▁to ▁the ▁W ey l ▁algebra ▁An , ▁and ▁the ▁two ▁offer ▁alternative ▁realiz ations ▁of ▁the ▁W ey l ▁map ▁of ▁the ▁space ▁of ▁polynomials ▁in ▁ ▁variables ▁( or ▁the ▁symmetric ▁algebra ▁of ▁a ▁vector ▁space ▁of ▁dimension ▁ 2 n ). ▁ ▁To ▁provide ▁an ▁explicit ▁formula , ▁consider ▁a ▁constant ▁Po isson ▁b ive ctor ▁ Π ▁on ▁ ℝ 2 n : ▁ ▁where ▁ Π ij ▁is ▁a ▁complex ▁number ▁for ▁each ▁i ,   j . ▁ ▁The ▁star ▁product ▁of ▁two ▁functions ▁ ▁and ▁ ▁can ▁then ▁be ▁defined ▁as ▁ ▁where ▁ ħ ▁is ▁the ▁reduced ▁Plan ck ▁constant , ▁treated ▁as ▁a ▁formal ▁parameter ▁here . ▁This ▁is ▁a ▁special ▁case ▁of ▁what ▁is ▁known ▁as ▁the ▁Bere zin ▁formula ▁on ▁the ▁algebra ▁of ▁symbols ▁and ▁can ▁be ▁given ▁a ▁closed ▁form ▁( which ▁follows ▁from ▁the ▁Baker – C amp bell – H aus dor ff ▁formula ). ▁The ▁closed ▁form ▁can ▁be ▁obtained ▁by ▁using ▁the ▁exponential : ▁ ▁where ▁ ▁is ▁the ▁multiplication ▁map , ▁, ▁and ▁the ▁exponential ▁is ▁treated ▁as ▁a ▁power ▁series : ▁ ▁That ▁is , ▁the ▁formula ▁for ▁ ▁is ▁ ▁As ▁indicated , ▁often ▁one ▁elimin ates ▁all ▁occurr ences ▁of ▁ ▁above , ▁and ▁the ▁formulas ▁then ▁restrict ▁naturally ▁to ▁real ▁numbers . ▁ ▁Note ▁that ▁if ▁the ▁functions ▁f ▁and ▁g ▁are ▁polynomials , ▁the ▁above ▁infinite ▁sums ▁become ▁finite ▁( redu cing ▁to ▁the ▁ordinary ▁W ey l - algebra ▁case