▁SU ( 2 ) ▁as ▁well , ▁and ▁moreover , ▁by ▁reli ance ▁on ▁the ▁comple ten ess ▁relation ▁for ▁Paul i ▁matrices , ▁▁ ▁This ▁is ▁the ▁starting ▁point ▁of ▁Schw inger ’ s ▁treatment ▁of ▁the ▁theory ▁of ▁quantum ▁angular ▁momentum , ▁pred ic ated ▁on ▁the ▁action ▁of ▁these ▁operators ▁on ▁ ▁F ock ▁states ▁built ▁of ▁arbitrary ▁higher ▁powers ▁of ▁such ▁operators . ▁For ▁instance , ▁acting ▁on ▁an ▁( unn ormal ized ) ▁F ock ▁eigen state , ▁▁▁ ▁while ▁▁▁ ▁so ▁that , ▁for ▁, ▁this ▁is ▁proportional ▁to ▁the ▁eigen state ▁, ▁▁ ▁Ob serve ▁ ▁and ▁ ▁. ▁ ▁Ant is ym metric ▁representations ▁of ▁Lie ▁al gebras ▁can ▁further ▁be ▁accommod ated ▁by ▁use ▁of ▁the ▁ferm ionic ▁operators ▁▁▁ ▁and ▁, ▁ ▁as ▁also ▁suggested ▁by ▁Jordan . ▁For ▁ferm ions , ▁the ▁commut ator ▁is ▁replaced ▁by ▁the ▁ant ic omm ut ator ▁, ▁▁▁▁ ▁Therefore , ▁ex ch anging ▁dis joint ▁( i . e . ▁) ▁operators ▁in ▁a ▁product ▁of ▁creation ▁of ▁anni h ilation ▁operators ▁will ▁reverse ▁the ▁sign ▁in ▁ferm ion ▁systems , ▁but ▁not ▁in ▁bos on ▁systems . ▁ ▁See ▁also ▁Hol stein – Prim ak off ▁transformation ▁Bor el - We il - B ott ▁Theorem ▁Current ▁algebra ▁Angular ▁momentum ▁operator ▁ ▁References ▁ ▁Category : Re presentation ▁theory ▁of ▁Lie ▁al gebras ▁Category : Math emat ical ▁physics ▁Category : The oret ical ▁physics <0x0A> </s> ▁Qu ir ina ▁Ali ppi - F