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Bounded local photon Weyl observable algebra

Statement

Assume Countable Choice and use the Minkowski coordinates and separation condition of Local covariant physical field-strength observables, so $η (x - y, x - y) = ∣ x - y ∣^{2} - (x^{0} - y^{0})^{2}$ . Use the Hilbert space and projection of Positive transverse photon one-particle space: $H = L^{2} (R^{3}, d^{3} k; C^{3})$ with pairing $⟨ u, v ⟩_{2} = \int \overline{u} \cdot v d^{3} k$ , the matrix $P_{T} (k) = I_{3} - n n^{T}$ for $n = k / ∣ k ∣$ and $k \neq 0$ , and its bounded multiplication operator $M_{P_{T}} : H \to H$ , $(M_{P_{T}} u) (k) = P_{T} (k) u (k)$ . The value at zero is immaterial in $L^{2}$ , and $H_{T} = Ran M_{P_{T}} = {u \in H : k \cdot u (k) = 0 a.e.}$ has the inherited Hilbert pairing. A spacetime region $O$ means an open subset of $R^{4}$ in its standard topology. A real antisymmetric test $h = (h^{a b})$ is supported in $O$ when each component $h^{a b} \in C_{c}^{\infty} (R^{4}; R)$ has support contained in $O$ . Two open regions $O_{1}, O_{2}$ are spacelike separated when $η (x - y, x - y) > 0$ for every $x \in O_{1}$ and $y \in O_{2}$ . Let $g_{h} \in H_{T}$ be the one-particle vector for such a test from Local covariant physical field-strength observables. The space $H_{T}$ is separable in its Hilbert norm topology: it is the range of $M_{P_{T}}$ on the separable space $H$ . Define $W_{O} (h) = W_{S e g a l} (\sqrt{2} g_{h}),$ where $W_{S e g a l}$ is the Weyl unitary of Self-adjoint Segal fields and explicit Weyl unitaries. It is bounded and unitary, and its one-parameter group has the self-adjoint field-strength closure as generator. For every open region $O \subseteq R^{4}$ , let $A (O)$ be the norm-closed unital $*$ -algebra generated by $W_{O} (h)$ for real antisymmetric tests supported in $O$ ; equivalently its bounded real and imaginary parts are included as observables. If $O_{1}, O_{2}$ are spacelike separated by the stated Minkowski condition, then $A (O_{1})$ and $A (O_{2})$ commute.

Facts & Assumptions

Given: Countable Choice; the local field-strength test vectors; the bosonic CCR; the countable-density result; and the exact Segal/Weyl-domain supplier.

[A1]

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

[F1]

A compact real antisymmetric test defines $F (h) = a (g_{h}) + a^{†} (g_{h})$ , and spacelike-separated field-strength tests commute. Local covariant physical field-strength observables

[F2]

The bosonic CCR on finite-particle vectors are $[a (u), a^{†} (v)] = ⟨ u, v ⟩_{2} I$ . Bosonic and fermionic Fock operators with exact domains

[F3]

Under Countable Choice, finite rational linear combinations of indicators of rational boxes form a countable dense subset of real $L^{2} (R^{3})$ . Rational box-step functions form a countable dense subset of $L^{p} (R^{n})$ for $1 \leq p < \infty$

[F4]

For separable $H$ , the Segal field is essentially self-adjoint on $D_{f i n}$ , its closure is the generator of $W_{S e g a l} (h)$ , and $W_{S e g a l} (u) W_{S e g a l} (v) = e^{- i Im ⟨ u, v ⟩_{2} / 2} W_{S e g a l} (u + v) .$ Self-adjoint Segal fields and explicit Weyl unitaries

[F5]

A finite product of at most countable sets is at most countable, by iterating the two-set product theorem. A product of two at most countable sets is at most countable

[F6]

A topological space is separable when it has an at most countable dense subset. Separability: the existence of an at most countable dense subset

[F7]

The operator $M_{P_{T}}$ on $H = L^{2} (R^{3}, d^{3} k; C^{3})$ is an orthogonal projection of norm at most one, and its range $H_{T}$ is a closed positive Hilbert space with the second-variable-linear integral pairing. Positive transverse photon one-particle space

Proof

Given: [A1], [F1]–[F4], and the second-variable-linear Fock pairing.

1.1A1F3F4F5F6F7given

Let $D$ be the real countable dense family in [F3]. By [F5], $D + i D$ is countable and dense in complex scalar $L^{2}$ , because the squared norm of a complex error is the sum of the squared real- and imaginary-part errors. Iterating [F5], its finite triple product gives a countable dense set $D_{3} \subset H$ . By [F7], $M_{P_{T}} (D_{3})$ is dense in $H_{T}$ : for $u \in H_{T}$ and $v \in D_{3}$ , $∥ M_{P_{T}} v - u ∥_{2} = ∥ M_{P_{T}} (v - u) ∥_{2} \leq ∥ v - u ∥_{2}$ . Its image is at most countable by enumerating the projected elements. By [F6], $H_{T}$ is separable, so [F4] applies to this positive Hilbert space with its second-variable-linear pairing and gives a bounded unitary $W_{S e g a l} (u)$ for every $u \in H_{T}$ .

2.1F1F2step 1.1algebra

From [F1]–[F2], on $D_{f i n}$ , $[F (h), F (k)] = [a (g_{h}) + a^{†} (g_{h}), a (g_{k}) + a^{†} (g_{k})] = (⟨ g_{h}, g_{k} ⟩_{2} - ⟨ g_{k}, g_{h} ⟩_{2}) I = 2 i Im ⟨ g_{h}, g_{k} ⟩_{2} I .$ For spacelike-separated supports the left side vanishes by [F1], hence $Im ⟨ g_{h}, g_{k} ⟩_{2} = 0$ .

2.2F1F4step 1.1

The supplier's one-parameter group generated by the closed Segal field gives $W_{S e g a l} (\sqrt{2} g_{h})$ with generator $\sqrt{2} Φ (g_{h}) = \overline{a (g_{h}) + a^{†} (g_{h})}$ , which is the closure of $F (h)$ from [F1]. Thus $W_{O} (h)$ is the bounded Weyl observable associated to this field strength. It is unitary, and its real and imaginary parts are bounded self-adjoint operators with norm at most one.

3.1F4step 2.1given∎

The Weyl relation [F4] gives $W_{O} (h) W_{O} (k) = e^{- i Im ⟨ \sqrt{2} g_{h}, \sqrt{2} g_{k} ⟩_{2} / 2} W_{S e g a l} (\sqrt{2} g_{h} + \sqrt{2} g_{k}) .$ By step 2.1 the phase is one for spacelike-separated tests, so $W_{O} (h)$ and $W_{O} (k)$ commute. Products, adjoints, and norm limits preserve commutation; therefore the norm-closed algebras of two spacelike-separated regions commute. Since $W_{S e g a l} (0) = I$ , these algebras are unital.

Bounded local photon Weyl observable algebra

Statement

Facts & Assumptions

Proof

Depends on

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