Manuscript IX-B@2 · IX-B-v2.tex · section · 8ce915d6

Off-shell insertion and fixed covariance

Fixed data and inherited inputs › Off-shell insertion and fixed covariance · lines 228–313

IX-B@2#eq:B, IX-B@2#eq:star, IX-B@2#eq:tensor, IX-B@2#eq:K

Let

M = R^{1 + 3}, M_{+} = {t > 0}, X = M_{+}^{3}, η = diag (- 1, 1, 1, 1),

and write $x = (t, x)$ , $y = (s, y)$ , $x^{'} = (t^{'}, x^{'})$ . Put

□ = - \partial_{t}^{2} + Δ_{x}, (C h)^{μ} = \partial_{ν} h^{μν} - \frac{1}{2} \partial^{μ} (η^{α β} h_{α β}), L = E_{0} C .

The inverse $E_{0}$ has zero value and zero first time derivative on $Σ_{0} = {t = 0}$ , and satisfies $□ E_{0} j = j$ . All configurations $(h, ϕ)$ are smooth and off shell. The insertion and middle observable are

B_{f} [h, ϕ] = - \int f (x) (L h)^{μ} (x) \partial_{μ} ϕ (x) d^{4} x, W_{g} [h] = \int g (y) h_{00} (y)^{2} d^{4} y,

where $f, g \in C_{c}^{\infty} (M_{+})$ .

We retain the VII-C Wick-symbol convention

F ⋆_{H} G = j \geq 0 \sum \frac{ℏ ^{j}}{j !} ⟨ F^{(j)}, H^{\otimes j} G^{(j)} ⟩ .

The sum is finite for finite polynomials; self-contractions inside a Wick symbol are excluded.

The fixed even blocks are

H_{hh α β ρ σ} = P_{α β ρ σ} Δ_{0}^{+}, H_{ϕϕ} = Δ_{m}^{+}, H_{h ϕ} = H_{ϕ h} = 0,

where

P_{α β ρ σ} = η_{α (ρ} η_{σ) β} - \frac{1}{2} η_{α β} η_{ρ σ},

and, for fixed $m \geq 0$ ,

Δ_{m}^{+} (x - y) = \int_{R^{3}} \frac{d ^{3} q}{( 2 π ) ^{3} 2 ω _{m} ( q )} e^{- i ω_{m} (q) (t - s) + i q \cdot (x - y)}, ω_{m} (q) = ∣ q ∣^{2} + m^{2} .

The following identities are inherited, not re-derived:

C_{x}^{μ} H_{hh ρ σ} = δ^{μ}_{(ρ} \partial^{x}_{σ)} Δ_{0}^{+}, H_{w w} = 0, w = L h .

They are VII-C's tensor cancellation and IX-A's fixed tensor lemma VIIC (prop:tensor) IXA (lem:tensor). Both sources retain the mixed reflected branch:

K (x, y) := H_{w h}^{0}_{00} (x, y) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot (x - y)} [(- \frac{t}{4∣ p ∣} + \frac{i}{8∣ p ∣ ^{2}}) e^{- i ∣ p ∣ (t - s)} - \frac{i}{8∣ p ∣ ^{2}} e^{i ∣ p ∣ (t + s)}] .

In particular, $H_{w w} = 0$ does not imply $K = 0$ .

Displayed formulas in this passage

IX-B@2#eq:B · line 247
$B_{f} [h, ϕ] = - \int f (x) (L h)^{μ} (x) \partial_{μ} ϕ (x) d^{4} x, W_{g} [h] = \int g (y) h_{00} (y)^{2} d^{4} y,$
IX-B@2#eq:star · line 257
$F ⋆_{H} G = j \geq 0 \sum \frac{ℏ ^{j}}{j !} ⟨ F^{(j)}, H^{\otimes j} G^{(j)} ⟩ .$
IX-B@2#eq:tensor · line 288
$C_{x}^{μ} H_{hh ρ σ} = δ^{μ}_{(ρ} \partial^{x}_{σ)} Δ_{0}^{+}, H_{w w} = 0, w = L h .$
IX-B@2#eq:K · line 298
$K (x, y) := H_{w h}^{0}_{00} (x, y) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot (x - y)} [(- \frac{t}{4∣ p ∣} + \frac{i}{8∣ p ∣ ^{2}}) e^{- i ∣ p ∣ (t - s)} - \frac{i}{8∣ p ∣ ^{2}} e^{i ∣ p ∣ (t + s)}] .$
unlabeled · line 231
$M = R^{1 + 3}, M_{+} = {t > 0}, X = M_{+}^{3}, η = diag (- 1, 1, 1, 1),$
unlabeled · line 237
$□ = - \partial_{t}^{2} + Δ_{x}, (C h)^{μ} = \partial_{ν} h^{μν} - \frac{1}{2} \partial^{μ} (η^{α β} h_{α β}), L = E_{0} C .$
unlabeled · line 267
$H_{hh α β ρ σ} = P_{α β ρ σ} Δ_{0}^{+}, H_{ϕϕ} = Δ_{m}^{+}, H_{h ϕ} = H_{ϕ h} = 0,$
unlabeled · line 273
$P_{α β ρ σ} = η_{α (ρ} η_{σ) β} - \frac{1}{2} η_{α β} η_{ρ σ},$
unlabeled · line 279
$Δ_{m}^{+} (x - y) = \int_{R^{3}} \frac{d ^{3} q}{( 2 π ) ^{3} 2 ω _{m} ( q )} e^{- i ω_{m} (q) (t - s) + i q \cdot (x - y)}, ω_{m} (q) = ∣ q ∣^{2} + m^{2} .$