nLab gauge parameter

under construction

Definition

(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

1. a vector bundle $\mathcal{G} \overset{gb}{\longrightarrow} \Sigma$ over spacetime $\Sigma$; the sections $\epsilon$ of which are to be called the gauge parameters;

2. a bundle morphism $R$ from the fiber product of its jet bundle with that of the field bundle to the vertical tangent bundle of $E$:

   $$\array{ J^\infty_\Sigma(\mathcal{G}) \times_\Sigma J^\infty_\Sigma(E) &\overset{R}{\longrightarrow}& V_\Sigma E \\ \downarrow & \swarrow_{} \\ E }$$

   such that

   1. $R$ is linear in the first argument.

   2. $R$ 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. )

   $$v_\epsilon \;\colon\; J^\infty_\Sigma(E) = \Sigma \times_\Sigma J^\infty_\Sigma(E) \overset{(j^\infty_\Sigma(\epsilon), id)}{\longrightarrow} J^\infty_\Sigma(\mathcal{G}) \times_\Sigma J^\infty_\Sigma(E) \overset{R}{\longrightarrow} V_\Sigma E$$

   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

$$v_\epsilon \;=\; \left( \epsilon^\alpha R^a_\alpha + \frac{d \epsilon^\alpha}{d x^\mu} R^{a \mu}_\alpha + \frac{d^2 \epsilon^\alpha}{d x^{\mu_1} d x^{\mu_2}} R^{a \mu_1 \mu_2}_\alpha + \cdots \right) \partial_{\phi^a} \,,$$

where the $R^{a \mu_1 \cdots \mu_k}_\alpha$ are smooth functions on the jet bundle of $E$ (prop. ).