l ▁correspond ence . ▁ ▁Mo yal ▁actually ▁appears ▁not ▁to ▁know ▁about ▁the ▁product ▁in ▁his ▁celebrated ▁article ▁and ▁was ▁cru cially ▁lack ing ▁it ▁in ▁his ▁legend ary ▁correspond ence ▁with ▁Di rac , ▁as ▁illustrated ▁in ▁his ▁bi ography . ▁The ▁popular ▁naming ▁after ▁Mo yal ▁appears ▁to ▁have ▁emer ged ▁only ▁in ▁the ▁ 1 9 7 0 s , ▁in ▁hom age ▁to ▁his ▁flat ▁phase - space ▁quant ization ▁picture . ▁ ▁Definition ▁The ▁product ▁for ▁smooth ▁functions ▁f ▁and ▁g ▁on ▁ ℝ 2 n ▁takes ▁the ▁form ▁ ▁where ▁each ▁C n ▁is ▁a ▁certain ▁bid iffer ential ▁operator ▁of ▁order ▁n ▁character ized ▁by ▁the ▁following ▁properties ▁( see ▁below ▁for ▁an ▁explicit ▁formula ): ▁▁▁ ▁< p ▁> ▁De formation ▁of ▁the ▁point wise ▁product ▁— ▁implicit ▁in ▁the ▁formula ▁above . ▁▁ ▁< p ▁> ▁De formation ▁of ▁the ▁Po isson ▁bra cket , ▁called ▁Mo yal ▁bra cket . ▁▁ ▁< p ▁> ▁The ▁ 1 ▁of ▁the ▁und e formed ▁algebra ▁is ▁also ▁the ▁identity ▁in ▁the ▁new ▁algebra . ▁▁ ▁< p ▁> ▁The ▁complex ▁conjug ate ▁is ▁an ▁an til ine ar ▁ant ia ut omorphism . ▁ ▁Note ▁that , ▁if ▁one ▁wishes ▁to ▁take ▁functions ▁val ued ▁in ▁the ▁real ▁numbers , ▁then ▁an ▁alternative ▁version ▁elimin ates ▁the ▁ ▁in ▁condition ▁ 2 ▁and ▁elimin ates ▁condition ▁ 4 . ▁ ▁If ▁one ▁restrict s ▁to ▁polynomial ▁functions , ▁the ▁above ▁algebra ▁is ▁is omorphic