Manuscript II@2 · II.tex · section · 50d64def

Quadratic action and conserved-current inverse

Locked kinematics, action, and contacts › Quadratic action and conserved-current inverse · lines 392–434

II@2#eq:PF, II@2#eq:K0, II@2#eq:D

For one species,

S_{2} [h] = \int d^{4} x (- \frac{1}{2} \partial_{λ} h_{μν} \partial^{λ} h^{μν} + \partial_{μ} h^{μν} \partial^{ρ} h_{ρ ν} - \partial_{μ} h^{μν} \partial_{ν} h + \frac{1}{2} \partial_{λ} h \partial^{λ} h), h = η^{μν} h_{μν} .

For two species,

S_{PF} = S_{2} [h^{1}] + S_{2} [h^{2}] .

The Hessian is

(K_{0} h)_{μν} = □ h_{μν} - \partial_{μ} \partial^{ρ} h_{ρ ν} - \partial_{ν} \partial^{ρ} h_{ρ μ} + \partial_{μ} \partial_{ν} h + η_{μν} (\partial^{ρ} \partial^{σ} h_{ρ σ} - □ h) .

The fixed propagator coefficient is

D (p) = - \frac{P}{p ^{2}}, P_{μν, ρ σ} = \frac{1}{2} (η_{μ ρ} η_{ν σ} + η_{μ σ} η_{ν ρ} - η_{μν} η_{ρ σ}) .

Displayed formulas in this passage

II@2#eq:PF · line 395
$S_{2} [h] = \int d^{4} x (- \frac{1}{2} \partial_{λ} h_{μν} \partial^{λ} h^{μν} + \partial_{μ} h^{μν} \partial^{ρ} h_{ρ ν} - \partial_{μ} h^{μν} \partial_{ν} h + \frac{1}{2} \partial_{λ} h \partial^{λ} h), h = η^{μν} h_{μν} .$
II@2#eq:K0 · line 411
$(K_{0} h)_{μν} = □ h_{μν} - \partial_{μ} \partial^{ρ} h_{ρ ν} - \partial_{ν} \partial^{ρ} h_{ρ μ} + \partial_{μ} \partial_{ν} h + η_{μν} (\partial^{ρ} \partial^{σ} h_{ρ σ} - □ h) .$
II@2#eq:D · line 423
$D (p) = - \frac{P}{p ^{2}}, P_{μν, ρ σ} = \frac{1}{2} (η_{μ ρ} η_{ν σ} + η_{μ σ} η_{ν ρ} - η_{μν} η_{ρ σ}) .$
unlabeled · line 407
$S_{PF} = S_{2} [h^{1}] + S_{2} [h^{2}] .$