Standard Heisenberg algebra for QM

Consider QM in nD. The Lie algebra over generated by the operators under the commutator is an example of a -graded Heisenberg algebra, with

for and otherwise, yielding the commutation relations

for .

Canonical realization

The irreducible representation of the Heisenberg algebra given by the Heisenberg module gives the vector space of polynomials in indeterminates with

which concurs with the realization of QM in nD position-space.