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Local covariant free field-strength and Weyl observables

Physical scope

Gauge-invariant observables of the specified positive transverse free-photon Fock model, smeared with compactly supported real tests; local commutativity applies to spacelike-separated supports, while the Coulomb-gauge potential is not claimed local.

Statement

Assume Countable Choice The Axiom of Countable Choice ( ${A C}_{ω}$ ) for the constructed Fock representation. In the adopted model Positive transverse free photon model, let $h^{a b} \in C_{c}^{\infty} (R^{1, 3}; R)$ be a real antisymmetric two-tensor test. By the natural field-strength normalization of Quantum field quantities and compatible SI normalizations, choose each component of $h^{a b}$ with unit $m^{- 2}$ . Since $f_{a b}$ has unit $m^{- 2}$ and $d^{4} x$ has unit $m^{4}$ , $f (h)$ below is dimensionless. The corresponding one-particle vector $g_{h}$ has profile unit $m^{3 / 2}$ under the coefficient formula of Smeared Coulomb photon operator distribution: its effective test $j^{i} = - \partial_{a} h^{a i}$ has unit $m^{- 3}$ , the Fourier integral contributes $m^{1}$ , and the factor $(2 ω)^{- 1 / 2}$ contributes $m^{1 / 2}$ . Thus its Hilbert norm under $d^{3} k$ of unit $m^{- 3}$ is dimensionless, as is the Weyl parameter $\sqrt{2} g_{h}$ . This is the test normalization adopted here, not a universal test-function unit. Here $f_{a b}$ is the natural-normalized field-strength distribution of the selected free model, corresponding to $F_{a b}$ in the mathematical free-photon suppliers and distinct from the SI tensor $F_{S I, a b}$ . Its smearing $f (h) = \frac{1}{2} \int f_{a b} (x) h^{a b} (x) d^{4} x$ is a well-defined operator-valued distribution on the common finite-particle domain $D_{f i n}$ , not a pointwise field operator. It is gauge invariant and covariant under the selected free mass-cone unitary action. For two such tests with spacelike-separated supports, $[f (h), f (ℓ)] = 0$ on $D_{f i n}$ . Let $g_{h}$ be the associated one-particle test vector and define the Weyl observable $W_{h} : = W_{S e g a l} (\sqrt{2} g_{h})$ as in Bounded local photon Weyl observable algebra. These are bounded unitaries of norm one; their real and imaginary parts are bounded Hermitian operators and commute for spacelike-separated supports. No locality claim is made for a Coulomb-gauge potential.

Facts & Assumptions

Given: Countable Choice; the positive transverse free-photon model; the current field-strength carrier; and the current bounded Weyl-observable carrier at their inspected conditional interfaces.

[A1]

Positive transverse free photon model: the selected physical state space is the bosonic Fock space over the positive transverse one-particle space, and its smeared field strengths and bounded Weyl observables are adopted within that free model. It supplies no pointwise field-strength operator. The postulate has a current escalated Step 1 record and is used conditionally.

[A2]

Countable Choice is assumed by the transverse Fock, Segal-field, and Weyl-domain constructions. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F4]

Quantum field quantities and compatible SI normalizations: coordinates have unit metre, the natural field strength has unit $m^{- 2}$ , and test-function units may be chosen so a specified smeared observable is dimensionless; no universal test unit is fixed.

[F5]

Smeared Coulomb photon operator distribution: the coefficient map is $g_{j} (k) = (2 π)^{- 3 / 2} (2 ω)^{- 1 / 2} P_{T} (k) \int j (x) e^{i ω x^{0} - i k \cdot x} d^{4} x$ , with $ω = ∣ k ∣$ and $d^{3} k$ as the one-particle measure; Local covariant physical field-strength observables uses this map with $j^{i} = - \partial_{a} h^{a i}$ .

[F1]

Local covariant physical field-strength observables: for a real compactly supported antisymmetric test, $F (h) = \frac{1}{2} \int F_{a b} h^{a b} = - A_{b} (\partial_{a} h^{a b})$ is a smeared operator on $D_{f i n}$ with one-particle vector $g_{h}$ ; it is gauge invariant and covariant, its commutators are supported in the closed causal cone, and spacelike-separated supports commute. The draft carrier has a current escalated Step 1 decision because its scalar causal-commutator supplier remains escalated through QFT-MF01/MF02.

[L1]

Bounded local photon Weyl observable algebra: it defines $W_{O} (h) = W_{S e g a l} (\sqrt{2} g_{h})$ , proves this operator is unitary with the field-strength closure as generator, and proves that the norm-closed unital $*$ -algebras generated in spacelike-separated regions commute. The draft carrier has a current escalated Step 1 decision because its field- strength prerequisite remains escalated; its Fock CCR and Segal-domain inputs have current ready records.

Proof

Given: [A1]–[A2], [F1], [F4]–[F5], and the conditional construction [L1].

1.1A1A2F1F4F5given

By [F4], the assigned units give the integrand $f_{a b} h^{a b} d^{4} x$ unit $m^{- 2} m^{- 2} m^{4} = 1$ , so $f (h)$ is dimensionless. By [F5], $j^{i} = - \partial_{a} h^{a i}$ has unit $m^{- 3}$ ; its spacetime Fourier integral has unit $m$ , and $(2 ω)^{- 1 / 2}$ has unit $m^{1 / 2}$ . Thus $g_{h}$ has profile unit $m^{3 / 2}$ under $d^{3} k$ of unit $m^{- 3}$ , making its Hilbert norm and $\sqrt{2} g_{h}$ dimensionless. The current operator construction is conditional on Countable Choice [A2]. The test $h$ is real, smooth, compactly supported, and antisymmetric, so it is an admissible input to [F1]. That carrier constructs $F (h) = \frac{1}{2} \int F_{a b} h^{a b} = - A_{b} (\partial_{a} h^{a b})$ as a smearing on $D_{f i n}$ and supplies its one-particle vector $g_{h}$ . Consequently the present $f (h)$ is exactly that free-model smearing under the stated natural-normalization identification; it does not assign an operator to $f_{a b} (x)$ at a point. Gauge invariance and covariance follow from the same conditional carrier.

1.2A1F1L1given

Suppose the compact supports of $h$ and $ℓ$ are spacelike separated. Their difference set is compact and disjoint from the closed causal cone, so it has positive Euclidean distance from that closed set. Choose bounded small open neighborhoods $O_{h}, O_{ℓ}$ of the two supports whose difference set remains outside the cone. By [F1], the smeared field-strength commutator vanishes on $D_{f i n}$ . By [L1], $W_{h} \in A (O_{h})$ and $W_{ℓ} \in A (O_{ℓ})$ , and these local algebras commute. Their adjoints and linear combinations belong to the same respective algebras; therefore the real and imaginary Hermitian parts of $W_{h}, W_{ℓ}$ commute. This is a conditional deduction in the selected free model, not an observed outcome.

2.1A1L1step 1.2algebra∎

By [L1], $W_{h}$ is unitary, so $∥ W_{h} ∥ = 1$ . The operators $(W_{h} + W_{h}^{*}) / 2$ and $(W_{h} - W_{h}^{*}) / (2 i)$ are self-adjoint, and the triangle inequality gives each norm at most one. They are therefore bounded local Hermitian observables in the adopted model.

Local covariant free field-strength and Weyl observables

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources