Quantum observables: algebra and calculus
We consider quantum observables as series in
non-commuting generators $\hat x$, $\hat p$.
The space of such series turns out to be an
infinite-dimensional associative and Lie algebra.
We present the concept of convergence for such series.
In this language quantum objects turn out to be
non-commutative analogs of classical ones.
We prove quantum analogs for several basic
theorems of classical mechanics.