Manuscript XVIII-B@2 · XVIII-B-v2.tex · definition · 27ed4f4b

Definition: The selected response

Frozen data and the selected response · lines 253–286

XVIII-B@2#def:response, XVIII-B@2#eq:Kubo, XVIII-B@2#eq:response

Let $Π^{R}_{ab c d}$ be a renormalized retarded splitting of the vacuum stress commutator, with convention

Π^{R}_{ab c d} (x) = \frac{i}{ℏ} Θ (x^{0}) ⟨[T_{ab} (x), T_{c d} (0)] ⟩_{Ω} away from x = 0.

For compact smooth symmetric $j, h$ , set

R_{ε} (j, h) = g^{2} \int_{M^{2}} j^{ab} (x) Π^{R}_{ab c d} (x - y) σ_{ε} (y^{0}) h^{c d} (y) d^{4} x d^{4} y = g^{2} ⟨ Π^{R}_{ab c d}, F_{j, σ_{ε} h}^{ab c d} ⟩, F_{j, H}^{ab c d} (r) = \int_{M} j^{ab} (y + r) H^{c d} (y) d^{4} y .

This is the linear response of the insertion $- g T (j)$ to the source $V_{σ_{ε} h}$ , hence it has quadratic source order. The corresponding first-order stress response is $- g Π^{R} * (σ_{ε} h)$ .

Displayed formulas in this passage

XVIII-B@2#eq:Kubo · line 257
$Π^{R}_{ab c d} (x) = \frac{i}{ℏ} Θ (x^{0}) ⟨[T_{ab} (x), T_{c d} (0)] ⟩_{Ω} away from x = 0.$
XVIII-B@2#eq:response · line 266
$R_{ε} (j, h) = g^{2} \int_{M^{2}} j^{ab} (x) Π^{R}_{ab c d} (x - y) σ_{ε} (y^{0}) h^{c d} (y) d^{4} x d^{4} y = g^{2} ⟨ Π^{R}_{ab c d}, F_{j, σ_{ε} h}^{ab c d} ⟩, F_{j, H}^{ab c d} (r) = \int_{M} j^{ab} (y + r) H^{c d} (y) d^{4} y .$

Definition: The selected response

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