Manuscript V-B@1 · V-B-v1.tex · section · 3fadc1ad

Representation and normalization

The external first retarded coefficient › Representation and normalization · lines 950–1029

V-B@1#eq:Q, V-B@1#eq:Feynman

Let

h = L^{2} (R^{3}, d^{3} p), H_{ϕ} = Γ_{s} (h), D_{ϕ} = n \geq 0 ⨁^{alg} S_{sym} (R^{3 n}) .

Use the auxiliary Minkowski vacuum and field

ϕ (x) = \int N_{p} (a (p) e^{- i ω_{p} t + i p \cdot x} + a^{†} (p) e^{i ω_{p} t - i p \cdot x}) d^{3} p,

ω_{p} = (∣ p ∣^{2} + m^{2})^{1/2}, N_{p} = (2 π)^{- 3/2} (2 ω_{p})^{- 1/2} .

For a real compact smooth $q$ ,

ϕ (q) = Φ (K q), (K q) (p) = N_{p} \int q (t, x) e^{i ω_{p} t - i p \cdot x} d^{4} x,

where $Φ (v) = a (v) + a^{†} (v)$ , with annihilation antilinear. The map $K$ takes compact smooth tests continuously to Schwartz vectors.

Let

T_{μν} (ϕ) = \partial_{μ} ϕ \partial_{ν} ϕ - \frac{1}{2} η_{μν} (\partial_{α} ϕ \partial^{α} ϕ + m^{2} ϕ^{2}) .

For real $g, f \in C_{c}^{\infty} (M)$ and a real smooth external tensor $h$ , define

V (g; h) = \frac{1}{2} \int g h^{μν} : T_{μν} : d^{4} x .

The perturbative convention is interaction $+ λ V$ and

R_{1, 1} (V, ϕ (f)) = i (T_{2} (V, ϕ (f)) - V ϕ (f)) .

For $H_{μν} = g h_{μν}$ , put

Q_{H} = - \partial_{μ} ((r H)^{μν} \partial_{ν}) - \frac{1}{2} m^{2} tr_{η} H .

It is formally self-adjoint and

Q_{H} = L_{H} + \frac{1}{2} (tr_{η} H) P .

The two-insertion normalization is fixed by vacuum Wick expansion and the scalar Feynman distribution

W_{m}^{F} (z) = Θ (z^{0}) W_{m} (z) + Θ (- z^{0}) W_{m} (- z),

where

W_{m} (z) = \int N_{p}^{2} e^{- i ω_{p} z^{0} + i p \cdot z} d^{3} p .

In these conventions,

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

The insertions are formed with off-shell labels before free-field representation, as in the retarded-product framework DF. We construct only this insertion pair.

Displayed formulas in this passage

V-B@1#eq:Q · line 1000
$Q_{H} = - \partial_{μ} ((r H)^{μν} \partial_{ν}) - \frac{1}{2} m^{2} tr_{η} H .$
V-B@1#eq:Feynman · line 1021
$P W_{m}^{F} = i δ, W_{m}^{F} (y - x) - W_{m} (y - x) = i G_{m, R} (x - y) .$
unlabeled · line 953
$h = L^{2} (R^{3}, d^{3} p), H_{ϕ} = Γ_{s} (h), D_{ϕ} = n \geq 0 ⨁^{alg} S_{sym} (R^{3 n}) .$
unlabeled · line 960
$ϕ (x) = \int N_{p} (a (p) e^{- i ω_{p} t + i p \cdot x} + a^{†} (p) e^{i ω_{p} t - i p \cdot x}) d^{3} p,$
unlabeled · line 966
$ω_{p} = (∣ p ∣^{2} + m^{2})^{1/2}, N_{p} = (2 π)^{- 3/2} (2 ω_{p})^{- 1/2} .$
unlabeled · line 971
$ϕ (q) = Φ (K q), (K q) (p) = N_{p} \int q (t, x) e^{i ω_{p} t - i p \cdot x} d^{4} x,$
unlabeled · line 980
$T_{μν} (ϕ) = \partial_{μ} ϕ \partial_{ν} ϕ - \frac{1}{2} η_{μν} (\partial_{α} ϕ \partial^{α} ϕ + m^{2} ϕ^{2}) .$
unlabeled · line 988
$V (g; h) = \frac{1}{2} \int g h^{μν} : T_{μν} : d^{4} x .$
unlabeled · line 993
$R_{1, 1} (V, ϕ (f)) = i (T_{2} (V, ϕ (f)) - V ϕ (f)) .$
unlabeled · line 1007
$Q_{H} = L_{H} + \frac{1}{2} (tr_{η} H) P .$
unlabeled · line 1013
$W_{m}^{F} (z) = Θ (z^{0}) W_{m} (z) + Θ (- z^{0}) W_{m} (- z),$
unlabeled · line 1017
$W_{m} (z) = \int N_{p}^{2} e^{- i ω_{p} z^{0} + i p \cdot z} d^{3} p .$