▁and ▁ob e ys ▁the ▁canonical ▁quant ization ▁relations ▁▁ ▁defining ▁the ▁Lie ▁algebra ▁for ▁so ( 3 ), ▁where ▁ ▁is ▁the ▁Le vi - C iv ita ▁symbol . ▁ ▁Under ▁gauge ▁transformations , ▁the ▁angular ▁momentum ▁transform s ▁as ▁ ▁The ▁gauge - in variant ▁angular ▁momentum ▁( or ▁" kin etic ▁angular ▁momentum ") ▁is ▁given ▁by ▁ ▁which ▁has ▁the ▁commut ation ▁relations ▁ ▁where ▁ ▁is ▁the ▁magnetic ▁field . ▁The ▁in equ ival ence ▁of ▁these ▁two ▁form ulations ▁shows ▁up ▁in ▁the ▁Ze eman ▁effect ▁and ▁the ▁Ah aron ov – B oh m ▁effect . ▁ ▁Un certain ty ▁relation ▁and ▁commut ators ▁▁ ▁All ▁such ▁n ont rivial ▁commut ation ▁relations ▁for ▁pairs ▁of ▁operators ▁lead ▁to ▁corresponding ▁uncertainty ▁relations , ▁involving ▁positive ▁semi - defin ite ▁expectation ▁contributions ▁by ▁their ▁respective ▁commut ators ▁and ▁ant ic omm ut ators . ▁ ▁In ▁general , ▁for ▁two ▁Herm it ian ▁operators ▁ ▁and ▁, ▁consider ▁expectation ▁values ▁in ▁a ▁system ▁in ▁the ▁state ▁, ▁the ▁var ian ces ▁around ▁the ▁corresponding ▁expectation ▁values ▁being ▁, ▁etc . ▁ ▁Then ▁▁ ▁where ▁ ▁is ▁the ▁commut ator ▁of ▁ ▁and ▁, ▁and ▁ ▁is ▁the ▁ant ic omm ut ator . ▁ ▁This ▁follows ▁through ▁use ▁of ▁the ▁C auch y – Sch war z ▁inequality , ▁since ▁, ▁and ▁; ▁and ▁similarly ▁for ▁the ▁shift ed ▁operators ▁ ▁and ▁ ▁. ▁( C f . ▁uncertainty ▁principle ▁deriv ations .) ▁ ▁Sub st