▁ 3 - dimensional ▁ ▁Lie ▁algebra ▁determined ▁by ▁the ▁commut ation ▁relation ▁ ▁is ▁called ▁the ▁He isen berg ▁group . ▁This ▁group ▁can ▁be ▁realized ▁as ▁the ▁group ▁of ▁ ▁upper ▁tri angular ▁matrices ▁with ▁ones ▁on ▁the ▁diagonal . ▁ ▁According ▁to ▁the ▁standard ▁mathematical ▁form ulation ▁of ▁quantum ▁mechan ics , ▁quantum ▁observ ables ▁such ▁as ▁ ▁and ▁ ▁should ▁be ▁represented ▁as ▁self - ad joint ▁operators ▁on ▁some ▁Hil bert ▁space . ▁ ▁It ▁is ▁relatively ▁easy ▁to ▁see ▁that ▁two ▁operators ▁satisfying ▁the ▁above ▁canonical ▁commut ation ▁relations ▁cannot ▁both ▁be ▁bounded . ▁C ertain ly , ▁if ▁ ▁and ▁ ▁were ▁trace ▁class ▁operators , ▁the ▁relation ▁ ▁gives ▁a ▁non zero ▁number ▁on ▁the ▁right ▁and ▁zero ▁on ▁the ▁left . ▁ ▁Altern ately , ▁if ▁ ▁and ▁ ▁were ▁bounded ▁operators , ▁note ▁that ▁, ▁hence ▁the ▁operator ▁norm s ▁would ▁satisfy ▁ ▁, ▁ ▁  ▁so ▁that , ▁ ▁for ▁any ▁n , ▁▁ ▁However , ▁▁ ▁can ▁be ▁arbitr arily ▁large , ▁so ▁at ▁least ▁one ▁operator ▁cannot ▁be ▁bounded , ▁and ▁the ▁dimension ▁of ▁the ▁underlying ▁Hil bert ▁space ▁cannot ▁be ▁finite . ▁If ▁the ▁operators ▁satisfy ▁the ▁W ey l ▁relations ▁( an ▁exponent i ated ▁version ▁of ▁the ▁canonical ▁commut ation ▁relations , ▁described ▁below ) ▁then ▁as ▁a ▁consequence ▁of ▁the ▁Stone – von ▁Ne umann ▁theorem , ▁ ▁both ▁operators ▁must ▁be ▁un bounded . ▁ ▁Still , ▁these ▁canonical ▁commut ation ▁relations ▁can ▁be ▁rendered