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

Frozen data and the selected response

Frozen data and the selected response · lines 192–252

XVIII-B@2#eq:Fourier, XVIII-B@2#eq:stress

Datum	Frozen choice
Spacetime	$M = R^{4}$ , $η = diag (1, - 1, - 1, - 1)$ .
Quantum field	One free real scalar, $(□ + m^{2}) ϕ = 0$ , $m > 0$ .
State	Minkowski Fock vacuum $Ω$ .
Stress insertion	Canonical minimally coupled tensor, vacuum Wick ordered; no improvement parameter is varied.
External source	A real compact smooth symmetric contravariant tensor.
Coupling	$g = κ /2$ , $V_{H} = - g T (H)$ .
Order	Connected vacuum source degree two; response proportional to $κ^{2} ℏ$ .
Preparation	Fixed smooth time ramp $σ_{ε}$ , fixed $ε > 0$ ; prepared source $H = σ_{ε} h$ .
Reference prescription	One subtraction at $z = 0$ in each reduced spectral form factor.
Allowed finite changes	Point-supported, scaling degree $\leq 8$ , real, parity-even, Lorentz-covariant, pair-symmetric, and transverse in both stress slots.
Calibration	The full momentum-space response has vanishing Taylor terms of total degrees two and four at $p = 0$ .
Operational output	A distributional response coefficient; no measurement instrument is constructed.

Indices are raised and lowered with $η$ . Write $p^{2} = η_{ab} p^{a} p^{b}$ and use

f (p) = \int_{M} e^{i p \cdot x} f (x) d^{4} x, f (x) = \int_{R^{4}} \frac{d ^{4} p}{( 2 π ) ^{4}} e^{- i p \cdot x} f (p) .

Thus $\partial_{a}$ has multiplier $- i p_{a}$ .

The stress tensor is

T_{ab} =: \partial_{a} ϕ \partial_{b} ϕ - \frac{1}{2} η_{ab} (\partial^{c} ϕ \partial_{c} ϕ - m^{2} ϕ^{2}) :, T (j) = \int_{M} j^{ab} T_{ab} d^{4} x .

Wick ordering is with respect to $Ω$ , so $⟨ T_{ab} ⟩_{Ω} = 0$ . The free field equation gives $\partial^{a} T_{ab} = 0$ as an operator-valued distribution on the usual finite-particle domain.

Displayed formulas in this passage

XVIII-B@2#eq:Fourier · line 227
$f (p) = \int_{M} e^{i p \cdot x} f (x) d^{4} x, f (x) = \int_{R^{4}} \frac{d ^{4} p}{( 2 π ) ^{4}} e^{- i p \cdot x} f (p) .$
XVIII-B@2#eq:stress · line 237
$T_{ab} =: \partial_{a} ϕ \partial_{b} ϕ - \frac{1}{2} η_{ab} (\partial^{c} ϕ \partial_{c} ϕ - m^{2} ϕ^{2}) :, T (j) = \int_{M} j^{ab} T_{ab} d^{4} x .$

Frozen data and the selected response

Displayed formulas in this passage

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