Manuscript X-E-ST@1 · X-E-ST.tex · theorem · 44614221

Theorem: Free operator and Green inverses

E3: the derived free graviton operator › Feynman-weight de Donder gauge fixing · lines 1087–1135 · proof included or linked

X-E-ST@1#thm:free, X-E-ST@1#eq:Pgr, X-E-ST@1#eq:grgreen, X-E-ST@1#eq:violation

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One has $P_{tr}^{2} = 1$ , and

G_{lin}^{μν} [l] \frac{δ ( S _{FP} + S _{gf} )}{δ l _{μν}} = - \frac{1}{2} □ (P_{tr} l)^{μν} + \partial^{(μ} C^{ν)} [l] - \frac{1}{2} η^{μν} \partial_{ρ} C^{ρ} [l], = - \frac{1}{4} □ (P_{tr} l)^{μν} =: (P_{gr} l)^{μν} .

Consequently

P_{gr} = - \frac{1}{4} P_{tr} □, Δ_{gr}^{R} = - 4 P_{tr} E_{0}^{R}, Δ_{gr}^{A} = - 4 P_{tr} E_{0}^{A} .

For a compact source $A$ ,

supp Δ_{gr}^{R} A \subset J_{η}^{+} (supp A), C^{μ} [Δ_{gr}^{R} A] = - 4 E_{0}^{R} \partial_{ν} A^{ν μ} .

Proof. In four dimensions trace reversal reverses the trace; applying it twice gives the identity. Substitution in (eq:Glin) gives the first line.

The gauge term varies as

δ S_{gf} = \int δ l_{μν} (- \frac{1}{2} \partial^{(μ} C^{ν)} + \frac{1}{4} η^{μν} \partial_{ρ} C^{ρ}) d^{4} x .

It cancels precisely the $C$ -terms in $\frac{1}{2} G_{lin}$ . This proves (eq:Pgr). The constant invertible trace-reversal factor gives the inverse formulas and support. Finally $P_{tr} Δ_{gr}^{R} = - 4 E_{0}^{R}$ , so differentiating proves the last identity.

Displayed formulas in this passage

X-E-ST@1#eq:Pgr · line 1090
$G_{lin}^{μν} [l] \frac{δ ( S _{FP} + S _{gf} )}{δ l _{μν}} = - \frac{1}{2} □ (P_{tr} l)^{μν} + \partial^{(μ} C^{ν)} [l] - \frac{1}{2} η^{μν} \partial_{ρ} C^{ρ} [l], = - \frac{1}{4} □ (P_{tr} l)^{μν} =: (P_{gr} l)^{μν} .$
X-E-ST@1#eq:grgreen · line 1103
$P_{gr} = - \frac{1}{4} P_{tr} □, Δ_{gr}^{R} = - 4 P_{tr} E_{0}^{R}, Δ_{gr}^{A} = - 4 P_{tr} E_{0}^{A} .$
X-E-ST@1#eq:violation · line 1110
$supp Δ_{gr}^{R} A \subset J_{η}^{+} (supp A), C^{μ} [Δ_{gr}^{R} A] = - 4 E_{0}^{R} \partial_{ν} A^{ν μ} .$
unlabeled · line 1123
$δ S_{gf} = \int δ l_{μν} (- \frac{1}{2} \partial^{(μ} C^{ν)} + \frac{1}{4} η^{μν} \partial_{ρ} C^{ρ}) d^{4} x .$

Theorem: Free operator and Green inverses

Displayed formulas in this passage

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