[[Heisenberg algebra]]
# Standard Heisenberg algebra for QM

Consider [[QM in nD]].
The Lie algebra over $\mathbb{C}$ generated by the operators $\{ -i\hbar, \hat{x}_{i},\hat{p}_{i} \}_{i=1}^n$ under the [[commutator]] is an example of a $\mathbb{Z}$-graded [[Heisenberg algebra]], with
$$
\begin{align*}
\mathfrak{l}_{0} &= \langle -i\hbar \rangle   & \mathfrak{l}_{-i}&= \langle\hat{x}_{i}\rangle & \mathfrak{l}_{i} &= \langle \hat{p}_{i} \rangle
\end{align*}
$$
for $1 \leq i \leq n$ and $\mathfrak{l}_{\pm i} = 0$ otherwise, yielding the commutation relations
$$
\begin{align*}
[\hat{x}_{i}, i\hbar] = [\hat{p}_{i}, i\hbar] = [\hat{x}_{i},\hat{x}_{j}] = [\hat{p}_{i},\hat{p}_{j}] &= 0 & [\hat{p}_{i}, \hat{x}_{j}] = -i\hbar\delta_{ij}
\end{align*}
$$
for $1 \leq i,j \leq n$.

## Canonical realization

The irreducible representation of the Heisenberg algebra given by the [[Heisenberg module]] $M(-i\hbar)$ gives the vector space $\mathbb{C}[x_{i}]_{i=1}^n$ of polynomials in indeterminates $\{ x_{i} \}_{i=1}^n$ with
$$
\begin{align*}
\hat{x}_{i} f &= xf & \hat{p}_{i}f &= -i\hbar \frac{ \partial }{ \partial x } f & {-i\hbar} f &= -i\hbar f
\end{align*}
$$
which concurs with the realization of [[QM in nD position-space]].

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