Manuscript Em01-F@3 · Em01-F-v3.tex · section · 225e59af

Action, background, field, and gauge

One action and its normalization dictionary › Action, background, field, and gauge · lines 668–740

Em01-F@3#eq:action

Fix $m \geq 0$ , $λ \in R ∖ {0}$ , and $ℏ > 0$ . Coefficient expansions are formal at $λ = 0$ . The metric variable and background are

g_{μν} = η_{μν} + λ γ_{μν}, (g, ϕ)_{background} = (η, 0) .

Use

Γ^{ρ}_{μν} = \frac{1}{2} g^{ρ σ} (\partial_{μ} g_{σ ν} + \partial_{ν} g_{σ μ} - \partial_{σ} g_{μν}),

R_{μν} = \partial_{ρ} Γ^{ρ}_{μν} - \partial_{ν} Γ^{ρ}_{μ ρ} + Γ^{ρ}_{ρ α} Γ^{α}_{μν} - Γ^{ρ}_{ν α} Γ^{α}_{μ ρ}, R = g^{μν} R_{μν} .

Let

d_{ν} (γ) = \partial^{μ} γ_{μν} - \frac{1}{2} \partial_{ν} tr_{η} γ, S_{gf} [γ] = \int d_{ν} (γ) d^{ν} (γ) d^{4} x .

The gauge-fixing functional is explicitly quadratic; no nonlinear gauge-fixing interaction is implicit.

With $g \in C_{c}^{\infty} (M; R)$ , define

S_{λ, g} [γ, ϕ] = - \frac{2}{λ ^{2}} \int - det g R [g] d^{4} x + S_{gf} [γ] + \frac{1}{2} \int (\partial^{μ} ϕ \partial_{μ} ϕ - m^{2} ϕ^{2}) d^{4} x + \frac{1}{2} \int g (x) [- det g (g^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}) - (\partial^{μ} ϕ \partial_{μ} ϕ - m^{2} ϕ^{2})] d^{4} x .

The action is evaluated on compact perturbations, or under compact local variations. Its linear gravitational term is a divergence.

The gravitational action is unswitched. The coordinate prescription below also has constant weight. The matter interaction is switched by $g$ . The model is used off shell; no existence result for an invariant switched Einstein jet is assumed.

Displayed formulas in this passage

Em01-F@3#eq:action · line 713
$S_{λ, g} [γ, ϕ] = - \frac{2}{λ ^{2}} \int - det g R [g] d^{4} x + S_{gf} [γ] + \frac{1}{2} \int (\partial^{μ} ϕ \partial_{μ} ϕ - m^{2} ϕ^{2}) d^{4} x + \frac{1}{2} \int g (x) [- det g (g^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}) - (\partial^{μ} ϕ \partial_{μ} ϕ - m^{2} ϕ^{2})] d^{4} x .$
unlabeled · line 673
$g_{μν} = η_{μν} + λ γ_{μν}, (g, ϕ)_{background} = (η, 0) .$
unlabeled · line 679
$Γ^{ρ}_{μν} = \frac{1}{2} g^{ρ σ} (\partial_{μ} g_{σ ν} + \partial_{ν} g_{σ μ} - \partial_{σ} g_{μν}),$
unlabeled · line 689
$R_{μν} = \partial_{ρ} Γ^{ρ}_{μν} - \partial_{ν} Γ^{ρ}_{μ ρ} + Γ^{ρ}_{ρ α} Γ^{α}_{μν} - Γ^{ρ}_{ν α} Γ^{α}_{μ ρ}, R = g^{μν} R_{μν} .$
unlabeled · line 699
$d_{ν} (γ) = \partial^{μ} γ_{μν} - \frac{1}{2} \partial_{ν} tr_{η} γ, S_{gf} [γ] = \int d_{ν} (γ) d^{ν} (γ) d^{4} x .$