Manuscript X-A-ST@1 · X-A-ST.tex · section · d9ccbde1

Variational recovery of coordinate covariance

Varied-action derivation: Route II › Variational recovery of coordinate covariance · lines 1398–1410

For four scalar functions $Z^{μ}$ , use the auxiliary action

S_{ε}^{aux} [q, Z] = \frac{1}{2} μ = 0 \sum^{3} \int A_{ε} [q]^{α β} \partial_{α} Z^{μ} \partial_{β} Z^{μ} d^{4} x .

The sum is over four scalar labels; it is not a spacetime-vector contraction.

Displayed formulas in this passage

unlabeled · line 1402
$S_{ε}^{aux} [q, Z] = \frac{1}{2} μ = 0 \sum^{3} \int A_{ε} [q]^{α β} \partial_{α} Z^{μ} \partial_{β} Z^{μ} d^{4} x .$