Classical charged-scalar action, current and stress variation

Let $\eta =\delta \phi$ be a compactly supported complex section. Direct differentiation gives ${\delta }_{\phi }{L}_{\phi }=2\mathrm{Re}(g^{ab}\overline{D_a\eta }{P}_b+V^{\mathrm{\prime }}\bar{\eta }\phi )$. The gauge-covariant product rule gives ${\mathrm{\nabla }}_a(\bar{\eta }{P}^a)=\overline{D_a\eta }{P}^a+\bar{\eta }{D}_a{P}^a$, so [F3] removes the compact divergence and $\int \overline{D_a\eta }{P}^a{\mathrm{v}\mathrm{o}\mathrm{l}}_g=-\int \bar{\eta }{D}_a{P}^a{\mathrm{v}\mathrm{o}\mathrm{l}}_g$. Multiplying by $-K_{\phi }\mathrm{/}(2c)$ yields the displayed scalar variation. If stationarity holds, then $\mathrm{Re}\int \bar{\eta }{E}_{\phi }{\mathrm{v}\mathrm{o}\mathrm{l}}_g=0$ for all compactly supported $\eta$. In a positively oriented chart, for every compactly supported real vector field $q=q^a{\mathrm{\partial }}_a$, its component $q^0$ is a real test function; choosing $\eta =q^0$ in the local scalar frame gives zero pairing of the covector $W_a=(\mathrm{Re}{E}_{\phi }){\delta }_a^0$ with $q$. [F5] implies $\mathrm{Re}{E}_{\phi }=0$. Choosing instead $\eta =i{q}^0$ gives zero pairing for $W_a=(\mathrm{Im}{E}_{\phi }){\delta }_a^0$, so [F5] implies $\mathrm{Im}{E}_{\phi }=0$. Thus $E_{\phi }=0$ pointwise. Conversely, $E_{\phi }=0$ makes the variation zero for every test section; hence stationarity is equivalent to $E_{\phi }=0$.

At fixed $g,\phi$, a real connection variation ${\alpha }_a$ gives ${\delta }_A{P}_a=-i\lambda \phi {\alpha }_a$ and ${\delta }_A\overline{P_a}=+i\lambda \bar{\phi }{\alpha }_a$. Their contraction is ${\delta }_A{L}_{\phi }=-2\lambda \mathrm{Im}(\bar{\phi }{P}^a){\alpha }_a$, so ${\delta }_A{S}_{\phi }=\frac{1}{c}{\int }_M{K}_{\phi }\lambda \mathrm{Im}(\bar{\phi }{P}^a){\alpha }_a{\mathrm{v}\mathrm{o}\mathrm{l}}_g$. This is the stated current.

At fixed lower $A$ and fixed $\phi$, the inverse-metric variation gives ${\delta }_g{L}_{\phi }=\mathrm{Re}(\overline{P_a}{P}_b)q^{ab}$. Combining this with [F2] in $S_{\phi }=-K_{\phi }(2c{)}^{-1}\int L_{\phi }{\mathrm{v}\mathrm{o}\mathrm{l}}_g$ yields ${\delta }_g{S}_{\phi }=-\frac{1}{2c}{\int }_M[K_{\phi }\mathrm{Re}(\overline{P_a}{P}_b)-\frac{K_{\phi }}{2}{g}_{ab}{L}_{\phi }]q^{ab}{\mathrm{v}\mathrm{o}\mathrm{l}}_g$, the displayed Hilbert tensor.

The covariant derivative of $j^a=K_{\phi }\lambda \mathrm{Im}(\bar{\phi }{P}^a)$ is $K_{\phi }\lambda \mathrm{Im}(\overline{P_a}{P}^a+\bar{\phi }{D}_a{P}^a)$. The first term is real; adding $V^{\mathrm{\prime }}\mathrm{\mid }\phi {\mathrm{\mid }}^2$ inside the imaginary part changes nothing because $V^{\mathrm{\prime }}$ is real. Therefore ${\mathrm{\nabla }}_a{j}^a=K_{\phi }\lambda \mathrm{Im}(\bar{\phi }(D_a{P}^a-V^{\mathrm{\prime }}\phi ))=K_{\phi }\lambda \mathrm{Im}(\bar{\phi }{E}_{\phi })$.

Raise the first index of $T^{\phi }$ and differentiate. Using [F6] to differentiate contractions and ${\mathrm{\nabla }}_b{L}_{\phi }=2\mathrm{Re}(\overline{P^a}{D}_b{P}_a)+2V^{\mathrm{\prime }}\mathrm{Re}(\bar{\phi }{P}_b)$, the product rule gives ${\mathrm{\nabla }}_a(T^{\phi }{)}^a{}_b=K_{\phi }\mathrm{Re}(\overline{E_{\phi }}{P}_b+\overline{P^a}(D_a{P}_b-D_b{P}_a))$. In a local trivialization, symmetry of the Levi-Civita symbols and commutation of scalar partial derivatives give $D_a{P}_b-D_b{P}_a=-i\lambda F_{ab}\phi$. Hence the second term is $-K_{\phi }\lambda \mathrm{Im}(\bar{\phi }{P}^a)F_{ab}=F_{ba}{j}^a$, proving the stated off-shell stress identity.

If $E_{\phi }=0$, steps 2.1–2.2 give ${\mathrm{\nabla }}_a{j}^a=0$ and ${\mathrm{\nabla }}_a(T^{\phi }{)}^a{}_b=F_{ba}{j}^a$. For the auxiliary Noether route, restrict to configurations for which the scalar density is absolutely integrable. Take $E$ to be the supplied associated complex line bundle (the trivial complex line in the global-potential branch), and let the functional domain consist of these finite-action triples. Compact smooth variations remain in this domain near zero; their density differences have common compact support, so differentiation of their coordinate integrals gives the real-linear partial splitting established in steps 1.1–1.3. The induced line-bundle gauge maps and horizontal connection-flow maps supply the smooth section transports of [F4], equal to the identity off the generating support. The covariant derivative transforms by the same phase as the section under gauge transport and as a covector under base transport; thus the scalar density is gauge invariant and pulls back under the base flow. Change of variables for its absolutely integrable density makes this real-valued functional invariant, so [F4] gives the same raised-index exchange equation. When only relative compact action differences are defined, steps 2.1–2.2 already prove both identities and the same sign directly, without invoking a real-valued global functional.