under construction
(gauge parameterized implicit infinitesimal gauge transformations)
Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. ). Then a collection of gauge parameters for $(E,\mathbf{L})$ is
a vector bundle $\mathcal{G} \overset{gb}{\longrightarrow} \Sigma$ over spacetime $\Sigma$; the sections $\epsilon$ of which are to be called the gauge parameters;
a bundle morphism $R$ from the fiber product of its jet bundle with that of the field bundle to the vertical tangent bundle of $E$:
such that
$R$ is linear in the first argument.
$R$ takes values in the sub-bundle of those evolutionary vectors which are infinitesimal symmetries of the Lagrangian (def. );
For every gauge parameter $\epsilon$ of compact support the composite of $R$ with the jet prolongation $j^\infty_\Sigma(\epsilon)$ (def. )
is an infinitesimal symmetry of the Lagrangian (def. ).
If the field bundle $E$ is a trivial vector bundle with field coordinates $(\phi^a)$ (example ) and also $\mathcal{G}$ happens to be a trivial vector bundle equipped with fiber coordinates $(e^\alpha)$ then this mean that $v_\epsilon$ is of the form
where the $R^{a \mu_1 \cdots \mu_k}_\alpha$ are smooth functions on the jet bundle of $E$ (prop. ).
Last revised on December 9, 2017 at 13:18:34. See the history of this page for a list of all contributions to it.