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Continuity and Klein–Gordon annihilation of scalar shell restriction

Statement

Assume Countable Choice The Axiom of Countable Choice ( ${A C}_{ω}$ ). For every $μ \geq 0$ , the pointwise restriction $K_{μ} f (p) = \int_{R^{4}} f (x) e^{i p \circ x} d^{4} x$ of Scalar future mass shell, invariant measure and test restriction maps $S (R^{4})$ continuously and linearly into $H_{μ}$ . It also annihilates the Klein–Gordon test operator: $K_{μ} ((\partial_{0}^{2} - Δ + μ^{2}) f) = 0 (f \in S (R^{4})) .$ The coordinates are the fixed length coordinates of that definition and $μ$ has inverse-length units.

Facts & Assumptions

Given: Countable Choice, the scalar shell and Fourier convention, and the Schwartz seminorm topology.

[A1]

Assume Countable Choice. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

The shell phase is $p \circ x = p^{0} x^{0} - p \cdot x$ , and $K_{μ} f (p) = F_{+} f (p)$ on $p = p_{μ} (k)$ . Scalar future mass shell, invariant measure and test restriction

[F2]

Schwartz space consists of smooth functions whose coordinate-weighted derivatives are bounded; these seminorms define its topology. Schwartz space and its seminorms

[F3]

The standard Fourier transform and inverse are continuous on Schwartz space and are an automorphism; the reflection and rescaling in [F1] preserve this property. Fourier transform is a topological automorphism of Schwartz space

[F4]

For the standard Fourier convention, Fourier transformation is continuous on Schwartz space and transforms derivatives into polynomial multiplication, with the stated $2 π$ factors. After the coordinate reflection and rescaling in [F1], the same derivative identity holds for $F_{+}$ . Fourier transform acts continuously on Schwartz space

Proof

1.1A1F1F2F3F4givenalgebra

Define the continuous Schwartz seminorm $q_{2} (f) = \sup_{p \in R^{4}} (1 + ∣ p ∣)^{2} ∣ F_{+} f (p) ∣ .$ Continuity follows from [F3,F4] and [F2]. On shell, $∣ F_{+} f (p_{μ} (k)) ∣ \leq q_{2} (f) (1 + ∣ k ∣)^{- 2}$ because $∣ p_{μ} (k) ∣ \geq ∣ k ∣$ . Hence $∥ K_{μ} f ∥_{H_{μ}}^{2} \leq \frac{q_{2} (f)^{2}}{2 (2 π)^{3}} \int_{R^{3}} \frac{(1 + ∣ k ∣)^{- 4}}{ω_{μ} (k)} d^{3} k .$ The integral is finite: in polar coordinates its radial integrand is $r^{2} (1 + r)^{- 4} / \sqrt{r^{2} + μ^{2}}$ , which is $O (r)$ near $0$ when $μ = 0$ , is $O (r^{2})$ near $0$ when $μ > 0$ , and is $O (r^{- 3})$ at infinity. Therefore $K_{μ} f \in H_{μ}$ , and the displayed seminorm bound proves continuity and linearity.

2.1A1F1F4step 1.1givenalgebra∎

By [F4], applying $\partial_{0}^{2} - Δ + μ^{2}$ before $F_{+}$ multiplies by $- (p^{0})^{2} + ∣ p ∣^{2} + μ^{2}$ . On the shell this multiplier is $- ω_{μ} (k)^{2} + ∣ k ∣^{2} + μ^{2} = 0$ . Restriction to the shell therefore gives the asserted Klein–Gordon identity.

Continuity and Klein–Gordon annihilation of scalar shell restriction

Statement

Facts & Assumptions

Proof

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