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Regular normal-ordering change is an exact star isomorphism

Statement

Let $W \in D^{'} (X \times X)$ , $X = R^{d}$ , and let $K \in D^{'} (X \times X)$ be symmetric under interchange of its variables. Define $Γ_{K} F (ϕ) = ⟨ K, F^{(2)} (ϕ) ⟩, α_{K} = \exp ⁣ (\frac{η}{2} Γ_{K}) = \sum_{j \geq 0} \frac{(η / 2)^{j}}{j!} Γ_{K}^{j}$ on $P_{r e g} (X) [η]$ . Each sum is finite on each polynomial because $Γ_{K}$ lowers degree by two. Then $α_{K}$ is invertible with inverse $α_{- K}$ , and $α_{K} (F ⋆_{W} G) = α_{K} (F) ⋆_{W + K} α_{K} (G)$ for all regular polynomial functionals $F, G$ . Thus the change from $W$ to $W + K$ is an exact algebra isomorphism on this regular class.

Facts & Assumptions

Given: [F1]–[F4], a symmetric continuous bidistribution $K$ , and the formal-parameter convention of [A1].

[A1]

The parameter $η$ is formal and fixed by complex conjugation.

[F1]

The regular coefficient kernels and their functional derivatives are smooth and compactly supported; compactly supported complex Riemann integrals are evaluated componentwise as defined in Regular polynomial field functional algebra, and the coefficient representation is unique (Tensor-product tests are dense in smooth compact kernel topology).

[F2]

Tensor products of distributions pair independently and iterated pairings associate (Tensor product distributions and iterated pairings).

[F3]

Distributional pairings with smooth common-compact-support parameter families are smooth, with derivatives passed through the pairing (Distribution pairing with smooth parameter families).

[F4]

For every continuous bidistribution $W$ , the regular contraction product $⋆_{W}$ is well defined and associative, with finite sums (Regular Wick-star contractions are well defined and associative).

Proof

Given: [A1]–[F4], $W$ , and symmetric $K$ .

1.1A1F1F2F3construct

For a homogeneous coefficient of degree $m$ , one application of $Γ_{K}$ contracts two derivative variables against $K$ , leaving a smooth compact kernel by [F1]–[F3]. It lowers the polynomial degree by two; therefore $Γ_{K}^{j} F = 0$ when $2 j > \deg F$ , so $α_{K} F$ is a finite sum in $P_{r e g} (X) [η]$ . The same finite argument defines $α_{- K}$ .

2.1A1F1step 1.1algebra

For this computation set $E_{f} (s, ϕ) : = \sum_{m \geq 0} s^{m} ℓ_{f} (ϕ)^{m} / m!$ as a formal generating series. Each second derivative contracts to $K (f, f) E_{f}$ . Coefficientwise this gives $α_{K} E_{f} (s) = \exp ⁣ (\frac{η s^{2}}{2} K (f, f)) E_{f} (s),$ where the exponential is its formal power series; each coefficient used below involves only finitely many derivatives.

3.1step 2.1F4algebra

On two auxiliary generators, the left side of the asserted identity has the scalar factor $\exp ⁣ (η s t W (f, g) + \frac{η}{2} K (s f + t g, s f + t g)) .$ The right side has factor $\exp ⁣ (\frac{η s^{2}}{2} K (f, f) + \frac{η t^{2}}{2} K (g, g) + η s t (W + K) (f, g)) .$ Symmetry and bilinearity of $K$ make these exponents equal.

4.1F1F2F3F4step 3.1algebra∎

Comparing coefficients of $s, t$ in step 3.1 first proves the identity for powers of linear functionals. Replacing $f$ by $\sum_{j} s_{j} f_{j}$ and comparing the coefficient of $s_{1} \dots s_{m}$ , and doing the same independently for $g$ , proves it for every finite product of distinct linear functionals. Each fixed coefficient is continuous in its smooth compact kernels by [F2]–[F4]; the density in [F1] therefore extends the identity to all $F, G \in P_{r e g} (X)$ . For an input of degree $d$ , the coefficient of $Γ_{K}^{r}$ in $α_{K} α_{- K}$ is multiplied by $\sum_{j = 0}^{r} (\binom{r}{j}) (- 1)^{r - j} = (1 - 1)^{r}$ , which is zero for $r > 0$ and one for $r = 0$ . The sums are finite because $Γ_{K}$ lowers degree by two, so $α_{K} α_{- K} = I$ ; the reverse composition is identical.

Regular normal-ordering change is an exact star isomorphism

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources