Manuscript V@1 · V.tex · section · e7ddf37d

Complete contraction list

The external-metric first retarded product › Complete contraction list · lines 2061–2117

V@1#eq:W, V@1#eq:Widentities, V@1#eq:contraction, V@1#eq:Wickexpansion

Use

W_{m} (z) W_{m}^{F} (z) = \int \frac{d ^{3} p}{( 2 π ) ^{3} 2 ω _{p}} e^{- i ω_{p} z^{0} + i p \cdot z}, = Θ (z^{0}) W_{m} (z) + Θ (- z^{0}) W_{m} (- z) .

With the stated Green convention,

W_{m} (z) - W_{m} (- z) = i Δ_{m} (z), P W_{m}^{F} = i δ, W_{m}^{F} (y - x) - W_{m} (y - x) = i G_{R} (x - y) .

For a numerical distribution $D (y - x)$ , put

C_{μν} [D] (y, x) = \partial_{μ} D (y - x) \partial_{ν} ϕ (y) + \partial_{ν} D (y - x) \partial_{μ} ϕ (y) - η_{μν} (\partial_{α} D (y - x) \partial^{α} ϕ (y) + m^{2} D (y - x) ϕ (y)) .

Derivatives of $D$ are in $y$ . If $N_{μν} (y, x)$ is the completely normal ordered expression formed from $T_{μν} (ϕ (y)) ϕ (x)$ , then

T_{2} (: T_{μν} (y) :, ϕ (x)) : T_{μν} (y) : ϕ (x) = N_{μν} (y, x) + C_{μν} [W_{m}^{F}] (y, x), = N_{μν} (y, x) + C_{μν} [W_{m}] (y, x) .

There is at most one cross-contraction: the second insertion contains only one scalar field. Contracting either differentiated factor in the stress gives the first two terms of (eq:contraction); the trace and mass terms give the rest. There is no double-contraction vacuum block. The uncontracted normal block has scalar degree three and will cancel in (eq:Rdef).

Displayed formulas in this passage

V@1#eq:W · line 2064
$W_{m} (z) W_{m}^{F} (z) = \int \frac{d ^{3} p}{( 2 π ) ^{3} 2 ω _{p}} e^{- i ω_{p} z^{0} + i p \cdot z}, = Θ (z^{0}) W_{m} (z) + Θ (- z^{0}) W_{m} (- z) .$
V@1#eq:Widentities · line 2074
$W_{m} (z) - W_{m} (- z) = i Δ_{m} (z), P W_{m}^{F} = i δ, W_{m}^{F} (y - x) - W_{m} (y - x) = i G_{R} (x - y) .$
V@1#eq:contraction · line 2082
$C_{μν} [D] (y, x) = \partial_{μ} D (y - x) \partial_{ν} ϕ (y) + \partial_{ν} D (y - x) \partial_{μ} ϕ (y) - η_{μν} (\partial_{α} D (y - x) \partial^{α} ϕ (y) + m^{2} D (y - x) ϕ (y)) .$
V@1#eq:Wickexpansion · line 2099
$T_{2} (: T_{μν} (y) :, ϕ (x)) : T_{μν} (y) : ϕ (x) = N_{μν} (y, x) + C_{μν} [W_{m}^{F}] (y, x), = N_{μν} (y, x) + C_{μν} [W_{m}] (y, x) .$