Manuscript X-C-ST@1 · X-C-ST.tex · section · b679954e

Switch derivatives before restriction

Formal duality and the two defects › Switch derivatives before restriction · lines 1363–1402

X-C-ST@1#eq:N

Let the unswitched coefficients be

Q_{1} [h] = L_{1, 1} [h], Q_{2} [h, k] = L_{2, 1} [h, k],

and put $N_{j} = Q_{j} - L_{j, g}$ . Explicitly,

N_{1} [h] u N_{2} [h, k] u = (1 - g) Q_{1} [h] u - (\partial_{μ} g) a_{1}^{μν} [h] \partial_{ν} u, = (1 - g) Q_{2} [h, k] u - (\partial_{μ} g) A_{2}^{μν} [h, k] \partial_{ν} u .

The $N_{j}$ need not have compact coefficients. Only their transformation combinations used below have compact coefficients.

The unswitched scalar Euler operator

E_{unsw} [q] u = \partial_{μ} (- det q q^{μν} \partial_{ν} u) + m^{2} - det q u

maps scalars to densities. The chain rule and change of variables give its naturality

E_{unsw} [F^{*} q] F^{*} = F^{♯} E_{unsw} [q] .

For formal jets this is an identity of Taylor coefficients; no global numerical inverse of $q$ is required.

Displayed formulas in this passage

X-C-ST@1#eq:N · line 1372
$N_{1} [h] u N_{2} [h, k] u = (1 - g) Q_{1} [h] u - (\partial_{μ} g) a_{1}^{μν} [h] \partial_{ν} u, = (1 - g) Q_{2} [h, k] u - (\partial_{μ} g) A_{2}^{μν} [h, k] \partial_{ν} u .$
unlabeled · line 1366
$Q_{1} [h] = L_{1, 1} [h], Q_{2} [h, k] = L_{2, 1} [h, k],$
unlabeled · line 1387
$E_{unsw} [q] u = \partial_{μ} (- det q q^{μν} \partial_{ν} u) + m^{2} - det q u$
unlabeled · line 1395
$E_{unsw} [F^{*} q] F^{*} = F^{♯} E_{unsw} [q] .$