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Explicit regular quadratic Wick star product

Example

Let $f, g \in D (R^{d})$ , let $ℓ_{f} (ϕ) = \int f (x) ϕ (x) d x$ , and let $W$ be any continuous bidistribution. For the regular quadratic functionals $F (ϕ) = ℓ_{f} (ϕ)^{2}$ and $G (ϕ) = ℓ_{g} (ϕ)^{2}$ , the finite contraction formula is $F ⋆_{W} G = ℓ_{f}^{2} ℓ_{g}^{2} + 4 η W (f, g) ℓ_{f} ℓ_{g} + 2 η^{2} W (f, g)^{2} .$ The formula is exact in the formal parameter $η$ .

Facts & Assumptions

[A1]

Let $d \geq 1$ , $X = R^{d}$ , $f, g \in D (X)$ , and define $ℓ_{f} (ϕ) = \int_{X} f (x) ϕ (x) d x$ and $ℓ_{g}$ analogously. Let $W \in D^{'} (X \times X)$ be a continuous complex-bilinear distribution, and let $η$ be formal.

[F1]

The first two derivative kernels of $ℓ_{f}^{2}$ are $2 ℓ_{f} (ϕ) f$ and $2 f \otimes f$ , and higher derivatives vanish; the same formulas hold for $g$ . The integrals defining $ℓ_{f}$ and $ℓ_{g}$ use the componentwise complex Riemann convention of Regular polynomial field functional algebra.

[F2]

A distribution is a continuous complex-linear functional on the test space with bilinear pairing (Distribution).

[F3]

For continuous bidistributions, the regular star product is the finite contraction sum $F ⋆_{W} G = \sum_{k} η^{k} ⟨ W^{\otimes k}, F^{(k)} \otimes G^{(k)} ⟩ / k!$ (Regular Wick-star contractions are well defined and associative).

Verification

Given: [A1]–[F3].

1.1givenA1F1F2F3algebra

Substituting the derivatives in [F1] into [F3], terms with $k \geq 3$ vanish, so the only contributions are $k = 0, 1, 2$ .

2.1step 1.1A1F1F2F3algebra

The $k = 0$ term is $ℓ_{f}^{2} ℓ_{g}^{2}$ . The $k = 1$ term is $η ⟨ W, (2 ℓ_{f} f) \otimes (2 ℓ_{g} g) ⟩ = 4 η W (f, g) ℓ_{f} ℓ_{g}$ .

3.1step 1.1F1F2F3algebra∎

The $k = 2$ term is $\frac{η^{2}}{2} ⟨ W^{\otimes 2}, (2 f \otimes f) \otimes (2 g \otimes g) ⟩ = 2 η^{2} W (f, g)^{2}$ , because the two independent pairings each equal $W (f, g)$ . Adding the three contributions proves the displayed formula.

Explicit regular quadratic Wick star product

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

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