 Ordered monomial basis of a Weyl algebra

The defining commutators $[{\partial }_i,x_j]={\delta }_{ij}$ reorder every word into the displayed monomials. Over a characteristic-zero field, the polynomial representation proves independence: iterated commutation with the coordinate multiplications extracts a highest derivative coefficient, multiplied by the nonzero factorial $\beta !$. Hence the representation is faithful and every polynomial-coefficient operator has a unique ordered expression.