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Transverse one-photon space and helicity projectors

Definition

Assume the Axiom of Countable Choice ${A C}_{ω}$ . Put $M = R^{3} ∖ {0}$ , $ω (k) = ∣ k ∣$ , and $d ν (k) = d^{3} k / (2 ω (k))$ . Define the unit sphere by $S^{2} = {n \in R^{3} : ∣ n ∣ = 1}$ with the subspace topology. For $j \in {1, 2, 3}$ and $ε \in {+ 1, - 1}$ , chart ${ε n_{j} > 0}$ by the other two coordinates in the open unit disk $B_{1}^{2} = {x \in R^{2} : ∣ x ∣ < 1}$ , with inverse $n_{j} = ε \sqrt{1 - \sum_{i \neq j} n_{i}^{2}}$ ; these six charts and their transition maps are smooth. Let $d Ω$ denote polar surface measure on this sphere. Polar coordinates give $d ν = (r / 2) d r d Ω$ , and every orthogonal map preserves $ν$ . The transverse one-photon Hilbert space $H_{T}^{ν}$ consists of the equivalence classes $f \in L^{2} (M, ν; C^{3})$ satisfying $k \cdot f (k) = 0$ for $ν$ -almost every $k$ , with pairing $⟨ f, g ⟩_{ν} = \int_{M} \overline{f (k)} \cdot g (k) d ν (k)$ , linear in $g$ . The preceding measure-isometry lemma gives the unitary $W f (k) = f (k) / \sqrt{2 ω (k)}$ from this space to the transverse space defined with measure $d^{3} k$ . The null set $k = 0$ is omitted throughout.

For $n \in S^{2}$ , define $P (n) = I - n n^{T}$ . Extend the real cross product complex-linearly by $(n \times v)_{i} = \sum_{j, ℓ = 1}^{3} ϵ_{i j ℓ} n_{j} v_{ℓ}$ , where $ϵ_{i j ℓ}$ is totally antisymmetric and $ϵ_{123} = 1$ , and set $h (n) = i n \times$ and $P_{σ} (n) = (P (n) + σ h (n)) / 2$ for $σ \in {+ 1, - 1}$ . Each $P_{σ} (n)$ is a rank-one orthogonal projector on the complex transverse plane $P (n) C^{3}$ , and $h$ has eigenvalue $σ$ on its range. The fibers $L_{σ, n} = ran P_{σ} (n)$ form smooth complex line subbundles of $S^{2} \times C^{3}$ . Here a smooth line subbundle means a family of one-dimensional complex fibers with smooth local unit sections. The common smooth domain for differential calculations is

$D = {f \in C_{c}^{\infty} (M; C^{3}) : k \cdot f (k) = 0 for every k \in M} .$

It is dense in $H_{T}^{ν}$ . For $0 < r_{0} < r_{1} < \infty$ , the mass-shell measure of the radial annulus is $ν ({r_{0} \leq ∣ k ∣ \leq r_{1}}) = π (r_{1}^{2} - r_{0}^{2})$ ; in particular $ν$ is sigma-finite.

Definition references

The mass-shell weighted transverse space matches the Lebesgue transverse space

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

Positive transverse photon one-particle space

Schwartz/distribution spaces and Hilbert Fock conventions

Facts & Assumptions

Given: ${A C}_{ω}$ ; the weighted and unweighted transverse Hilbert spaces; $S^{2}$ with its outward orientation; and the Euclidean coordinate and measure conventions.

[A1]

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

[F1]

The map $W f = f / \sqrt{2 ∣ k ∣}$ is a unitary from the weighted transverse space with $d ν = d^{3} k / (2 ∣ k ∣)$ onto the published transverse space with $d^{3} k$ , preserving the second-variable-linear pairing. The mass-shell weighted transverse space matches the Lebesgue transverse space

[F2]

Under ${A C}_{ω}$ , $C_{c}^{\infty} (R^{3}; C)$ is dense in $L^{2} (R^{3}, d^{3} k; C)$ ; the componentwise statement gives density for $C^{3}$ -valued functions. Compactly supported smooth functions are dense in W^{k,p}(R^n)

[F3]

Polar coordinates give $d^{3} k = r^{2} d r d Ω$ and the polar surface measure of $S^{2}$ has total mass $4 π$ . Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma Zonal and quadratic sphere moments for the polar surface measure on $S^{2}$

[F4]

Orthogonal transformations of $R^{3}$ preserve polar surface measure on $S^{2}$ . Agreement with the existing polar sphere measure

[F5]

If $0 < r < R$ , there is a smooth function $ρ : R^{3} \to [0, 1]$ that equals $1$ on ${\overline{B}}_{r} (0)$ and has support in $B_{R} (0)$ . A smooth bump between concentric Euclidean balls

Proof

Given: [A1], [F1]–[F5], and the definitions in the Statement.

