Manuscript X-G@1 · X-G.tex · section · 01566e14

The signed free operators

Frozen conventions and a support correction › The signed free operators · lines 406–444

X-G@1#eq:Cstar, X-G@1#eq:P, X-G@1#eq:DR

Put

P u = u - \frac{1}{2} η tr_{η} u, C^{μ} [u] = \partial_{ν} (P u)^{ν μ} .

Then

P^{2} = id, C K = □ .

The formal adjoint of $C$ is

(C^{*} v)_{μν} = - \partial_{(μ} v_{ν)} + \frac{1}{2} η_{μν} \partial_{ρ} v^{ρ} = - \frac{1}{2} (P K v)_{μν} .

Parentheses include the factor $1/2$ .

Let $A = \frac{1}{2} G_{lin}$ . The identities copied from X-E are

A + \frac{1}{2} C^{*} C = P_{gr} = - \frac{1}{4} P □, A K = 0.

The metric Green operators are

Δ_{R}^{gr} = - 4 P E_{0}^{R}, Δ_{A}^{gr} = - 4 P E_{0}^{A} .

Thus the metric equation is equivalent, through a constant invertible fibre map, to a normally hyperbolic equation. We do not change its coefficient to $+ 4$ .

The scalar action Hessian is $- P_{m}$ , with retarded inverse $- E_{m}^{R}$ . Its equation inverse is $E_{m}^{R}$ . These two uses must not be conflated.

Displayed formulas in this passage

X-G@1#eq:Cstar · line 418
$(C^{*} v)_{μν} = - \partial_{(μ} v_{ν)} + \frac{1}{2} η_{μν} \partial_{ρ} v^{ρ} = - \frac{1}{2} (P K v)_{μν} .$
X-G@1#eq:P · line 428
$A + \frac{1}{2} C^{*} C = P_{gr} = - \frac{1}{4} P □, A K = 0.$
X-G@1#eq:DR · line 433
$Δ_{R}^{gr} = - 4 P E_{0}^{R}, Δ_{A}^{gr} = - 4 P E_{0}^{A} .$
unlabeled · line 409
$P u = u - \frac{1}{2} η tr_{η} u, C^{μ} [u] = \partial_{ν} (P u)^{ν μ} .$
unlabeled · line 414
$P^{2} = id, C K = □ .$