Physics theoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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State differences remove common local subtraction

Physical scope

A conditional identity for two Hadamard states on the same fixed smooth background under one identical local renormalization prescription. It does not assert independence of the absolute renormalized stress tensor from finite local curvature terms.

Statement

Let $g$ be one fixed smooth globally hyperbolic Lorentzian four-metric and let $ω_{1}, ω_{2}$ be two positive Hadamard states for the same real scalar Klein-Gordon operator. Use one identical prescription: the same local parametrix $H_{μ}$ , bidifferential operator $D_{a b}$ , scale $μ$ , overlap convention, and state-independent curvature correction $C_{a b}$ . Then, on each convex normal neighborhood, $T_{a b}^{r e n} [g, ω_{1}] - T_{a b}^{r e n} [g, ω_{2}] = \lim_{y \to x} D_{a b} (x, y) (W_{ω_{1}} - W_{ω_{2}}) (x, y),$ where the difference on the right is smooth and its coincidence limit exists. Adding the same state-independent local curvature tensor to both stress prescriptions leaves this difference unchanged.

Statement references

Hadamard and renormalized stress contract

Facts & Assumptions

Given: The same fixed metric, two positive Hadamard states for the same operator, and one common renormalization prescription as stated above.

[F1]

Hadamard and renormalized stress contract: for each Hadamard state the supplied local parametrix $H_{μ}$ is state-independent, each remainder $W_{ω} - H_{μ}$ is smooth, $D_{a b}$ has finite order on smooth biscalars, and $T_{a b}^{r e n}$ is the coincidence limit of $D_{a b} (W_{ω} - H_{μ})$ plus the same local tensor $C_{a b}$ .

[A1]

The two states use the same $H_{μ}$ , $D_{a b}$ , $μ$ , overlap convention, and $C_{a b}$ ; this is the identical-prescription hypothesis in the Statement.

Proof

1.1givenF1A1algebra

For $i = 1, 2$ , [F1] and [A1] give $W_{ω_{i}} - H_{μ} \in C^{\infty} (U \times U)$ on each prescribed neighborhood $U$ . Their difference is therefore smooth: $(W_{ω_{1}} - H_{μ}) - (W_{ω_{2}} - H_{μ}) = W_{ω_{1}} - W_{ω_{2}} .$ Since $D_{a b}$ is finite order and maps smooth biscalars to smooth tensor-valued functions, its application to this difference is smooth near the diagonal and has a coincidence limit.

2.1givenF1A1step 1.1algebra

Apply [F1] separately to both states and subtract. The common $C_{a b}$ cancels, and linearity of $D_{a b}$ gives $T_{a b}^{r e n} [g, ω_{1}] - T_{a b}^{r e n} [g, ω_{2}] = \lim_{y \to x} (D_{a b} (W_{ω_{1}} - H_{μ}) - D_{a b} (W_{ω_{2}} - H_{μ})) = \lim_{y \to x} D_{a b} (W_{ω_{1}} - W_{ω_{2}}),$ where existence of the final limit follows from step 1.1.

3.1givenstep 2.1algebra∎

If a common state-independent local curvature tensor $K_{a b} [g]$ is added to both prescriptions, their difference becomes $(T^{r e n} [g, ω_{1}] + K [g]) - (T^{r e n} [g, ω_{2}] + K [g])$ , which equals the difference in step 2.1 by cancellation. This establishes only independence of common local additions to the difference, not independence of either absolute stress tensor.

Depends on

Hadamard and renormalized stress contract