Positive Energy Representations Of Gauge Groups I: Localization
 

This is the first in a series of papers on projective positive energy representations of gauge groups. Let $\Xi \rightarrow M$ be a principal fiber bundle, and let $\Gamma_{C}(M,\mathrm{Ad}(\Xi))$ be the group of compactly supported (local) gauge transformations. If $P$ is a group of "space-time symmetries" acting on $\Xi \rightarrow M$, then a projective unitary representation of $\Gamma_{C}(M,\mathrm{Ad}(\Xi))\rtimes P$ is of positive energy if every "timelike generator" $p_{0} \in p$ gives rise to a Hamiltonian $H(p_{0})$ whose spectrum is bounded from below.