= Ordered monomial basis of a Weyl algebra
{title2=$\{x^\alpha\partial^\beta\}$}
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.
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