Manuscript II-B@1 · II-B-v1.tex · section · bfd2eb3c

Pauli–Fierz normalization

The frozen response problem › Pauli–Fierz normalization · lines 489–526

II-B@1#eq:PF, II-B@1#eq:K0

For one species define

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} .

This is the massless spin-two quadratic normalization FierzPauli. Its 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) .

Use the fixed propagator coefficient

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-B@1#eq:PF · line 492
$S_{2} [h] = \int d^{4} x (- \frac{1}{2} \partial_{λ} h_{μν} \partial^{λ} h^{μν} + \partial_{μ} h^{μν} \partial^{ρ} h_{ρ ν} - \partial_{μ} h^{μν} \partial_{ν} h + \frac{1}{2} \partial_{λ} h \partial^{λ} h), h = η^{μν} h_{μν} .$
II-B@1#eq:K0 · line 505
$(K_{0} h)_{μν} = □ h_{μν} - \partial_{μ} \partial^{ρ} h_{ρ ν} - \partial_{ν} \partial^{ρ} h_{ρ μ} + \partial_{μ} \partial_{ν} h + η_{μν} (\partial^{ρ} \partial^{σ} h_{ρ σ} - □ h) .$
unlabeled · line 517
$D (p) = - \frac{P}{p ^{2}}, P_{μν, ρ σ} = \frac{1}{2} (η_{μ ρ} η_{ν σ} + η_{μ σ} η_{ν ρ} - η_{μν} η_{ρ σ}) .$