Manuscript VI@1 · VI.tex · proposition · b5b60afb

Proposition: Algebraic observable quotient

The linearized interface and a sourced nonempty domain › What linearization at the chosen background gives · lines 629–662 · proof included or linked

VI@1#prop:linear-quotient, VI@1#eq:linear-quotient

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The quotient

E_{lin} = \frac{{( J , c ) \in C _{c} ^{\infty} ( S ^{2} T M ) \oplus C _{c} ^{\infty} ( M ) : \partial _{μ} J ^{μν} = 0 }}{P _{0} ( C _{c} ^{\infty} ( S ^{2} T ^{*} M ) \oplus C _{c} ^{\infty} ( M ) )}

is well defined. Its elements induce gauge-invariant linear functionals on free solutions.

Proof. The linearized Bianchi identity implies $K_{0}^{*} P_{0} = 0$ , so the denominator is contained in the numerator. The free Hessian is formally self-adjoint: it is the second variation of the quadratic action, and its explicit expression also verifies this by integration by parts. Consequently, a test in the denominator pairs to zero with every solution of $P_{0} (a, u) = 0$ .

A numerator test annihilates every smooth pure gauge:

\int J^{μν} (K ξ)_{μν} d^{4} x = - 2 \int ξ_{ν} \partial_{μ} J^{μν} d^{4} x = 0.

All integrations by parts are justified by compactness of the tests.

Displayed formulas in this passage

VI@1#eq:linear-quotient · line 632
$E_{lin} = \frac{{( J , c ) \in C _{c} ^{\infty} ( S ^{2} T M ) \oplus C _{c} ^{\infty} ( M ) : \partial _{μ} J ^{μν} = 0 }}{P _{0} ( C _{c} ^{\infty} ( S ^{2} T ^{*} M ) \oplus C _{c} ^{\infty} ( M ) )}$
unlabeled · line 656
$\int J^{μν} (K ξ)_{μν} d^{4} x = - 2 \int ξ_{ν} \partial_{μ} J^{μν} d^{4} x = 0.$

Proposition: Algebraic observable quotient

Displayed formulas in this passage

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