Quantum Observables: An Algebraic Aspect

Quantum observables are represented as series in noncommuting generators $\hat x$ and $\hat p$. The space of such series turns out to be an infinite-dimensional associative algebra and a Lie algebra. The concept of convergence is presented for such series. In this language, quantum objects turn out to be noncommutative analogues of classical objects. Quantum analogues are proved for several basic theorems of classical mechanics.