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

Scalar representation and the switched vertex

The external-metric first retarded product › Scalar representation and the switched vertex · lines 1949–2031

V@1#eq:scalar-domain, V@1#eq:massshell, V@1#eq:vertex, V@1#eq:Q, V@1#eq:QL, V@1#eq:switch

From now on $m > 0$ . Let

h_{ϕ} = L^{2} (R^{3}, d^{3} p), H_{ϕ} = Γ_{s} (h_{ϕ}),

and define the dense domain

D_{ϕ} = n = 0 ⨁^{alg} S_{sym} (R^{3 n}) .

There is no fixed common particle ceiling. The scalar field is

ϕ (x) = \int \frac{d ^{3} p}{( 2 π ) ^{3/2} 2 ω _{p}} (a (p) e^{- i ω_{p} t + i p \cdot x} + a^{†} (p) e^{i ω_{p} t - i p \cdot x}), ω_{p} = ∣ p ∣^{2} + m^{2} .

For real compact smooth $q$ ,

(K q) (p) = \frac{1}{( 2 π ) ^{3/2} 2 ω _{p}} \int q (t, x) e^{i ω_{p} t - i p \cdot x} d^{4} x, ϕ (q) = Φ (K q),

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

Take an external real $h \in V_{fin}$ and real $g, f \in C_{c}^{\infty} (M)$ . The switch is always compact. The minimal stress is

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

Vacuum normal ordering is fixed once and for all. The vertex, without the perturbative parameter, is

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

The interaction is $+ κV (g; h)$ .

For the switched tensor $H_{μν} = g h_{μν}$ , define

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

It is formally self-adjoint and is the first variation of the scalar-quadratic vertex. Expanding derivatives gives

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

On a free scalar solution,

Q_{g h} ϕ = g L_{h} ϕ - (\partial_{μ} g) h ˉ^{μν} \partial_{ν} ϕ .

For spatial TT $h$ ,

Q_{g h} v = - \partial_{i} (g h^{ij} \partial_{j} v) .

The letter $H_{μν}$ here is a switched metric coefficient. It is unrelated to the defect covector $H$ .

Displayed formulas in this passage

V@1#eq:scalar-domain · line 1957
$D_{ϕ} = n = 0 ⨁^{alg} S_{sym} (R^{3 n}) .$
V@1#eq:massshell · line 1975
$(K q) (p) = \frac{1}{( 2 π ) ^{3/2} 2 ω _{p}} \int q (t, x) e^{i ω_{p} t - i p \cdot x} d^{4} x, ϕ (q) = Φ (K q),$
V@1#eq:vertex · line 1998
$V (g; h) = \frac{1}{2} \int g h^{μν} : T_{μν} (ϕ) : d^{4} x .$
V@1#eq:Q · line 2006
$Q_{H} = - \partial_{μ} (H ˉ^{μν} \partial_{ν}) - \frac{1}{2} m^{2} tr_{η} H .$
V@1#eq:QL · line 2014
$Q_{H} = L_{H} + \frac{1}{2} (tr_{η} H) P .$
V@1#eq:switch · line 2019
$Q_{g h} ϕ = g L_{h} ϕ - (\partial_{μ} g) h ˉ^{μν} \partial_{ν} ϕ .$
unlabeled · line 1952
$h_{ϕ} = L^{2} (R^{3}, d^{3} p), H_{ϕ} = Γ_{s} (h_{ϕ}),$
unlabeled · line 1965
$ϕ (x) = \int \frac{d ^{3} p}{( 2 π ) ^{3/2} 2 ω _{p}} (a (p) e^{- i ω_{p} t + i p \cdot x} + a^{†} (p) e^{i ω_{p} t - i p \cdot x}), ω_{p} = ∣ p ∣^{2} + m^{2} .$
unlabeled · line 1990
$T_{μν} (ϕ) = \partial_{μ} ϕ \partial_{ν} ϕ - \frac{1}{2} η_{μν} (\partial_{α} ϕ \partial^{α} ϕ + m^{2} ϕ^{2}) .$
unlabeled · line 2025
$Q_{g h} v = - \partial_{i} (g h^{ij} \partial_{j} v) .$