Manuscript Em02-E@1 · Em02-E-v1.tex · proposition · f3795050

Proposition: First free contraction of the quadratic insertion

The quantum input gate and the first contraction target › The contraction count · lines 1642–1670 · proof included or linked

Em02-E@1#prop:count, Em02-E@1#eq:prospective-contraction

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In a Gaussian free representation, the quadratic insertion

Q_{f} = \frac{1}{2} ⟨ H_{f}, η \otimes η ⟩

has at most one complete pair contraction. If a two-point matrix $H^{A B}$ is specified, the prospective regulated scalar contraction is

\frac{ℏ}{2} A, B \sum ⟨ H^{A B}, H_{f, δ, R}^{A B} ⟩,

with the slot convention fixed by the chosen representation. The linear insertion has no complete self-contraction.

Proof. A complete pair contraction consumes two field factors. There are two in the quadratic insertion and only one possible pair. The factorial coefficient is the $1/2$ in the Hessian representation. A linear insertion contains no pair. For the smoothed compact kernel at finite regulators, the pairing in (eq:prospective-contraction) is a defined distributional pairing whenever $H$ is supplied.

Displayed formulas in this passage

Em02-E@1#eq:prospective-contraction · line 1651
$\frac{ℏ}{2} A, B \sum ⟨ H^{A B}, H_{f, δ, R}^{A B} ⟩,$
unlabeled · line 1645
$Q_{f} = \frac{1}{2} ⟨ H_{f}, η \otimes η ⟩$

Nearby in this section

section: The contraction count