Heisenberg algebra

Standard Heisenberg algebra for QM

Consider QM in nD. The Lie algebra over ℂ generated by the operators { −𝑖ℏ,ˆ𝑥𝑖,ˆ𝑝𝑖}𝑛𝑖=1 under the commutator is an example of a ℤ-graded Heisenberg algebra, with

𝔩0=⟨−𝑖ℏ⟩𝔩−𝑖=⟨ˆ𝑥𝑖⟩𝔩𝑖=⟨ˆ𝑝𝑖⟩

for 1 ≤𝑖 ≤𝑛 and 𝔩±𝑖 =0 otherwise, yielding the commutation relations

[ˆ𝑥𝑖,𝑖ℏ]=[ˆ𝑝𝑖,𝑖ℏ]=[ˆ𝑥𝑖,ˆ𝑥𝑗]=[ˆ𝑝𝑖,ˆ𝑝𝑗]=0[ˆ𝑝𝑖,ˆ𝑥𝑗]=−𝑖ℏ𝛿𝑖𝑗

for 1 ≤𝑖,𝑗 ≤𝑛.

Canonical realization

The irreducible representation of the Heisenberg algebra given by the Heisenberg module 𝑀( −𝑖ℏ) gives the vector space ℂ[𝑥𝑖]𝑛𝑖=1 of polynomials in indeterminates {𝑥𝑖}𝑛𝑖=1 with

ˆ𝑥𝑖𝑓=𝑥𝑓ˆ𝑝𝑖𝑓=−𝑖ℏ𝜕𝜕𝑥𝑓−𝑖ℏ𝑓=−𝑖ℏ𝑓

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

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