .

In particular, for $H = f \otimes h$ ,

F_{f, h} (z) = \int f (y + z) h (y) d^{4} y, χ (f, h) = ⟨ R, F_{f, h} ⟩ .

Displayed formulas in this passage

VII-H@1#eq:joint-relative · line 586
$F_{H} (z) = \int H (y + z, y) d^{4} y .$
VII-H@1#eq:correlation-test · line 595
$F_{f, h} (z) = \int f (y + z) h (y) d^{4} y, χ (f, h) = ⟨ R, F_{f, h} ⟩ .$
unlabeled · line 574
$E = D (M; R) .$
unlabeled · line 579
$R (x, y) = R (x - y) .$
unlabeled · line 591
$⟨ R, H ⟩ = ⟨ R, F_{H} ⟩ .$