nLab gauge parameter

under construction


(gauge parameterized implicit infinitesimal gauge transformations)

Let (E,L)(E,\mathbf{L}) be a Lagrangian field theory (def. ). Then a collection of gauge parameters for (E,L)(E,\mathbf{L}) is

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

  2. a bundle morphism RR from the fiber product of its jet bundle with that of the field bundle to the vertical tangent bundle of EE:

    J Σ (𝒢)× ΣJ Σ (E) R V ΣE 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. RR is linear in the first argument.

    2. RR takes values in the sub-bundle of those evolutionary vectors which are infinitesimal symmetries of the Lagrangian (def. );

    For every gauge parameter ϵ\epsilon of compact support the composite of RR with the jet prolongation j Σ (ϵ)j^\infty_\Sigma(\epsilon) (def. )

    v ϵ:J Σ (E)=Σ× ΣJ Σ (E)(j Σ (ϵ),id)J Σ (𝒢)× ΣJ Σ (E)RV ΣE 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 EE is a trivial vector bundle with field coordinates (ϕ a)(\phi^a) (example ) and also 𝒢\mathcal{G} happens to be a trivial vector bundle equipped with fiber coordinates (e α)(e^\alpha) then this mean that v ϵv_\epsilon is of the form

v ϵ=(ϵ αR α a+dϵ αdx μR α aμ+d 2ϵ αdx μ 1dx μ 2R α aμ 1μ 2+) ϕ a, 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 α aμ 1μ kR^{a \mu_1 \cdots \mu_k}_\alpha are smooth functions on the jet bundle of EE (prop. ).

Last revised on December 9, 2017 at 18:18:34. See the history of this page for a list of all contributions to it.