▁respect ▁to ▁the ▁internal ▁indices ▁ ▁without ▁changing ▁the ▁( in verse ) ▁metric . ▁This ▁is ▁the ▁origin ▁of ▁the ▁ ▁gauge ▁in var iance . ▁Now ▁if ▁one ▁is ▁going ▁to ▁operate ▁on ▁objects ▁that ▁have ▁internal ▁indices ▁one ▁needs ▁to ▁introduce ▁an ▁appropriate ▁derivative ▁( cov ari ant ▁derivative ), ▁for ▁example ▁the ▁cov ari ant ▁derivative ▁for ▁the ▁object ▁ ▁will ▁be ▁ ▁where ▁ ▁is ▁the ▁usual ▁Le vi - C iv ita ▁connection ▁and ▁ ▁is ▁the ▁so - called ▁spin ▁connection . ▁Let ▁us ▁take ▁the ▁configuration ▁variable ▁to ▁be ▁ ▁where ▁ ▁and ▁. ▁The ▁dens it ized ▁drei - be in ▁is ▁the ▁conjug ate ▁momentum ▁variable ▁of ▁this ▁three - dimensional ▁SU ( 2 ) ▁gauge ▁field ▁( or ▁connection ) ▁, ▁in ▁that ▁it ▁satisfies ▁the ▁Po isson ▁bra cket ▁relation ▁ ▁. ▁ ▁The ▁constant ▁ ▁is ▁the ▁Im mir zi ▁parameter , ▁a ▁factor ▁that ▁ren ormal izes ▁Newton ' s ▁constant ▁. ▁The ▁dens it ized ▁drei - be in ▁can ▁be ▁used ▁to ▁re ▁construct ▁the ▁metric ▁as ▁discussed ▁above ▁and ▁the ▁connection ▁can ▁be ▁used ▁to ▁re construct ▁the ▁extr insic ▁curv ature . ▁Ash te kar ▁variables ▁correspond ▁to ▁the ▁choice ▁ ▁( the ▁negative ▁of ▁the ▁imag inary ▁number ), ▁ ▁is ▁then ▁called ▁the ▁ch iral ▁spin ▁connection . ▁The ▁reason ▁for ▁this ▁choice ▁of ▁spin ▁connection ▁was ▁that ▁Ash te kar ▁could ▁much ▁simplify ▁the ▁most ▁troubles ome ▁equation ▁of ▁canonical ▁general ▁relativ