▁commut ation ▁relations ▁but ▁not ▁the ▁W ey l ▁relations . ▁( Th ese ▁same ▁operators ▁give ▁a ▁counter example ▁to ▁the ▁na ive ▁form ▁of ▁the ▁uncertainty ▁principle .) ▁These ▁technical ▁issues ▁are ▁the ▁reason ▁that ▁the ▁Stone – von ▁Ne umann ▁theorem ▁is ▁form ulated ▁in ▁terms ▁of ▁the ▁W ey l ▁relations . ▁ ▁A ▁discrete ▁version ▁of ▁the ▁W ey l ▁relations , ▁in ▁which ▁the ▁parameters ▁s ▁and ▁t ▁range ▁over ▁, ▁can ▁be ▁realized ▁on ▁a ▁finite - dimensional ▁Hil bert ▁space ▁by ▁means ▁of ▁the ▁clock ▁and ▁shift ▁matrices . ▁ ▁General izations ▁ ▁The ▁simple ▁formula ▁ ▁valid ▁for ▁the ▁quant ization ▁of ▁the ▁simplest ▁classical ▁system , ▁can ▁be ▁generalized ▁to ▁the ▁case ▁of ▁an ▁arbitrary ▁L agr ang ian ▁. ▁We ▁identify ▁canonical ▁coordinates ▁( such ▁as ▁ ▁in ▁the ▁example ▁above , ▁or ▁a ▁field ▁ ▁in ▁the ▁case ▁of ▁quantum ▁field ▁theory ) ▁and ▁canonical ▁moment a ▁▁ ▁( in ▁the ▁example ▁above ▁it ▁is ▁, ▁or ▁more ▁generally , ▁some ▁functions ▁involving ▁the ▁derivatives ▁of ▁the ▁canonical ▁coordinates ▁with ▁respect ▁to ▁time ): ▁ ▁This ▁definition ▁of ▁the ▁canonical ▁momentum ▁ens ures ▁that ▁one ▁of ▁the ▁E uler – L ag range ▁equations ▁has ▁the ▁form ▁ ▁The ▁canonical ▁commut ation ▁relations ▁then ▁amount ▁to ▁ ▁where ▁ ▁is ▁the ▁Kr one cker ▁delta . ▁ ▁Further , ▁it ▁can ▁be ▁easily ▁shown ▁that ▁ ▁Using ▁, ▁it ▁can ▁be ▁easily ▁shown ▁that ▁by ▁mathematical ▁induction ▁ ▁G au ge