Standard Heisenberg algebra for QM

Consider QM in nD. The Lie algebra over $ℂ$ generated by the operators ${- 𝑖 ℏ, ˆ 𝑥 𝑖, ˆ 𝑝 𝑖} 𝑛 𝑖 = 1$ under the commutator is an example of a $ℤ$ -graded Heisenberg algebra, with

𝔩 0 = ⟨ - 𝑖 ℏ ⟩ 𝔩 - 𝑖 = ⟨ ˆ 𝑥 𝑖 ⟩ 𝔩 𝑖 = ⟨ ˆ 𝑝 𝑖 ⟩

for $1 \leq 𝑖 \leq 𝑛$ and $𝔩 \pm 𝑖 = 0$ otherwise, yielding the commutation relations

[ˆ 𝑥 𝑖, 𝑖 ℏ] = [ˆ 𝑝 𝑖, 𝑖 ℏ] = [ˆ 𝑥 𝑖, ˆ 𝑥 𝑗] = [ˆ 𝑝 𝑖, ˆ 𝑝 𝑗] = 0 [ˆ 𝑝 𝑖, ˆ 𝑥 𝑗] = - 𝑖 ℏ 𝛿 𝑖 𝑗

for $1 \leq 𝑖, 𝑗 \leq 𝑛$ .

Canonical realization

The irreducible representation of the Heisenberg algebra given by the Heisenberg module $𝑀 (- 𝑖 ℏ)$ gives the vector space $ℂ [𝑥 𝑖] 𝑛 𝑖 = 1$ of polynomials in indeterminates ${𝑥 𝑖} 𝑛 𝑖 = 1$ with

ˆ 𝑥 𝑖 𝑓 = 𝑥 𝑓 ˆ 𝑝 𝑖 𝑓 = - 𝑖 ℏ 𝜕 𝜕 𝑥 𝑓 - 𝑖 ℏ 𝑓 = - 𝑖 ℏ 𝑓

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

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