Manuscript X-D-ST@1 · X-D-ST.tex · section · e0cf8fdb

Switch derivatives are not suppressed

Gauge passage and the defect ledger › Switch derivatives are not suppressed · lines 1496–1533

X-D-ST@1#eq:N, X-D-ST@1#eq:switch-defect

Write

Σ_{F} = κ S_{1} + κ^{2} S_{2} .

Let $Q_{j}$ be the corresponding unswitched Euler coefficients, and set $N_{j} = Q_{j} - L_{j}$ . Thus

N_{1} u N_{2} u = (1 - g) Q_{1} u - (\partial_{μ} g) a^{μν} \partial_{ν} u, = (1 - g) Q_{2} u - (\partial_{μ} g) A^{μν} \partial_{ν} u .

Unswitched scalar-density naturality gives

S_{1} S_{2} = (g - 1) Q_{1} [K ξ] + (\partial_{μ} g) a_{1}^{μν} [K ξ] \partial_{ν}, = - (N_{2}^{'} - N_{2}) - N_{1}^{'} X_{ξ} + D_{ξ} N_{1} .

The latter expression includes derivatives such as

- ξ^{ρ} (\partial_{ρ} \partial_{μ} g) a^{μν} \partial_{ν} u

arising from $D_{ξ} N_{1}$ . Thus second derivatives of the switch are not discarded.

The combinations $S_{j}$ have compact generator coefficient support even though $1 - g$ , and in general $g - σ_{ε}$ , do not have compact support. If $g = 1$ near the generators, then $S_{1} = S_{2} = 0$ .

Displayed formulas in this passage

X-D-ST@1#eq:N · line 1504
$N_{1} u N_{2} u = (1 - g) Q_{1} u - (\partial_{μ} g) a^{μν} \partial_{ν} u, = (1 - g) Q_{2} u - (\partial_{μ} g) A^{μν} \partial_{ν} u .$
X-D-ST@1#eq:switch-defect · line 1511
$S_{1} S_{2} = (g - 1) Q_{1} [K ξ] + (\partial_{μ} g) a_{1}^{μν} [K ξ] \partial_{ν}, = - (N_{2}^{'} - N_{2}) - N_{1}^{'} X_{ξ} + D_{ξ} N_{1} .$
unlabeled · line 1499
$Σ_{F} = κ S_{1} + κ^{2} S_{2} .$
unlabeled · line 1521
$- ξ^{ρ} (\partial_{ρ} \partial_{μ} g) a^{μν} \partial_{ν} u$