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Time-ordered Ward contact relation

Statement

Assume the compact finite-graph model and the Axiom of Choice. Fix a vertex $x$ , pairwise distinct real times $t_{1}, \dots, t_{n}$ with $n \geq 1$ , bounded operators $O_{j}$ , and a normalized physical vector $ψ$ . Assume each $O_{j}$ maps $D (G_{x})$ into itself and has a bounded commutator with $G_{x}$ . Let $O_{j} (t_{j}) = U (t_{j})^{†} O_{j} U (t_{j}), G_{x} (t) = U (t)^{†} G_{x} U (t),$ where $U (t) = e^{- i t H / ℏ}$ . Define ordinary time ordering by placing the factor with the larger time to the left; $G_{x}$ is bosonic, so moving it past an insertion introduces no grading sign. Then $G_{x} (t) = G_{x}$ , and, as a distribution in $t$ , $\frac{d}{d t} ⟨ ψ, T ⁣ [G_{x} (t) \prod_{j = 1}^{n} O_{j} (t_{j})] ψ ⟩ = \sum_{j = 1}^{n} δ (t - t_{j}) ⟨ ψ, T ⁣ ([G_{x}, O_{j} (t_{j})] \prod_{i \neq j} O_{i} (t_{i})) ψ ⟩ .$ The products and commutators are evaluated on the common domains just specified. This is a finite-regulator contact identity, not a claim about continuum renormalized time-ordered products.

Facts & Assumptions

Given: The exact finite-graph Hamiltonian, its gauge group and physical subspace, the Axiom of Choice, pairwise distinct insertion times, and domain-preserving bounded insertions with bounded gauge-generator commutators.

[A1]

The Axiom of Choice states that every family of nonempty sets has a choice function. The Axiom of Choice

[F1]

Exact regulated gauge and infinitesimal Ward identities: time-evolved bounded insertions preserve their gauge charges and $D (G_{x})$ ; the time evolution commutes with the gauge group.

[F2]

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

Proof

Given: [A1], [F1], [F2], and the domain hypotheses in the Statement.

1.1A1F1F2given

By the Axiom of Choice [A1], the self-adjoint $H$ generates the unitary group in [F2]. Its commutation with every gauge transformation implies commutation with the self-adjoint generators $G_{x}$ on $D (G_{x})$ ; the domain-preserving statement is established in [F1]. Hence $G_{x} (t) = G_{x}$ . Each $O_{j} (t_{j})$ is bounded and preserves $D (G_{x})$ by [F1], so every time-ordered product in the Statement is defined on $ψ$ .

2.1step 1.1given

Between two successive insertion times, the ordering of the fixed $O_{j} (t_{j})$ and the single variable-time factor $G_{x} (t)$ does not change. Since $G_{x} (t) = G_{x}$ , the ordered operator product and its expectation are constant on each such open interval.

2.2givenstep 1.1algebra

When $t$ increases through $t_{j}$ , the factor $G_{x}$ moves from the right of $O_{j} (t_{j})$ to its left. The right-limit minus the left-limit of the ordered product is therefore the same ordered product with that factor replaced by $G_{x} O_{j} (t_{j}) - O_{j} (t_{j}) G_{x} = [G_{x}, O_{j} (t_{j})]$ ; all other insertion orders remain fixed because the $t_{i}$ are distinct. The commutator is defined on the required domain and bounded by assumption.

3.1step 2.1step 2.2algebra∎

A piecewise constant locally integrable function has distributional derivative equal to the sum of its jumps times delta distributions: for every compactly supported smooth $φ$ , integration by parts on the intervals separated by $t_{j}$ gives $- \int F (t) φ^{'} (t) d t = \sum_{j} (F (t_{j} +) - F (t_{j} -)) φ (t_{j})$ . Substituting the jumps from step 2.2 proves the displayed identity.

Time-ordered Ward contact relation

Statement

Facts & Assumptions

Proof

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