# Definition The *Heisenberg Algebra* $H$ is defined as the [[span]] of $\{p,q,1\} \subset \mathcal{C}(\mathbb{R}^2)$, where $\mathcal{C}(P)$ is the [[Lie algebra]] of [[Observable (Classical Mechanics)|observables]], and $P = \mathbb{R}^2$ is the [[Phase Space]]. The only non-trivial [[Poisson Bracket]] is $ \{p,q\} = 1 $ $H$ is closed under the Poisson bracket and thus is a [[Lie Subalgebra]] of $\mathcal{C}(\mathbb{R}^2)$. # Another Basis If we treat $H$ as a complex [[vector space]] we can consider another [[basis]] for it: $\{Q, P, 1\} = \mathcal{C}(\mathbb{R}^2)$ where $ \begin{align} Q &\equiv \frac{p + iq}{\sqrt{2}},\\ P &\equiv \frac{p - iq}{\sqrt{2}}. \end{align} $ and $ \{Q, P\} = 1 $ which makes $Q$ and $P$ a [[Canonical transformation|canonical transformation]].