1.1givenalgebra

Since $∣ n ∣ = 1$ , $P = I - n n^{T}$ satisfies $P^{*} = P = P^{2}$ and is the orthogonal projection onto $n^{⊥}$ ; writing $z = P z + n (n \cdot z)$ gives $∥ P z ∥^{2} = ∥ z ∥^{2} - ∣ n \cdot z ∣^{2}$ , so $∥ P ∥ \leq 1$ . Write $C_{n} v = n \times v$ , whose matrix is real skew-symmetric. Expanding with $\sum_{k} ϵ_{i j k} ϵ_{ℓ m k} = δ_{i ℓ} δ_{j m} - δ_{i m} δ_{j ℓ}$ gives $(C_{n}^{2} v)_{i} = n_{i} (n \cdot v) - v_{i} = - [P (n) v]_{i}$ . Antisymmetry also gives $n \cdot C_{n} v = 0$ and $C_{n} n = 0$ , so $P C_{n} = C_{n} P = C_{n}$ . Consequently $h (n)^{*} = h (n)$ , $h (n)^{2} = P (n)$ , and $P h = h P = h$ .

1.2F3F4givenalgebra

Applying the polar-coordinate formula in [F3] to the density $(2 r)^{- 1}$ gives $d ν = (r / 2) d r d Ω$ . Orthogonal maps preserve $d Ω$ by [F4] and fix $r$ , so they preserve $ν$ .

1.3F3givenalgebra

Applying [F3] to the radial density $(2 r)^{- 1}$ gives $ν ({r_{0} \leq ∣ k ∣ \leq r_{1}}) = \frac{1}{2} \int_{r_{0}}^{r_{1}} r d r \int_{S^{2}} d Ω = π (r_{1}^{2} - r_{0}^{2}) .$ The annuli ${1 / m \leq ∣ k ∣ \leq m}$ have finite measure and cover $M$ , so $ν$ is sigma-finite.

1.4A1F1F2choose

Given $f \in H_{T}^{ν}$ , put $u = W f \in H_{T}^{0}$ by [F1]. Write $u = (u_{1}, u_{2}, u_{3})$ . For each of the three components, [F2] and [A1] give a compactly supported smooth $v_{j}$ with $∥ u_{j} - v_{j} ∥_{L^{2} (d^{3} k)} < ε / (3 \sqrt{3})$ . The three component choices are permitted by Countable Choice, and $v = (v_{1}, v_{2}, v_{3})$ then satisfies $∥ u - v ∥_{L^{2} (d^{3} k)} < ε / 3$ .

1.5F3F5givenalgebra

For each $m$ , [F5] supplies a smooth cutoff $ρ_{m}$ equal to $1$ on ${\overline{B}}_{1 / m} (0)$ and supported in $B_{2 / m} (0)$ ; set $χ_{m} = 1 - ρ_{m}$ . By [F3], $λ_{3} (B_{2 / m} (0)) = 4 π \int_{0}^{2 / m} r^{2} d r = (4 π / 3) (2 / m)^{3}$ . Since $v$ is bounded, $∥ χ_{m} v - v ∥_{L^{2} (d^{3} k)} \leq ∥ v ∥_{\infty} \sqrt{λ_{3} (B_{2 / m} (0))} ⟶ 0.$ Choose $m$ large enough that this norm is below $2 ε / 3$ . The field $χ_{m} v$ is smooth, compactly supported away from $0$ , and can therefore be projected with the smooth matrix $P (k / ∣ k ∣)$ .

2.1step 1.1algebra

The operators $P_{σ}$ are self-adjoint and $P_{σ}^{2} = \frac{P + 2 σ h + h^{2}}{4} = P_{σ}, P_{+} P_{-} = 0, P_{+} + P_{-} = P .$ Moreover $h P_{σ} = σ P_{σ}$ . The trace of $P$ is $2$ and the trace of $h = i C_{n}$ is $0$ , so $tr P_{σ} = 1$ ; a finite-dimensional orthogonal projection has rank equal to its trace, hence each range is a complex line.

3.1step 2.1construct

The matrices $P_{σ} (n)$ depend smoothly on $n$ . For any $n_{0}$ , choose $v \in C^{3}$ with $P_{σ} (n_{0}) v \neq 0$ . Continuity makes $P_{σ} (n) v$ nonzero on a neighborhood of $n_{0}$ , and normalizing this smooth vector gives a smooth unit local section. Thus the ranges form the stated smooth line subbundles.

4.1F1step 1.1step 1.4step 1.5construct∎

Set $g_{m} (k) = P (k / ∣ k ∣) χ_{m} (k) v (k)$ . It is smooth, transverse, and compactly supported away from $0$ . Since $P u = u$ for the transverse field $u$ and $∥ P (k / ∣ k ∣) ∥ \leq 1$ by step 1.1, $∥ g_{m} - u ∥_{L^{2} (d^{3} k)} \leq ∥ χ_{m} v - v ∥_{L^{2} (d^{3} k)} + ∥ v - u ∥_{L^{2} (d^{3} k)} .$ The choices in steps 1.4 and 1.5 make the right side less than $ε$ . The inverse image $W^{- 1} g_{m} = \sqrt{2 ∣ k ∣} g_{m}$ is smooth and compactly supported away from $0$ , hence belongs to $D$ . As $W$ is unitary by [F1], $D$ is dense in $H_{T}^{ν}$ .

Transverse one-photon space and helicity projectors

Definition

Definition references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources