Manuscript X-A@1 · X-A.tex · section · dc7f5c0a

Convention sheet

Convention sheet · lines 464–508

The first deliverable is a sheet. Every later identity refers to it.

Item	Declaration
Spacetime	$M_{T} = [0, T] \times R^{3}$ , $Σ = {t = 0}$ , $M_{T}^{\circ} = (0, T) \times R^{3}$ ; $η = diag (1, - 1, - 1, - 1)$ , $□ = \partial_{t}^{2} - Δ$
Fields	Smooth real symmetric $h_{μν}, k_{μν}$ on a neighbourhood of $M_{T}$ ; formal real scalar $ϕ = ϕ_{0} + κ ϕ_{1} + κ^{2} ϕ_{2}$ ; no quotient, no decay, no field equation
Metric	$q = η + κh + κ^{2} k$ in $C^{\infty} [κ] / (κ^{3})$ ; formal only
Action	Switched minimally coupled scalar (eq:action), mass $m \geq 0$ ; no gravitational action is used, the metric is off shell
Switching	$g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1])$ multiplies the curved-minus-flat scalar density before variation (eq:Ag); $g$ does not enter the coordinate prescription
Regulator	$σ_{ε} (t, x) = χ (t / ε)$ , $χ$ explicit (eq:chi), $0 < ε < T$ ; enters only the coordinate prescription, through the auxiliary density (eq:Aeps)
Gauge algebra	$g_{U} = C_{c}^{\infty} (U; T M)$ , $U$ open, relatively compact, $U \subset P_{g} \cap {ε < t < T}$ , $P_{g} = int {g = 1}$ ; all parameters and all derivatives vanish outside $U$ , hence near $Σ$ and in the edge zone ${0 \leq t \leq ε}$
Cauchy data	$X^{μ} ∣_{Σ} = x^{μ}$ , $\partial_{t} X^{μ} ∣_{Σ} = δ^{μ}_{0}$ ; positive-order scalar coefficients have zero data; gauge transformations are the identity near $Σ$
Observable	$Φ_{g, ε} = ϕ \circ X_{q}^{- 1}$ smeared by one $f \in C_{c}^{\infty} (M_{T}^{\circ}; R)$ in label space
Orders	Through $κ^{2}$ ; no quantum coefficient