Manuscript X-C-ST@1 · X-C-ST.tex · proposition · b679954e

Proposition: Three-term variation

Three-term variation and the completed identity › Proof of record · lines 1541–1623 · proof included or linked

X-C-ST@1#prop:variation, X-C-ST@1#eq:variation, X-C-ST@1#eq:var1, X-C-ST@1#eq:var2

For arbitrary smooth classical backgrounds and arbitrary compact interior generators,

e [q^{'}] - e [q] = P_{m} ((F^{*} - 1) s [q]) + R_{q^{'}} Θ_{F} [q] - R_{q^{'}} (Σ_{F} [q] s [q]) (mod κ^{3}) .

The first term is coefficientwise in $P_{m} D$ . The other two are the dressed coordinate and switch defects.

Writing

Z = X_{ζ} + \frac{1}{2} X_{ξ}^{2}, L^{'}_{1} = L_{1, g} [h^{'}],

the coefficient formulae are

e^{'}_{1} - e_{1} = e^{'}_{2} - e_{2} = P_{m} (X_{ξ} E_{A} f) + θ_{1} - S_{1} E_{A} f, P_{m} (X_{ξ} E_{A} e_{1} + Z E_{A} f) + θ_{2} - S_{2} E_{A} f - S_{1} E_{A} e_{1} - L^{'}_{1} E_{A} (θ_{1} - S_{1} E_{A} f) .

Proof. Let $s^{'} = s [q^{'}]$ . By Lemma lem:advanced,

E_{g} [q] s = c [q], E_{g} [q^{'}] s^{'} = c [q^{'}], P_{m} s = e [q], P_{m} s^{'} = e [q^{'}] .

The defect definitions give

E_{g} [q^{'}] F^{*} s = F^{♯} c [q] + Σ_{F} s = c [q^{'}] - Θ_{F} + Σ_{F} s .

Subtracting,

E_{g} [q^{'}] (s^{'} - F^{*} s) = Θ_{F} - Σ_{F} s .

The right-hand side is a compact formal source: $Θ_{F}$ has support in $K$ , and the coefficients of $Σ_{F}$ have compact generator support.

The difference $s^{'} - F^{*} s$ is advanced-type. Indeed, $s, s^{'}$ are advanced-type, and each coefficient of $(F^{*} - 1) s$ has compact generator support. No assumption that an unexpanded diffeomorphism preserves the completion causal relation is needed.

Advanced uniqueness therefore yields

s^{'} - F^{*} s = E_{q^{'}}^{A} (Θ_{F} - Σ_{F} s) .

Subtract $s$ and apply $P_{m}$ . Using $P_{m} E_{q^{'}}^{A} = R_{q^{'}}$ proves (eq:variation).

Every coefficient of $(F^{*} - 1) s$ is a smooth compact function, so its $P_{m}$ -image lies in $P_{m} D$ . Finally expand

R_{q^{'}} = 1 - κ L^{'}_{1} E_{A} + O (κ^{2}), s_{0} = E_{A} f, s_{1} = E_{A} e_{1} .

The degree-one and degree-two coefficients are exactly (eq:var1) and (eq:var2).

Displayed formulas in this passage

X-C-ST@1#eq:variation · line 1545
$e [q^{'}] - e [q] = P_{m} ((F^{*} - 1) s [q]) + R_{q^{'}} Θ_{F} [q] - R_{q^{'}} (Σ_{F} [q] s [q]) (mod κ^{3}) .$
X-C-ST@1#eq:var1 · line 1564
$e^{'}_{1} - e_{1} = e^{'}_{2} - e_{2} = P_{m} (X_{ξ} E_{A} f) + θ_{1} - S_{1} E_{A} f, P_{m} (X_{ξ} E_{A} e_{1} + Z E_{A} f) + θ_{2} - S_{2} E_{A} f - S_{1} E_{A} e_{1} - L^{'}_{1} E_{A} (θ_{1} - S_{1} E_{A} f) .$
X-C-ST@1#eq:var2 · line 1564
$e^{'}_{1} - e_{1} = e^{'}_{2} - e_{2} = P_{m} (X_{ξ} E_{A} f) + θ_{1} - S_{1} E_{A} f, P_{m} (X_{ξ} E_{A} e_{1} + Z E_{A} f) + θ_{2} - S_{2} E_{A} f - S_{1} E_{A} e_{1} - L^{'}_{1} E_{A} (θ_{1} - S_{1} E_{A} f) .$
unlabeled · line 1559
$Z = X_{ζ} + \frac{1}{2} X_{ξ}^{2}, L^{'}_{1} = L_{1, g} [h^{'}],$
unlabeled · line 1579
$E_{g} [q] s = c [q], E_{g} [q^{'}] s^{'} = c [q^{'}], P_{m} s = e [q], P_{m} s^{'} = e [q^{'}] .$
unlabeled · line 1585
$E_{g} [q^{'}] F^{*} s = F^{♯} c [q] + Σ_{F} s = c [q^{'}] - Θ_{F} + Σ_{F} s .$
unlabeled · line 1591
$E_{g} [q^{'}] (s^{'} - F^{*} s) = Θ_{F} - Σ_{F} s .$
unlabeled · line 1605
$s^{'} - F^{*} s = E_{q^{'}}^{A} (Θ_{F} - Σ_{F} s) .$
unlabeled · line 1617
$R_{q^{'}} = 1 - κ L^{'}_{1} E_{A} + O (κ^{2}), s_{0} = E_{A} f, s_{1} = E_{A} e_{1} .$

Proposition: Three-term variation

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