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Exact regulated gauge and infinitesimal Ward identities

Statement

Assume the finite-graph model of Exact compact U(1) rotor plus finite CAR data and the Axiom of Choice. Let $ψ \in H_{p h y s}$ be a normalized vector. A bounded operator $O$ has gauge charge $q = (q_{x})_{x \in V} \in Z^{V}$ when $U (α) O U (α)^{†} = e^{i \sum_{x} q_{x} α_{x}} O .$ Then $⟨ ψ, O ψ ⟩ = 0$ whenever $q \neq 0$ , and any finite product of bounded gauge-charge eigenoperators has zero expectation when the sum of its charge vectors is nonzero.

For each $x$ , if $O$ maps $D (G_{x})$ into itself and $[G_{x}, O]$ extends to a bounded operator, then $⟨ ψ, [G_{x}, O] ψ ⟩ = 0.$ The same assertions apply to $O (t) = U (t)^{†} O U (t)$ : the finite-regulator time evolution preserves the gauge charge and the domains $D (G_{x})$ . These are exact operator identities for the stated regulator; no trace or path-integral identity and no continuum Ward–Takahashi statement is asserted.

Facts & Assumptions

Given: The finite-graph gauge representation, $H_{p h y s}$ , the self-adjoint gauge generators, and the Axiom of Choice.

[A1]

The Axiom of Choice applies to every family of nonempty sets. The Axiom of Choice

[F1]

Gauss operators, gauge group and nonempty physical projection: $U (α)$ is strongly continuous, $H_{p h y s}$ is its fixed subspace, and the joint Fourier/occupation basis diagonalizes every $G_{x}$ with integer eigenvalues.

[F2]

Global unitary dynamics in the Gauss sector: under the Axiom of Choice, $U (t)$ is a strongly continuous unitary group that commutes with every $U (α)$ and preserves $H_{p h y s}$ .

Proof

Given: [A1], [F1], [F2], and a normalized $ψ \in H_{p h y s}$ .

1.1F1givenalgebra

Physicality gives $U (α) ψ = ψ$ for every $α$ . If $O$ has charge $q$ , then $⟨ ψ, O ψ ⟩ = ⟨ U (α) ψ, O U (α) ψ ⟩ = ⟨ ψ, U (α)^{†} O U (α) ψ ⟩ = e^{- i \sum_{x} q_{x} α_{x}} ⟨ ψ, O ψ ⟩ .$ If $q \neq 0$ , choose a vertex $x$ with $q_{x} \neq 0$ and set $α_{x} = π / q_{x}$ , with all other components zero. The phase is $- 1$ , forcing $⟨ ψ, O ψ ⟩ = 0$ .

1.2F1givenalgebra

For $ψ \in H_{p h y s} \subseteq D (G_{x})$ and $O ψ \in D (G_{x})$ , self-adjointness of $G_{x}$ gives $⟨ ψ, [G_{x}, O] ψ ⟩ = ⟨ G_{x} ψ, O ψ ⟩ - ⟨ ψ, O G_{x} ψ ⟩ = 0.$ If the commutator is initially defined on $D (G_{x})$ , its bounded extension has the same value on these vectors.

1.3A1F1F2algebra

Fix $g \in Z^{V}$ and define the joint character projection $Q_{g} = \frac{1}{(2 π)^{∣ V ∣}} \int_{T^{V}} e^{- i \sum_{x} g_{x} α_{x}} U (α) d α .$ Since the joint basis diagonalizes the integer $G_{x}$ , $Q_{g}$ is the orthogonal projection onto the joint eigenspace with eigenvalue vector $g$ . The unitary $U (t)$ commutes with every $U (α)$ by [F2], hence with every $Q_{g}$ . The exact domain is $D (G_{x}) = {ϕ : \sum_{g \in Z^{V}} g_{x}^{2} ∥ Q_{g} ϕ ∥^{2} < \infty};$ commutation and unitarity therefore show that $U (t)$ preserves $D (G_{x})$ and commutes there with $G_{x}$ .

2.1F1step 1.1algebra

A product $O_{1} \dots O_{k}$ of charge eigenoperators with charges $q^{(j)}$ transforms with charge $\sum_{j} q^{(j)}$ , by inserting $U^{†} U = I$ between adjacent factors. Applying step 1.1 to that product proves the finite-product selection rule.

3.1F2step 1.3step 1.1step 1.2∎

Gauge covariance of $O (t)$ follows from commuting $U (t)$ through the transformation law of $O$ . Unitary conjugation preserves boundedness, and the domain relation in step 1.3 carries any $D (G_{x})$ invariance and bounded commutator of $O$ to $O (t)$ . The selection and infinitesimal identities already proved thus apply at every time.

Exact regulated gauge and infinitesimal Ward identities

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

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